Talk:Lift (force)/Archive 8

Limits of current human knowledge
Looking over the article and all the archived discussions, one basic thought occurs to me: Humans do not fully and clearly understand why airfoils generate lift. There are various competing explanations, and there is no broad, general, and complete consensus among scientists as to why airfoils generate lift. This is an astounding fact. I think this aspect of current human understanding should be mentioned in the introduction to the article. It is remarkable with the wide-spread use of airfoils throughout history (age of sail, age of flight) that humans do not fully and clearly understand why the airfoil works. --Westwind273 (talk) 18:06, 20 March 2013 (UTC)


 * I rather strongly disagree with the assessment that "there is no broad, general, and complete consensus among scientists as to why airfoils generate lift." In fact I would say just the opposite, that the science and engineering of airfoils and lift is a mature subject that is well understood and "settled" science.  If you have gotten the opposite impression from the article, I apologize.


 * Where there is disagreement is in how to explain these rather complicated and sophisticated ideas in layman's terms. This disagreement over pedagogy is quite distinct from any controversy over the underlying science.


 * I am now more inclined to agree with J Douglas McLean's statement upthread that "the quote by John D. Anderson gives the misleading impression that there are disagreements over the science itself, not just the qualitative explanations" and will remove that material. Mr. Swordfish (talk) 19:02, 20 March 2013 (UTC)


 * I agree that when read by a layman, the article certainly gives the impression that lift is not "settled" science. To the layman, it certainly reads as if there are various competing explanations for the underlying science.  For example, this introductory section definitely gives the impression of various competing explanations of the underlying science:  "There are several ways to explain how an airfoil generates lift. Some are more complicated or more mathematically rigorous than others; some have been shown to be incorrect. For example, there are explanations based directly on Newton’s laws of motion and explanations based on Bernoulli’s principle. Either can be used to explain lift."  Does the Newton explanation conflict with the Bernoulli explanation?  Newton and Bernoulli are not alternative explanations of lift; rather they are both in effect simultaneously to create lift, aren't they?  Then why present them as two alternatives?  It is relatively easy for the layman to understand the Bernoulli effect, but the key question that the article never explains clearly to a layman is:  Why does the air on the top flow faster than the air on the bottom?  This is the key point.  Later the article says "Explaining lift while considering all of the principles involved is a complex task and is not easily simplified."  To me, this is a cop out as far as Wikipedia is concerned.  The article is basically saying "We're going to punt on any attempt to explain this to a layman, and instead divert into scientific mumbo-jumbo that you will never understand."  Is it really so hard to explain why the air on top is flowing faster than the air on bottom?  Overall I am quite disappointed in this Wikipedia article.  --Westwind273 (talk) 19:29, 21 March 2013 (UTC)

I think the whole concept of "multiple explanations" is the wrong way to approach this topic. It leads to deep confusion in the layman. If it is settled science, there should be one explanation which can be presented in greater or lesser levels of depth. --Westwind273 (talk) 19:38, 21 March 2013 (UTC)
 * I think that the problem being described here is the one about the way science is taught (in some schools). First there are phenomena in the real world, then there are scientific theories and mathematical models that attempt to explain the phenomena. There's nothing after that - no point at which physical phenomena stop being what they were and start following the dictates of the mathematical models, rather than their own inscrutable processes. That does not mean that science is wrong or badly understood, but that the scientific models and equations we have, are all we have, apart from the mysterious wonders of nature itself. People try to use similar arguments to rubbish climate science - "If this is all just a theory, then let's wait until it's been sorted out", etc. Lift is settled science - jet airliners fly every day to within tolerances very close to those their designers intended - but there isn't one simple explanation. We just have to get used to that. --Nigelj (talk) 20:35, 21 March 2013 (UTC)


 * >... there should be one explanation which can be presented in greater or lesser levels of depth.


 * Alas, the world is not that simple. Almost every scientific principle admits multiple explanations. For instance, there are at least nine ways to prove the fundamental theorem of algebra, and this is an area that has been "settled" for centuries.  As another example, many problems in Newtonian physics can be approached in multiple ways - conservation of energy, conservation of momentum, direct application of newton's laws, the principle of least action, etc.  These are just different ways of looking at the same problem.


 * Much of the controversy surrounding lift is due to some individuals thinking that there is "one true explanation" that is correct and therefore all other explanations must be incorrect. Unfortunately, one of the most common explanations actually is incorrect;  were this not the case I surmise that the differing explanations would simply sit quietly side-by-side as they do in most topics.


 * I agree that the sentence "Explaining lift while considering all of the principles involved is a complex task and is not easily simplified." adds little to the article, and we have contemplated removing it. Maybe its time has come?  Opinions form other editors?


 * Finally, to answer your question "Why does the air on the top flow faster than the air on the bottom?" there's a very simple answer:


 * When air follows a path that is curved, the pressure is lower on the inside of the curve than on the outside. So, there's a region of lowered pressure on the top side of the wing.  When the air flows from atmospheric pressure into this lower pressure region, there is more pressure behind than in front resulting in a net force on the air which speeds it up.  (the last sentence is basically Bernoulli's principle in a nutshell)


 * Perhaps something like this should be added to the article. I've never considered "why does the air speed up" to be a "key point" since lift can be explained (albeit incompletely) without even mentioning the speed changes.  But if our readers are coming here looking for the answer to that question, perhaps we should present it. Mr. Swordfish (talk) 20:42, 21 March 2013 (UTC)


 * I agree with Mr Swordfish, and I disagree with Westwind273 when he writes Humans do not fully and clearly understand why airfoils generate lift. I disagree for two reasons.  Firstly, science does not address questions about why the universe operates the way it does. That is a question for theology. Science merely observes the universe and attempts to determine principles and laws which describe these things. As a simple example, scientists understand and apply Newton's first law of motion but science does not bother with the question of why a force is necessary to cause an object to accelerate. Theologians might explain that a force is necessary because God dictated that it would be so, but scientists find Newton's observations, and his laws of motion, to be entirely adequate. Similarly, Bernoulli observed the flow of fluids and the interchange of speed and pressure and described it all in his famous principle; but he didn't bother to include any speculation about why it is so. That would not be science.
 * Secondly, Westwind273 has grasped John Anderson's statement that scientists and engineers disagree over what is the most fundamental way to explain the phenomenon of aerodynamic lift. Westwind273 has misunderstood Anderson to be referring to some remarkable mystery about aerodynamic lift. If it is true that humans do not fully and clearly understand these things, then that is equally true of any scientific phenomenon. Westwind273 should also be writing that humans do not fully and clearly understand magnetism, electricity, meteorology, thermodynamics, and so on. There is nothing specially intractable about aerodynamic lift which is a splendid application of Bernoulli's principle, Kutta's observations, Newton's third law of motion, the principle of momentum and so on. Each person contemplating the phenomenon of aerodynamic lift, whether that person is a student pilot, college student, professional engineer or research aerodynamicist, must choose whichever of these scientific principles is most satisfactory for him. There is no one true explanation for any scientific phenomenon, and certainly not aerodynamic lift. Dolphin  ( t ) 23:28, 21 March 2013 (UTC)

I think it's understandable that Westwind273 got the mistaken impression from the current article that there is a lack of consensus on the science of lift. And I largely agree with the responses by Mr. Swordfish, Nigelj, and Dolphin, but with some quibbles below. I have a suggested revised version of the article in my sandbox User:J_Doug_McLean/sandbox that I think would avoid the mistaken impression that Westwind273 got. It sets the record straight on the science and also answers the question of why the flow over the upper surface speeds up a little differently from the way Mr. Swordfish did. I'd appreciate it if Westwind273 would read my suggested version and provide feedback.

The answer by Mr. Swordfish to the question "Why does the air on the top flow faster than the air on the bottom?" is a good start, but it needs to be added that the cause-and-effect relationship between the pressure field and the velocity field is circular, as explained in my suggested revision. Both flow curvature and changes in flow speed are caused by differences in pressure, and the differences in pressure are sustained by the changes in flow direction and speed.

The issue of cause-and-effect and "why" has come up before, and I still don't entirely agree with the hard-line take on that topic by Dolphin. I agree that for some fundamentals like Newton's second law we don't understand the "why". But at other levels, I think it's perfectly reasonable to talk about causation and "why", as in saying that a force causes an acceleration. So "why" does something accelerate? Because of a force. "Why" are force and acceleration related the way they are? We don't know. J Doug McLean (talk) 02:07, 6 April 2013 (UTC)


 * My opinion is that this article has long suffered from the fact that we editors are sometimes too close to the subject matter and get caught up in meta-discussions about the material instead of just presenting the material in a srtraightforward manner. We've all read the great Bernoulli/Newton debates of the 90s, and while this is interesting to folks who already know quite a bit about the subject I do not think it is helpful to re-hash that debate here.  Perhaps it belongs in its own wikipedia article, but the purpose of this article is to introduce the basic ideas in a manner understandable to the lay reader.  It seems to me that when we write about how we're we're going to explain the actual subject, we've made the artcle harder to understand by encapsulating it in a second layer of meta-analysis.


 * The result is that some readers, such as Westwind, get the mistaken impression that there is a lack of scientific consensus. Doug has suggested that we add language to correct that impression, eg; The mathematical theories are scientifically rigorous, are supported by empirical observations, and have been agreed upon by the scientific and engineering communities since the early 20th century.  My take is that this should go without saying, of course the theories are rigorous and well accepted - having to state it explicitly seems like we "doth protest too much".


 * To put a finer point on it, everything in science has multiple explanations; all science is rigorous (otherwise it wouldn't be science). There is no need to apologize for choosing one explanation over another, or to have to make assurances that the science is actually science.


 * So, I would advocate editing the article to downplay the Bernoulli/Newton "controversy", and to avoid giving the impression that these are two or more competing theories. As an example, take a look at the Bernoulli's_principle article itself - it states that BP can be derived from either conservation of momentum or by directly integrating newton's 2nd law.  To my eyes, it does so without implying that one is correct and the other is wrong, or that there's any controversy over which derivation is correct.  The subtext is that both are correct; somehow that subtext seems to be missing in this article.  I do think that this article should present both explanations, but I think we can do so without making such a fuss over it.  In fact, the more I think about the subject, the more I am persuaded that it's really one big explanation that fits together in harmony, rather than multiple competing theories.


 * By way of analogy, this subject reminds me of the blind men and the elephant. It's a great parable, but here we're writing an article describing elephants, not the arguments of the blind men. Statements like "some say the elephant is like a snake and some say it is like a wall" doesn't really advance the article.


 * In other matters, Doug has pointed out a definite shortcoming of the current article - the airfoil affects the flow over a wide area around it - and this fact along with the reasons why it occurs (ie the self re-enforcing interplay between pressure and fluid motion) should be incorporated into the article. I'll take a swat at this in coming days, most likely just stealing Doug's text.  Mr. Swordfish (talk) 21:20, 8 April 2013 (UTC)
 * The blind men and the elephant is spot on. Avoid the arguments between the blind men, and ignore those who ask "Why is there an elephant in the room anyway?" This is an article about lift, and the main conceptual and mathematical models that enable pilots and engineers to understand and manipulate lift. People who want to know why, or how did we get here, are looking for different topics, for which there may or may not be articles at this time. --Nigelj (talk) 22:33, 8 April 2013 (UTC)

I understand the urge to simplify things by downplaying the "meta-analysis", but I think it would be a mistake.

If we were the only ones who had ever discussed alternate explanations of lift, and if the controversies were limited to our little talk-page circle, then I'd agree that those discussions would be out of place in the article. But that's not the situation. The public folklore on this topic is full of misconceptions and erroneous explanations. And the controversies have been out there in public view for decades. I'll bet many of the potential readers of this article have already read a Bernoulli explanation (likely based on longer path length) or deflection explanation of lift and have probably also read somewhere that one or the other is wrong. And they're also very likely to have already gotten the impression from somewhere that the science is unsettled.

So the fact that the science on this particular topic is in good shape doesn't speak for itself. It needs to be spelled out. And given what's out there in the popular culture, we'd be short-changing the reader if we left out discussion of the pros and cons of the alternate explanations (and the outright errors in some versions). Whether we like it or not, these issues are now part of the topic of lift. Just the straight facts will not be enough to enable a reader to see through the fog.

Mr. Swordfish (Mr. Swordfish), you suggest that we should present "both explanations" of lift in the same way that the Bernoulli's_principle article presents its two alternate derivations. I don't entirely agree. Both derivations of Bernoulli are actually correct and self-sufficient. Our two simpler explanations of lift are both correct to some extent but also have significant shortcomings. The shortcomings shouldn't be swept under the rug.

Mr. Swordfish, you also say you're persuaded that the two simpler explanations really amount to "one big explanation that fits together in harmony". If you follow that idea to its logical conclusion and try to write it out as an explanation of lift, including the ideas of flow influence over a wide area and mutual interaction between pressure and velocity, what you arrive at is my "more comprehensive explanation" in my proposed revised version in my sandbox User:J_Doug_McLean/sandbox. The hard work is already done, including integrating the "one big" explanation with the simpler ones in the pedagogically favored order.

My proposed revision also attempts to put the mathematical theories and the various qualitative explanations in perspective in the new sections "The understanding of lift as a physical phenomenon", "Popular physical explanations of lift" and "Shortcomings of the popular explanations". Not all the material in these sections is new, but I think organizing it in this way makes things clearer. It explains the science in a way that would prevent misunderstandings like the one expressed by Westwind273, and it provides the necessary "meta-analysis" of the qualitative explanations. J Doug McLean (talk) 22:33, 17 April 2013 (UTC)


 * I get the science vs theology thing. For example, we know a lot about how gravity works, but we don't actually know why it works.  We feel confident that there are such things as photons, but are there such things as gravitons?  We don't know.  But I think the explanation of lift should be much closer to science than theology.  I strongly agree with the statements above that the article's explanation bends strongly toward circular reasoning:  Why does the air move faster on top?  Because there is lower pressure.  Why is there lower pressure?  Because the air is moving faster.  This circular reasoning really turns off the layman reader, and I think it is reasonable to ask science to avoid this kind of circular reasoning without resorting to theology.  The air on top is being made to both go faster and become lower pressure, but why?  The best I can figure is that it is some sort of combination of surface cohesion for the molecules closest to the top of the wing, and a whiplash effect for the molecules a bit further away from the wing.  This surface cohesion and whiplash combo creates both the lower pressure and faster airflow on top.  Is this correct?


 * I think the authors of this article need to realize how uniquely odd this article is in beginning the explanation with the "There are several ways to explain..." paragraph. I challenge you to find any other scientific article on Wikipedia that starts an explanation in this way.  For example, look at the Wikipedia articles on nuclear fission or freezing.  The explanation is straightforward and does not wade into this "several different ways" explanation.  --Westwind273 (talk) 04:50, 30 January 2014 (UTC)


 * I was not saying that the current article "bends strongly toward circular reasoning", and I don't think it does. What I was saying is that the article should be revised so as to state explicitly that the cause-and-effect relationship between pressure and velocity in an airfoil flow is circular.  Circular cause-and-effect is not the same thing as "circular reasoning", which generally refers to a false argument that purports to establish something that was assumed a priori and for which there is no support other than a circular argument.  The circular cause-and-effect relationship between pressure and velocity in fluid flows isn't in the circular-reasoning category because it's the way the physics actually works.


 * Of course Newton's second law is the key physical principle here. In many applications of Newton's second law, it's appropriate to think of the force as the "input" and the motion as the "output", but that way of thinking misses part of the picture in continuum fluid flows.  The motion of a local parcel of fluid does depend on the net force exerted on it by all the adjacent parcels in contact with it, consistent with Newton's second law.  But that force depends on the motions of the adjacent parcels, which depend on the motions of the parcels adjacent to them, and so on.  Because we're dealing with the motions of many parcels, all interacting with their adjacent neighbors, we effectively have a situation in which the motions depend on the forces, and the forces depend on the motions, i.e. circular cause-and-effect.


 * So of course science should avoid circular reasoning, but circular cause-and-effect between pressure and velocity is a fact of life in aerodynamic flows, and a good qualitative explanation of lift needs to make that clear.


 * One way to summarize the relationship between pressure and velocity is as follows: The pressure gradient causes a fluid parcel to accelerate (consistent with Newton's second law), and the combination of the parcel's inertia and acceleration causes the pressure gradient to be sustained.  The necessity of acceleration to sustain the pressure gradient involves Newton's third law.  When the pressure gradient is nonzero, a fluid parcel experiences a net pressure force exerted by it's neighbors.  A net force on a fluid parcel can exist only if the parcel pushes back, consistent with Newton's third law.  In the effectively inviscid flow outside the boundary layer, a fluid parcel can push back only through the combination of its inertia and acceleration.


 * I'll admit that the idea that the parcel's acceleration "sustains" the pressure gradient isn't that east to grasp intuitively. Does "sustain" mean the same thing as "cause" in this case?  I think the answer is yes, but a bit indirectly.  I found it helpful to think of a simple example from solid mechanics.


 * Think of a square block of wood, a couple of inches on a side, resting on a rigid table. Center your thumb on the top of the block and press downward.  The forces exerted on the external surface (your thumb pushing down on the top and the table pushing up on the bottom) cause the stress field throughout the interior of the block to be altered.  For one thing, there will be a non-uniform distribution of vertical compression stress, likely more concentrated directly under your thumb and more spread out at the bottom of the block where it presses on the table.  This will be balanced by a non-uniform distribution of shear stress such that the net force on any parcel of material in the interior is zero, consistent with Newton's second law.  At the local level at any point in the interior, the only cause we can identify for the compression-stress gradient is the shear-stress gradient.  Locally, the two stresses are engaged in a mutual interaction, i.e. circular cause-and-effect.  At the global level (the whole block), the cause of the whole non-uniform compression-stress field is the forces applied at the surface.  There is circularity at this level as well because the distribution of compression stress applied by your thumb depends on the deformation of your thumb, which depends on the distribution of stress.  This is not circular reasoning, just circular cause-and-effect associated with a mechanical interaction.


 * We can apply the same line of reasoning to an airfoil flowfield. Because the shear stress in most of the field is insignificant, the compression-stress gradient (i.e. the pressure gradient) acting on any fluid parcel must be balanced by fluid acceleration instead.  At the local level, the interaction between the pressure gradient and the acceleration is mutual, or circular, just as it was with the two interacting stress gradients in the solid.  At the global level, the cause of the non-uniform pressure field (and thus also of the non-uniform velocity field) is the force applied to the flow by the airfoil, acting at the airfoil surface.  As in the case of the wood block, the pressure field in the flow is just a state of stress in the interior of the domain, resulting from the application of forces at the boundaries, and distributed in a manner consistent with the laws of motion throughout the field.


 * I think this answers the questions that crop up a little later in the exchange between Westwind and Swordfish: Why does the flow above the upper surface follow a curved path?  And why is there a pressure gradient associated with that curvature?  At the local level, the answer is in the mutual interaction between pressure and velocity that I've already described.  At the global level, the ultimate cause of the non-uniform pressure and velocity fields is the force exerted on the flow by the airfoil.  This sounds unsatisfyingly circular because that force is just the equal-and-opposite reaction to the lift force that we're trying to explain.  But that's how the physics works.  The force exerted by the airfoil makes the flow non-uniform, and the non-uniform flow exerts force on the airfoil.  The cause-and-effect is circular even at this global level, but it's all tied together by the facts that the airfoil shape and angle of attack impose a boundary condition on the velocity at the surface and that Newton's second law applies throughout the field.  This is circular cause-and-effect, but not circular reasoning.


 * On a related issue raised by Westwind273, the following of the curved surface by the flow has nothing to do with "surface adhesion". Air molecules don't adhere in significant numbers to solid surfaces, and air can't be put in tension.  The background atmospheric pressure is high enough that the pressure at the airfoil upper surface, though lower than ambient, is still strongly positive in an absolute sense.  So the flow follows a curved path and is able to follow the convex upper surface because it is pushed from above by higher pressure.  There is no pulling from below.  And the following of the curved surface has nothing to do with viscosity either.  J Doug McLean (talk) 19:45, 3 April 2014 (UTC)


 * If readers see circular reasoning when they are trying to find a satisfying explanation for lift, it is most likely because the question "Why does the air move faster on top?" is not a particularly serious scientific question. (I think the best answer to this question would be "Because air observes the laws of physics"; but this is unlikely to satisfy many of the people who ask "Why does the air move faster on top?") It is a bit like Kepler's laws of planetary motion. Science is absolutely fascinated that the planets move in such a regular, repeatable and predictable manner that Kepler was able to postulate three laws that accurately summarise their motion; but science is not at all interested in the question "Why do the planets obey Kepler's laws of planetary motion?" If someone (presumably a layman) set about trying to give a fundamental explanation as to why the planets move in a regular, repeatable and predictable manner, it is my guess that he would end up presenting circular reasoning.
 * Similarly, the air moves around an airfoil in such a regular, repeatable and predictable manner that we can see its motion is consistent with Newton's laws of motion, the Kutta condition, Bernoulli's principle etc. If the air did not move faster across the top of an airfoil, it would demonstrate a flaw in these fundamental laws of physics.
 * The scientific approach to lift is firstly to select Newton's laws of motion or the Kutta condition or Bernoulli's principle, explain it in some detail and then present the phenomenon of lift on an airfoil as a practical example of Newton or Kutta or Bernoulli. If it is done in this way, the question "Why does the air move faster on top?" doesn't arise because the air moving faster on top is exactly what the Kutta condition predicts for airfoils and lots of other bodies with sharp edges.
 * If Lift (force) attempts to answer this question, or give the impression it is attempting to answer this question, there is grounds for amending the article to avoid that impression. Dolphin  ( t ) 05:48, 30 January 2014 (UTC)


 * Dolphin, I'm going to disagree with you here. Why does the air move faster on top? is perfectly reasonable scientific question, and the article does explain it, although in a somewhat roundabout (but not circular) manner. It's not presented in this order in the article, but the pieces are there to fit together the following explanation:


 * Why does the air move faster on top?
 * Because there is a region of low pressure along the top of the wing. According to Bernoulli's principle when air flows into a region of low pressure it speeds up. This is because there is more pressure behind than in front which results in a net force on the air molecules. Consequently they accelerate to a higher speed.
 * Ok, so why is there a region of low pressure on top of the wing?
 * So, why does the air follow a path that is curved?
 * It is deflected by the wing, with the geometry of the flow path dependent on the shape of the wing and the angle of attack. It's obvious why it is deflected downward by the bottom the wing - the wing is solid and there is nowhere else for the air to go. Along the top, the air follows the surface of the foil, resulting in a curved path.
 * Why does the air follow the surface of the foil instead of just continuing on in a straight line?
 * At this point we're getting beyond the scope of the article, but this is usually explained using viscosity.
 * I don't think Why does the air move faster on top? is central to the topic of lift, so I wouldn't re-structure the article to answer it. But the answer is there if the reader is willing to hunt for it.  And the reasoning is not circular.


 * Regarding Kepler's laws of planetary motion, they can be derived from Newton's laws (including the law of gravity), although since Kepler died a dozen years before Newton was born he obviously didn't derive them that way. So a reasonable non-circular explanation of Kepler's laws would be to start with Newton and proceed from there - Kepler's laws are true because they are a logical extension of Newton's laws. Of course, this begs the question of Why are Newton's laws true?, and the answer to that is that you've got to start somewhere and the reason we accept them (without a logical proof starting from more fundamental assumptions) is the millions of observations and experiments confirming them.


 * All this said, I do agree that the various physical phenomenon surrounding lift (a net force, pressure differences, speed differences, air changing direction etc.) can be explained starting from basic principles. And I think the present article does this.  Mr. Swordfish (talk) 15:54, 30 January 2014 (UTC)


 * Mr Swordfish wrote: Because the air is following a path that is curved. Euler's equation, which is derived directly from Newton's laws says that whenever a fluid follows a curved path there are pressure differences, with lower pressure on the inside of the curve and higher pressure on the outside.  The flow turning causes a region of low pressure along the top of the wing.
 * This is what I don't understand. Your explanation of the first and third questions here helped me a lot, but it is this middle second one that still seems unexplained to me.  Specifically, why does air that follows a curved path have lower pressure on the outside?  I know that Euler's equation says that it does, but why?  Is it simply that the outside path is longer and therefore the molecules get spread out over a greater distance?  This seems remarkably close to the equal-transit-time theory, which we know is false.  Is the answer to this theology?  I think it should not be.  I read the Wikipedia article on Euler's equation for fluid dynamics, and it did not help me.  --Westwind273 (talk) 20:37, 30 January 2014 (UTC)


 * Sorry, I was being a bit imprecise. When I said "Euler's equation" I meant the one referenced in this article, which is different than the one treated at Euler's_equations_(rigid_body_dynamics).  So, it's unsurprising that that article didn't shed much light on it.  Mathematically, the equation in question  (dp/dr = rho*v^2/R) is derived by just writing an expression for centripetal force, applying F = ma and doing a bit of algebra. Babinsky's paper ( http://iopscience.iop.org/0031-9120/38/6/001/pdf/pe3_6_001.pdf ) has a concise derivation at the very end.


 * To get an intuitive notion of why it's true, ie why there is less pressure on the inside of the curve, imagine riding on a train between two cushions pressing equally on you from the sides - as the train goes around a curve, your body will press on the outside cushion and pull away from the inside cushion; less pressure on the inside and more pressure on the outside. The tighter the turn and the faster the train is moving, the greater these pressure differences become.  The analogy only goes so far, but air following a curved path experiences the same forces.  It's just a consequence of the centripetal force necessary to make the air follow a curve. Mr. Swordfish (talk) 22:28, 30 January 2014 (UTC)


 * Yes, this is what I was trying to understand. It is quite similar to what I was saying before.  The overall airflow follows the curve of the wing on top because of the tendency of the air molecules to adhere to the wing, once they come in contact with it.  But at the same time, there is a kind of whiplash centripetal force that is throwing the air molecules away from the top of the wing.  The result is a powerful spreading out of the air molecules in the area above the wing, resulting in lower pressure there and the resulting lift.  Does this sound right?  Could the article be modified to give some kind of explanation like this to the layman?  --Westwind273 (talk) 00:38, 31 January 2014 (UTC)


 * There is no force "throwing the air molecules away from the top of the wing". However, there is inertia which would make the air follow a straight line in the absence of any force. Since there is a force pulling the air down towards the wing surface, the air changes direction.


 * I'm not sure what you mean by "a powerful spreading out of the air molecules in the area above the wing", but if you mean the air becomes less dense (fewer air molecules per unit volume) that is incorrect. In a first approximation, the air is incompressible meaning the density remains constant throughout the airflow.  This is not strictly true since air does compress at sufficiently high airspeeds, but trying to explain lift via changes in air density is barking up the wrong tree. Mr. Swordfish (talk) 14:58, 31 January 2014 (UTC)


 * I think the authors of this article need to realize how uniquely odd this article is in beginning the explanation with the "There are several ways to explain..." paragraph. I challenge you to find any other scientific article on Wikipedia that starts an explanation in this way. 


 * Two that come to mind immediately are derivation of Bernoulli's principle which states: "The Bernoulli equation for incompressible fluids can be derived by either integrating Newton's second law of motion or by applying the law of conservation of energy" and the proofs of the fundamental theorem of algebra.  Granted, the latter doesn't call a lot of attention to the fact that there are dozens of different lines of reasoning that can be used to arrive at the result, it just presents them.  The phrases "yet another" and "still another" occur numerous times.


 * A quick search or two ( https://en.wikipedia.org/w/index.php?title=Special%3ASearch&profile=default&search=several+ways+to+explain&fulltext=Search, https://en.wikipedia.org/w/index.php?title=Special%3ASearch&profile=default&search=multiple+explanations&fulltext=Search ) turns up thousands of wikipedia articles that present multiple explanations. Which is unsurprising, since there are usually several ways to explain any phenomena. Insisting on only presenting one when the literature presents several would violate WP:NPOV Mr. Swordfish (talk) 16:25, 30 January 2014 (UTC)
 * Hi Mr Swordfish. Thank you for taking the time to write such detailed explanation on the subject. I don’t have any major objection to what you have written. I think my only objection is that your explanations, regardless of their correctness, are not answering the question “Why does the air move faster on top?” but that is not because there is anything wrong with your answer; it is because the question is not a good one.
 * Let me illustrate my thinking. Person A might ask “Why do ripe apples fall to the ground?” Person B might answer “Because of Newton’s law of universal gravitation.” Person A might be entirely satisfied with this answer, but I don’t find it a satisfying answer at all. Ripe apples were falling to the ground for many millions of years before Newton was born so Newton cannot possibly be part of the explanation of falling apples. (Science is fascinated that ripe apples, and all other unrestrained objects, always fall towards the center of mass of the Earth with predictable initial acceleration, but science has little or no interest in why.)
 * I would prefer the conversation go like this: “We observe that ripe apples always fall to the ground. Is Newton’s law of universal gravitation consistent with this observation?” To which the answer is “Yes, Newton’s law of universal gravitation appears to be consistent with this observation.” (Albert Einstein made some observations and found that Newtonian mechanics were not consistent with the observations, so Einstein developed a replacement system of mechanics that more closely match his observations. The universe does not change its ways in order to behave in accordance with our laws!)
 * In your explanation of why there is lower pressure on the inside of a curve than on the outside, you have strayed too close to suggesting it is because of Euler’s equation. The pressure gradient across curved streamlines existed for millions of years before Euler’s birth so Euler’s equation is not an explanation of why this pressure gradient has always existed. However, it is reasonable to say “We observe a pressure gradient across curved streamlines. Is there any scientific principle that matches this observation?” To which the answer is “Yes, Euler’s equation is consistent with these observations”. Euler’s equation has stood the test of time and we confidently use it to predict pressure gradients and the curvature of streamlines, but we should not imagine Euler or his equation is an answer to the question “Why is there a pressure gradient across curved streamlines?” That is not a sound question for a scientist. Dolphin  ( t ) 06:23, 31 January 2014 (UTC)


 * My background is mathematiics, so I'm used to working with axiomatic systems. To me, Newton's laws of motion are like axioms, and if I can deduce that something logically follows from the axioms I'm satisfied - "why does X happen?  because it's logical consequence of Newton's laws" - is a perfectly acceptable explanation of why in my book.  I understand that not everyone will agree, and that I'll never get an answer to why the axioms (Newton's laws) are correct.


 * If someone asks me "why is the sky blue" or "why are there infinitely many prime numbers" or "why is a catamaran harder to tack than a monohull?" I'll try to give them an answer based on scientific principles and logic. I don't scold them for asking an unsound question, or say that we can never know why about anything.  Agree that we can never really know why, but we can explain complex things in terms of simpler, easier to understand concepts.  Anyway, we've kind of gotten away from discussing the article, so I'll let you have the last word if you care to.  Mr. Swordfish (talk) 15:47, 31 January 2014 (UTC)

I have been grappling with the difficulty I see whenever Wikipedia pretends to answer scientific questions that begin with “Why” such as "Why does an airfoil generate lift?"

It is true that Wikipedia (and others) give answers in response to these questions but I don’t believe the question they are answering is the one beginning with “Why”. Superficially, it appears that an answer has been given to the Why question, but philosophically I doubt it.

I would prefer it if Wikipedia answered the question “How does an airfoil generate lift?” It is easy to give a genuine answer to this question by referring to the flow pattern:
 * When a symmetric airfoil is moving relative to the surrounding air but it isn’t generating lift, the flow patterns above and below the airfoil are identical. At each point in the atmosphere above the airfoil, the flow speed and air pressure are identical to the flow speed and air pressure at the corresponding point below the airfoil. The force of the air pressure acting on the top surface of the airfoil is equal to the force acting on the bottom surface. The two forces balance and no lift is generated. But when a symmetric airfoil is generating lift, the air is approaching the airfoil with a non-zero angle of attack and the flow pattern above the airfoil is significantly different to the flow pattern below the airfoil. At each point in the atmosphere above the airfoil, the flow speed and air pressure are different to the flow speed and air pressure at the corresponding point below the airfoil. In particular, the streamlines adjacent to the top surface are closer together than the streamlines adjacent to the bottom surface. The air is moving past the top surface faster than it is moving past the bottom surface and the force of the air pressure acting on the top surface is less than the force acting on the  bottom surface. The two forces don’t balance and the resultant is called lift.

I think this kind of response is an answer to the question “How does an airfoil generate lift?” rather than the question “Why does an airfoil generate lift?” Everyone agrees on the answer; it is the question that still perplexes us! Dolphin ( t ) 06:31, 26 February 2014 (UTC)

++++ The OP stated that humans do not understand why airfoils generate lift.

Now, "Humans understand lift" could mean several different things. The answer depends upon the meaning assigned to the proposition.

But the common interpretation in the scientific community isn't at all that vague. It might be stated: "broadly accepted scientific theory (in this case, mainly the laws of Newtonian fluid dynamics) accounts for the all the salient observed facts concerning lift, without arbitrary ad hoc assumptions, as confirmed by repeatable experiments." If we were to accept this interpretation, then it would remain only to ask ourselves, do we agree or not? I think that all experts on fluid dynamics would agree that the proposition is true. If so, then the article should reflect that accepted dogma, until and unless it is superseded by a new paradigm.

It is understandable that some participants in the discussion are not experts: they don't know Napier's equation, they don't understand what it does and doesn't say about causality, they don't understand the relationship in science between theoretical models and experiments, etc. But the article should not reflect their ignorance...it should reflect humankind's current best understanding of the subject.

That's just my opinion. Do you agree? Mark.camp (talk) 00:33, 27 March 2014 (UTC)
 * I agree.
 * There is a temptation to imagine that to fully and clearly understand why airfoils generate lift, the explanation must be esoteric, complex, with lots of advanced math. There is no reasonable ground to imagine that. Different people will prefer different explanations of lift, and all those different explanations can be legitimate. For example, in one of John D. Anderson's books he goes looking for the simplest explanation of lift and concludes that it is based on the observation that the pressure on the upper surface of an airfoil is different to the pressure on the lower surface, resulting in an aerodynamic force; lift is the component of the aerodynamic force perpendicular to the vector representing the relative motion between the airfoil and the free stream of fluid moving past it. That is a simple explanation but it is entirely legitimate as an explanation of how (or why) airfoils generate lift. Many people, including those who embrace Anderson's simple explanation, are entitled to object to the OP's suggestion that Humans do not fully and clearly understand why airfoils generate lift. Dolphin  ( t ) 05:41, 27 March 2014 (UTC)

+ + + + + + I agree that the science allows multiple formulations of explanation or presentation. This an inherent characteristic of Newtonian physics; it is a set of definitions, assumptions, and equations, and the equations can be transformed mathematically without changing their meaning. Often, for example, one can present the predictions of science, in a given case, in terms of forces, or alternatively in terms of energies. You can present Newton's original way of describing the evolution of a system, or you can use the form developed later, the law of least action, to describe the very same system evolution. As a final example, you can present Maxwell's laws in differential form or in integral form. Same science, same math, different presentation.

But that is an entirely different question from that of presenting the explanation of lift as the result of net force on the body, as Anderson does. This is a *stage* in the explanation, not a *form* of the explanation. I think we would all agree that it is a necessary introductory stage. But it is only the beginning. Any curious reader will be happy to understand at this simple level but will immediately want an explanation of the pressure field and velocity field itself: "OK, I understand that if that is the velocity field, then that is the pressure field, and the conclusion is that there is lift.  But now I want to know WHY those are the fields". The next stage, I think all of us would perhaps agree, is to proceed to a very simple model (called ironically "complex" potential) that accounts for a particular solution for the two fields that is consistent with lift (and an infinite number of others consistent with the boundary condition but NOT consistent with the observed lift!). But again a persistent reader will say, "ok, I see how ONE of the infinite number of possibilities yields the observed lift, but you have not explained why nature always settles on JUST THAT ONE, the one that happens to produce the observed lift." The next stage, I think you will agree, is to allow a slightly more realistic model...still far from the truth, still no turbulence, no separation, no chaos...but one which can account for the solution (Kutta condition, other approximations concerning effective shape of the foil rather than its actual shape) which nature has been proven experimentally to choose, very approximately.

A good explanation for this article, I think, would allow the reader to pick his own level of advance. If he didn't understand the first stage about net force, he could understand just that much and be satisfied to stop. If he had the next obvious question, he could get a clear exposition of that. If he still were curious, he could pursue the next level, which allows for friction and thus the Kutta condition. If he were still curious, he could move on to boundary layer, and then to turbulence, and then to separation, and then to chaos....

Mark.camp (talk) 00:36, 29 March 2014 (UTC)


 * I agree that while this article is very informative, it does not really contain a clear and simple explanation of the obvious forces at work.


 * As for why air 'moves faster over the upper surface of the wing', there are some inbuilt misconceptions here. It only moves faster in relation to the lower surface air and not for the reason most people think.


 * '''The air does not 'speed up' over the upper surface - in fact relative to the static air, it slows down...

'''


 * If we look at the Cambridge University Wind Tunnel video  (yes wind tunnels have some limitations) and watch the pulsed smoke (with the viewpoint of the wing moving and the air still) we can see that the air under the lower surface is pushed forwards (i.e. in the direction of the wing's travel) and downwards as the wing pushes into the static air. (This is a large component of lift hence the very significant convex on the underside of SC wings-particularly at the root)


 * The upper surface air is pushed up and over the leading edge by the pressure bubble/bow wave under the leading edge and as it turns the corner and changes direction (as it must for fluid continuity) the rapid change of direction and the rapid equalisation following the mechanical displacement effect of the wing, cause the pressure change. (This is why high speed wings (SC and others) have minimal upper surface camber-to minimize the pressure drop from angular change induced into the air and thus reduce standing shock waves.)


 * If you watch the video and the vertical columns of pulsed smoke you will see that the air streamlines near the wing are pulled forwards (i.e. slowed down) not sped up over the wing. Follow a single column of streamlines and watch how they are displaced by the wing.


 * In particular, look at the reference streamlines near the wind tunnel wall and compare them with their counterparts above and below the wing. You will see that being undisturbed, they have 'travelled further' than the streamlines near the wing. The 'speed differential between upper and lower is only a relative thing. The pressure differentials are caused by the wing pushing the air forwards and downwards under it and bending the upper air through an angular change relative to the AOA of the upper surface. F=Ma.


 * Pressure differentials are an effect of the angular change of the static air not the cause of lift. No angular change= no pressure diff.


 * There are pieces of evidence to support the idea that this is important for conceptual understanding:


 * First: view a flat plate wing in a wind tunnel or using NASA's FoilSim. Flat plates have a better L:D ratio than aerofoils and do not have any camber, so this simplifies things. Observe that flat plates generate lift in exactly the same way as an aerofoil but stall earlier. This demonstrates simple angular mechanical intervention. (Oddly unless I have misread it, this article claims that flat plates create more drag than streamlined aerofoils yet simple simulations like NASA's Foilsim show this to be incorrect as does basic common sense because flat plates have much less 'wetted' area means that for a start, they must produce less form drag. All that is left is induced drag and that is AOA/angular flow change dependent.


 * Second: Think of the air as a fluid (which it how it behaves subsonically) If you imagine any of the functions or effects of an aerofoil occurring under water, all of s sudden it is easier to conceive. We don't ask why the water follows the curve of the upper surface AOA because it is self evident. After all-what is the alternative?


 * It seems to me that unless you are an aerodynamicist and fiddle at the edges of the obscure, wings are not that complicated. Newton explains them just fine. I don't go into Bernoulli as it is derived from the 2nd law anyway and causes more confusion than clarity.


 * I have found the bigger problem in teaching undergraduate pilots is debunking all of the silly 'theories' like ETT that are still taught in flying schools!


 * Just my two cents Completeaerogeek (talk) 06:09, 22 October 2014 (UTC)

Proposed changes to early sections of the integration
This integration is a good start. With this as a starting point, I'm proposing some moves and revisions for the early sections. I've installed them in my sandbox User:J_Doug_McLean/sandbox for your review, and I'll explain my rationale here.

I propose moving my new section "The understanding of lift as a physical phenomenon" back to a prominent position, just after the overview. It doesn't fit with the rest of the material in the "A more comprehensive physical explanation" section and would be much less effective there. Discussions over the years on this talk page have convinced me that there's widespread misunderstanding of the status of the hard science and how the qualitative explanations relate to it, so I think this context-setting section should come before the explanations start. I also propose adding words to the effect that the qualitative explanations cannot provide quantitative information for engineering. That's a property of all qualitative explanations, so I think it belongs here rather than in its current place as a limitation of flow deflection.

The integration's "Description of lift on an airfoil" section seems to me to mix explanation and description in a way that's not coherently organized and thus not as easy to follow as it could be (e.g. several of the subheadings under "Newton's laws: lift and the deflection of the flow" don't fit there.). I think the whole section could benefit from reorganization.

So I'm proposing rearranging this as two main sections. The first is "Simplified physical explanations of lift on an airfoil", which gives the deflection explanation first, in keeping with the AAPT advice. I put the points covered by the current "Flow on both sides of the wing" in with the deflection explanation. I kept separate headings for limitations of deflection and Bernoulli, but I'm proposing changes to what's included in those sections. Failure to produce quantitative information is no longer a limitation of deflection, as it's been covered above. I also propose a change under the limitations of Bernoulli (see below).

I propose putting everything else in the current "Description..." section under "Basic attributes of lift", with the current "Summary" statement moved to the front, and keeping the subheadings other than "Flow on both sides of the wing", the main points of which are would now be covered under flow deflection. I'd really prefer to put this section between "Overview" and "The understanding of lift as a physical phenomenon", but putting it after the simplified explanations, so that flow deflection can come earlier, is a compromise I can live with.

I look forward to feedback on these proposed changes. J Doug McLean (talk) 20:01, 20 June 2014 (UTC)


 * Apologies for taking some time off from this article. We didn't make the July 1st deadline, but hopefully we can get things moving again in the next several weeks.


 * I think the section https://en.wikipedia.org/wiki/User:J_Doug_McLean/sandbox#The_understanding_of_lift_as_a_physical_phenomenon is a fine section and don't disagree with anything in it. However, I do not think it deserves to be the opening section fight after the overview.  I recognize that opinions may vary about this, but my issue is that the section is rather meta- that is, it talks about the explanation rather than simply giving it.  At some point some amount of meta-analysis about the explanation is in order, but I prefer to cut to the chase and go right in to the explanation. Mr. Swordfish (talk) 20:29, 22 July 2014 (UTC)


 * I have integrated this section into my draft later in the article. Mr. Swordfish (talk) 19:42, 23 July 2014 (UTC)

Proposed changes to "Mathematical theories"
The integration makes some changes to the new section "Mathematical theories of lift". The added explanation of getting velocity vectors from CFD and then the pressure from Bernoulli isn't applicable to CFD in general, only to potential-flow methods. I've taken a crack at fixing that and at dealing with the repetition regarding the Kutta condition.

I look forward to feedback on these proposed changes. J Doug McLean (talk) 20:01, 20 June 2014 (UTC)


 * I'm ok with this. I've replaced this section in my draft with the version from your draft. Mr. Swordfish (talk) 20:42, 22 July 2014 (UTC)

Proposed deletion or substitution of two figures
The figure currently illustrating streamtube pinching, with the caption "Streamlines around an airfoil in a wind tunnel. ..." seems to me to be contradictory. The horizontal bars appear to be intended to represent wind-tunnel walls, but the streamlines don't appear to be constrained by the walls. And wind-tunnel walls are not relevant to the explanation anyway. I propose, as I show in my sandbox, replacing this figure with the flow animation, with a caption tailored to the streamtube pinching explanation.

The figure caption "Uniform flow plus vortex flow (circulation) gives the total flow below" is not technically accurate. To get the flow around the NACA 0012 you would need to add a particular distribution of vortex strength along the chord (not just simple circulation), as well as a distribution of sources and sinks to represent airfoil thickness. I propose just deleting this figure. J Doug McLean (talk) 06:13, 21 June 2014 (UTC)


 * I agree that the diagram depicting streamtube pinching leaves something to be desired, but it is what's available in the public domain. That said, I think it does a better job of depicting streamtube pinching than the animated picture (https://upload.wikimedia.org/wikipedia/commons/9/99/Karman_trefftz.gif).   Ideally, we'd find a better diagram. I'll see what I can turn up.


 * Agree that the other diagrams are not ideal either. The idea was lifted from one of Anderson's books, but I did the graphics and I'm a terrible graphic artist.  I've deleted them from my draft since they may give a too-simplified impression.  I do think that a picture representation of the idea of vortex flow + steady flow == total flow helps with a layman's understanding of circulation.  But maybe it doesn't need to be in this article.


 * In coming days I hope to take a good look at both versions of the opening sections and attempt further integration. What would be nice would be to get a third (or fourth or more) editor(s) to help with this. Mr. Swordfish (talk) 20:58, 22 July 2014 (UTC)


 * I found a better picture for the streamtube pinching. Streamlines around a NACA 0012.svg I think this depicts it better than the animated diagram, and it doesn't have the issues you mention that the old/current picture have. Mr. Swordfish (talk) 14:14, 23 July 2014 (UTC)

Lift does NOT only apply to airfoils - they are a special case
This is my first ever contribution to Wikipedia - so I apologise in advance if I breach any guidelines.

My concern is that this whole article suffers from a major distortion in its emphasis. Lift is defined as being the force perpendicular to the motion of an object relative to a fluid. The object does not need to be an airfoil (wing shape). But this whole article is utterly preoccupied with airfoils. And as a consequence with Bernoulli's principle (and its various possible explanations). Therefore the article largely duplicates the existing Bernoulli Principle article. They should be merged together.

The article also - as a consequence of the above - completely ignores other mechanisms whereby Lift is generated, and all the real-world examples. EG the rudder of a boat; a kite; the tail of a fish or dolphin, a scuba diver's fins, a weathervane  etc etc. There are plenty of examples of flat objects generating lift purely because of their angle of attack, nothing to do with Bernoulli and airfoils. These should be the first topic in an article on Lift. Airfoils should be mentioned as a special case with a link to the existing articles on Airfoil and Bernoulli. To ignore the most simple and straightforward source of Lift is bizarre. — Preceding unsigned comment added by 94.11.120.238 (talk) 00:39, 4 September 2014 (UTC)
 * Welcome to Wikipedia, and thanks for your contribution. The only deficiency with your first edit is that you forgot to sign your "name" by typing four tildes - see WP:SIGN. This should be done at the end of all contributions to a Talk page. (You will notice a Bot has signed your "name" for you.)
 * But this whole article is utterly preoccupied with airfoils. I disagree. The word "airfoil" is not used until the seventeenth sentence. Many textbooks concede that even a flat plate can experience lift when it is behaving like an airfoil. This article is about lift and how it is generated. It is reasonable to clarify the meaning of lift by describing it as a force experienced by airfoils and other objects when they are behaving like airfoils.
 * The article also completely ignores other mechanisms whereby lift is generated, ... I disagree. The seventh sentence in this article states "Lift is also exploited in the animal world, and even in the plant world by the seeds of certain trees." (Or are you alluding to the Magnus effect?)
 * To ignore the most simple and straightforward source of lift is bizarre. You forgot to state what it is you believe is the "most simple and straightforward source of lift." Please clarify.
 * Please read the article carefully and if you find a sentence or paragraph that states or implies lift is produced only by airfoils, or predominantly by airfoils, please let us know by replying here on this Talk page. Dolphin  ( t ) 06:38, 4 September 2014 (UTC)

Release candidate?
I've spent some time further integrating Doug's draft into my draft, adopting much of his organizational structure and copying entire sections. At this point I think we have a release candidate.

The immediate question is whether it is an improvement over the current article. If so, we should replace the current article now and move forward with further improvements in-place. No article is ever "done" on wikipedia; refinements, additions, and improvements will likely be made in coming months and years.

If the draft is not an improvement then let's discuss how to improve it so that it is better than the current article.

My own opinion is that it's a stronger article. Many thanks to Doug for his effort and patience.

The draft is at https://en.wikipedia.org/wiki/User:Mr_swordfish/Lift Please take a look and share your opinion here. Mr. Swordfish (talk) 14:02, 25 July 2014 (UTC)


 * Great work! I have had a quick look at the release candidate and left a couple of comments at User talk:Mr swordfish/Lift. I will try to do more in coming days. Dolphin  ( t ) 06:38, 26 July 2014 (UTC)


 * I agree that it's a stronger article, and by that criterion it qualifies for release. But I'd still like to advocate for some further changes that I think would improve it further, if you'll bear with me.


 * The first has to do with the section "The understanding of lift as a physical phenomenon". We have agreement that it should be included, but not on where to put it.  I still think it should precede the qualitative explanations because it seems to me that when things aren't put in perspective at the start, the article ends up giving a misleading impression.  If you just "go right in to the explanation" as you prefer, the article gives the impression, intended or not, that that's how lift is understood, and that the rest, including the mathematical theory, is just filling in the details.  I think that's a misleading picture of how we really understand lift.  And I don't think that reading the meta-analysis later in the article, assuming the reader even gets that far, will be very effective in undoing the impression.  Better, I think, not to give the impression in the first place.


 * The mathematical theory is the bedrock of human understanding of lift. The qualitative explanations are secondary, really just attempts to square the theory with our intuition.  And that's a hard thing to do, given that the continuum flow in effect consists of innumerable little parcels of fluid moving in concert to get around the airfoil, each one obeying the 2nd law in a mutual interaction with its neighbors.  The theory handles this complexity correctly by requiring the solution of a set of PDEs, but our intuition doesn't do so well when faced with an entire flowfield.  For one thing, how the pressure field comes about in such a flow is very difficult to grasp intuitively.  The popular explanations either avoid the question altogether (the deflection explanation) or do it badly (the Bernoulli explanations), and even my "more comprehensive explanation" is shaky on this point.


 * So I feel strongly that before we dive into the qualitative explanations, we owe the reader a heads-up on where such explanations stand in the overall scheme of things, and why. "The understanding of lift..." attempts to do this, but it may not be entirely satisfactory as it stands.  It should perhaps be beefed up to make it clear that the difficulty of the problem dooms the qualitative explanations to fall short of being completely satisfying, not just that they don't produce numbers and that there's been disagreement on what to include in them.  In my sandbox User:J_Doug_McLean/sandbox I've added a sentence to try to do that.


 * There are times when some meta-analysis up front makes things better for the reader, and I think this is one of those times.


 * And the section heading should remain "The understanding of lift as a physical phenomenon", not "Understanding lift as a physical phenomenon". The former implies we're talking about the understanding held by the community at large (which is what this section is doing), while the latter implies a concentration on changing the understanding held by the reader.  I think the difference is significant.


 * Next are a couple more issues with headings:


 * The content in "Description of lift on an airfoil" isn't really description; it's explanation. I think "Simplified physical explanations of lift on an airfoil" is more consistent with the content.


 * I think the heading "Methods to determine lift on an airfoil" promises more than we deliver. To apply either "Lift coefficient" or "Pressure integration" you have to know something a priori that's tantamount to knowing the lift.  So these are really just relationships for converting one form of knowledge about lift to another; they don't really "determine" lift.  I propose deleting the heading "Methods to determine lift on an airfoil" and promoting "Lift coefficient" and "Pressure integration" to sections in their own right.


 * Most of the material in the section "Kutta-Joukowski theorem" is now covered in "Circulation and the Kutta-Joukowski theorem" in the "Mathematical theories of lift" section. I propose integrating some material from "Kutta-Joukowski theorem" into "Circulation and the Kutta-Joukowski theorem", moving the description of the Magnus effect to "Lift forces on bluff bodies", and deleting "Kutta-Joukowski theorem".  I've taken a crack at this in my sandbox User:J_Doug_McLean/sandbox.


 * Then a few technical issues:


 * Under "Flow deflection and Newton's laws", the statement "The resulting force upwards is equal to the time rate of change of momentum of the air downwards" is problematic on two counts, in spite of the fact that it has a citable source.


 * The first problem with the statement is that for it to be true, the downward force exerted by the airfoil on the air surrounding it would have to be the only force being exerted on "the air". There are many possibilities for how we can define the body of air we're considering, and this condition (no other force but the lift) isn't met in general.  The airfoil exerts a downward force on the inner boundary of the body of air surrounding it (at the airfoil surface), but the surrounding environment exerts unbalanced pressure forces on the outer boundary of the body of air.  This problem cannot be eliminated just by increasing all the dimensions of the "box" of air we consider, even to the limit of infinity.  As the box is made larger, the pressure disturbances at the outer boundary get weaker, but the area over which they act gets larger, and the integrated force remains comparable to the lift.  How much of the lift is accounted for by this pressure force rather than by momentum change depends on the proportions of the box.  For example, for a box that is very large horizontally compared to its vertical dimension, practically all of the lift is accounted for by pressure at the outer boundary, and practically none by momentum changes.  Only in the limit as the vertical dimension of the box becomes large relative to the horizontal is it true that lift is accounted for by momentum changes.


 * The other problem is that most such analysis in fluid mechanics deals with boxes whose boundaries are fixed in space. The time rate of change of momentum in such a box is zero in steady flow, and momentum changes must be assessed in terms of fluxes in and out, not the time rate of change.


 * Thus the simple explanation in terms of flow deflection is correct only if it's couched in vague terms such as "for the airfoil to deflect the flow downward, it must exert a downward force on the air". The more specific statement "lift is equal to the time rate of change of momentum of the air" is not correct in general.  I recommend deleting this sentence.


 * Under "Limitations of deflection/turning", the only limitation mentioned is that this explanation doesn't produce numbers. This limitation is a characteristic of all of the qualitative explanations and is now included in "The understanding of lift as a physical phenomenon".  I recommend substituting the paragraph under "Limitations of the flow-deflection explanation" in my sandbox version User:J_Doug_McLean/sandbox, which discusses some limitations specific to deflection.


 * Under "Increased flow speed and Bernoulli's principle", the first paragraph needs to stipulate that Bernoulli's principle requires steady flow.


 * In that same section, I recommend deleting the second paragraph. The statement "Bernoulli's principle does not explain why the air flows faster over the top of the wing" isn't true.  On the contrary, Bernoulli's principle tells us that if the air flows faster, it is because of the lower pressure.  It's just that that didn't help the originators of the Bernoulli explanations in boot-strapping their way toward an explanation of where the low pressure comes from, and they had to find other reasons for the faster flow.


 * We're getting close, but I'd appreciate it if you'd consider the above changes. Thank you for your patience.


 * J Doug McLean (talk) 01:58, 27 July 2014 (UTC)


 * A lot to respond to at once, so I'll break it up into bullet points;


 * Placement of the "The understanding of lift as a physical phenomenon" section - As I've stated before I think it s a good addition to the article, but leading with it seems off-putting to the general audience. And really, fundamentally I think our disagreement here stems from a different idea of who the intended audience is.  For the layperson who knows little about the subject (i.e. the vast majority of wikipedia readers) the section would make little sense without first providing some context.  I am open to moving it up in the article, say, between "Basic attributes of lift" and "A more comprehensive physical explanation."


 * Section heading - I don't see a big difference in meaning between "The understanding of lift as a physical phenomenon" and "Understanding lift as a physical phenomenon" to my eyes, the former merely has two extraneous words. But I can see how you would parse it differently than I, so I've reverted the heading to your original.


 * Section heading - "Simplified physical explanations of lift on an airfoil" is fine by me, I'll implement that change too.


 * Section heading - propose deleting the heading "Methods to determine lift on an airfoil" and promoting "Lift coefficient" and "Pressure integration" to sections in their own right. that sounds reasonable. I'll do it and see how it looks.


 * "The resulting force upwards is equal to the time rate of change of momentum of the air downwards" is problematic - I'm gong to punt on this one for now. Let me give it some thought and attention and I'll get back to you.


 * "Limitations of deflection/turning" - I think you make a reasonable criticism, but I also think there's a bit of strawman there - the basic deflection explanation does not refer only to forces "exchanged at the airfoil surface, where the air and the airfoil are actually in contact". The diagrams clearly show air being deflected at some distance from the foil, not just at the surface.  Agree that it doesn't  explain why the air is deflected a distance away from the foil, or how the force manifests itself as a pressure difference, but my take is that it doesn't have to. For instance, it also doesn't explain why the air moves faster over the top or any of a hundred other related phenomena. It does explain where the lift force comes from and that's the point of the exercise.  We have to be careful that we don't give the misleading impression that deflection is wrong or incorrect and I think the typical wikipedia user could get that impression from what you have written.  I'll take a stab at addressing your concerns by adding some of these issues to the list of limitations.


 * Redundancies in K-J theory section. Your proposal sounds fine.  I'll take a look at integrating your changes into my version.


 * Under "Increased flow speed and Bernoulli's principle", the first paragraph needs to stipulate that Bernoulli's principle requires steady flow. Ok, I'll add something to that effect.


 * "Bernoulli's principle does not explain why the air flows faster over the top of the wing" isn't true.  Yeah, that sentence has always bothered me too. Once you know that the pressure is reduced, BP tells you that the speed is faster. So it does explain why. Equal transit time doesn't explain why the air goes faster, and most explanations based on BP do not explain (correctly anyway) why the air goes faster. I'll look at changing that sentence.


 * It's now Friday, August first, and I think we are close enough that we can go live with the revision as-is. We can continue to discuss outstanding issues afterwards.  I'll make the changes outlined above and unless I hear objections I'll replace the current article with the draft early next week.Mr. Swordfish (talk) 15:19, 1 August 2014 (UTC)


 * UPDATE: I've now completed the above edits. One issue I see is in response to I propose deleting the heading "Methods to determine lift on an airfoil" and promoting "Lift coefficient" and "Pressure integration" to sections in their own righ... I propose integrating some material from "Kutta-Joukowski theorem" into "Circulation and the Kutta-Joukowski theorem", moving the description of the Magnus effect to "Lift forces on bluff bodies", and deleting "Kutta-Joukowski theorem".


 * I've done this in my draft (with the exception of the treatment of the Magnus effect - will take a look at that next) A question: does it make sense for Lift Coefficient and Pressure Integration to have their own sections, or does it make more sense for them to be sub-sections under "Mathematical theories of lift"?  I'm in favor of the latter, but could be convinced otherwise.  I'm going to move them under the math section pending further discussion.  Mr. Swordfish (talk) 18:29, 1 August 2014 (UTC)

Doug McLean wrote: ''...the statement "The resulting force upwards is equal to the time rate of change of momentum of the air downwards" is problematic... ''

I have to say that I was surprised by this, so much so that I needed to take a few days to think about it before responding. And the reason for the surprise is that the statement is merely a combination of Newton's 2nd and 3rd laws with dp/dt replacing ma. This should be about as uncontroversial as it gets. In the simple model where all we consider is the air flow and the foil, it follows directly from Newton's laws. Of course, if the air is being accelerated by something other than its interaction with the foil then that additional acceleration will not contribute to the lift force, but it seems clear to me from the context that we're not talking about that scenario. I've added some language to clarify:


 * The resulting force upwards is equal to the time rate of change of momentum of the air deflected downwards by the foil.

Agree that this total momentum change is difficult to calculate or measure. But in theory at least it must be equal to the lift force.

BTW, there's not just one cite for it, three others are included elsewhere: https://en.wikipedia.org/wiki/User:Mr_swordfish/Lift#cite_note-7 https://en.wikipedia.org/wiki/User:Mr_swordfish/Lift#cite_note-23 https://en.wikipedia.org/wiki/User:Mr_swordfish/Lift#cite_note-32

Mr. Swordfish (talk) 18:33, 4 August 2014 (UTC)


 * I'm also surprised by Doug's comment. Perhaps he is alluding to the idea that the time rate of change of momentum is equal to the aerodynamic force and not just its vertical component, lift. Dolphin  ( t ) 22:28, 4 August 2014 (UTC)


 * No. The problem has nothing to do with whether we consider the total force exerted by the foil or only the lift component.  See my response to Mr. Swordfish above. J Doug McLean (talk) 00:19, 7 August 2014 (UTC)


 * Your reasoning regarding placement of "The understanding of lift as a physical phenomenon" puzzles me. You argue that leading with it would be "off-putting to the general audience" and that it would "make little sense without first providing some context".  But providing context is what this section is intended to do.  It seems to me that launching directly into the deflection explanation without the context provided by "The understanding of lift as a physical phenomenon" gives a mistaken impression to the reader, to be remedied only later in the article:  "Oh, by the way, those explanations we gave you early on aren't the real story on how lift is understood."


 * As for making "little sense" to an audience without prior knowledge, I don't see it that way. The section is brief and to the point, and, I think, easy to understand.  I assume the target audience of an article in a general encyclopedia is literate adults, not children.  If I were the reader, I'd welcome the context-setting up front.  This is an encyclopedia article, not a mystery story.


 * So how do we decide this? Let's try a little meta-analysis of the arguments pro and con.  I've argued that logical exposition of the subject matter favors having "The understanding of lift as a physical phenomenon" precede the simplified explanations, so as to put them in context with general understanding of lift.  You haven't offered a counter-argument to this but have instead brought  up other issues:  "It's -meta."  "It's potentially difficult for an audience without prior knowledge to understand."  I think I've offered effective rebuttals to these arguments.


 * Regarding my proposed passage in "Limitations of deflection/turning", I think your argument that there's a strawman there is unjustified. I do say that the only forces referred to are those "exchanged at the airfoil surface, where the air and the airfoil are actually in contact", which is true.  I don't say that those forces are all that the explanation refers to.  I think the passage should be included.  It doesn't imply that deflection is incorrect, just that it leaves a gap in that it doesn't explain how a deeper swath of flow is deflected than is touched by the airfoil.


 * Regarding "The resulting force upwards is equal to the time rate of change of momentum of the air downwards", I thought I made it clear in my previous posting what the problem with this statement is, but your response indicates that you don't agree that the force exerted by the airfoil is not generally the only force exerted on "the air" as a result of the lift. Let's look at this further.


 * A crucial question raised by the statement is what is meant by "the air". Again, any body of air that you choose to define as "the air" surrounding the airfoil must have both an inner boundary where the airfoil contacts it and an outer boundary where the surrounding environment contacts it.  As a result of the lift there is generally an unbalanced pressure force on the outer boundary, so that the force exerted by the airfoil on the inner boundary isn't the only force resulting from the lift.  With some detailed bookkeeping this pressure force can be quantified.  If we put the outer boundary far enough from the foil, the idealized model of a uniform flow plus a vortex suffices, and we can draw general conclusions.  Please read my previous comments where I explain how the percentage of the airfoil's force that is offset by the pressure force on the outer boundary depends on the proportions of the outer-boundary box.


 * Anyway, for most ways of defining what is meant by "the air", the statement is untrue. For example, the reader might reasonably assume that "the air" refers to the whole atmosphere.  Given this definition of "the air", the downward force exerted on the air by the airfoil is completely offset by a distribution of over-pressure on the ground (see the famous figure 150 in Prandtl and Tietjens for what this pressure footprint looks like in 3D), so that the net force exerted on the air as a result of the lift is zero.  Then the time rate of change of the integrated vertical momentum of the air must be zero as well.  It's just Newton's second law, as you say.  So again, the problem with the statement in the current draft is that it's not generally true, because it doesn't account for all the forces.


 * Your proposed clarification, i.e. limiting the statement to "the air deflected downwards by the foil", doesn't fix the problem. The subset of the air that's actually experiencing downward acceleration is still a body of air that an outer boundary can be drawn around.  That body will still in general have a net pressure force on its outer boundary, so that the downward force exerted by the foil will not be the only force acting on that body of air.  Thus even your clarified version of the statement isn't generally true.


 * There is one way to define "the air" so that the statement is true, but I think it's too specialized and complicated to be appropriate for this article. Draw the outer boundary of the box so that the vertical dimension is much larger than the horizontal dimension.  In the limit as the vertical dimension goes to infinity relative to the horizontal dimension, the net vertical pressure force on the outer boundary vanishes.  Then all of the lift is accounted for by the momentum transfer and none by the outer-boundary pressure force.  But we're not done yet.  To observe the momentum transfer as a time rate of change, we have to take another special step.  The outer boundaries of the box must be assumed to be moving with the flow so that the box is not gaining or losing fluid anywhere along the boundary (This is different from the usual approach to control-volume analysis, in which box boundaries are fixed in the reference frame of the body).  Only for this very special definition of "the air" can we make the statement that the lift is equal to the time rate of change of the integrated downward momentum of "the air".  Unless we're willing to add these specialized qualifications to the statement (along with an appropriate citation), I think we should delete it.


 * This isn't just a quibble about rigor. The statement L = dp/dt is actually untrue for many reasonable ways of defining what is meant by "the air", i.e. it's untrue for anything other than an infinitely tall vertical sliver with boundaries that move with the flow.


 * The three sources you mention all make an error that's easy to make, i.e. applying the second law to a body of air, but without adequately defining what is meant by "the air" and without identifying all the relevant forces. The idea that "the air" generally has lift-related forces acting on it other than that exerted by the foil seemingly didn't occur to them. J Doug McLean (talk) 00:19, 7 August 2014 (UTC)


 * There seem to be three remaining areas of disagreement;


 * Placement of the "The understanding of lift as a physical phenomenon" section. At this point, I don't think either of us are going to be swayed by the others opinion. My sense is that we have a disagreement over the intended audience and how best to serve that audience.  My view is that the section makes a lot of sense to those who are already familiar with the material, but that it's "inside baseball" for 99% of the audience. My editorial sense is that we shouldn't lead with it. This will have to be resolved by seeing what the other editors think or getting a third opinion.  I do take exception to your characterization "Oh, by the way, those explanations we gave you early on aren't the real story on how lift is understood."  The simplified explanations are every bit as "real" as the more thorough explanations.  Replace real with full and I'll agree with you.  But the article is quite upfront about the fact that the simplified explanations are not the "full story".


 * "Limitations of deflection/turning" I've added a few lines reflecting the issues you bring up. Please take a look.


 * "The resulting force upwards is equal to the time rate of change of momentum of the air downwards"    I have to say that I am not able to follow your line of argumentation. And even if I  could and agreed with it, it wouldn't matter for our purposes here as editors.  Discussions on this page are not citable.  Our job as editors is to reflect what is published in reliable sources.  In reading literally hundreds of articles on this subject, I have never run across a single one refuting this assertion.  Meanwhile, there are several reliable sources that support the statement.  Perhaps if there was some disagreement in the literature we could present it as a controversy.  But unless we have some reliable published source we are bound to present what's been  published. See WP:TRUTH for more details.


 * Mr. Swordfish (talk) 18:33, 7 August 2014 (UTC)


 * My arguments on the "rate of change of momentum" statement are all supported by citable sources. A minimal but sufficient set of them will be quoted below.  I didn't bring them up before because I was advocating for deleting the statement, and I didn't think that would require citations.


 * Because the statement doesn't specify what it means by "the air", a reader would and should expect it to be true for any reasonable assumption as to what "the air" encompasses. However, it is well established in the aerodynamics literature that the statement is false for most of the assumptions the reader might make, i.e. it is false if "the air" refers to the whole atmosphere or to any subset of it that isn't very tall compared to its width.  If the statement failed only in exceptional circumstances I wouldn't press the issue.  But it fails for the most obvious assumption the reader is likely to make, i.e. that "the air" refers to the whole atmosphere.  So the problem is serious.


 * Because the statement has been shown by reliable sources to be contradicted in relevant situations, it has been effectively refuted, and letting it stand "as is" would be inaccurate and inconsistent with "what's been published". I think that leaves us two options:


 * 1) Delete the statement and the citation.  It isn't crucial to the deflection explanation, which is most often stated without the quantitative assertion "[Lift] is equal to...." anyway.  We have ample evidence from the mainstream aerodynamics literature that the statement is faulty, justifying our deleting it.


 * 2) Keep the statement but add the clarification that's needed to make it clear when it's true and when it's not.  Here's a rough draft of what I think that would have to look like:


 * In the text of the deflection explanation:
 * The resulting force upwards is equal to the time rate of change of momentum of the air downwards[AAPT citation]. This statement assumes that all of the lift can be accounted for by a momentum change in "the air", which is true only if "the air" refers to a region that is very tall relative to its width.  For the atmosphere as a whole, or for a subset of it that is not tall compared to its width, part or all of the lift is accounted for by pressure differences on the top and bottom of the body of air in question, reducing the proportion accounted for by the momentum change[note].


 * In the notes section:
 * [note]For the atmosphere as a whole, the integrated time rate of change of vertical momentum due to the lift on a wing is zero [Lanchester, 1907], and the lift is reacted entirely by a pattern of overpressure on the ground[Prandtl and Tietjens, 1934]. For regions that are subsets of the atmosphere, the proportions of the lift that are accounted for by momentum change and by pressure differences depend on the size and shape (vertical dimension compared to horizontal dimension) of the region.  Only if the vertical dimension is very large relative to the horizontal dimension are the pressure differences negligible, leaving the lift entirely accounted for by the change in momentum[Lissaman, 1996][McLean, 2012].  (Section numbers and quotes would be added to these citations, and other citations could be added.  I mention only the ones that come immediately to mind, but I think even just these would be sufficient.)


 * I expect you'll agree that the second option is too complicated and technical to be appropriate for this article. I'd argue that deleting the statement and the citation is the better option.


 * I wouldn't advocate presenting this as a "controversy" because I don't think it amounts to one. The "con" arguments are from the mainstream aerodynamics literature, where they are supported by rigorous math.  The "pro" statements you've cited are not supported by rigorous analysis and are all from "The Physics Teacher", which is not a mainstream source of information on aerodynamics.  The error made by the statement isn't something esoteric about which experts might disagree; it's basic:  It is wrong to apply Newton's second law to just a subset of the forces exerted on a body.  In addition to the force exerted by the foil, the air around an airfoil generally has unbalanced pressure forces acting on it.  Not being aware of these pressure forces is understandable in this case.  Authors of articles in "The Physics Teacher" are not typically mainstream experts on aerodynamics.


 * A relevant quote from WP:TRUTH: "To know where we have a dispute and where a simple mistake, consider whenever the author is really an expert on the topic (and not an expert on another topic, making a brief reference to something beyond his area of expertise)...."  So we editors are not just cyphers.  We are expected to exercise judgment as to the relative authoritativeness of our sources.  Weighing what's been published in the mainstream aerodynamics literature against the statement in question, I think we'd be on firm ground deleting the statement and the citation.


 * Your proposed changes to "Limitations of deflection/turning move in the right direction, but not far enough, in my opinion. And the first and second sentences have a jarring relationship.  The first sentence deals with the failure to produce quantitative results.  The second begins with "In particular," implying it is about to home in on a particular aspect of that issue, but then deals only with the incompleteness of deflection as a qualitative explanation, which is a separate issue.  I'd replace "In particular" with "Furthermore".  The third sentence deals with issues that are treated further later in the article, so some tie-in would be good.  Here's a shot at fixing the whole paragraph, with some rearranging to keep the quantitative and qualitative issues separate:


 * This simple explanation, while correct in as far as it goes, is not sufficiently detailed to support the precise calculations required for engineering. Quantitative predictions require a mathematical theory as described below under "Mathematical theories of lift."[31][32][33]


 * Furthermore, this explanation does not explain pressure and velocity variations in the vicinity of the airfoil or how the airfoil can impart downward turning to a much deeper swath of the flow than it actually touches.[Reference McLean (2012), Section 7.3.3] "A more comprehensive physical explanation" given below attempts to address these issues in a qualitative way.


 * On an earlier question, I don't see "Pressure integration" and "Lift coefficient" as belonging in the mathematical-theory section. I think they would fit well in "Basic attributes of lift", with the material in "Pressure integration" merged into the current "Pressure differences", and the material in "Lift coefficient" merged into the current "Air speed and air density".  I've tried this out in my sandbox User:J_Doug_McLean/sandbox, and I think it works well.


 * J Doug McLean (talk) 06:44, 10 August 2014 (UTC)


 * This discussion on change of momentum raises some interesting questions. It seems there's no controversy over the physics, but you disagree with the way some authors have concisely described the fulfilment of Newton's second and third laws. I appreciate that a quantitative integration of momentum change would require bounds to be defined carefully to avoid incorrect results, but our reader is not asked to do so. I agree this level of detail would be unnecessary for the article.


 * Does the reader need any concept of packets of air which are subject to changes in either momentum or pressure? Isn't all fluid pressure ultimately caused by the change in momentum of particles of fluid as they strike their container?


 * If the change of momentum of the air deflected downwards by the foil is understood to refer only to air which has its momentum changed downwards because of the movement of the foil, what other subset of 'the air' is included in the description which shouldn't be? Burninthruthesky (talk) 11:21, 10 August 2014 (UTC)


 * It's not just that I disagree with the concise statement. It's that the statement, taken at face value, has been refuted by authoritative sources.  But you raise interesting questions.


 * Yes, all pressure in gases is caused by changes in momentum of molecules striking and rebounding from the surface. In liquids, it's more complicated because molecules are in constant contact with their neighbors and can transmit force and exert pressure on the surface without changes in momentum.  So the answer to your question is yes, but only for gases, not fluids in general.


 * The molecular momentum change involved in gas pressure must be assessed very close to the surface. The only molecules that can be included are those between the last collision with another molecule before striking the surface and the first collision after rebounding.  At sea level the region of interest would be about a micron (several mean free paths) thick, and only a fraction of the molecules in that region would qualify.  That's a very limited subset of the air surrounding the airfoil.


 * We could make the AAPT statement true by limiting it to that small subset of the molecules surrounding the foil. But I see several serious drawbacks to casting the statement in that form:


 * 1. Aerodynamics generally deals with fluid motion as if the fluid were a continuum rather than individual molecules because it is very difficult to gain understanding or make predictions at the flowfield level with the molecular approach.


 * 2. The continuum description is more generally applicable.  Gases and liquids behave very differently at the molecular level but practically identically at the continuum level (for low Mach number in the gas).


 * 3. Lift is an aerodynamic (and hydrodynamic) phenomenon.  In the continuum approach to aerodynamics, "momentum" refers to the bulk momentum of the flowing fluid.  The explanation of pressure in terms of molecular momentum, as you're proposing, refers to the thermal momentum of the molecules, not the bulk momentum of the fluid.  This kind of explanation doesn't distinguish between aerodynamic and aero-static situations.  Even in still air we could say that the pressure force on a portion of a surface is equal to the time rate of change of thermal momentum of the air molecules near the surface (the right subset of them).  But this is a situation in which the air has no bulk momentum in the conventional aerodynamic sense.


 * 4. The statement in the AAPT article refers to bulk momentum in the conventional continuum aerodynamic sense and deals with momentum changes taking place over distances measured in airfoil chords, not molecular mean free paths. If we were to recast it in the limited molecular sense, we'd need a different citation.


 * The upshot: I don't think that interpreting the AAPT statement in terms of molecular momentum is a good solution to the questions it raises.  I still think we should just delete it.  J Doug McLean (talk) 20:34, 10 August 2014 (UTC)

Thank you for your detailed reply. I now understand the texts refer to momentum at the continuum level rather than the molecular level.

No doubt your argument is well supported by sources, but I don't see a refutation of the statement. On the contrary, I see a description of how to prove mathematically that the statement is true; identifying which subset of the air is subject to downwards momentum changes and avoiding errors such as applying Newton's second law to a subset of forces, or summing action and reaction in the calculation of a single force. We agree that that calculation is beyond the scope of the article.

I'm still not convinced that the cited non-specialist authors have made a mistake. I should point out that Chris Waltham's article does mention that, "to do this more correctly, we would box in the wing with a control volume of infinite vertical thickness". It happens all the time in science that a problem needs to be described and understood at the basic level before more rigorous treatement is attempted. Mr. Swordfish makes a good point that there's a risk of losing the reader by expecting too much knowledge too early in the article. Burninthruthesky (talk) 08:07, 12 August 2014 (UTC)

Also, Eastlake says: "In the interest of generalization, it is appropriate to recognize that the isolated wing is not the only type of flow-field geometry. When there are other surfaces nearby, such as walls in flow through ducts or the ground, those other surfaces can and do change the momentum of the flow as well."

Would it help the article to clarify that it refers to an isolated wing/foil? Burninthruthesky (talk) 16:33, 12 August 2014 (UTC)


 * My preference is to keep the section under discussion as brief and to the point as possible, without adding a lot of qualifying language that obscures the simple meaning. It's an introductory section aimed at the lay reader after all. I'd rather remove the sentence than replace it with a multi-paragraph treatment of how to compute the infinite integrals to make the statement true.   I don't think the statement is essential to the presentation and if there is consensus to delete I will reluctantly go along.


 * That said, I still think its a fairly straghtforward re-statement of Newtons' 2nd and 3rd laws and should be uncontroversial. Granted, when attempting to precisely elaborate on it, what one means by "the air" can be thorny. In a very simple model where all that is present is a uniform infinite fluid flow and a single airfoil, it has to be true.  Add other things to the model such as gravity or the ground or other phenomena that affect the air and one has to be much more careful to obtain that result via the calculations.  But it seems clear to me from the context that we're talking about a simple model rather than a more complex one.


 * Adding the word "isolated" probably won't make the sentence any clearer to the lay reader, but I'll go along with that if necessary. I really don't want to add a paragraph of introductory language to set up the simple statement of F = dp/dt.  Mr. Swordfish (talk) 15:52, 13 August 2014 (UTC)


 * I agree. The section describes a principle of physics, not how to model a real-world example. I suppose the article wouldn't suffer greatly from the removal of that sentence, but I don't agree that it is false at face value. Burninthruthesky (talk) 17:33, 13 August 2014 (UTC)


 * On that note, I think the context was clearer and flowed more naturally from, "Whenever airflow changes direction", before "by the foil" was //en.wikipedia.org/w/index.php?title=User:Mr_swordfish/Lift&diff=next&oldid=619617044 added. As Doug McLean said, it didn't help. Burninthruthesky (talk) 07:13, 14 August 2014 (UTC)


 * Agreed. I'll remove "by the foil" Mr. Swordfish (talk) 11:38, 14 August 2014 (UTC)


 * >Doug wrote:Here's a shot at fixing the whole paragraph [limitations of deflection/turning], with some rearranging to keep the quantitative and qualitative issues separate:


 * I adopted this language in the draft. I will look at moving the "Pressure integration" and "Lift coefficient" as per your suggestions. I think we are closing in on a releasable article. Mr. Swordfish (talk) 15:57, 13 August 2014 (UTC)


 * I think the draft is an improvement and look forward to the release. Burninthruthesky (talk) 17:33, 13 August 2014 (UTC)


 * I've now moved the "Pressure integration" and "Lift coefficient" sections from the "Mathematical theories of lift" to "Basic attributes of lift". However I kept them as their own sub-sections - my take is that they are important enough on their own to merit their own sub-section.  However, I did retain most of Doug's edits to the material. Mr. Swordfish (talk) 20:22, 13 August 2014 (UTC)


 * I'd like to take one more try at easing your reluctance to delete the L = dp/dt statement. The statement's problems really are more serious than either of you have given them credit for.


 * The conclusion that "In a very simple model where all that is present is a uniform infinite fluid flow and a single airfoil, it has to be true" isn't supported by the math unless you make stipulations about the shape of the region you're talking about. To evaluate either the pressure forces or the momentum fluxes in an infinite atmosphere, you have to evaluate the integrals over some finite box and then take the limit as the dimensions of the box go to infinity.  Even in the case without a ground plane or any other surface, the L = dp/dt statement is true only if the ratio of the vertical height of the box to the horizontal width is infinite, a very specialized condition.  It is untrue for any other shape of box.  Thus simply adding "isolated" wouldn't by itself  make the statement true in general.


 * I'll mention two examples with an infinite atmosphere, in which the statement fails, both from the mainstream literature. In both cases the results hold as the size of the box goes to infinity.


 * 1. For a "pancake" box (infinite ratio of width to height) all of the lift shows up as integrated pressure differences between the top and bottom, whether there is a ground plane or not.  The only difference is that with a ground plane all of the integrated force comes from the pressure excess on the bottom (the ground), while without a ground plane the force is equally divided between the pressure excess on the bottom of the box and a pressure deficit on the top.


 * 2. For a square box centered on the foil, once the sides of the box are longer than about ten airfoil chords, effectively half the lift is accounted for by the pressure differences between the top and bottom of the box, and half by the change in momentum.  As the dimensions go to infinity, "effectively half" converges to "exactly half".  The same goes for a circular box.


 * I realize that this is counterintuitive. How can the pressure differences infinitely far from the foil account for half the lift?  Well, for the small disturbances far from the foil the pressure perturbations are proportional to the velocity perturbations associated with the circulation, which die off as 1/r.  The area over which these must be integrated is proportional to r.  The upshot is that in the limit as the box size goes to infinity the integrated pressure force on the outer boundary of the box goes to a constant value equal to half the lift.  So the force exerted by the foil is not the only force exerted on the air in the box.  No matter how large the box is made, the air outside it exerts an unbalanced pressure force on the air inside.


 * So I still maintain that reliable sources have shown that the L = dp/dt statement is false at face value. There is a wide range of reasonable interpretations of what is meant by "the air" for which it fails, even in the simple "isolated" case.  It isn't a "fairly straightforward re-statement of Newtons' 2nd and 3rd laws" because meeting the requirements for applying Newton's second law (the force used in the equation must be the resultant of all the forces, not just a subset) still requires a very special shape for the box.


 * I disagree with the statement "It happens all the time in science that a problem needs to be described and understood at the basic level before more rigorous treatment is attempted", at least when it comes to aerodynamics. Trying to understand aerodynamics at the basic level without support of rigorous analysis is too error-prone.  J Doug McLean (talk) 19:29, 14 August 2014 (UTC)


 * Thank you for your time and patience in explaining your point. It's been an interesting discussion and I've learned a lot:


 * In an infinite (square) universe, the forces between the airfoil, the air, and the environment can all be calculated by integration. Changing the bounds of integration changes the results, although the facts and forces in the situation remain the same.


 * It can be shown that the airfoil exerts a force on the air which is equal to the time rate of change of momentum of the air downwards.


 * It can also be shown that the environment exerts a force on the air which is equal to the time rate of change of momentum of the air upwards.


 * In accordance with Newton's third law, the upwards and downwards forces on the air are equal and opposite. As a result, the net change in momentum of the air as a whole is zero.


 * It is not correct to say, "half the lift is accounted for by the pressure differences between the top and bottom of the box, and half by the change in momentum." The lift is reacted entirely by a pattern of overpressure on the ground [Prandtl and Tietjens]. If you had it both as an overpressure on the ground and as momentum in the air, that would be double bookkeeping [ [McLean] ].


 * I still see no reason to remove the disputed statement but if I've misunderstood any of the above, I am happy to be corrected. Burninthruthesky (talk) 07:00, 15 August 2014 (UTC)


 * My take is that if the integration results are dependent on the relative dimensions of the box as the limit goes to infinity, then the results are not physical but rather an artifact of the model. We've seen this before: a well known rule of thumb for real world air foils is that the downwash angle is approximately equal to one half the angle of attack.  This can be measured without recourse to taking integrals over an infinite area.  But when you model it as 2-D airflow, in the limit as the span -> infinity the downwash angle becomes zero.  This is an artifact of the model, not a reflection of the physics.  I think that's what's going on here.


 * I agree with both Doug and Burninthruthesky that it's incorrect to say that half the lift comes from momentum change and half comes from pressure differences. Granted, it may be possible to integrate in such a way that two terms appear that can be interpreted as momentum change and pressure difference, and depending on the relative dimensions of the box it can be 50-50, 0-100, or 100-0.  My understanding of the physics is that all of the lift force can be ascribed to momentum change, and that all of the lift force can be ascribed to pressure differences.  This half-and-half nonsense appears in some popularizations (one variant says "the foil generates lift on the bottom by Newtons law and  on the top by Bernoulli's principle").


 * I'm with Burninthruthesky in seeing no reason to remove the statement. Mr. Swordfish (talk) 19:43, 15 August 2014 (UTC)


 * Both of you ( Mr swordfish and Burninthruthesky) seem still to think that my arguments against the L = dp/dt statement don't really apply to an infinite atmosphere and can be dismissed. The counterarguments you offer, however, aren't supported by the math or by reliable sources, only intuition.  Intuition is not a reliable guide on this issue, as I'll try to show.  You're probably tired of my arguments on this issue, but I intend to persist as long as the arguments you present for dismissing them are erroneous.


 * The "artifact of the model" argument doesn't hold water for either example.


 * The example of the airfoil "rule of thumb" is comparing apples and oranges, I think. The downwash angle behind "real world airfoils" equals roughly half the angle of attack (relative to the zero-lift line), when evaluated at a location about a quarter chord behind the trailing edge.  That close to the airfoil, the downwash isn't affected much by aspect ratio, even as it goes to infinity, so the rule of thumb should apply regardless of aspect ratio.  In 2D airfoil theory, the predicted near-field flow agrees with the rule of thumb, and the predicted downwash angle goes to zero only far behind the airfoil.  So I think you're comparing the rule of thumb for the downwash angle near the foil with what 2D theory predicts for the downwash angle far away, and unjustly blaming the discrepancy on the 2D theory.  And on the centerline far behind a 3D wing, the downwash angle goes to zero as aspect ratio goes to infinity, both in theory and in the real world.


 * In the example of the integration of pressures and momentum changes in boxes surrounding an airfoil, the results are indeed "dependent on the relative dimensions of the box", even as the box size goes to infinity. But this is not because the flow model used is "not physical".  It is because an infinite atmosphere is an artificiality.  For boxes of finite size, no matter now large, the different results for different relative box dimensions reflect physical reality.  The fact that the differences persist as the dimensions go to infinity isn't an "artifact" of the math.  It reflects the fact that the momentum aspect of the physics is actually ill posed on an infinite domain.  See comments below on Burninthruthesky's first bullet item.


 * The fact is that the partition of the force into pressure differences and momentum changes actually does depend on the shape of the box, no matter how large. So you haven't refuted my objections to the L = dp/dt statement.


 * To repeat and address the bullet points:


 * Burninthruthesky wrote * In an infinite (square) universe, the forces between the airfoil, the air, and the environment can all be calculated by integration. Changing the bounds of integration changes the results, although the facts and forces in the situation remain the same.


 * It's true that what's going on physically doesn't change depending on how we choose to model it. But to quantify the forces and momentum changes, we have no choice but to calculate them by integration, and to do that you have to specify a domain.  Then if the integration is done correctly, the results reflect the "facts and forces in the situation" in that domain.  As I've pointed out, how much of the lift is accounted for by pressure differences and how much by momentum changes depends on the shape of the domain, even as the dimensions of the domain go to infinity.


 * Our intuitions resist this, and we tend to assume that there must be a single correct answer for the entire infinite domain. But our intuitions about infinite domains are often wrong, and they're wrong in this case.  The double (2D) or triple (3D) integral for the net vertical momentum due to a lifting airfoil or wing in an infinite atmosphere is non-convergent, which means that the vertical momentum is indeterminate.  And you can't make quantitative statements about an indeterminate quantity.  Still, one thing these integrals tell us about the "facts and forces in the situation" is true:  The balance between pressure differences and momentum changes is different in domains of different shapes, no matter how large the domains are.


 * This non-convergence doesn't matter in the real world because there is always a ground plane. Even for a semi-infinite space above a ground plane, the integrals converge, and the integrated vertical momentum in the whole atmosphere is zero.  For finite subsets of the atmosphere, the shape of the box has strong effects on the results, as I've argued all along.  See comment on fourth bullet below for what happens when the height above the ground is large compared to the box size.


 * The non-convergence of the momentum integral in an infinite domain and convergence in a semi-infinite domain is not reported in the aerodynamics literature, other than in my book, as far as I know. It's based on a careful analysis of the far-field behavior of the integral by a colleague, a qualified mathematician.


 * Back to the infinite-atmosphere case: What you say about the effects of "Changing the bounds of integration" could benefit from some clarification.  The results of the integration change drastically with the shape of the domain, but if the shape is held constant, the results cease to change with the size of the domain once it is large enough.


 * Burninthruthesky wrote * It can be shown that the airfoil exerts a force on the air which is equal to the time rate of change of momentum of the air downwards.


 * No, not in general. It can be shown only for a domain with an infinite ratio of height to width.


 * Burninthruthesky wrote * It can also be shown that the environment exerts a force on the air which is equal to the time rate of change of momentum of the air upwards.


 * This is not true for any domain that contains a lifting airfoil. Remember, a force is equal to a time rate of change in momentum only if it's the only force (or the vector sum of all the forces) acting on the object in question.  When there's a lifting airfoil, the force exerted by the environment isn't the only force acting on the air in the domain.


 * Burninthruthesky wrote * In accordance with Newton's third law, the upwards and downwards forces on the air are equal and opposite. As a result, the net change in momentum of the air as a whole is zero.


 * Are you referring to the forces exerted on the air by the foil and by the surrounding environment? If so, what you're saying isn't a correct application of Newton's third law.  The third law refers to the forces exchanged between two objects, not to the forces exerted by two objects (the foil and the environment) on a third (the air).  There is no reason two separate forces acting on a third object must be equal and opposite.


 * With this bullet and the two previous, you've argued yourself into a contradiction: The net change of momentum of the air is zero.  Thus according to the first of the three bullets, the lift is zero, which I don't think is what you were assuming.


 * Burninthruthesky wrote * It is not correct to say, "half the lift is accounted for by the pressure differences between the top and bottom of the box, and half by the change in momentum." The lift is reacted entirely by a pattern of overpressure on the ground [Prandtl and Tietjens]. If you had it both as an overpressure on the ground and as momentum in the air, that would be double bookkeeping.


 * I think I made it clear that the statement you quote applies to the case of a square or circular box without a ground plane. And it is correct for that case.  It's also correct if there is a ground plane, provided the distance to the ground plane is large compared to the dimensions of the box, so that the box and it's environs are effectively in free air.  Of course it's not true if the box gets close to the ground plane, and especially if the bottom of the box is the ground plane.  You're right to say that would be double bookkeeping.


 * No reason "to remove the disputed statement"? As I've said, the statement is inaccurate, to say the least, unless we add clarification.  It's been shown to be false for the most obvious interpretation the reader is likely to make of  what "the air" means.  This is from reliable sources, and no one here has effectively rebutted it.  We all agree that the required clarification would be inappropriate for the article.  To me, this adds up to a compelling reason to delete it.


 * J Doug McLean (talk) 05:19, 16 August 2014 (UTC)


 * To my mind, if you calculate something twice using two different methods, you should get the same answer, otherwise there is a problem with the calculation. If a physical fact is mathematically proven as true, then it is a fact. The conclusion I draw from argument you have made on this page is that the statement is true. I have not seen any reliable source, nor any reference to one, which says it is not.


 * I think the physics between the three bodies is pretty simple. The air is given some momentum by the wing. That momentum reverses direction as it rebounds from the surface, causing an overpressure. Momentum is a vector quantity. Equal and opposite vectors add up to zero. Burninthruthesky (talk) 17:15, 16 August 2014 (UTC)


 * Most of the time you'd be right in saying that if you calculate something by two different methods you should get the same answer. But that's true only if the "something" you're trying to calculate has a definite value.  A non-convergent integral, which is what we have in the case of the vertical-momentum integral in an infinite domain, doesn't have a definite value.  A classic symptom of non-convergence (also called non-existence) in a double or triple integral on an infinite domain is that when you try to calculate it you get different values depending on the order or the relative rates with which you take the dimensions of the box to infinity.  In a case like this, there is indeed a "problem with the calculation", but it's not what you're thinking.  The problem is that no correct way to do the calculation exists when the thing you're trying to calculate doesn't have a definite value.  This isn't just math.  What it says about the physics is true.  The vertical momentum due to a lifting foil in an infinite atmosphere is indeterminate.  So I stand by my statement that the vertical-momentum aspect of the physics is ill-posed on an infinite domain.  And this means that The Statement is problematic if the atmosphere is assumed to be infinite.


 * Your conclusion that The Statement is "true" is unfounded. You can't make a blanket statement that something is true if there are common situations in which it is untrue.  The situations for which The Statement is untrue are well documented in the literature.  In my entry of 10 August I cited a sufficient set of reliable sources.  Do you disagree with these sources?  If so, what are your specific objections?


 * I think it's clear from this discussion that my understanding of the quote you gave from Prandtl and Tietjens is quite different to yours. Burninthruthesky (talk) 09:50, 17 August 2014 (UTC)


 * I see a couple of problems with your discussion of the physics of three bodies. First, if the momentum imparted by the foil actually "reverses direction" in its interaction with the ground, the downward force on the ground would be twice the lift (Say the lift has a value of +1, for which the foil imparts downward momentum to the air with a value of -1.  If the ground reversed the direction of that, it would go to +1, and the imparted change in momentum would be +2).  Second, it's true that equal and opposite vectors add to zero, but in this case the vectors are equal and opposite only because you supposed them to be so.  It isn't required by Newton's third law as you implied in the fourth bullet of your previous posting. J Doug McLean (talk) 05:31, 17 August 2014 (UTC)


 * I'm saying the air presses down on the ground with a total force equal to lift. The ground presses back with an equal and opposite force. That is Newton's Third law. It may be true that Newton's Third law alone is not sufficient to prove conservation of momentum, but that's besides the point. Burninthruthesky (talk) 08:34, 17 August 2014 (UTC)


 * A further thought: Even if I agreed with you that The Statement was true for an infinite atmosphere, I'd argue that the right decision would still be to delete it.  We have an explanation (deflection) that works fine without it and is usually stated without it.  Why add a statement that's true only in a fictional abstract situation (an infinite atmosphere) and not in the real world where lift actually takes place (the real atmosphere with a ground plane)?  The infinite-atmosphere case plays a role in the mathematical theories, but that's no reason to add something that's true only in that case to an explanation for the general reader.


 * J Doug McLean (talk) 16:10, 16 August 2014 (UTC)


 * The onus is on you to obtain consensus for your proposal. Burninthruthesky (talk) 18:46, 16 August 2014 (UTC)


 * That's what I've been trying to do. But I'd say the onus isn't just on me.  The Statement:  "The resulting force upwards is equal to the time rate of change of momentum of the air deflected downwards" is a paraphrase of a quote from a cited source in one of the notes in the current article.  It doesn't appear in the body of the current article.  Putting it in the body of the article, as Mr swordfish proposes in his draft replacement is a change relative to the current article.  Technically speaking, Mr swordfish may be more obliged to obtain consensus to add The Statement to the body of the article than I am to obtain consensus to omit it.  But this shouldn't be a legalistic game.  It's supposed to be a cooperative effort.


 * I've put some effort into trying to convince you and Mr swordfish that The Statement is technically sloppy and that without proper clarification it raises more problems than it's worth. I've offered detailed technical arguments backed by citable sources, and detailed rebuttals to your counterarguments.  Neither of you has really attempted to rebut my arguments in any detail.  Instead, you offer general observations on how you think the physics ought to work.  When I've rebutted these, you've either tried another tack or simply restated your previous general assertion, but you haven't pointed out specific faults in my arguments.  I'm open to being corrected, but I feel like one of the reasons this discussion is at a stalemate is that you're not really engaging with me on the technical details.


 * I've just argued that even if The Statement were true in a fictional abstract situation, it is still untrue in the real world, and that's enough to justify omitting it. What, specifically, do you disagree with in that argument?  If this is to be a cooperative effort, the onus is on you as well. J Doug McLean (talk) 05:31, 17 August 2014 (UTC)


 * I now understand the statement can be proven true even in a square flowfield, and at least to a good approximation in most real-world situations. You argue that in a square atmosphere, the ground reaction only supports half the lift. This cannot be true; if it were, the sky would fall. That is more than a sufficient rebuttal of your argument, in addition to the others I've given. Personally I don't have a strong opinion on the inclusion of the sentence, but I am quite satisfied from this discussion that it is true. Please WP:LISTEN to Mr. Swordfish. Burninthruthesky (talk) 08:34, 17 August 2014 (UTC); edited 11:51, 20 August 2014 (UTC)

Having had some more time to consider, I begin to understand how some of the misconceptions in this discussion relate to the citations provided.

For the atmosphere as a whole, the integrated time rate of change of vertical momentum due to the lift on a wing is zero.

When the momentum of a parcel changes from mv to -mv, the change in momentum is mv - -mv = 2.

The total momentum is mv + -mv = 0. There is no change.

Therefore, there is no contradiction between, "For the atmosphere as a whole, the integrated time rate of change of vertical momentum due to the lift on a wing is zero", and, "The resulting force upwards is equal to the time rate of change of momentum of the air downwards".

This citation does not imply that changes in momentum do not happen in the atmosphere, or that the AAPT statement can sometimes be false.

the lift is reacted entirely by a pattern of overpressure on the ground

Reacted does not mean produced. The airfoil is fully supported by the air. The air is fully supported by the ground. Therefore the air's reaction to the airfoil and the ground's reaction to the air are both equal to lift.

The overpressure is caused by the ground's reaction to the momentum imparted to the air by the airfoil. It cannot exist in isolation, that would breach Newton's Third law.

'''For regions that are subsets of the atmosphere, the proportions of the lift that are accounted for by momentum change and by pressure differences depend on the size and shape (vertical dimension compared to horizontal dimension) of the region. Only if the vertical dimension is very large relative to the horizontal dimension are the pressure differences negligible, leaving the lift entirely accounted for by the change in momentum'''

I can see why calculating this would create challenges, although I have not seen the details. As agreed, it is wrong to apply Newton's second law to a subset of forces, which is why any calculation would have to account only for the force relevant to that calculation.

It makes sense that a region with large vertical dimensions relative to the horizontal would account mainly for momentum changes due to the airfoil because that region contains the entire airfoil and a relatively small proportion of the ground. For similar reasons, a horizontal box would account mainly for ground reaction.

Enlarging the region to a square box would not achieve the isolation described above. As a result, the downwards forces on the air will enter the calculation as well as the upwards forces. This is incorrect. I would expect such a calculation to give a result of zero, since the net change of momentum in the atmosphere as a whole is zero.

This citation does not imply that changing the bounds of a calculation has any effect on forces in the physical world.

I understand the objection that external pressure differences will slow down the air as it travels downwards and prevent it from keeping all of the momentum it has been given by the wing. However, a slowing down of the air is still a negative change of momentum. It is not correct to say that the statement does not apply when there are other forces involved. It does apply, it's just more difficult to quantify.

By the way, I have found an additional [source] which is more specifically related to aerodynamics. Hope this helps. Burninthruthesky (talk) 11:51, 20 August 2014 (UTC)

As Mr. Swordfish says, there is not yet a clear consensus on this issue. Unfortunately I am not able to engage in all the technical details since I am not a subject expert. My ideas may be naïve but you are of course welcome to discuss them. I am sorry if I have created the impression otherwise. Burninthruthesky (talk) 07:42, 26 August 2014 (UTC)

I am disappointed to see another week of embarrassing silence. The issue should be resolved, not as an exercise in schadenfreude, but in order to progress the volume of knowledge which we have all worked towards.

I just found the following quote: '''lift is accounted for either by pressure or by momentum flux, depending on the proportions of the control volume. ''' [McLean p.434]

This seems clearer than the final citation given above, confirming that the control volume can account entirely for the force exerted on the air by the airfoil or for the reaction from the environment, not 'proportions of the lift'. I hope this settles any remaining doubt or confusion.

I don't think anyone disputes that it is possible to define a region of 'the air' which only accounts for the downward change in momentum caused by the foil, and also a different region which only accounts for overpressure caused by the reaction from the environment. The fact remains that the environment only touches the outer boundary of the air, it does not contact the foil and cannot contribute to the lift force.

The question is whether the wording, "The resulting force upwards is equal to the time rate of change of momentum of the air downwards" is sufficient to identify the region of air referred to, without necessarily describing it in detail. I say it does, because there is only one correct way to calculate 'rate of change of momentum of the air downwards'.

Do we now have consensus for the proposed addition? Burninthruthesky (talk) 09:03, 4 September 2014 (UTC); edited 10:56, 4 September 2014 (UTC)


 * I don't see any evidence that anyone has changed their mind. And I have little expectation that further discussion will alter that fact. A couple of weeks have gone by without hearing from the lone dissenter, so the answer is a qualified "maybe". I'm not going to add the sentence, but I won't object if someone else does. Mr. Swordfish (talk) 16:16, 4 September 2014 (UTC)


 * It does shake your confidence a little to be told you're wrong, repeatedly by someone who is in a position to know, doesn't it? On the other hand, we know that an argument from authority is fallacious. Even experts can make mistakes, sometimes very serious ones.


 * I think what would help us most now would be the considered opinion of an aerodynamics expert. Burninthruthesky (talk) 16:41, 4 September 2014 (UTC)

Mr. Swordfish wrote: ''if there is consensus to delete I will reluctantly go along... In the interest of moving forward, I have removed it''

Thank you for demonstrating your willingness to compromise by removing it, and for your patience awaiting further discussion. I agree that now looks unlikely to happen. I was going to wait until after my holiday to post this, but I think we've waited long enough for a resolution to this dispute.

I see a quality argument that the statement is in keeping with guidance from the AAPT. I see no legitimate concern in opposition, so I am re-adding it. Burninthruthesky (talk) 05:10, 7 September 2014 (UTC)

L=-dp/dt
In the discussion above we have failed to reach consensus (yet) about the inclusion of the sentence "The resulting force upwards is equal to the time rate of change of momentum of the air deflected downwards."

In the interest of moving forward, I have removed it from the draft and would like to proceed with publishing the draft in it's current state. We can continue to discuss the issue, but I don't want to hold up publication pending the resolution of what I think is a fairly minor issue in the greater scheme of things. Have we reached consensus on publishing the article as-is? If so, I'll make the switch. Mr. Swordfish (talk) 19:56, 19 August 2014 (UTC)
 * I agree. I'm in favor of you making the switch. Well done! Dolphin  ( t ) 06:12, 20 August 2014 (UTC)
 * It is clear how much time and effort you (J Doug McLean and Mr. Swordfish) have both put into improving the clarity, coverage, and accuracy of the article. Thank you. I haven't been able to review the draft in as much detail as you, but I think it would be a shame to let it go stale because of a disagreement over one or two sentences. It makes sense to move into the article space and iron out any remaining details in the normal course of Wikipedia editing. Burninthruthesky (talk) 08:28, 20 August 2014 (UTC)


 * It's now been a week, and seeing no opposition I'm making the draft live. We can continue to discuss and improve that article moving forward. Mr. Swordfish (talk) 14:49, 26 August 2014 (UTC)

The Statement (L = -dp/dt) should be deleted
I've been away from internet access for several weeks and thus not able to participate in the discussion. I see that in the interim Mr swordfish installed the revised article, which I support, and that Burninthruthesky has added to it The Statement, "The resulting force upwards is equal to the time rate of change of momentum of the air deflected downwards", which I don't support.

Two general lines of argument have been put forward in support of The Statement, an earlier one by Mr swordfish and a more recent one by Burninthruthesky. Both of these lines of argument are contradicted by what actually happens in lifting flows, as I'll show below. Burninthruthesky has put forward several rebuttals of my arguments against The Statement, but none of these rebuttals is consistent with the physics, as I'll also show.

Thus in the interests of technical accuracy and consistency with the published specialist sources The Statement should either be deleted, or the required clarification should be added, as I've discussed in earlier posts. I may be the "lone dissenter", but my "con" position is supported by the physics as described in the mainstream aerodynamics literature, while the "pro" position is supported only by the statement itself in an article in a journal for physics teachers, and by intuitive and insufficiently rigorous arguments on this page, which I believe to be erroneous and for which I know of no citable source.

Let's look at the failings of the "pro" arguments.

Limiting integration to "the air deflected downwards"
Some time ago Mr swordfish argued, and Burninthruthesky later agreed, that The Statement must be true if the rate of change of momentum is integrated only over the air that is being "deflected downwards", i.e. that is undergoing downward acceleration. But this idea has no support in the physics or in the literature.

In a steady flowfield around a lifting airfoil, consider the air that is currently undergoing downward acceleration and thus contributing to the integrated rate of change. This is a body of air around which a boundary can be drawn and for which we could in principle calculate the integrated rate of change of vertical momentum. But we can arrive at a qualitative assessment of Mr swordfish's idea more easily by looking at the forces exerted on this body of air. For The Statement to be true, the total force exerted on "the air deflected downwards" would have to be equal to the force exerted on it by the airfoil, and the net force exerted on it by its other surroundings would have to be zero.

For simplicity, assume the inner boundary of this body of air is everywhere in contact with the airfoil surface, though that may not always be true. Assume further that at this inner boundary the airfoil exerts a downward force on the air equal to the lift. What we need to evaluate now is whether the force exerted on the air by its surroundings at the outer boundaries is zero as required if The Statement is to be true. What do these outer boundaries look like? In general they will take the form of two curves fanning forward from near the leading edge, one upward and one downward, and two curves spreading aft from the trailing edge, one upward and one downward. In the far field, these curves will approach +-45-degree lines configured like a letter X (see fig 7.3.23 in my book). The air being accelerated downward is thus contained within two generally fan-shaped regions, one above the airfoil, and one below, with the combined regions forming a general hourglass shape with the airfoil spanning the waist. For our purposes, the key characteristic of these regions is that the boundaries have extensive projected horizontal area on which the pressure differences in the field will act and thus exert an integrated vertical pressure force. The theory gives us no reason to expect that the integrated vertical pressure force on this entire outer boundary is zero, and in fact it isn't. The boundaries of the upper region are immersed in lower-than-ambient pressure, and the boundaries of the lower region are immersed in higher-than-ambient pressure, resulting in an unbalanced pressure force. Thus the total force exerted on "the air deflected downwards" is not equal just to the downward force exerted by the airfoil.

So I stand by my earlier argument that The Statement is true only for a box that is very tall compared to its width, and is not generally true for "the air deflected downwards". J Doug McLean (talk) 01:19, 17 September 2014 (UTC)


 * I accept your point that 'the air deflected downwards' includes more air than just a slim column. However, for the reasons you have given, there is only one correct way to calculate 'the rate of change of momentum of the air deflected downwards'. The resulting force is always equal to lift, regardless of what other incorrect calculations could be made. Burninthruthesky (talk) 15:18, 17 September 2014 (UTC)


 * If we take 'the air deflected downwards' to be a small box, then the assumption that its surroundings exert no force on it is false: in accelerating downwards, it experiences an increase in lift from the box above, which in turn experiences an increase in downforce. But the counterargument to "The Statement" actually asks us to; "consider the air that is currently undergoing downward acceleration and thus contributing to the integrated rate of change. This is a body of air around which a boundary can be drawn" This boundary, around the total volume undergoing acceleration, is in fact arbitrarily tall. The assumption is true for an arbitrarily tall volume, and consequently the counterargument is not valid. &mdash; Cheers, Steelpillow (Talk) 13:28, 22 September 2014 (UTC) [Updated 09:00, 23 September 2014 (UTC)]
 * To cite a source on this. According to McLean, "...we define a control volume ... with vertical outer boundaries at the front and back, extending to infinity vertically. Conservation of momentum requires that the lift on the airfoil be balanced by the forces and momentum fluxes at these outer boundaries. Because the boundaries are vertical, there is no net vertical force contribution by the pressure, and the lift must be accounted for by the net flux of vertical momentum into the control volume." This is saying the same thing: when we consider the total lift, the pressure component disappears and we are left with just the momentum change. &mdash; Cheers, Steelpillow (Talk) 12:46, 23 September 2014 (UTC)
 * Actually, I think we are all agreed on the physics, the issue seems more of what aspect to present where. "The Statement" was made as part of an introductory discussion and I can see no issue with placing it there. It is only later, when the air under consideration is in a finite box, that we need to qualify it, and this article has not gone there. &mdash; Cheers, Steelpillow (Talk) 12:54, 23 September 2014 (UTC)

Lift manifested as a rate of change of momentum in the neighborhood of the airfoil, with that momentum being removed elsewhere by interaction with the ground
Burninthruthesky has argued that the airfoil imparts downward momentum to air in some region surrounding it at a rate equal to the lift and that that momentum is then canceled in the far field in its interaction with the ground, resulting in the overpressure on the ground. This idea is intuitively appealing, but it isn't consistent with the detailed physics of the flow around an airfoil. See, for example, my argument above regarding the fan-shaped regions of "the air deflected downwards", and the fact that even in the near field that body of air will generally have unbalanced pressure forces acting on it in addition to the force exerted on it by the airfoil.

Or consider the air in a square box surrounding the airfoil, where the box is at least several chords in size but small compared to the distance from the ground, so that the flow in the box is as if the airfoil were in free air. In this case only half the lift is manifested as a rate of change of momentum, and the other half as pressure differences on the top and bottom of the box (Burninthruthesky took issue with this result for a square box, but that was the result of a misunderstanding on his part, as I discuss below).

A further counterargument: Removing downward momentum from the air requires upward acceleration. If the overpressure on the ground were a result of downward momentum being removed from the air in the neighborhood of the ground, there should be a region of upward acceleration of the air overlying the area of overpressure on the ground. But this isn't what we see. For an airfoil many chords above the ground, the overpressure on the ground is centered directly under the airfoil (The 2D version of the overpressure distribution is qualitatively just a 2D version of the 3D drawing in Prandtl and Tietjens). The dominant central portion of this region of overpressure, where the overpressure is strongest, is overlain by air that is accelerating downward, not upward. Only the weaker parts of the overpressure distribution, well ahead of the airfoil and behind, are overlain by air that is accelerating upward (This can be shown based on the model of the flow as a uniform flow with a lifting vortex and the image of the lifting vortex under the ground superimposed, which would be valid for large height compared to the chord. For smaller heights, some details would differ, but not the overall conclusion). Thus Burninthruthesky's simple momentum explanation for the overpressure is not consistent with the real pressure and velocity fields. J Doug McLean (talk) 01:19, 17 September 2014 (UTC)


 * I never specified the nature of the region of air accelerating upwards. I simply inferred its existence from the facts that the airfoil accelerates air downwards and the net rate of change of momentum of the atmosphere as a whole is zero. I see no evidence that the momentum explanation I gave is inconsistent with yours. Burninthruthesky (talk) 15:18, 17 September 2014 (UTC)
 * What happens in the far field is wholly irrelevant to the way in which forces are generated and momentum is changed locally. This sub-topic is equally irrelevant to the main discussion. &mdash; Cheers, Steelpillow (Talk) 13:36, 22 September 2014 (UTC)

Rebutting the rebuttals of the "con" arguments
Burninthruthesky has put forward several rebuttals of the "con" arguments, but none of these rebuttals is consistent with the physics. I'll address some of the main points here.

Burninthruthesky wrote: *I now understand the statement can be proven true even in a square flowfield, and at least to a good approximation in most real-world situations.


 * It can't be proven for a square box because it isn't true for a square box. You haven't proven it, and you haven't offered a citable source that proves it. J Doug McLean (talk) 01:19, 17 September 2014 (UTC)
 * If it's possible to perform a calculation on a subset of the atmosphere, it is likewise possible to perform a calculation on a subset of a square box. That is my interpretation of the citations you have given. Let's not forget as well that The Statement itself is already cited. Burninthruthesky (talk) 15:18, 17 September 2014 (UTC)

Burninthruthesky wrote: *You argue that in a square atmosphere, the ground reaction only supports half the lift. This cannot be true; if it were, the sky would fall.


 * This is a misinterpretation of my argument. As I wrote originally and later reiterated, for a square box, equal partition of the effect of lift between momentum changes and pressure differences is the result for the case where there is no ground plane, or the ground plane is far away compared to the size of the box.  With equal partition, the half of the lift accounted for by pressure differences is equally split between the pressure excess at the bottom of the box and the pressure deficit at the top, i.e. one quarter each.


 * I never said that the partition is equal when the bottom of the box is the ground plane. If the airfoil is centered in a square box that is large compared to the airfoil chord, and the bottom of the box is the ground plane, it can be shown that half the lift is accounted for by the pressure excess on the part of the ground plane that forms the bottom of the box, about 14% is accounted for by the pressure deficit on the top, and about 36% is accounted for by the change in momentum.  To find all of the lift accounted for by the overpressure on the ground, you must integrate over the entire ground plane, not just the part inside the square box. J Doug McLean (talk) 01:19, 17 September 2014 (UTC)


 * So, we agree that this calculation does not fully account for either of the two forces on the air. Burninthruthesky (talk) 15:18, 17 September 2014 (UTC)

Burninthruthesky wrote: *Therefore, there is no contradiction between, "For the atmosphere as a whole, the integrated time rate of change of vertical momentum due to the lift on a wing is zero", and, "The resulting force upwards is equal to the time rate of change of momentum of the air downwards".


 * If you simply assume, as you did, that the airfoil imparts downward momentum at a rate equal to the lift, and that the ground takes it away at the same rate, then the total rate adds up to zero as it should, and there is no contradiction. But the fact that the assumed rates add up to the right sum doesn't prove that the assumed rates were correct to start with.  In a real airfoil flow, the rate of momentum change is equal to the lift only for a very restricted definition of "the air" (a tall "sliver" of a box).  For any other definition of "the air", including for "the air deflected downwards", the rate of momentum change is not equal to the lift, and there is a contradiction.  See my arguments above. J Doug McLean (talk) 01:19, 17 September 2014 (UTC)


 * As I said, the environment only touches the outer boundary of the air, it does not contact the foil. You seem to be implying that the rigidity of the ground actually contributes to the lift, rather than simply reacting to it. That is not what is said in the citation you gave from Prandtl and Tietjens. If the ground reaction cannot contribute to lift, the foil must be entirely supported by momentum changes. So I believe it is absolutely true that 'the airfoil imparts downward momentum to the air at a rate equal to the lift.' I disagree with your opinion that it would be 'reasonable' for anyone attempting a calculation of the rate to assume that air subject to other forces should be included. Burninthruthesky (talk) 15:18, 17 September 2014 (UTC); edited 07:13, 20 September 2014 (UTC)

Burninthruthesky wrote: *I don't think anyone disputes that it is possible to define a region of 'the air' which only accounts for the downward change in momentum caused by the foil, and also a different region which only accounts for overpressure caused by the reaction from the environment.


 * If you mean one region surrounding the airfoil in which the force exerted by the airfoil is reflected only in momentum changes, and another region farther away in which the environmental pressures act on the imparted momentum, then I dispute it. In general, there is no support in the physics or in the literature for the idea that these effects can be separated into different regions in that way.  You haven't shown us any example of a region that meets your requirement that it "only accounts for the downward change in momentum caused by the foil". J Doug McLean (talk) 01:19, 17 September 2014 (UTC)


 * You have already given an example: "Only in the limit as the vertical dimension of the box becomes large relative to the horizontal is it true that lift is accounted for by momentum changes." Burninthruthesky (talk) 15:18, 17 September 2014 (UTC)


 * I've shown that any rectangular box that isn't very tall compared to its width doesn't satisfy the requirement, and that the body of "the air deflected downwards" doesn't either, because unbalanced pressure forces on the outer boundaries of "the air deflected downwards" are acting even in the near-field of the airfoil. Again, the only way to see lift manifested only as a change in vertical momentum is to confine your view to a box that is very tall compared to its width.

The bottom line
Those are the technical issues as I see them. I think it's clear from what's written in the mainstream literature that The Statement isn't true unless it is explicitly stated that "the air" refers to a region of air in the form of a very tall "sliver". And because this level of detail isn't appropriate for the article, it would be better just to delete The Statement. As for restricting it to the air undergoing downward acceleration (i.e. by changing "momentum of the air downwards" to "momentum of the air deflected downwards"), that doesn't fix the problem either.

However, in one way the "air deflected downwards" issue is beside the point. Even if adding "deflected" did fix the problem, it would be an unwarranted extrapolation from the source material and thus constitute original research. The AAPT article says nothing about integration at all, let alone about the idea of restricting the integration to the air undergoing downward acceleration.

But the bottom line is that no viable defense of either version of The Statement has been given. The Statement in either form is inconsistent with what's known from the mainstream literature, and it should be deleted.

If you still disagree with me, I'd suggest you seek a second specialist opinion. I'm not going to recommend a particular expert to you because I'd be open to the accusation of seeking a friendly witness. The arguments I've made here are pretty basic aerodynamics, based on the standard control-volume framework for analyzing momentum balance in fluid flows, and using a well-accepted model for 2D lifting flow (uniform flow plus a vortex, plus an image vortex if there is a ground plane). If another aerodynamics specialist agrees to look at this, I'd expect him/her to look at the same sources, apply the same models, and reach the same conclusions that I have. J Doug McLean (talk) 01:19, 17 September 2014 (UTC)


 * I appreciate a reply with some clarification. However, I see no significant new material or progress towards resolving the issues raised.


 * I see no response to the problem I pointed out, with a fresh citation which clarifies that the proportion of momentum and pressure accounted for by the control volume is dependent on its aspect ratio. My understanding is that the force on the foil is caused entirely by momentum changes, and the force on the ground is caused entirely by pressure, and both forces are always equal in magnitude to lift. A control volume which accounts for a mixture of these effects will therefore account for a mixture of the equal and opposite forces on the air. This would not be a calculation of the lift force. As you say in your book, "You can apply the standard procedures for evaluating integrals and, without making any procedural error, obtain a wrong answer."


 * You are now requesting a 'specialist opinion', knowing I am not a specialist. I would welcome an opinion from an alternative specialist, but unfortunately I don't know how, or whether, that can be arranged. More importantly, I agree with your earlier statement, that it "isn't something esoteric about which experts might disagree; it's basic: It is wrong to apply Newton's second law to just a subset of the forces exerted on a body." I have studied physics at degree level and consider myself qualified to comment on the Newtonian mechanics between three bodies. You have described in detail how it is possible to choose a region which mathematically eliminates the reaction force in order to prove that the force on the airfoil is equal to the rate of change of momentum, yet you claim this statement is not generally true. The sources you have offered do not appear to support your assertion. Burninthruthesky (talk) 15:18, 17 September 2014 (UTC)


 * I prefer to understand lift as a reaction to the force accelerating the air downwards: F=ma. This is dimensionally equivalent to the rate of momentum change over time, dp/dt, with both expressions having dimension MLT-2 (Mass x Length / Time squared). The statement given in the article is perfectly valid - no amount of sophistry is going to overcome such basic physics. &mdash; Cheers, Steelpillow (Talk) 13:50, 22 September 2014 (UTC)

22 Sept 2014 - I've remained silent for the past week to give others a chance to express their opinions. Welcome back, Doug. I am pleased to see that your absence was only temporary. My take on the current situation:


 * We still haven't reached consensus on the inclusion of The Statement. I do not think a temporary absence by one of the main editors should be interpreted as achieving consensus, so in the interest of fairness and following the wikipedia protocols I am (reluctantly) removing The Statement until we reach consensus to add it.  Whether this is temporary or permanent remains to be seen.


 * I have posted requests for assistance at the parent project pages to solicit wider opinion. I see that Steelpillow has already responded. (welcome!)   Hopefully, we'll get some other views.  The next step would be filing an RFC, but let's see what the folks from the project pages have to say.

Mr. Swordfish (talk) 14:30, 22 September 2014 (UTC)
 * I have only looked at the extensive walls of text here in a very cursory way, but my initial impression is that this wording is actually a bit confusing, independent of the question of whether it is rigorously true. Why is it being described as a rate of change in momentum rather than a force? The more intuitive wording would be something like, "The force applied upward is equal to the [net downward force applied to air deflected by the wing/foil]". The current wording requires that you convert "change in momentum over time" to "mass times acceleration" in your head just to get the units correct. 0x0077BE [talk/contrib] 16:49, 22 September 2014 (UTC)


 * Yeah, it's quite a wall, isn't it? Anyway, The American Association of Physics Teachers has this pedagogical recommendation:


 * "...lift on an airfoil should be explained simply in terms of Newton’s Third Law, with the thrust up being equal to the time rate of change of momentum of the air downwards." (https://en.wikipedia.org/wiki/Lift_%28force%29#cite_note-7)


 * So, that's where the language comes from. It seems pretty clear to me, but maybe I'm too close to the issue.  Mr. Swordfish (talk) 17:17, 22 September 2014 (UTC)
 * I suppose that a strict derivation from Newton's Third Law would require one to identify the net force on the air as, say, F. One then points out that by said law, F and L are equal and opposite. Both F = ma and F = dp/dt are then valid derivations of F. I guess which of them one uses will depend on whether momentum or acceleration is more to the fore in the subsequent treatment. Here, I did not notice any subsequent treatment, so I'd suggest that we describe both relationships, making it clear that they are equivalent. &mdash; Cheers, Steelpillow (Talk) 17:57, 22 September 2014 (UTC)
 * Yes, I would agree with this - I am seeing this entirely out of context, and it's not obvious to me why we're comparing a force to a change in momentum over time. I guess it's because the statement is conveying two ideas simultaneously - the fact that the two forces (the force downward on the air and the force upward on the wing) are equal, and the fact that forces are changes in momentum over time. I think it would work fine if it were broken out more clearly: "The force applied upward is equal to the [net downward force] - or, equivalently, the change in momentum over time - [applied to air deflected by the wing/foil]." Of course, the question is still open, in my opinion, as to why we're bringing momentum into this at all. If there was some reasoning for this particular choice in the original source, does it still apply here if, says, we're not actually discussing the momentum specifically?  0x0077BE  [talk/contrib] 18:40, 22 September 2014 (UTC)
 * I came here from the physics project. Walls of text and sometimes missing signatures and/or inconsistent indents mean that I've only skimmed the controversy. To help resolve the problem of whether the statement, or something like it, should be in the article, we should appeal one of the pillars of Wikipedia, verifiability. If mainstream reliable sources assert that the statement is true, then the statement is verified and with due weight that explanation deserves a place in the article, along with citations to said sources. If there are reliable source that claim the statement isn't true, then that controversy should be reported with due weight and cited sources. It doesn't matter whether I or any other editor think the statement is true or false or incomplete, only the reliable sources matter. Indeed, our personal takes on the subject could become original research if we stray too far from the sources.
 * Given the pillar of verifiability, perhaps we can resolve this controversy by giving an accounting of the reliable sources for and against. Then inclusion of the statement would be based on judging the quality and weight of the sources, not our personal takes on the subject. I confess to not parsing all the verbiage to extract those sources. Could they be summarized here? --Mark viking (talk) 18:38, 22 September 2014 (UTC)
 * Yes, I would also appreciate a summary of the sources, and if someone who understands it better could clarify in a succinct way the particular nature of the conflict, that would be appreciated. I'm starting to parse out the particular objections, but it'd be much easier if this were done RfC style, with a simple summary of the conflict. 0x0077BE [talk/contrib] 18:49, 22 September 2014 (UTC)
 * I've re-signed some text which I broke up using inline replies. I hope that helps a little. Burninthruthesky (talk) 07:44, 23 September 2014 (UTC)
 * The suggestion of presenting a controversy was discussed, then rejected, 'I wouldn't advocate presenting this as a "controversy" because I don't think it amounts to one' on 10 August. Burninthruthesky (talk) 13:35, 23 September 2014 (UTC)
 * Thank you for adding signatures--it does help in following the conversation--and thank for the pointer to the controversy discussion. --Mark viking (talk) 17:45, 23 September 2014 (UTC)

Summary of the nature of the conflict:

As part of the recent extensive rewrite the following statement (which came to be known as "The Statement") was added :


 * The resulting force upwards is equal to the time rate of change of momentum of the air deflected downwards.

The Statement is supported by several RS cites (see below).

One of the contributors (a well respected expert in the field) described it as "problematic":


 * "..the statement "The resulting force upwards is equal to the time rate of change of momentum of the air downwards" is problematic on two counts, in spite of the fact that it has a citable source.


 * The first problem with the statement is that for it to be true, the downward force exerted by the airfoil on the air surrounding it would have to be the only force being exerted on "the air". There are many possibilities for how we can define the body of air we're considering, and this condition (no other force but the lift) isn't met in general. The airfoil exerts a downward force on the inner boundary of the body of air surrounding it (at the airfoil surface), but the surrounding environment exerts unbalanced pressure forces on the outer boundary of the body of air. This problem cannot be eliminated just by increasing all the dimensions of the "box" of air we consider, even to the limit of infinity. As the box is made larger, the pressure disturbances at the outer boundary get weaker, but the area over which they act gets larger, and the integrated force remains comparable to the lift. How much of the lift is accounted for by this pressure force rather than by momentum change depends on the proportions of the box. For example, for a box that is very large horizontally compared to its vertical dimension, practically all of the lift is accounted for by pressure at the outer boundary, and practically none by momentum changes. Only in the limit as the vertical dimension of the box becomes large relative to the horizontal is it true that lift is accounted for by momentum changes.


 * The other problem is that most such analysis in fluid mechanics deals with boxes whose boundaries are fixed in space. The time rate of change of momentum in such a box is zero in steady flow, and momentum changes must be assessed in terms of fluxes in and out, not the time rate of change."

Much more has been written, but that's the gist. On the other side, it is asserted that L=dp/dt is simply a re-statement of Newton's 2nd and 3rd laws.

Summary of sources supporting The Statement:


 * "Most of the texts present the Bernoulli formula without derivation, but also with very little explanation. When applied to the lift of an airfoil, the explanation and diagrams are almost always wrong. At least for an introductory course, lift on an airfoil should be explained simply in terms of Newton’s Third Law, with the thrust up being equal to the time rate of change of momentum of the air downwards." Cliff Swartz et al. Quibbles, Misunderstandings, and Egregious Mistakes - Survey of High-School Physics Texts THE PHYSICS TEACHER Vol. 37, May 1999 pg 300 http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PHTEAH000037000005000297000001&idtype=cvips&doi=10.1119/1.880292&prog=normal


 * "Birds and aircraft fly because they are constantly pushing air downwards: L = dp/dt Here L is the lift force and dp/dt is the rate at which downward momentum is imparted to the airflow." Flight without Bernoulli Chris Waltham THE PHYSICS TEACHER Vol. 36, Nov. 1998 http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/fly_no_bernoulli.pdf


 * "Now let’s move on to conservation of momentum: the force exerted on a fluid equals the time rate of change (derivative with respect to time) of its linear momentum. If you exert a force on something, you change its momentum. If you don’t exert a force on something, its momentum stays unchanged or is conserved. This is Newton’s laws, if you choose to call it that. When an airfoil is producing lift, that force does in fact change the vertical component of the airflow’s linear momentum, and the drag force changes the horizontal component of the airflow’s linear momentum. ...Measuring lift by measuring the increase in downward vertical velocity in the flow coming off the trailing edge of the airfoil is conceptually possible. This downward velocity is definitely there and is known as downwash. I have never heard of anyone actually measuring it with sufficient precision to calculate lift, not because it is physically unsound but because it is not a practical experiment." Charles N. Eastlake An Aerodynamicist’s View of Lift, Bernoulli, and Newton THE PHYSICS TEACHER Vol. 40, March 2002 http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf


 * "The Rate of change of momentum of a body is equal to the resultant force acting on the body, and takes place in the direction of the force." An Introduction To Fluid Mechanics : CIVE1400 SCHOOL OF CIVIL ENGINEERING http://www.efm.leeds.ac.uk/CIVE/FluidsLevel1/Unit03/T5.html

I am unaware of any cites directly refuting The Statement. Perhaps Doug or someone can provide some specific references. Mr. Swordfish (talk) 19:45, 22 September 2014 (UTC)

Can we just prepend "Where no other force but the lift is imposed (which is rare), ..."? I can see merit in the statement, at least as a theoretical exercise. Gryllida (talk) 03:44, 23 September 2014 (UTC)


 * It's a bit cheeky of me, but I think it may help focus on the real problem here, so I would offer this quote (which I also posted above) from McLean in support of The Statement: "...we define a control volume ... with vertical outer boundaries at the front and back, extending to infinity vertically. Conservation of momentum requires that the lift on the airfoil be balanced by the forces and momentum fluxes at these outer boundaries. Because the boundaries are vertical, there is no net vertical force contribution by the pressure, and the lift must be accounted for by the net flux of vertical momentum into the control volume." This is saying the same thing: when we consider the total lift, the pressure component disappears and we are left with just the momentum change. So I think we are all agreed on the physics, the issue seems more of what aspect to present where. "The Statement" was made as part of a "Simplified physical explanations of lift on an airfoil" and I can see no issue with placing it there. It is only later, when the air under consideration is in a finite box (e.g. ground effect), that we need to qualify it, and this article has not gone there. &mdash; Cheers, Steelpillow (Talk) 12:54, 23 September 2014 (UTC)


 * I also keep thinking we're all agreed on the physics, only to find myself contradicted. I think the issue of the shape of 'the air' would only occur to someone with an in-depth knowledge of aerodynamics. They are not the target audience of this section.


 * Even still, I think an expert reader should understand what the article means and take a reasonable interpretation. It has been said above that, 'Not being aware of these pressure forces is understandable in this case. Authors of articles in "The Physics Teacher" are not typically mainstream experts on aerodynamics.' This was refuted with evidence that the cited authors are perfectly well aware of the need to remove the effects of other forces from any mathematical treatment. (See the discussion on 12 August.) Burninthruthesky (talk) 13:25, 23 September 2014 (UTC)
 * Regarding the physics of the situation, I'm unclear about this point regarding "other forces" on the wing. The example of gravity, in particular, seems strange; generally in a physics textbook (or other educational material), you would talk about the contribution from each component independently unless they have some sort of non-linear dependence on one another, so if I were to throw a ball up in the air, I would say that I've impelled it with a certain amount of force (F), and I, in turn, am absorbing an equal force in the opposite direction (-F). The actual velocity of the ball can't be calculated without taking into account other forces acting on the ball (e.g. wind resistance, gravity), but that's independent of the force imparted to it its interaction with me. I'm wondering if we can solve this problem by being slightly more specific (or deciding that the wording already contains sufficiently specific terms of art) with the phrasing, to indicate that when we say "lift" we're only talking about the force on the wing created by the deflection of air, and that when we say "momentum of the air", we're specifically talking about the change in momentum as a result of the deflection. If I'm not mis-understanding the objection here, I'd say we're at least 90% of the way towards having the appropriate specificity anyway. 0x0077BE [talk/contrib] 18:04, 23 September 2014 (UTC)


 * To clarify the point about 'other forces', I think the main objection is that as the wing pushes air downwards, that air experiences a push upwards in the form of pressure from the environment. The air spreads out as it goes, so the reaction force from the environment is generally exerted over a wider area than the wing. As I understand it, this is why a calculation over a wide box of air mainly represents the pressure from the environment, and a tall box mainly represents the force on the foil.


 * I think there has been some confusion in this discussion caused by use of the term 'lift' to refer to the force on the foil, the reaction from the environment, and even a combination of the two. I absolutely agree that when we say "lift" in this context we're only talking about the force on the wing created by the deflection of air and I think the wording in the article is quite clear. Burninthruthesky (talk) 07:47, 24 September 2014 (UTC)
 * Glad to know that we seem to be on the same page about the lift portion. I still don't quite understand the objection about the volume of air. Clearly the wing is imparting some net downward force on something, and the only thing around it is the air. If the air which has been deflected by the wing then loses that momentum from interactions with the environment, that doesn't change anything about the balance of forces in the wing-air system (which sounds like what we're talking about in The Statement). Unless there's something that's being deflected that's not accounted for in The Statement, this seems like a simple restatement of Newton's laws, and as such I see no reason to believe that the sources are being inaccurate when they make this claim. 0x0077BE  [talk/contrib] 14:04, 24 September 2014 (UTC)
 * I agree. To clarify the objection, if you measured the rate of change of momentum of a wide volume of air, your calculation would account for more environmental pressure than momentum. So you would get a rate of change which is not equal to the force on the foil. I would say that's because this isn't a calculation of the rate of change of momentum deflected downwards, and I think that's the crux of the disagreement. Burninthruthesky (talk) 14:27, 24 September 2014 (UTC); edited 14:31, 24 September 2014 (UTC)
 * The issue remains that no simple rewording will overcome the objection, "in the interests of technical accuracy and consistency with the published specialist sources The Statement should either be deleted, or the required clarification should be added". (I understand the 'required clarification' is that listed under 'In the notes' on 10 August. Nobody supports this.) I fully agree that if The Statement is inaccurate, it should be deleted. But we need evidence. Burninthruthesky (talk) 13:30, 24 September 2014 (UTC); edited 13:51, 24 September 2014 (UTC)


 * Agree that if The Statement is inaccurate it should be deleted. Also agree that we need evidence, and unfortunately I haven't been able to follow the con arguments.  I also haven't seen any reliable source backing the con argument, only a lot of WP:OR posted to the talk page.  From a verifiability standpoint all the evidence is in favor of The Statement. Mr. Swordfish (talk) 17:07, 24 September 2014 (UTC)

The evidence against The Statement

I thank Mr swordfish for reverting the addition of The Statement until this is resolved.

I think there is ample citable evidence against The Statement, and I'll list some formal citations here. But I don't expect that to be enough. I've already described this evidence several times in my own words. Though I didn't include formal citations, I thought I made it clear that these arguments were based on published sources and were not just my own work. Still, the other participants, principally Mr. Swordfish, Burninthruthesky, and Steelpillow, have not been convinced that this evidence means what I think it does. So in addition to listing citable sources, I'll try again to convince you that this published evidence actually does contradict The Statement. First, to summarize the basic line of argument against:

The Statement in either of its forms is not "simply a re-statement of Newton's 2nd and 3rd laws" as asserted in Mr. Swordfish's summary of the "pro" evidence. This is clear from the published flowfield analyses I'll cite below and from the basic statement of the 2nd law cited by Mr. Swordfish: "The Rate of change of momentum of a body is equal to the resultant force acting on the body, and takes place in the direction of the force." (An Introduction To Fluid Mechanics : CIVE1400 SCHOOL OF CIVIL ENGINEERING http://www.efm.leeds.ac.uk/CIVE/FluidsLevel1/Unit03/T5.html) Note that in this basic physics context "resultant force" means "net force" or "vector sum of all the forces". Thus the L = dp/dt statement, where L is the total lift force exerted on the foil, is a correct application of the 2nd law only if L is actually the magnitude of the "resultant force" acting on "the air", which can be so only if the integrated force exerted on the air by its other surroundings is zero. The evidence shows that this is true only if "the air" is defined as a region that is infinitely tall compared to its width, and is untrue for every other shape of region that has been looked at, large or small. Thus The Statement is not true in general, but only true in one special case. Analyzing all of the forces, to be sure we have properly identified the "resultant force", isn't "sophistry", as Steelpillow would have it. It's basic physics.

In looking at the evidence it will be important to distinguish between The Original Statement from the AAPT papers, and The Revised Statement proposed for inclusion in the article. The Original Statement is not specific as to what body of air is to be integrated over to determine the time rate of change of momentum, referring to it simply as "the air" in Swartz's version and "the airflow" in Waltham's version. The Revised Statement attempts to be more specific, adding the word "deflected", specifying that the integration is to be limited to the region of the air that is experiencing downward acceleration at the current instant.

All of the mainstream published work I'm aware of that analyzes the momentum balance in the flow around a lifting airfoil predates the publication of AAPT papers and therefore does not specifically address The Original Statement per se. However, much of this work arrives at conclusions that contradict The Original Statement and thus effectively refute it.

What do I mean by "contradict"? We have a very specific statement, L = dp/dt, where L is the lift force exerted on the foil by the air, and dp/dt is the rate of change of momentum downward, integrated over some volume of "the air" surrounding the foil. The problem is that "the air" is not specifically defined, implying that you don't have to be very specific in choosing the body of air to find an integrated dp/dt that's equal to L. However, the analyses cited below find only one very specific choice for the region of "the air" that gives the stated result, while all of the other choices give values of dp/dt that are not equal to L.  Thus, as I've argued before, we have a statement that claims to be true in a very general way, but that is actually true only for one special case. The Original Statement is thus inaccurate unless it is modified to be more specific.

Here are the citable sources I know of:

On basic control-volume analysis of the rate of change of momentum in a moving fluid:


 * ''Shapiro, A. H. 1953.  The Dynamics and Thermodynamics of Compressible Fluid Flow. New York:  The Ronald Press Company.  Section 1.5

This analysis shows that for a steady flow the integrated time rate of change of momentum of fluid parcels passing through the interior of a control volume is equal to the integrated (net) flux of momentum through the boundary. This is a basic ingredient in the other analyses cited below.

For the atmosphere as a whole, including a ground plane:


 * ''Prandtl, L., and O. G. Tietjens. 1934.  Applied Hydro- and Aeromechanics.  New York:  Dover Publications.  Derivation in connection with figure 150.

I don't have a copy at hand, so I can't provide a quote, but this is the classic analysis showing that the pressure pattern on the ground constitutes a downward force on the ground, and thus an upward force on the atmosphere, equal to L. The net force on the atmosphere due to the lift, (i.e. the vector sum of the forces exerted by the wing and the ground) is therefore zero, so that the integrated rate of change of vertical momentum for the atmosphere as a whole must be zero. Thus for the most obvious assumption a reader is likely to make regarding what is meant by "the air" (i.e. the atmosphere as a whole), The Original Statement is false.

The Prandtl and Tietjens analysis is for the 3D case. It is easy to show that the same overall conclusion applies in 2D. A citable source for the 2D analysis probably exists, but I don't know of one offhand.

For a circular region centered on the airfoil:


 * ''Durand, W. F., ed. 1932.  Aerodynamic Theory, vol. 1. New York:  Dover Publications.  Sections B. V. 6 and B. V. 7.

This is a control-volume analysis of the flow around a 2D lifting body of arbitrary cross-section in an infinite atmosphere, using a circle of large radius as the outer boundary of the volume. It shows that in the far field the flow is independent of the details of the body, and that significant contributions to the pressure and the momentum fluxes at the outer boundary come only from the combination of the uniform flow and the bound vortex. It arrives at a derivation of the Kutta-Joukowski theorem in equation 7.3. Equation 5.6 shows that the flux of vertical momentum across the outer boundary, and thus the time rate of change of vertical momentum in the air in the interior, is equal to only half the lift. Equation 6.6 shows that the integrated vertical pressure force on the outer boundary is upward and equal to half the lift. The net force on the air due to the lift is therefore downward and equal to half the lift, and Newton's second law is satisfied. It is explicitly stated that this result holds regardless of how large the radius of the circle is made. Thus a large circle is another example of a region of "the air" for which a reader might reasonably expect The Original Statement to apply, but for which it is in fact false.

For rectangular control volumes:


 * ''Lissaman, P. B. S. 1996.  The facts of lift.  AIAA 1996-161.  Section titled "Lift in thin slices:  the two dimensional case".

Lissaman assumes an infinite atmosphere with no ground plane and summarizes the results of his analysis as follows: "For a large rectangular control surface, part of the lift is attributable to pressure and part to momentum, depending on the aspect ratio of the surface.  For a square control surface the contributions on the surface due to momentum and pressure are equal; for a tall, long vertical surface the contributions are mainly momentum, while for a streamwise, long, flat, horizontal surface the lift is primarily due to pressure.  This illustrates that it doesn't make much sense to attribute the lift on an airfoil to either pressure or momentum effect, unless one takes a control surface on the actual airfoil surface, when the lift is indisputably due only to pressure."

According to Lissaman's results, if "the air" is taken to be the air in a rectangular box surrounding the airfoil, The Original Statement isn't even close to being true unless the box is a tall, slender sliver, and even then it isn't strictly true until the vertical dimension of the box is taken to infinity. Steelpillow quotes the section of my book that describes the result for the infinitely tall, slender sliver, the only control-volume shape for which The Original Statement has been shown to be true, and interprets it as being "in support of The Statement". A balanced recounting of what my book says would also quote the discussion in connection with figure 8.5.4, which deals with other control-volume shapes for which The Original Statement isn't true.

To me, the evidence cited above makes it clear that The Original Statement is inaccurate as it stands.

The Revised Statement was an attempt to fix this, the idea being, as I understand it, that the integrated dp/dt would be equal to the lift if the integration were limited to the region of air undergoing downward acceleration. However, no citable source has been put forward for this revision, so The Revised Statement isn't a viable candidate for inclusion. Even so, the idea is intriguing, and I looked at it in some detail. I plotted the far-field pressure distributions along the boundaries of this hourglass-shaped region of "the air" (the +-45-degree lines in the far field and the arbitrary horizontal lines at the top and bottom of the hourglass) and convinced myself that the integrated pressure force on the complete hourglass boundary cannot be zero, even if the height of the hourglass is taken to infinity, and thus that the rate of change of momentum of "the air deflected downward" cannot be equal to the lift. Of course this is just my own work. So the proposed revision has no citable evidence either for or against.

Now to respond to some of the recent arguments put forward on the "pro" side:

Burninthruthesky and Steelpillow have made several statements aimed at countering the general control-volume line of argument. But the gist of what they say, if I understand it correctly, isn't consistent with how the momentum balance in a control volume works.

The basic principle is that if you integrate the pressure and the momentum fluxes over the entire boundary of a control volume, the sum must be zero for a steady flow, provided a consistent sign convention is followed. This is a vector relationship, and in the examples I've discussed it has been applied to the vertical component of the pressure force combined with the flux of vertical momentum. Consider a control volume that completely surrounds a 2D airfoil. The airfoil surface forms the inner boundary of the control volume. The lift is exerted in the form of pressure on the airfoil surface, and thus can be calculated by integrating the vertical component of the pressure force over the inner boundary. And because the airfoil is impermeable, the momentum flux integrated over the inner boundary is zero. Because the sum of the integrals over the entire boundary (inner and outer) must be zero, the lift can also be calculated from the appropriately summed integrals of the vertical component of the pressure force and the flux of vertical momentum over the outer boundary of the volume.

This kind of analysis is valid for any control volume, large or small, and of any shape, as long as it completely surrounds the airfoil, and the boundary is piecewise smooth. As the cited evidence shows, the two outer-boundary integrals (pressure and momentum-flux) "account for" different portions of the lift depending on the shape of the control volume and on its disposition relative the ground plane, if there is one. This is not contradictory. It does not imply that the overall process of lift production is different depending on what control volume you choose, which would of course be incorrect. Neither does it imply that we're somehow subdividing the actual physical lift force. All that's being subdivided is the manifestations of the lift in the surrounding air. This subdivision merely reflects the fact that, even in the same flow, different regions see different balances between pressure and momentum flux. And regardless of the relative magnitudes of the two outer-boundary integrals, their sum always equals the lift. So Steelpillow is mistaken when he says "when we consider the total lift, the pressure component disappears and we are left with just the momentum change." Likewise Burninthruthesky is mistaken when he says "A control volume which accounts for a mixture of these effects will therefore account for a mixture of the equal and opposite forces on the air. This would not be a calculation of the lift force." On the contrary, the sum of the outer-boundary integrals is always equal to the total lift force.

In another argument aimed at countering the control-volume analyses Burninthruthesky quotes my book as saying "You can apply the standard procedures for evaluating integrals and, without making any procedural error, obtain a wrong answer". This takes the words out of context and reaches a wrong conclusion. That passage refers to integrals that are non-convergent on an infinite domain. The control-volume analyses I've been discussing here all involve integrals that converge, and the results correctly reflect physical reality.

Another idea put forward by Burninthruthesky and seconded by  0x0077BE  goes as follows (paraphrased as I understood it):


 * Given that the foil and the ground are not in contact with each other, but communicate only through their contact with the air, the foil can transmit a force to the ground only by imparting momentum to the air at a rate equal to the force, and the force is transmitted to the ground when the air gives up that momentum in an interaction with the ground.

This would be a physically realistic possibility if the air were a projectile that flew between the foil and the ground, such that momentum was the only way for the air to "remember" the effect of the foil after leaving contact with the foil. But the air is not a projectile flying between the foil and the ground. It is an extended mass that moves as if it were a continuous material, and it is in constant, simultaneous contact with both the foil and the ground. Taking on momentum and giving up momentum is not the only way the air can transmit a force. It also transmits force through the pressure field.

As an illustration of this, consider replacing the air with a brick resting on the ground, and replacing the foil with your hand pressing downward on the top of the brick with a force F. The brick transmits F to the ground through its distribution of internal stresses, and the brick remains at rest with zero rate of change of momentum. True, the air around an airfoil is unlike the brick in the sense that the air has some freedom to move, and some change of momentum takes place, at different rates depending on what region of the air you look at. But the air is similar to the brick in the sense that it carries an internal distribution of stress (the pressure field). Because the non-uniform pressure field pervades the entire airspace and can transmit force between different portions of the air, Newton's second law does not require the integrated rate of change of momentum of the air, or any portion of the air, to be equal to the force applied by the foil, any more than it requires the rate of change of momentum of the brick to be equal to F.

To summarize: Arguments and rebuttals put forward on the "pro" side are not convincing. Citable sources in the mainstream aerodynamics literature provide sufficient evidence that The Original Statement is inaccurate and should not be added to the article. J Doug McLean (talk) 22:31, 27 September 2014 (UTC)
 * I'm sorry, but I am fully convinced that the statement is accurate. I think your objections lie in a practical analysis of specific control volumes of air, but I'm still failing to see how this well-cited statement (which comes from proper secondary sources, mind you) is anything but a simple restatement of conservation of momentum. It is implicit in these types of statement that you are talking about only the relevant quantities during the relevant time period, e.g. the momentum imparted on the foil and the momentum imparted on the air. These two quantities will definitely balance out, whether the force imparted on the air is dissipated by compressing the fluid, increasing the total pressure, or by heating due to friction, the instantaneous change in momentum in the air-foil system balances. My point about the air being the only thing around wasn't about how the pressure is transmitted to the ground, in fact the exact opposite, I was suggesting that the ground and the environment can't possibly matter when we're talking about the instantaneous balance of forces.
 * It sounds like what you are saying is that the measured change in momentum of a given volume of air will not necessarily match the lift on the foil because the energy can be dissipated in other ways. I'm suggesting that it doesn't matter where that momentum goes, just that it is initially transferred between the air and the foil. The standard way of teaching or explaining the physics of a system is to simplify it it to its most basic form, so with no further clarification you should assume that L = -dp/dt refers to only the relevant (interacting) parts of the system. The later consequences of the momentum imparted into the air may be relevant to airfoil design, but I don't think they invalidate this statement. 0x0077BE [talk/contrib] 14:33, 29 September 2014 (UTC)
 * I agree with 0x0077BE. If The Statement is inherently wrong, it should not require such an immense wall of text to rebut it. In its context within the article, it is perfectly satisfactory. That is to say, the one "special case" in which it is wholly accurate happens to be the one special case that the article is introducing at that point. The furthest one might go at this stage in the article would be to add a caveat that it applies strictly only to unbounded flow, but really, the discussion is not even at that level of sophistication at this stage. &mdash; Cheers, Steelpillow (Talk) 16:05, 29 September 2014 (UTC)
 * The Statement is mathematically true when the correct control volume is considered, and I agree with 0x0077BE and Steelpillow that this is the only one which applies in the context under discussion. The truth of The Statement is unaffected by the fact it is possible to 'look at' other regions of the air and consider the same physics in terms of pressure. I think it's been well established that other analyses have no relevance here. Burninthruthesky (talk) 10:43, 30 September 2014 (UTC)


 * 0x0077BE asserts that "the ground and the environment can't possibly matter when we're talking about the instantaneous balance of forces." If the exchange of equal-and-opposite forces between the foil and the air were all we were talking about, I'd agree.  But it isn't, and I don't.  The Statement also deals with the rate of change of momentum of the air, which can be defined only by a definite integral over some spatial domain, i.e. a "control volume".  The foil exerts its force on the air at the surface, but the resulting change of momentum of the air is spread over a substantial area above and below the foil.  Thus it isn't consistent with the physics of a continuum flow to think of the momentum as being "initially transferred between the air and the foil" and then "going" elsewhere.  At any given instant, the imparting of downward momentum isn't just taking place at the surface.  It takes place over a wide area.   And integration over a small area close to the surface isn't going to see enough of it to make The Statement true.  The "correct control volume", as Burninthruthesky refers to it, must, at a minimum, be fairly large and extend some distance above and below the foil.


 * The trouble is that any control volume large enough to contain a rate of change of momentum equal to the lift is also large enough to have substantial pressure forces imposed on it by its surroundings. So even in free air, "local pressure variations" come into play, and the "environment" matters.  For The Statement to be true, a "correct control volume" must be large, but the air in it must behave as if the force exerted on it by the foil were the only force acting on it, i.e. the integrated pressure force on its outer boundary must be zero.  Only one such control volume has been identified in the literature:  the infinitely tall sliver analyzed by Lissaman (1996).  If this is the control volume that 0x0077BE and Burninthruthesky are referring to, then maybe we've reached some level of agreement on the physics.  But I would still disagree with the contention that the context within the article makes this clear.  If The Statement is true only for the air in an infinitely tall sliver, then that needs to be spelled out.


 * If 0x0077BE and Burninthruthesky think The Statement is true for some control volume other than the infinitely tall sliver, they need to tell us specifically what control volume that is and provide citable sources for their assertion. I don't think the "air deflected downward" qualifies because there's no citable source for it that I know of, and it doesn't make The Statement true anyway, as far as I can tell, because its outer-boundary pressure integral isn't going to be zero (See my earlier discussion of the hourglass shape of this region of air).


 * Now I see that Steelpillow has added to the article a new version of The Statement, with the added stipulation that it applies to "free flow conditions", and not "in a restricted space". This fix is deficient on two counts.


 * First, the restriction to "free flow conditions" is not sufficient to make The Statement true. The only situation for which it has been found to be true is the infinitely tall sliver (Lissaman, 1996).  For regions of any other shape, there is always an unbalanced pressure force on the outer boundary, even as the size of the region is taken to infinity.  Note that the analyses for a circular control volume (Durand) and rectangular control volumes (Lissaman) assumed an infinite atmosphere (unbounded flow).  The evidence is clear that "local pressure variations" come into play in both the near and far fields, even in an unbounded flow.  The only proven way to eliminate outer-boundary pressure forces is to limit integration to the infinitely tall sliver.  A control volume that is infinite in both directions doesn't do it, because the outer-boundary pressure contribution doesn't vanish in that case.


 * Second, the assertion that "free flow conditions" is a sufficient restriction isn't supported by any citable source. Lissaman doesn't support this assertion because he makes it clear that if the aspect ratio of the control volume isn't infinite, the outer-boundary pressure will account for part of the lift.


 * This addition to the article is both inaccurate and unsubstantiated, and should be reverted.


 * To summarize the situation as I see it:


 * The Original Statement referring simply to "the air" has a reliable source in the AAPT papers. But other sources in the mainstream literature find that for most of the assumptions a reader would be likely to make as to what is meant by "the air", The Original Statement is contradicted.  The Original Statement is therefore inaccurate unless it is made more specific.


 * Neither "free flow conditions" nor "the air deflected downward" is acceptable as a fix for The Original Statement. The only acceptable fix is a stipulation that it is true only for Lissaman's infinitely tall sliver control volume, including a discussion of how that is the only way to make the outer-boundary pressure contribution negligible.  This is the only fix that is both true and verifiable.


 * But the context here is the simplest explanation (flow deflection) in a section titled "Simplified physical explanations of lift". The flow-deflection explanation works just fine without The Statement.  It is supposed to be qualitative and shouldn't need to include a quantitative mathematical statement.  And consider the audience.  The AAPT papers are advising teachers on how to explain lift to physics students.  Our audience is the general public, not physics students.  As I've said before, the best option is simply to leave The Statement out.  J Doug McLean (talk) 19:38, 4 October 2014 (UTC)
 * I think your first sentence clearly concedes the point we were trying to make: "If the exchange of equal-and-opposite forces between the foil and the air were all we were talking about, I'd agree. But it isn't, and I don't." - that is exactly what we're talking about. The presumption in physics is that you are talking about the basic, ideal case; in this case, we're talking about the foil and the air with which it is interacting. The momentum in the system is conserved. I think there's general consensus at this point for inclusion of the statement (given that it is well-cited, and I see no citations from you explicitly stating that this commonly-taught concept is not true, just statements where you need to be an expert to see how it applies, and there's no support here for the idea that they are being applied appropriately).


 * I get the impression that you are arguing based on either a more complicated model of the system than the one being presented here (i.e. one where momentum gained by the air can be damped from the environment) or you have a different definition of momentum than the rest of us. By Newton's 3rd law, whatever the foil is interacting with must have a force applied to it equal and opposite to the one applied to the foil, and change in momentum over time is equivalent to force. Regardless of the appropriate "control volume" (which only you have brought up, and the rest of us feel is irrelevant to this very simple statement), what exactly is having a downward force applied to it (instantaneously), if not air? And if it is air, is it not valid to say that whatever air is having the downward force applied to it, its instantaneous change in momentum as a function of time (which is what dp/dt is anyway) is equal to the force applied upwards on the lift? 0x0077BE  [talk/contrib] 20:39, 4 October 2014 (UTC)
 * I am concerned that "indefinitely large" and "infinite" heights appear to lead to mathematically conflicting solutions: to me, that suggests a flaw in someone's model somewhere. 0x0077BE also makes an excellent point: The Statement is directly verifiable in cited sources while its repudiation is not - that requires synthesis (WP:SYNTH) by the editor, which Wikipedia abhors. As such we are bound to present The Statement as current encyclopaedic knowledge. I concede that my phrase "in free flow conditions" may conceivably be bettered, but the statement it qualifies seems to be unavoidable in one form or another. We have a three-to-one consensus on that, I personally think it's time to accept that and move on. &mdash; Cheers, Steelpillow (Talk) 22:17, 4 October 2014 (UTC)


 * A vote is not a consensus. I realize that unanimity isn't required, but satisfying all editors' legitimate concerns is supposed to be a goal.  I've cited ample evidence that my concerns about The Statement are legitimate.  To simply add The Statement without some detailed qualification, as three of you advocate, would be to ignore these concerns and would be inconsistent with the published evidence.


 * The current majority view of the evidence assigns all the weight to the AAPT papers and none to the mainstream sources that address the same question of how much momentum is imparted to "the air" by a lifting foil. This is backwards.  A quality-of-the-argument analysis of this body of evidence would assign the weighting the other way around.


 * The Statement in the AAPT papers simply says dp/dt = L, where dp/dt is the rate of change of the vertical momentum of "the air", and implies that this is a straightforward application of Newton's second law. But what is meant by "the air" is not defined, and no supporting analysis is given to establish whether or under what conditions this meets the requirements of a proper application of the second law, i.e. no evidence is given that L actually constitutes the resultant of all the forces acting on "the air".  Thus the AAPT statement is both ambiguous and technically sloppy.


 * In contrast, the mainstream sources I've cited all define precisely what they mean by "the air", and they all apply the established, rigorous method for applying the second law in fluid flows, i.e. control-volume analysis. All of the relevant forces acting on the air are included in these analyses.  The results of these rigorous analyses should be regarded as more reliable than the sloppy statement in the AAPT papers.


 * For the examples of circular and square control volumes, the cited analyses arrive at the result dp/dt = 0.5L, in direct contradiction to The Statement. I'd say a statement in the form of a mathematical equation constitutes "explicit" refutation of The Statement for these particular definitions of "the air".  And comparing what different sources say about precisely the same question is not "synthesis".


 * 0x0077BE feels that control-volume analysis is "irrelevant", and insists that we're just talking about "the air with which the foil is interacting". But this idea isn't consistent the physics or with the mainstream published sources.  How can "the air with which the foil is interacting" be defined in any other way than as the air in some defined volume?  It can't just be the air in direct contact with the surface.  That would be a layer of zero thickness and thus zero mass and zero momentum.  You have to consider a non-zero volume just to find a non-zero momentum for any body of air.  And once you do that you have to consider the other forces acting on that volume besides the force exerted by the foil.  0x0077BE says the "change in momentum over time is equivalent to force", but that's true only if "force" refers to the resultant of all the forces acting on the body.  And the mainstream sources have found that the force exerted by the foil on the air constitutes the resultant force only for the air in an infinitely tall sliver.


 * An extended control volume is a "more complicated model" than you'd apparently like to consider, but it's the minimum level of complication that makes physical sense for an extended substance like "the air". The mainstream sources I cited use this model because it's the appropriate way to apply the second law in such calculations.  This is what the minimum "basic, ideal case" looks like in fluid flows.


 * Whether you have trouble seeing "how it applies", or are unsure whether it is "being applied appropriately", or have concerns about "a flaw in someone's model somewhere" is beside the point. These analyses are the most reliable published evidence we have available to us.


 * The published evidence taken as a whole supports my position that The Original Statement is not sufficiently specific and is contradicted by several citable examples. To be true, it needs to be qualified, and as I've said before, there is only one qualifying statement that could be added to it that is verifiable.  If we're "bound" to include The Statement at all, then I think we're also bound to include the qualifier that it has been found to be true only for the air in an infinitely tall sliver, and that for regions of other shapes momentum accounts for only part of the lift, with outer-boundary pressure accounting for the rest.
 * J Doug McLean (talk) 00:10, 8 October 2014 (UTC)

As I pointed out on 12 August, one of the cited authors refers explicitly to a control volume later on in his discussion, but still sees fit to introduce The Statement without qualification. He references ''Bradley Jones. Elements of Practical Aerodynamics''. This online edition introduces a Newtonian explanation on page 12.

A control volume analysis is a far more advanced concept than Newton's second law, which is what is being described at this point in the article structure. Even so, an analysis which eliminates reaction forces and accounts only for momentum confirms that Newton's second law is fulfilled.

Likewise, to find all of the momentum which accounts for lift, you must integrate over an infinitely tall sliver, not any other region. The citations which have been provided for other analyses are therefore irrelevant.

The Statement is cited and I see no legitimate concern to prevent its inclusion. Burninthruthesky (talk) 08:00, 8 October 2014 (UTC); edited 11:14, 11 November 2014 (UTC)


 * The bottom line? "Consensus on Wikipedia does not mean unanimity" - see WP:CONSENSUS. We have exhaustively discussed a minority view and given it due weight but have nevertheless established a clear majority consensus. The process between us here is now exhausted. Anybody who still wants to make an issue of it may seek wider support through the mechanisms mentioned on WP:CONSENSUS. &mdash; Cheers, Steelpillow (Talk) 11:01, 8 October 2014 (UTC)

Moving forward...
A lot or bits have been spilled about the statement L = -dp/dt and I don't want to re-hash it all here. But as a summary and to (perhaps) draw the matter to a close I offer the following observations:


 * The Statement is correct and supportable, but only with certain limiting assumptions that are not generally applicable.
 * Adding language to indicate the limited circumstances where it is true would distract from the discussion at this point in the article - this introductory section should be fairly simple.
 * The article reads fine without it, and a good argument can be made that this qualitative intro section is not the place for a quantative statement.
 * I don't think we have received consensus to include it, even though there's really only one holdout. Wiki policy is to leave out material unless there's consensus to include it, so I think we should follow that.
 * At this juncture, I don't think that further discussion among the current active participants on the talk page will get us to consensus.
 * If we are going to take it further, I think the right approach is to do a "request for Comment" or a "request for close".
 * My own preference is to just leave it out and move on. That seems to be the de-facto result at this point and I'm fine with that.

All that said, let's return to the article as it currently appears. I'm not really happy with the following paragraph:


 * Newton's second law, F=ma, tells us that the lift force exerted on the air is equal to its mass times its downward acceleration. This is often more conveniently expressed as the rate of momentum change over time.[28] This analysis is accurate for an airfoil in free flow conditions, but in a restricted space (such as when flying in ground effect) local pressure variations can also come into play.

I don't really know what we're trying to express with this paragraph, other than talking about the rate of momentum change while carefully avoiding saying The Statement. I think we either need to make it say something more direct/concrete or simply remove it. As it stands, I don't think it adds much to a laypersons understanding. Other thoughts? Mr. Swordfish (talk) 17:15, 10 November 2014 (UTC)


 * First of all, WP:CONSENSUS does not require unanimity, nor necessarily a majority in any discussion or vote, see also WP:CLOSE on closing discussions.
 * Secondly, there is a big difference between an approximation which is sometimes useful and a fundamental principle which is sometimes a sufficiently close approximation. This introductory section needs to be introducing the principles, not discussing their validity as approximations. The concluding apologia about restricted spaces was introduced in an attempt to address concerns expressed by one editor, based on struggles with the approximation. My own view is that the apologia is out of place in introducing the principle and should be deleted. (furthermore, contrary to the first bullet point above, the principle is indeed an adequate approximation for most purposes: it is mathematically equivalent to F=ma and nobody is challenging that).
 * Thirdly, the first part of the existing paragraph, introducing F=ma, is saying something very important. If the mathematically equivalent rate of momentum change is common in textbook discussions then it is useful to introduce it here, otherwise not. If it is to stay, it needs appropriate citation - I do not find Richard Feynman (currently cited) an adequate source in this context, as he was not an aerodynamicist.
 * Finally, if momentum change is to be introduced here, do we need to give the equation F=-dp/dt or are the existing words sufficient? my own view is that, unlike F=ma, the equation is not sufficiently well-known to be recognisable by the layman and should be left out.
 * &mdash; Cheers, Steelpillow (Talk) 19:15, 10 November 2014 (UTC)


 * I agree it seems unlikely this discussion will achieve unanimity, and also that unanimity is not a requirement. WP:CONSENSUS says, "The quality of an argument is more important than whether it represents a minority or a majority view". I believe articles should not be allowed to suffer through an argument to moderation, which is a risk when editors try to avoid a version of an article just because it provoked an argument, or leave a compromise which pleases nobody. My view on how WP:CONSENSUS should apply to this discussion hasn't materially changed (see 7 September).
 * Just to clarify, the [//en.wikipedia.org/w/index.php?title=Lift_%28force%29&oldid=624501848#Flow_deflection_and_Newton.27s_laws 7 September revision of the article] only cites Feynman to support a generic statement of Newton's second law (which is all it does support), and provides further citations to support more specific statements relevant to lift. As far as I'm aware, the equation F=-dp/dt has not at any time been mentioned in the article body (only the notes, which I think is appropriate.) Burninthruthesky (talk) 11:14, 11 November 2014 (UTC)


 * Correct, the equation F=-dp/dt has never been in the article (at least in recent memory - maybe it was a long time ago before I started participating), but it was expressed in English as "The resulting force upwards is equal to the time rate of change of momentum of the air deflected downwards" which is the same thing. I don't think anyone is advocating using the formula in the article. I'm not, although I have no problem using it here on the talk page as shorthand.


 * I think Steelpillow has a good point in saying " If the mathematically equivalent rate of momentum change is common in textbook discussions then it is useful to introduce it here, otherwise not." To answer that, there are several articles that indicate that rate of momentum change is fundamental to the understanding of lift, the most prominent is the AAPT's pedagogical advice "...lift on an airfoil should be explained simply in terms of Newton’s Third Law, with the thrust up being equal to the time rate of change of momentum of the air downwards." (https://en.wikipedia.org/wiki/Lift_%28force%29#cite_note-7) There are other cites, included elsewhere on this page. Physicists often use dp/dt and ma interchangeably, although I'm not sure how commmon it is among engineers. Looking at aerodynamic textbooks my impression is the idea of rate of momentum change is not explored very thoroughly, if at all.  The thing is, trying to expand rate of momentum change into useful engineering is basically a dead end.  Aerodynamics textbook writers are eager to get to the useful engineering bits so they don't do much with this idea.  It's the sort of thing that appeals to a physicist taking a broad view of the physics, not someone who is trying to solve a particular problem.


 * So, do we treat rate of momentum change? The AAPT thinks we should.  One editor found it "confusing".  I can go either way - leave it out, or present a simple short declarative non-apologetic statement.  Mr. Swordfish (talk) 14:51, 11 November 2014 (UTC)


 * If you're asking my opinion, yes, I would put The Statement back in (actually I did, on 7 September). Nobody is convinced that it is in fact, "contradicted by several citable examples" and I think the argument that, "The AAPT thinks we should" far outweighs any other.
 * I agree with Steelpillow there is a "clear majority consensus" here for including it in the article. Wikipedia policy says we should be WP:BOLD in updating the encyclopedia.
 * If you still think a more formal method of dispute resolution is more appropriate at this point, you are entitled to start one. Personally, I think the time and effort spent on this discussion became excessive some time ago, which is why I am reluctant to ask yet more editors to become involved. Burninthruthesky (talk) 07:47, 12 November 2014 (UTC); edited 11:04, 12 November 2014 (UTC)
 * I have posted a note at Wikipedia talk:WikiProject Aviation, asking for an uninvolved editor to close this whole discussion. &mdash; Cheers, Steelpillow (Talk) 10:43, 12 November 2014 (UTC)


 * Closing this discussion would be premature. Mr. Swordfish has proposed two options, either of which would involve changes from the current version of the article, i.e. to leave The Statement out, or to include it in "non-apologetic" form.  His question hasn't really been answered.


 * I agree with Mr. Swordfish's third bullet item, i.e. that "this qualitative intro section is not the place for a quantitative statement". This by itself is a sufficient reason to leave The Statement out.  Yes, Newton's second law is "something very important", but the version of this explanation of lift that was in place before The Statement was added already explained the importance of the second law in a qualitative way.


 * The other compelling reason is that The Statement in non-apologetic form misrepresents the momentum balance as being simpler than it really is. Mr. Swordfish has it right in his first bullet item:  The Statement dp/dt = -L is true only "with certain limiting assumptions" (i.e. that dp/dt is integrated over a region that is very tall compared to its width).   For integration over a region of any other shape in free air, dp/dt ranges from zero to -L, depending on the shape of the region.  And for the atmosphere as a whole with a ground plane, dp/dt = 0.  Details and citations from the mainstream literature are in my post of 27 September.


 * Note that all but one of these analyses assumes an unbounded atmosphere (Only the domains of integration are finite). So Steelpillow's "concluding apologia", which implies that The Statement's only problem is with "restricted spaces", doesn't address the problem.  Steelpillow is also mistaken in characterizing the problem as one of "approximation", unless one regards dp/dt = 0 or dp/dt = -0.5L as "a sufficiently close approximation" for dp/dt = -L.


 * The question addressed by The Statement can be expressed symbolically as "dp/dt = ?" The AAPT papers give the answer as "-L", without specifying any qualifying assumptions or providing any supporting analysis (Mentioning a "control volume" is not the same as actually presenting a control-volume analysis).  The mainstream sources I've cited state their assumptions explicitly, explain their analyses in detail, and find answers of "0", "-0.5L", and "-L", depending  on the shape of the domain of integration.  This is more complicated than the AAPT answer, and it seems to offend the physical intuition of a majority of this group, but this is what mainstream aerodynamics sources say dp/dt is for the air surrounding a lifting foil.  The majority's speculations to the contrary (and their protestations that the mainstream results are somehow not "relevant") are not supported by specific analysis in any citable source.


 * The majority advocates relying solely on the AAPT papers and ignoring what the mainstream sources say on the same question ("dp/dp = ?"). To me, this seems like a case of Truth by consensus.


 * The momentum balance associated with lift is complicated. The fact that it's caused such confusion on this page indicates to me that the article should include some discussion of it.  But the "Simplified physical explanations" section isn't the right place.  I think the right approach would be to leave The Statement out of that section and add a new subsection, titled "Momentum balance in lifting flows", that briefly explains the findings of the control-volume analyses and mentions how the AAPT statement fits in with those findings.  This might fit well after "Pressure integration" or after "Circulation and the Kutta-Joukowski theorem".  I'd be willing to draft it if there's support for it.
 * J Doug McLean (talk) 23:52, 13 November 2014 (UTC)
 * FYI I have found a simpler way to rebut your view here. You accept that F=ma and every physicist tells us that ma=dp/dt. The statement that -dp/dt=0 is mathematically equivalent to the statement that ma=0. Now, when we stand back and look at a large chunk of atmosphere as a plane flies through, this is reasonable enough - the atmosphere does not accelerate down on us more and more with every passing plane, it soon decelerates and swirls back up again. But nobody claims that ma=0 apples to the local airflow over the wing. Yet in claiming that dp/dt=0 you are effectively claiming that it does. No, only once the craft has passed and the trailing vortex brought the air back up again will either ma or dp/dt return to zero. And that does not contribute to the lift. This article is about the lift, not about how the atmosphere restores itself afterwards. OTOH those learned models you quote do address the restoration. Your whole analysis does not belong here. I am sure that it has a place on Wikipedia, though not in the present article.
 * There is clear consensus for the statement to remain but for its apology to go. This is based on reasoned argument and sources, not on ultimate truth by consensus as you suggest. There is nothing new in this your latest journey round the old circle, like I say the discussion is exhausted.&mdash; Cheers, Steelpillow (Talk) 09:05, 14 November 2014 (UTC)
 * I agree. I think a formal closure is for the best. Thank you. Burninthruthesky (talk) 16:26, 14 November 2014 (UTC)
 * The total downward momentum imparted to air within the atmosphere is the same whether someone makes an integration which accounts for all of it correctly or not. This is basically what I've been trying to say since August.


 * There are sound logical reasons for saying analyses above are irrelevant. For example, see my comment on 8 October for an explanation of why only one of the cited analyses can answer the question of, "how much momentum is imparted to the air by a lifting foil". To reiterate, the other analyses do not account for all of the momentum imparted to the air. The one that does, proves that, "The resulting force upwards is equal to the time rate of change of momentum of the air deflected downwards". The correct interpretation of this statement (The Statement) is true whether proof is stated or not.


 * Other interpretations place undue weight on the meaning of two words. If "the air" meant, "air passing the outer boundary of an integration region of an arbitrary size and shape", it would be correct to say The Statement is not specific enough. That is not what it says. Integration is not mentioned. Even so, the tall control volume which shows that dp/dt = -L is entirely contained within the atmosphere, and so the momentum it finds obviously still exists within the atmosphere. Nobody here is claiming that "the whole atmosphere" is deflected downwards. As I understand it, neither are the cited authors. Burninthruthesky (talk) 09:56, 15 November 2014 (UTC); last edited 10:46, 28 November 2014 (UTC)

Extended Discussion
There has been a request at WP:ANI for formal closure of protracted discussions, going on for months (more than a year). Unfortunately, due to the excessive length of the discussion, I don't see what sort of closure is desired. If there is a content dispute, I suggest that it be taken to one of the processes for dispute resolution, such as a Request for Comments or moderated dispute resolution at the noticeboard. If there isn't a content dispute, just excessive talk, then we can box (archive) the discussions. What is the question? Robert McClenon (talk) 04:03, 20 November 2014 (UTC)
 * There has been extended discussion of a comment made on 27 July, 'the statement "The resulting force upwards is equal to the time rate of change of momentum of the air downwards" is problematic'. The latest attempt to conclude the discussion is in the section above. Burninthruthesky (talk) 08:02, 20 November 2014 (UTC)
 * There was one dissenting editor who felt that the issue of momentum was being improperly treated. I have just made the change that consensus seemed to be pointing to - see diff. Technically I should have waited until someone like Robert arbitrated on the discussion, but it sprawled on in such an incomprehensible tangle so that I thought creating a diff would be the best marker in the sand. If Robert can close the discussion accordingly, that would be brilliant. Otherwise I guess we'll have to wait and see whether anybody still wishes to contest my edit: I'd suggest the best thing would be for an aggrieved party to do as Robert says and take it to dispute resolution rather than return to ANI just yet. HTH. &mdash; Cheers, Steelpillow (Talk) 11:29, 20 November 2014 (UTC)
 * I have also reordered a couple of topics on this talk page, so that the relevant discussion now starts at and runs continuously down to here. &mdash; Cheers, Steelpillow (Talk) 11:40, 20 November 2014 (UTC)
 * And there's more in, which I've re-ordered as well. (Some earlier discussion of sources relevant to the discussion started before that). Burninthruthesky (talk) 16:52, 20 November 2014 (UTC)
 * An uninvolved editor has now opened a closure request at Administrators' noticeboard/Requests for closure. Apparently there's quite a backlog, so it might take a while before anyone arrives to close the case. &mdash; Cheers, Steelpillow (Talk) 15:55, 20 November 2014 (UTC)
 * Well, the recent major re-write of the article that took place about three months ago took over a year and a half to see the light of day. Then we've  spent the last three months arguing about one sentence.  I'm not in any particular hurry and am willing to be patient.  (c:   BTW, I support your edit from yesterday, at least as an interim step. If we don't get a ruling from a non-involved closing editor for a while I'm ok with that.  The article as it exists now is a big improvement over what was here before Doug showed up to help with things.  Mr. Swordfish (talk) 01:27, 21 November 2014 (UTC)
 * I've boxed the extended discussions, since there doesn't appear to be any continuing disagreement. If anyone disagrees, they can add their comments at WP:AN and request closure review.  The original comment, that humans cannot fully explain why airfoils generate lift, was philosophical rather than scientific.  A mathematical explanation is sufficient for most scientific purposes.  Robert McClenon (talk) 03:23, 21 November 2014 (UTC)
 * Thank you all for your helpful responses to my request. Burninthruthesky (talk) 10:20, 21 November 2014 (UTC)


 * Well, I see some boxes around a couple of discussions with notes attached saying something to the effect that "the issue seems to have been resolved" but I'm still at a loss regarding what to do with The Statement ("The resulting force upwards is equal to the time rate of change of momentum of the air downwards"). Is there consensus to include it?  Is there consensus to exclude it?  Should we file another request for closure (this time on the right notice board clearly spelling out the issue)?  Maybe do an RFC first?  Mr. Swordfish (talk) 19:55, 24 November 2014 (UTC)
 * There is a strong consenus to keep it in some form of words. Cogent arguments have been made to avoid the actual equation, and nobody has spoken in its defence. This has been made clear before and the article has been edited accordingly. This is why the independent editor who closed the L=-dp/dt topic said it appeared to have been resolved. I accept that judgement. IMHO we don't need to formally close anything more, it is time to move on. &mdash; Cheers, Steelpillow (Talk) 22:26, 24 November 2014 (UTC)
 * Fair enough. I'm satisfied with the present wording (Newton's second law, F=ma, tells us that the lift force exerted on the air is equal to its mass times its downward acceleration. This is often more conveniently expressed as the rate of momentum change over time.) and am happy to move on.  Mr. Swordfish (talk) 18:54, 25 November 2014 (UTC)


 * I just came across the following quote from Lanchester:


 * The basis of the Newtonian method is found in the principle of the conservation of momentum, which may be taken as corollary to the third law of motion as written: When force acts on a body the momentum generated in unit time is proportional to the force. p.2


 * I see he then goes on to explain deficiencies of the Newtonian method. I don't have time to read it all in detail. This seems to concur with a lot of what's already been discussed.


 * For the time being, I too am quite happy with the current state of the article. I've no doubt it will continue to improve and evolve in future. Burninthruthesky (talk) 10:28, 27 November 2014 (UTC)

Reopening the old "Lift = -dP/dt" debate
(Zapletal writes ->) This is a REWRITTEN version of my previous 2 posts, which were repeatedly deleted, apparently because of presumed "personal attack".

To those who did the deleting, PLEASE READ THIS THROUGH before deleting again. If you wish to delete again, then please explain why.

(First post from 27 November 2014.) Briefly,
 * This issue is not resolved.
 * "The Statement" is incorrect.
 * Doug McLean is right. Lanchester also explained this position very well in 1894, and then in his book of 1907.
 * I suggest the editors who agree with "TS" should study the history of Fluid Dynamic Lift theory.
 * The AAPT is negligent in its promotion of "The Statement" (see below), and is most definitely negligent in suggesting that F->dP/dt is Newton's THIRD.
 * NII, paraphrased, is "F CAUSES P-dot". Teaching that NII is "F=ma" is plain WRONG, and consequently leads to mistakes.
 * IMO "The Statement" should be IN the article, but with an explanation for why it is so WRONG. Else future generations will have an ever worsening understanding of FDL.

(Doug, I may write more in few weeks, if I have time. You are NOT ALONE. "Hourglass" thinking is good. NB the "upwash" always present in front of the aerofoil. FWIW, a single vortex in unbounded fluid is as likely as a magnetic-monopole, because it implies infinite angular-momentum and energy, hence difficulty with calculations. So calcs that include a ground-plane (= image vortex underground) work better. And there is most definitely a "wavelike" motion in lifting flows that explains most of above (see Lanchester, 1907).) (End first post.)

(Second post from 29 November.) Burninthruthesky, are you aware of the particular "Newtonian method" that Lanchester is discussing in your above quote, namely the assumed type of fluid? Are you aware of the "deficiencies" of that method, namely that it gives the wrong results? Are you aware that Lanchester, who was the first to develop the current, accurate, "circulation" theory of lift (in 1894) did so by adopting the "Principle of No Momentum"?

In short, the notion that lift is a direct consequence of "rate of change of momentum of the fluid downwards" was known to be flawed in the 1700s. Some 120 years ago, Lanchester (and then Kutta, Zhoukowsky, Prandtl, et al) fixed this problem, and came to the much more accurate explanation of "circulation theory" by abandoning "The Statement".

So, why do you, together with 0x0077BE and Steelpillow, want to reinstate this flawed model? (End second post, Zapletal.)101.170.42.165 (talk) 10:10, 30 November 2014 (UTC)


 * Our answers may be found in the long debate which sprawled across many topic headings above. Since you are sufficiently prejudiced against our view to twice post insults to us here and yet a third time at User talk:J Doug McLean, there appears little point in us engaging with your rhetoric. Since you also appear to be new to Wikipedia, I would strongly suggest that you check out WP:Five pillars and especially WP:CIVIL. This will explain why I deleted your insulting posts from here. I reply now and offer my advice only out of politeness, but I am unlikely to respond further unless there is clear evidence that this conversation might not simply rehash the same old same old but actually, and with some semblance of decorum, go somewhere encyclopedic. &mdash; Cheers, Steelpillow (Talk) 09:47, 2 December 2014 (UTC)


 * In posts since 15 November Steelpillow and Burninthruthesky have put forward what amount to new arguments to effect that the classical control-volume analyses are irrelevant. So the following rebuttals are not the "same old same old".


 * Steelpillow wrote:


 * [T]he atmosphere does not accelerate down on us more and more with every passing plane, it soon decelerates and swirls back up again. But nobody claims that ma=0 apples to the local airflow over the wing. Yet in claiming that dp/dt=0 you are effectively claiming that it does. No, only once the craft has passed and the trailing vortex brought the air back up again will either ma or dp/dt return to zero. And that does not contribute to the lift. This article is about the lift, not about how the atmosphere restores itself afterwards. OTOH those learned models you quote do address the restoration.  Your whole analysis does not belong here.


 * First, this misrepresents what I claimed, which was only that dp/dt = 0 for a pancake control volume that is very wide compared to its height and thus extends far from the foil. This is not the same as claiming dp/dt = 0 applies to the "local airflow".  But are you thinking dp/dt = -L applies to the "local airflow"?  That isn't true  either.  Sources have found dp/dt = -L only for the tall sliver control volume.  For practical purposes the height doesn't have to be infinite, but it does have to be quite large.  To get 99% of the "right" answer, i.e. dp/dt = -0.99L, the total height must be about 64 widths.  So to find the answer advocated in The Statement you have to look beyond the "local airflow".


 * But the main flaw in your reasoning is the idea that there are two separate processes going on, the imparting of downward momentum by the foil and the "restoration" ("afterwards") by the atmosphere, and that only the first "contributes" to lift. I've not seen any reputable source on aerodynamics that describes lifting flow in those terms.  And your two-process model isn't consistent with the following features of the flow:


 * 1) In an integrated sense, half the upward acceleration (positive dp/dt) in the field happens ahead of the foil, in air that has not yet been accelerated downward.  Is this part of your purported "restoration" process?


 * 2) There is substantially more downward acceleration (negative dp/dt) in the field than just -L.  We can see this by looking at an hourglass-shaped control volume that contains negative dp/dt almost exclusively, i.e. that concentrates on "the air deflected downward".  Using the same mathematical velocity-field expressions used in the published classical analyses, it is easy to show that an hourglass-shaped control volume with a total height of about 64 chords contains dp/dt = -1.8L.  Based on your model of the flow, to what would you attribute the excess -0.8L?  On the other hand, it's easy to explain in terms of the interaction between the pressure and velocity fields, and it's consistent with the citable analyses.


 * So your two-process concept is seriously flawed, and so is the idea that anything other than downward momentum is irrelevant to lift. All of the fluid accelerations in the field are essential to maintaining the pressure differences acting at the airfoil surface.  In a continuum flow these pressure differences at the surface must be accompanied by an extended pressure field that has the two-lobed form sketched in the section "A more comprehensive physical explanation".  The lobes of low and high pressure extend beyond the leading and trailing edges of the foil, and the resulting upward accelerations ahead of the foil and behind are essential to sustaining the pressure field, as Lanchester explained in 1907.  So your assertion that upward accelerations don't contribute to lift is unfounded.


 * The momentum balance in the field around a lifting foil is not irrelevant as you claim. Some discussion of it belongs in the article.  I intend to draft new subsection "Momentum balance in lifting flows".  Then we'll let the community decide whether it belongs or not, based on the detailed merits and on the citable sources.  You don't have sole veto authority.


 * Burninthruthesky wrote:


 * The total downward momentum imparted to air within the atmosphere is the same whether someone makes an integration which accounts for all of it correctly or not.


 * and


 * To reiterate, the other analyses do not account for all of the momentum imparted to the air. The one that does, proves that, "The resulting force upwards is equal to the time rate of change of momentum of the air deflected downwards". The correct interpretation of this statement (The Statement) is true whether proof is stated or not.


 * Regarding the integrated negative dp/dt in the air, there is no basis for saying that -L is "all of it", or that it represents the "total". The tall sliver control volume doesn't find "all of it".  The hourglass analysis finds that considerably more negative dp/dt than -L "exists" in the atmosphere if you look in the right place (see above).  No, there is not just one correct "total" that somehow applies to "the air" in general.  There are many different correct totals for the "time rate of change of momentum of the air" depending on what part of the air you integrate over.  The mainstream published analyses make this clear.  And the idea that they are irrelevant isn't consistent with the physics.


 * Burninthruthesky wrote:


 * Other interpretations place more weight of meaning on the words "the air" than they can bear. If "the air" meant "an integration region of an arbitrary size and shape", it would be correct to say The Statement is not specific enough. That is not what it says. Integration is not mentioned.


 * True, "integration" isn't explicitly mentioned, but it is inescapable. The Statement makes a specific mathematical claim about a quantity ("the time rate of change of momentum of the air") that for a continuum fluid in non-uniform motion has no logical definition other than by integration over some fixed volume.  The idea that The Statement refers to something other than an integrated quantity defies logic.


 * Burninthruthesky also cites Lanchester's discussion of "the Newtonian method" and its deficiencies, concluding that it "seems to concur with a lot of what's already been discussed." Actually, Lanchester's "deficiencies" section provides no support for The Statement, but instead supports what I've been arguing all along. The "Newtonian medium" (a hail of projectiles that don't interact with each other) is a poor model for flows of real fluids.  In a Newtonian hail of projectiles, pressure exists only where the particles impact a surface.  There is no pressure internal to the fluid and thus no pressure field, and as the classical analyses show, the pressure field is crucial to assessing the momentum balance in lifting flows.  Your other citation, Bradley Jones. Elements of Practical Aerodynamics, also introduces a Newtonian flow model, but doesn't get the details right (e.g. he finds a sin(alpha) lift curve for an inclined plate, where Newton found sin squared) and is remiss in not discussing the deficiencies.


 * I'm posting this because it didn't seem right to leave the latest round of erroneous reasoning unanswered.  Zapletal's joining the discussion means that in a pure numerical sense the "consensus" is not as "strong" as before.  Our opposition is not based on "prejudice", but on the fact that the majority position is inconsistent with established physics.


 * J Doug McLean (talk) 19:45, 5 December 2014 (UTC)


 * I'm a bit reluctant to re-engage with this debate since it has gone on so long, but having thought about it some more and re-reading the arguments I'm now persuaded to Doug's position - The Statement is problematic and misleading. (I have to confess to being mislead by it myself) That said, I don't think it is actually wrong but it requires a bit of clarification and specific language to treat momentum transfer in a technically correct manner.


 * Upthread, Doug suggested including a section on momentum transfer. I support this. If we're going to treat The Statement and momentum transfer in the article we'll need a paragraph or two. My hunch is that if we go about writing it we'll probably be able to reach consensus since we'll have the space to be more nuanced and won't be constrained by an 'up or down' ruling on The Statement.


 * Thanks to Doug for asking tough questions and making me think this through more thoroughly - I hadn't really given it the thought it deserved. I'll probably have more to say on Monday. The article is fine as it is, and we don't have a deadline.  Mr. Swordfish (talk) 20:35, 5 December 2014 (UTC)


 * What we think, is of no importance. This is:
 * "Thus the lift of the wing is equal to the rate of transport of downward momentum of this air." &mdash; Clancy, L.J.; Aerodynamics, Pitman 1975, page 76.
 * Wikipedia relies on Verifiability, not truth (WP:NOTTRUTH). Need I say more? &mdash; Cheers, Steelpillow (Talk) 20:43, 5 December 2014 (UTC)


 * Yes, I'm aware of the policy on "verifiability, not truth". But that policy doesn't mean that our job as editors is to take statements from sources out of context, without comparing them with what other sources say, or without exercising some judgment as to what the statement represents.


 * Clancy's statement, for example, was made in the context of his discussion of a very crude model for the flow around a 3D wing, in which the flow is seen as a stream of circular cross-section, like that from a fire hose (see his fig 5.21), that is uniformly deflected downward by its interaction with the wing. This model is similar to the Newtonian flow model I discussed above in the sense that it takes no account of the non-uniform pressure field around the wing.  This deficiency of the Newtonian model has been discussed by Lanchester and other citable sources.


 * The horseshoe-vortex model is a higher-fidelity model for the flow around a 3D wing. Lissaman (1996) and others have used this model and arrived at the same conclusion as for a 2D foil, i.e. that The Statement is true only for the special case of a tall sliver (tall slab in 3D).


 * But even if we were to take Clancy's statement at face value, it's only one view among many in citable sources.  The classical control-volume analyses I've cited arrive at a more nuanced view, showing that The Statement is valid only for a special case.  If we keep The Statement in its current unapologetic form, we'll be presenting a biased view of what the established sources say on this topic.  What I'm saying here isn't just my own "truth".  It's all verifiable in citable sources.


 * Thanks, BTW, to Mr. Swordfish. I appreciate the positive feedback.


 * J Doug McLean (talk) 01:26, 6 December 2014 (UTC)


 * J Doug McLean, thank you for listening, and addressing the issues I raised. My understanding of the evidence presented is that "lift is accounted for either by pressure or by momentum flux, depending on the proportions of the control volume." We must consider only citable sources, and you said that they do not find more negative dp/dt than -L.


 * Lanchester says there are "circumstances in which Newtonian theory is applicable and those in which it is not". This is discussed fully in §8. It was suggested above that an isolated airfoil (outside ground effect) is one of those circumstances where the method does give results "within measure of the truth", as Lanchester puts it. Why do you say that 'Lanchester's "deficiencies" section provides no support for The Statement'? Burninthruthesky (talk) 11:23, 6 December 2014 (UTC)


 * Yes, I know of no citable source for the hourglass analysis. I raised that example only as an argument against other original-research arguments on this page.  If discussion of the control-volume results is added to the article, as I advocate, it should of course be limited to the citable sources, all of which have found dp/dt between zero and -L.


 * At the top of §5 Lanchester says "It is evident from the above that the theory of the Newtonian medium is capable of giving results within measure of the truth, when applied to real fluids." The only thing "the above" refers to, as far as I can tell, is the drag of a plate normal to the flow.  I don't see anywhere that Lanchester applies the Newtonian method to the lift of a plate or foil at angle of attack, or where he says it gives reasonable results for that case.  Newton's own results for an inclined plate are not "within measure of the truth".  Newton's lift curve (proportional to sin squared) has zero slope at zero alpha, where foils in real flows have lift slopes in the neighborhood of two pi.


 * In §8, the only example he cites where the Newtonian method is "applicable" is the flow through the disc of a screw propeller. My reason for saying that 'Lanchester's "deficiencies" section provides no support for The Statement' is his conclusion of §5:  "It is thus apparent that no momentum is imparted to an actual fluid in the sense that it is imparted to the Newtonian medium".  That and the fact that nothing else in §5 supports dp/dt = -L.


 * J Doug McLean (talk) 22:49, 6 December 2014 (UTC)


 * Lanchester §8 says the Newtonian method is applicable in cases "where there are well-defined conditions on which to compute the amount of fluid dealt with per second". We agree there are well-defined conditions which show dp/dt = -L.


 * He discusses aerodynamic support of an aerofoil in §112. He says, "Under ordinary conditions" "the weight may be regarded as in no part statically supported." He goes on to clarify the exception of ground-effect.


 * I support the proposal to add more description of momentum/pressure further down the article, along with keeping a simple description of the Newtonian method at the beginning of the article, in line with WP:UPFRONT. Burninthruthesky (talk) 07:50, 7 December 2014 (UTC)


 * Yes, there are "well-defined conditions" for which dp/dt = -L. But as I've maintained all along, those conditions are narrow (literally), and The Statement in its unapologetic form is therefore misleading.


 * Lanchester's §112 amounts to a verbal derivation of the tall sliver control-volume analysis and arrives at the same result published by Lissaman (1996). So it doesn't support an unapologetic version of The Statement any more than Lissaman did.


 * The simple qualitative flow-deflection explanation based on Newton's laws belongs up front. I think the quantitative statement dp/dt = -L doesn't, whether it comes from the tall sliver control volume or from the "Newtonian flow model".  See my response to Steelpillow's post of 7 December.


 * J Doug McLean (talk) 01:42, 10 December 2014 (UTC)

(Zapletal Writes->) (Just noticed that Doug and others posted while I was writing this. Will read above and may reply tomorrow...)

Steelpillow, the following is a long post covering the above issues. It is broken up with sub-headings to improve readability. Please make yourself comfortable.

1. MY PREVIOUS POSTS HERE. - Yes, I don't spend much time on these Wikipedia Talk pages. This is mainly because I find the formatting very difficult to read, and even harder to write.

The "wall of words" makes it very difficult to know who is writing what. Especially so when one author's long post, with name only at the bottom, is then broken up with many other author's posts. For this reason I have recently got into the habit of bracketing my posts with my name, as above/below here, simply so that I can more easily navigate this alphabet soup.

Regarding entering the text, whatever happened to WYSIWYG!? IMO, most every other forum on the web does "communication" much better than Wiki, ... but that is another matter.

Anyway, I have been here before. First back in Feb 2011, on archived Talk page 5, headings "Is Vicosity Necessary..." and "Correction Required...". Also on archived Talk page 7, July 2013 "This Article is Still Deeply Flawed". And a few other places...

I have not read through all the archived Talk pages yet (for reasons given above), but I noticed that Doug McLean has been here from near the beginning, on page 1. Sadly, it has taken Doug some 8 years, and apparently much, much effort, to improve this article. And still he meets with unreasonable obstruction. I can understand why he hasn't responded recently.

~o0o~

2. "THE STATEMENT" (IE. L = -dP/dt) SO FAR. - Having quickly read Talk page 1, I see that exactly this issue was being argued ad nauseam back then. Both there, and on this page, the pattern is the same.

The posts arguing "against TS" are usually long. They often contain definitions of the terms used, or at least an attempt at such. They are somewhat technical, and appear capable of giving quantitative results (ie. real numbers), though the actual "maths" isn't always included. They are supported by many "Reliable Sources" going back many years. And they appear to be prepared to clarify their case in even more detail, if requested.

Conversely, the posts arguing "for TS", namely that "Lift is fundamentally due to downwash", tend to be much shorter. They are usually bereft of well-defined terms, or detailed maths. They have a decidedly ideological flavour (ie. "...it MUST be so..."). They quote as their RSs bodies such as the AAPT, who are, quite literally, "dilettantes" in this particular field. And they seem to be in a great rush to meet "consensus" (ie. in their favour), and "move on".

I have read all the "for" arguments on this page, and, despite the difficulty of keeping track of who is saying what, I have not found a single convincing "for" argument. More on this below.

~o0o~

3. MY REASONS FOR BEING HERE. - My bluntness in my above, much deleted, post was a result of the above point. Namely, that this issue has been clarified so many times over, that it does not warrant yet another Talk page "wall of words". Nor should the Article revert, yet again, to a version that promotes misleading ideas such as "TS".

My aim in writing these and previous posts here, has always been to improve the educational prospects of future generations. That is, I have tried to encourage a higher standard of this particular Wiki Article. To explain this again, I quote here from my July 2013 post.

''Furthermore, the notion that an Encyclopedia is simply "an assembly of information taken from reliable published sources" is codswallop. There are today countless "reliable published sources" offering many and varied explanations of Fluid Dynamic Lift. Unfortunately, a great many of these explanations are pure bulldust. A random sampling of these sources does not constitute an encyclopedia. Worse yet, a biased sampling, which is perfectly acceptable under your definition, is a disservice to society in general.''

''In short, the editors of good encyclopedias filter out the nonsense. If this does NOT happen, then very soon we will have to believe that the world is flat, voodoo is real, and the star signs predict our future. That is, any and all of the urban myths and superstitions that can be found in some "reliable published source" somewhere, will quickly spread throughout society. This is because these myths are usually easier to understand than the more difficult truths, so are more often repeated. Soon after, the difficult truths become outnumbered, and eventually disappear.''

''This particular article is a good example of the above process. The "Bernoulli vs Newton" explanations, the "Coanda effect", and "Lift is because of downwash," are all examples of the dumbing-down of this phenomena. An appropriate quote by Theodore Von Karmann is, "When you are speaking to technically illiterate people you must resort to the Plausible Falsehood instead of the Difficult Truth.". ''

(Extra emphasis added.)§

~o0o~

4. WIKIPEDIA AND EDUCATION. - My general feeling about Wikipedia is that it is hastening our society's descent into The Next Dark Ages. It is doing so by encouraging academic standards that amount to "the lowest common denominator". The fact that anyone, at anytime, can change Articles, or even delete other peoples' Talk posts, without any good reason, means that any notion of eventually achieving high standards is wishful thinking.

To explain by example, in the "Golden Age" of antiquity (~ 6th to 3rd centuries BC) it was widely considered that the Earth is a sphere spinning on its axis, and together with the other Planets it orbits the Sun, with the fixed Stars at a much greater distance. The Pythagoreans believed this. Aristarchus clearly put this case. Eratosthenes even measured very accurately the Earth's diameter, and its distance from the Sun.

Such "difficult ideas" were allowed to spread back then because the national pastime was debating stuff, any stuff, in the town square. In fact, the first subjects taught at school, the "Trivium" of Grammar, Rhetoric, and Logic, were intended to hone those skills. ROBUST DEBATE WAS ENCOURAGED! Others' arguments were not simply deleted. Instead, they were critically analysed, dissected, deconstructed, and ultimately shown to be either bereft of reason, or perhaps, maybe, worth considering...

But all this was hard work. So, within a few centuries the Ptolemaic System of Astronomy took hold, and the great-unwashed were taught that the Earth was flat, and at the centre of the Universe. So much easier for the great-unwashed to learn! And so much easier for the developing Priesthood to teach, especially when embellishments like the Four-Elephants-And-Then-Turtles-All-The-Way-Down could be added willy-nilly.

So, what has this rant got to do with "TS, L = -dP/dt"?

Well, if any supporter of "TS" can point to a single argument in its favour that they think is valid, in all of these walls of words, then I will happily explain where said argument is flawed. Please note that my response may necessarily have to be robust, in order to get the message through. For example, a single flaw in an argument can make the argument, as a whole, INVALID.

~o0o~

5. TWO CASES TO THINK ABOUT. - The following two examples are intended to improve readers' understanding of this issue. These are NOT intended to prove that, in this particular field of Fluid Dynamic Lift (= FDL), "L = -dP/dt" is wrong. So please don't say "Oh, but that's different, so it doesn't count...". Rather, the intent here is that if the reader can understand the following, then they will better be able to grasp why "TS" is wrong.

~o0o~

5.1. Consider a "lighter-than-air" airship, such as a hot-air balloon or zeppelin. (This example also mentioned on Talk page 1.)

Ask yourself whether any downward force acting on such an airship, such as its gravitational weight, or a rope pulling it downwards, needs to be counter-acted (NIII) by an upward inertial reaction force that MUST come from a never-ending "downward rate of change of momentum" of the surrounding atmosphere (NII).

The answer, of course, is that the airship can stay aloft with NO MOVEMENT of the atmosphere at all. It is AEROSTATIC Lift. This works because the amosphere's gravitational mass, in concert with the Earth's gravitational mass and the consequent (hypothesized) gravitational field, creates a PRESSURE GRADIENT in the atmosphere.

That is, the pressure at lower altitudes is greater than the pressure at higher altitudes. As a result of this pressure gradient, the air-pressure forces on the under-surface of the airship are greater than the air-pressure forces on the over-surface. As I hope is obvious, the net sum (or integration) of these surface pressure forces amounts to a Lift force on the airship.

In a similar way, in FDL the Lift force is nothing more than the integration of the fluid pressure forces acting on the surface of the lifting body, typically an aerofoil (2-D) or wing (3-D).

However, a big difference is that in this "buoyancy" example here, the pressure gradient is due to the fluid's GRAVITATIONAL mass embedded in a gravitational field, whereas in FDL the pressure gradients are due to the fluid's INERTIAL mass, and its behaviour in Lanchester's "sustaining wave". More below.

~o0o~

5.2. Consider a wave on the ocean. Better yet, consider a surfer riding such a wave. Or since some readers here are interested in yachts, consider a yacht surfing the wave. (And to be a bit more rigorous, perhaps we should be considering "solitary waves", aka "solitons", travelling along a canal.)

Ask how the wave can "Lift" its massive self ABOVE the natural water-level, whilst also travelling vast distances. Ask how the wave can also Lift its passengers. Ask whether the wave, or its passengers, must constantly be "throwing mass downwards" to sustain these Lifts.

Well, in a sense, yes, the wave does "throw down mass". The wave has mass just aft of its peak that is moving, and indeed accelerating, downward. It even has mass just in front of its peak that is moving upward, but accelerating downward. But further forward and rearward the accelerations, and "changes of momentum", are upwards. Importantly, the net "changes of vertical momentum" are ZERO (not including "unsteady" effects...).

Nevertheless, despite zero net momentum change, the "wave" has an energy that is greater than if the water were still. The wave is also very identifiable as an "entity" (see definition below). And, unfortunately, its behaviour is rather difficult to understand, both physically and mathematically, or at least so say most students who study it.

Perhaps most relevantly here, the wave is almost impossible to understand when viewed in a reference frame that travels with the wave. Here it appears as a body of water that is standing high up above its natural level, in complete defiance of gravity and all common sense! (Looking closer, we see the water particles flowing rapidly through the wave, from one side to the other, but that doesn't help the explanation at all.)

Interestingly, almost all standard explanations of FDL are done in the same reference frame as above, namely moving with the aerofoil. IMO, this is the unique reference frame where it is hardest to make sense of what is happening. But the maths is a bit easier, which is why it is taught that way.

~o0o~

6. LANCHESTER'S "SUSTAINING WAVE" - Much to say here. Perhaps it is most important to note that the essence of this explanation is really very simple, but it is also rather subtle.

For example, the whole issue of "cause and effect of pressure-velocity", much argued on these pages, only really becomes clear when you take it right back to the start. Namely, back when the aerofoil was first stationary, and then started to move, and thus created "wavelike" motions in the fluid around it, with the fluid of these "waves" now having higher energy than the quiescent fluid a moment before. And then there is the issue of "hydrodynamic impulse", which is necessary to understand when doing the detailed maths, but it obfuscates the "gist" of the FDL explanation.

Anyway, keeping in mind that this is just a brief (!) post on a Talk page. My dictionary defines a "wave", amongst many other meanings, as, "14. Physics - an energy-carrying disturbance propagated through a medium or space by a progressive local displacement of the medium or a change in its physical properties, but without any overall movement of matter....

When FDL is viewed in the more natural reference frame of the stationary-bulk-fluid, with the aerofoil moving through it, then the above wavelike nature of the flow becomes startlingly obvious. Well, with the clarification that an FDL wave leaves fluid particles directly under the path of the aerofoil shifted slightly forwards, and those above shifted slightly rearward. Different movements when no Lift, but still significant shifts, EVEN IN INVISCID FLUID. (Other very different "waves" also do these small, permanent, shifts.)

Also obvious in this view is a pressure field that moves with the wave and aerofoil, and looks a bit like Doug's Figure in the Article. Away from the small near-field distortions, the isobars are all circles tangent to the aerofoil, and with centres lying on a vertical line through the aerofoil. Highest positive pressure directly under the aerofoil. Lowest pressure above it. Vaguely similar to the pressure field of point 5.2 above.

But necessary in this moving picture is that the aerofoil CONSTANTLY exerts a downwards force on the fluid, equal to -L. If the aerofoil vanishes, or perhaps disintegrates, and this downforce also vanishes, then the flow pattern changes, and the "bound vortex" starts looping the loop.

Much of this behaviour was explained in Lanchester's 1907 book, and also in other classics such as Lamb's "Hydrodynamics". Simple usage of the circulation/vortex theory equations (ie. due to Kutta and Zhoukowsky for 2-D, or Prandtl for 3-D) confirms these behaviours.

Interestingly, the Lillienthal brothers (two "farm-boys" who decided to fly) also spoke of a wave that supported the wing. And the notion of "vortex lift" was used by Rayleigh in his 1877 paper "On the Irregular Flight of a Tennis Ball", although at that time it was considered somewhat speculative. But it seems quite certain that school-boys (well, tertiary level) in the mid-to-late 1800s where drawing with pen and paper these wavelike motions of fluids, both with and without vortex Lift.

Unfortunately, back then those workers had to work quite hard to make such "dynamic" phenomena visible. The Potential Flow methods (ie. Div.V = Curl.V = 0) are quite straightforward, but neverthless quite arduous with only pen and paper. And a "moving picture" requires many such individual drawings. Perhaps, if I ever get the free time, I will post somewhere on the interweb such visualisations of the above "sustaining wave", showing the particle displacements and velocities, and also the pressure maps. (I have these, but not yet in a form suitable for You-Tubing, or whatever...)

Bottom line here, is that this "wavelike" nature of FDL is what allows flight to be so efficient. In the limit of very long wings (ie. getting close to 2-D), the Lift can be sustained with almost NO ENERGY INPUT. See "Caspian Sea Monster" for a good example. Conversely, any "Lift due to downwash" implies a constant exependiture of energy to maintain the Lift. See helicopter fuel bills for a good example.

In a sense, this Lanchesterian FDL is quite magical, which might be why only a small number of people ever understand it.

~o0o~

7. "DOWNWASH" - Very briefly, in planar-aerofoil-flow (aka "infinitely long wing", or "2-D flow", or REAL WING flying between two real parallel walls), THERE IS NO NET DOWNWASH. More explicitly, the upwash in front of the wing is exactly equal to the downwash behind it. (This covered immediately above, and also throughout these Talk pages.)

In "3-D flow" (ie. a finite-span wing inside a large volume of fluid), there is again no net downwash (as explained in many posts...). BUT, there IS MORE DOWNWASH of fluid directly behind the wing than upwash in front of it. So there is a "delta-of-downwash" from front to rear of wing. Note that there is upwash to the rear-sides of the wing, which gives the zero sum.

But, importantly, this downwash is NOT directly related to FD-LIFT.

In 3-D this delta-downwash is related to DRAG. Specifically "induced" (the old word), or "wing-tip vortex" (newer term) drag.

In a loose sense the wing has to fly in descending fluid. Thus its Lift vector, always perpendicular to the onset flow, is tilted rearward, giving a Drag component. The forwards force required to overcome this vortex-drag x the forwards travel of wing (= force x distance) = work required from engine = extra energy put into the swirling fluid behind the wing (ie. downward directly behind wing, up at the two rear-sides).

As a further brief explanation, consider that for a given amount of Lift, wings of greater span have a lesser difference between the upwash in front, and downwash behind, them (ie. lesser delta-downwash). So the amount of "downwash" can vary, while the Lift stays the same. So it is not possible to quantitatively assign a certain amount of "downwash", to account for a certain amount of lift.

Indeed, for very long wings, as seen on gliders, or Burt Rutan's Voyagers, the amount of delta-downwash "near" the aerofoil is quantifiably MUCH, MUCH TOO SMALL to account for the Lift. Similar considerations explain why large diameter helicopters are much more efficient (ie. require less fuel/sec/Lift-force) than smaller diameter rockets or "jet-packs". (This is usually called "actuator disc theory".)

As a last argument against the idea that "Lift happens because the wing pushes the fluid down". Consider the all too common (and very misleading!) picture of an aerofoil with fluid streamlines approaching it horizontally, then departing with "downwash" behind the aerofoil. According to "TS", or "Downwash Theory", the Lift should be able to be calculated by integrating the vertical momentum in the flow just behind the aerofoil (taken through an infinitely high wall).

As explained many times on these pages, the above calculation only gives about half the Lift, because the other half comes from the UPWASH in front of the aerofoil. And it also gives unrealistically large Drag.

In short, if "TS" were true, then flight would be very expensive.

~o0o~

8. TWO MORE RELIABLE SOURCES. - Other than the writings of the abovementioned founders of this work, and also, of course, Thompson (aka Kelvin), Helmholtz, et al, here are two other references worth reading.

~o0o~

8.1. Felix Klein (eminent German mathematician of the time, and teacher of Prandtl) wrote a short but very important paper in 1910, "Uber die Bildung von Wirbeln in reibungslosen Flussigkeiten.". This is usually referred to as the "Kaffeeloffel", or "coffee-spoon" experiment. It is in German, and I have yet to see a translated version, only English summaries of it.

If anyone knows where to get an English translation, then please advise.

This paper gives a good explanation of how the starting and bound vortexes required for FDL can be created INSIDE an INVISCID fluid domain, albeit with NO fluid particles rotating (ie. no fluid "vorticity", though this depends on definition of the fluid's "boundaries").

~o0o~

8.2. David Bloor has more recently (2011) written a book "The Enigma of the Aerofoil...". This is a sociological study of the contrasting approach to understanding FDL taken by the English "Cambridge School" and the German "Gottingen School" in the early 1900s.

Briefly, despite Lanchester's seminal work, the English ignored him and insisted (almost religiously!) that Stokes' Viscous Equations and the "discontinuous flow" postulated by Rayleigh, MUST BE USED to explain FDL. Conversely, the Germans were inspired by Lanchester's much simpler ideas, and ran with them, eventually arriving at the very accurate 2-D and 3-D equations that describe "vortex Lift".

Keep the above paragraph in mind when reading Lanchester's 1907 book. Lanchester was largely self-taught in this field, and was very much alone in developing his ideas. Sadly, to get his book published it seems he had to sully his main ideas by paying lip-service to viscosity and Rayleigh's "discontinous flow" hypothesis.

In fact, the Cambridge School only accepted the inviscid model in the late 1920s, after Glauert, of German parentage but Cambridge education, visited Prandtl, then wrote his "Elements of Aerofoil..." book. However, even this kept in it a certain amount of the "viscosity" lip-service.

It seems that nothing much has changed in the English speaking world's understanding of FDL. Except that now it might be even worse!

~o0o~

9. AND - ... much more to say. For example, NS-Eqns should NOT feature prominently in this Article because they misleadingly suggest that viscosity is somehow important, even though the much simplified Euler-Eqns can grasp the "essence" of FDL.

But this post is already too long.

So, last request. Please don't edit the above, or put inline comments in it. IMO that makes it much harder for later readers to follow the arguments. "Cut-and-paste" makes it easy enough to extract any of the above, should you want to then criticize it, demolish it, whatever...

(End Zapletal)101.170.170.152 (talk) 05:13, 6 December 2014 (UTC)


 * As I have said before, there is much to be said about this but a simple sentence in an introductory section on the Newtonian model is the wrong place to try and say it all. Your view that there is a fundamental battle between viewpoints in the history of the subject is a fascinating one and I wonder whether the battle per se has been documented. It it has, then we should present it either here or in an associated article. But if it has not, then Wikipedia cannot present it.
 * Returning to the dp/dt picture as stated by Clancy, I would suggest that the justification runs as follows:
 * From Newton, every action has an equal and opposite reaction
 * Therefore there must be a reaction on the air, -L
 * From Newton's F=ma, -L = ma or, L = -ma
 * Therefore the air against which the wing acts must accelerate downwards
 * ma = dp/dt
 * Therefore L = -dp/dt
 * Which step(s) contain flaws? And more importantly for Wikipedia, where are those sources which explicitly say that it is flawed? If any has yet been cited with exactitude, I have missed it. &mdash; Cheers, Steelpillow (Talk) 09:55, 6 December 2014 (UTC)


 * The flaw in the reasoning is assuming that -L is the only force acting on the air. Let's consider the case of a plane flying straight and level at a steady speed: the lift is exactly balanced by the weight (L=-W) so the net force on the plane in the vertical direction (L+W) equals zero. Thus, there is no change in momemtum of the plane and dp/dt=0.  Turning our attention to the air, there are forces acting on the air other than -L.  If -L were the only force acting on the air then the statement would be correct.  But there are other forces acting on the air, and while the total force F equals ma or dp/dt, -L is not the total force, F!=-L, so you can't say -L=dp/dt.


 * In the above paragraph "the air" means the entire atmosphere. If we restrict ourselves to a subset of the atmosphere, then there is a region for which it is true that dp/dt = -L.  But it's not clear from the statement that  "the air" refers to that particular subset rather than to the atmosphere as a whole.


 * Regarding citations, we don't need a citation showing something is false to leave it out of the article. We do have many reliable sources that deal with momentum transfer to the air and we need to look carefully at all of them. Some who make the statement (or something similar) are not very specific about what they mean by "the air", and the ones that do specify what they mean by "the air" are careful to point out that -L=dp/dt only applies to a specific subset of the air, not the entire atmosphere.


 * I have to say that until recently I was applying a similar line of reasoning to the one you give above and thought that -L was the only force acting on the air or at least that the other forces were negligible. This is not a valid assumption.  I also thought that "the air" in the statement was the entire atmosphere.  It's not.  I'm still optimistic that we can arrive at consensus on how treat the issue in the article - given its own sub-section and a bit more space to describe the assumptions I think we will find language we can agree on.  Mr. Swordfish (talk) 15:16, 8 December 2014 (UTC)


 * Indeed it is not necessary to find a reference refuting an unsupported statement before removing it, but that is beside the point. This statement is now supported by two reliable citations, and it most certainly is necessary to find sufficient references refuting these sources before the statement can be removed. Bear in mind too the points I raise below. I too would be happy to see Doug and Zapletal's approach being given proper encyclopedic treatment - in its own section. Provided we adhere to WP:VERIFICATION and similar, we should be fine. &mdash; Cheers, Steelpillow (Talk) 16:06, 8 December 2014 (UTC)


 * We seem to have a disagreement on policy:


 * Mr. Swordfish says "Regarding citations, we don't need a citation showing something is false to leave it out of the article."


 * Steelpillow says "This statement is now supported by two reliable citations, and it most certainly is necessary to find sufficient references refuting these sources before the statement can be removed."


 * Mr. Swordfish's version seems to me to be more in line with common sense, but what is the actual Wikipedia policy? In my reading of the policy pages I've seen nothing saying that a citable source is needed to justify exclusion or removal of anything from an article, supported or not.  Can [User:Steelpillow|Steelpillow]] steer me to where it says that?


 * J Doug McLean (talk) 01:42, 10 December 2014 (UTC)

(Zapletal Writes->)

Steelpillow, your flaw is in step 3. Doug has covered this before (ie. there is only an "mA" when there is an unbalanced force acting on the body, which in this case is the very large and slippery volume of fluid with a groundplane below it to carry the "-L"). I also hinted at this flaw in my point 5 above "Two Cases..." (eg. the airship exerts a large downward force "-L" on the air, yet there is NO "mA" of the air, anywhere).

Lanchester introduces his "Principle of No Momentum" in section 5, page 5, of his 1907 book. Is he not a reliable enough source?

(Added:) To quote Lanchester (page 9) "As a whole, the fluid, in the previous section, does not gain or lose momentum any more than does a cast-iron pillar supporting a load."

(End Zapletal.) 101.171.127.232 (talk) 12:01, 6 December 2014 (UTC)


 * We've seen that no net vertical momentum is imparted to the atmosphere as a whole by an airplane in steady level flight. However, there are regions of both upward and downward momentum in the field [McLean p.431]. There can be no doubt this momentum exists, and that it is communicated from region to region. Burninthruthesky (talk) 13:19, 6 December 2014 (UTC)


 * Lanchester's "Principle of No Momentum" refers to net momentum in the entire field. He wasn't ruling out momentum in local areas.


 * What you say is correct, but to say that momentum is "communicated from region to region" is potentially confusing in that someone might think you're referring to some kind of action-at-a-distance. Actually, momentum is convected (carried along) by fluid parcels and can thus be carried from one region to another.  Otherwise, it can be exchanged only locally, by the pressure or the viscous stress between fluid parcels in contact with each other.


 * J Doug McLean (talk) 22:49, 6 December 2014 (UTC)


 * Somewhere in the previous debates I mentioned the distinction between a principle and its application in practice. I do find the airship analogy dangerous, perhaps spurious. The situations are utterly different. Airship lift is static, aerofoil lift dynamic, and this topic is confined to the dynamic situation. Analogy with a static situation can at best confuse. We look at the net force on any mass. In the case of the airship, the downforce exerted locally is counterbalanced by an upward pressure from below, net force is zero and there is no related momentum anywhere in the system. In the case of a dynamic flow over an aerofoil, the downforce is exerted through pressure changes brought on by the passage of the aerofoil through the air. The foil rides a "wave" of higher pressure beneath and of lower pressure above. Momentum circulates around this wave. In the foil, the lift forces generated exactly balance gravity. But in the air there is no such locally counteracting force and -L = ma should in principle apply.
 * Lanchester's point, that in the long run no net momentum gets transferred, is of course correct. There is no persistent "downwash" (ignoring the inefficiencies of the average helicopter rotor). But unlike a cast-iron pillar, or the underside of an airship, the business end of the air is highly dynamic. The localised downflow is rapidly corrected, but it has by then done its job. And during that local downward deflection, it briefly gained momentum. Only part of the overall flow pattern supports the aerofoil, the rest is an accompanying mechanism imposed by the laws of physics. And in that part that actually supports the wing, the air is being deflected downwards. Depending on which bits of the overall flow one considers, and which specific effects, the net downward momentum will vary substantially. Typically the momentum at the trailing edge will comprise not merely downward momentum imparted by the reaction to lift, but also upward momentum imparted by other mechanisms, some if not all of which have been exhaustively set out. Thus, in practice, the application of dp/dt is quite complex.
 * So the true debate here should not be, "how does it all work?" but, "what aspects should a simple introduction to the Newtonian model of dynamic lift be talking about?" And should it be discussing the basic principle or the complications we meet in practice?" Depending on one's answer to this last, I would suggest that it should be introducing the basic principle, and to the relevant localised airflow. I would also suggest that from this perspective my Step 3 is valid as a principle. It is only when one moves beyond the basic principle of lift to the more practical analysis of airflow overall that it becomes trite. &mdash; Cheers, Steelpillow (Talk) 14:54, 6 December 2014 (UTC)


 * I think Steelpillow is trying to frame the debate in the wrong terms. The article subsection in question is not about "a simple introduction to the Newtonian model of dynamic lift".  It is about the popular qualitative "flow-deflection" explanation of lift, which in its usual form in sources other than the AAPT papers makes no quantitative claim regarding the rate of change of momentum.  The flow-deflection explanation has shortcomings, but it belongs here, provided the shortcomings are explained.  The Newtonian model of lift has been thoroughly discredited by numerous sources.  See my comment above regarding the zero-versus-two-pi discrepancy in lift slope.  The Newtonian model doesn't belong in this section.  Perhaps some discussion of the Newtonian model and its deficiencies would be appropriate in the "Mathematical theories" section.


 * No, the imparting of momentum locally to the flow around the foil at the rate -L isn't a "basic principle" unless you accept the grossly unrealistic Newtonian flow model. In a real continuum flow, the non-uniform pressures in the field apply a "locally counteracting force", so that dp/dt = -L is not true in the local flow around the foil.  It is true only if you consider a control volume that is slender and very tall, so as to eliminate the pressure force.  The pressure field is not just a "complication we meet in practice".  It is a basic feature of the flow when realistically modeled.  Steelpillow's Step 3 is not valid as a principle.  I agree with Zapletal that Step 3 is the flaw.


 * J Doug McLean (talk) 22:49, 6 December 2014 (UTC)
 * I cannot fathom this argument at all, it appears full of straw men and patent absurdities. The article section is titled "Simplified physical explanations of lift on an airfoil" and the subsection "Flow deflection and Newton's laws". It is not possible to deflect a flow without invoking Newton's laws. To suggest that a discussion of the Newtonian model does not belong in a section titled "Flow deflection and Newton's laws" is patently absurd. This debate has always been about the content of said section. Zapletal suggested that someone put the Newtonian case so that it could be criticised. I offered my understanding of Clancy's statement of it, focusing as he did on the airflow region in which lift (the topic of the article) is generated. Apparently this is "framing the debate in the wrong terms". I'm sorry, but such rhetoric just heralds a return to the same old same old.
 * Besides seeking refutations of Clancy's position, I emphasised the importance of relevant citations. I had hoped for verifiable quotations clearly relevant to Newtonian flow deflection and explicitly countering Clancy's position, yet the silence on this point is becoming deafening. No relevant citations, no progress. &mdash; Cheers, Steelpillow (Talk) 11:15, 7 December 2014 (UTC)


 * I think the reason my argument seems "absurd" to you is that you're failing to make the distinction between "the Newtonian model of dynamic lift" and "Newton's laws". They're not the same thing.  I apologize if I didn't make that sufficiently clear.


 * In the aerodynamics literature, "the Newtonian model" refers to a specific flow model, i.e. the modeling of the flowing fluid as a hail of bullets that interact only with the foil and not with each other. Just because Newton's second law is applied to the model doesn't mean the results will be correct.  Because the flow model itself is unrealistic, it leads to a wrong answer for dp/dt in the near field of a lifting foil.


 * Clancy's fire-hose model is wrong about near-field dp/dt in the same way. It isn't a bullet model, but it has a similar failing:  It assumes that only a limited part of the flow interacts with the foil and doesn't interact with the rest of the surrounding flow.  Thus it doesn't account for the role of the pressure field in the momentum balance, as I've already pointed out.  Clancy's model gets dp/dt = -L for the flow in the near field of the foil.  We know this is wrong for reasons I've explained several times and Mr. Swordfish explained above.  And we have citable sources for the fact that it's wrong, i.e. the classical control-volume analyses I've cited.  How are these sources not seen as "countering Clancy's position"?  And how are they not "clearly relevant to Newtonian flow deflection"?  They clearly show that the Newtonian model's answer for dp/dt in the near field is wrong, and they make it clear why it's wrong.


 * You are mistaken when you refer to the air that Clancy's statement focuses on as "the airflow region in which lift is generated". In Clancy's statement, "this air" refers to the air inside the fire-hose stream, which is a narrow region of the flow.  In the real flow there's not enough momentum imparted to the air in a region of that size to account for all of the lift.  In reality, lift is generated by the flow over a much wider region, and the momentum-balance picture is more complicated.


 * I think this subsection of the article should just be about the popular qualitative flow-deflection explanation based on Newton's laws. This simple explanation has many sources:  NASA websites, "Stick and Rudder" by Wolfgang Langewieshe, and others.  It avoids getting into the quantitative question of dp/dt, and I think that's the appropriate level of detail for this section of the article.  Newton's laws are central to this explanation, but the Newtonian model doesn't belong as part of it.  The Newtonian model is unnecessarily quantitative, and erroneously quantitative at that.


 * If this is "a return to the same old same old", it's because you continue to push for giving a prominent place to the same old discredited model.


 * J Doug McLean (talk) 01:42, 10 December 2014 (UTC)

(Zapletal Writes->)

Steelpillow, you wrote:

1. '' "I do find the airship analogy dangerous, perhaps spurious. The situations are utterly different..." ''

Just before making that analogy I wrote:

'' "5. TWO CASES TO THINK ABOUT. - ... please don't say "Oh, but that's different..." ..." ''

The first "airship" example was intended to make clear that a Lift force can be created in a fluid by pressure forces alone, so -dP/dt is NOT absolutely essential. The second "wave" example hinted at how dynamic effects can create such Lifting pressure zones.

~o0o~

2.  "Only part of the overall flow pattern supports the aerofoil ... in that part that actually supports the wing, the air is being deflected downwards. [etc...]" 

That whole paragraph is wrong, or at the least, very misleading. All parts of the flow pattern depend on all other parts of the flow pattern. That is why it is called "Continuum Mechanics".

Any smallish volume of fluid near the aerofoil (ie. any fluid that might be said to "actually support the wing"), necessarily has very small mass. There is simply NO WAY that this small mass can cause ALL of the Lift via its -dP/dt.

Instead, this small volume of fluid is responsible for a VERY SMALL part of the Lift, via its downward dP/dt. The rest of the Lift comes from the PRESSURE FIELD surrounding this small volume. That pressure field (established when the aerofoil first started to move, or accelerated) is also related to the momentum changes of all the other parts of the fluid, more than half of which are now UPWARD dP/dts. ~o0o~

3. '' "Thus, in practice, the application of dp/dt is quite complex ... So the true debate here should [be, how do we give] a simple introduction to the Newtonian model of dynamic lift ... introducing the basic principle ... to the relevant localised airflow..." ''

What you are suggesting is that this article should teach a FALLACIOUS model of Lift, for purely IDEOLOGICAL reasons. (This shouting is just to make this point clear, so I don't have to repeat it.)

Specifically, your qualitative "Newtonian Model" is incapable of giving any quantitative results. To get any results at all, you must first know the Lift, then ARBITRARILY assume an amount of mass to do the Lifting (ie. your "relevant localised airflow"), after which you get an "average" acceleration value, but NO IDEA of what the actual flow field looks like.

This is an utterly useless approach to understanding FDL (ie. it is of zero help in designing aerofoils), that is taken simply to support an ideological viewpoint (ie. that "Lift MUST be due to Downwash..."), with that viewpoint being WRONG anyway (ie. as many others here now agree, there is NO NET change of vertical momentum).

As I wrote earlier,

 "In short, the editors of good encyclopedias filter out the nonsense. If this does NOT happen, then very soon we will have to believe that the world is flat, voodoo is real, and the star signs predict our future." 

To which we might have to add, "... and, because Newton said so, all FD-Lift is due to Downwash". And maybe after that ETT can make a comeback.

~o0o~

4. '' "Besides seeking refutations of Clancy's position, I emphasised the importance of relevant citations. I had hoped for verifiable quotations clearly relevant to Newtonian flow deflection and explicitly countering Clancy's position, yet the silence on this point is becoming deafening. No relevant citations, no progress." ''

Again, your deafness seems decidedly ideological in nature.

As already quoted and cited many times, the major part of Lanchester's book is directed at refuting "Newtonian flow deflection". All of the "Circulation/Vortex Theory of Lift", hugely successful for 100+ years, and which has countless citations from umpteen reliable sources, refutes it.

If anyone is seriously interested in this Article, then they should study this body of knowledge more deeply before trying to impose a false ideology upon the next generation.

I will be away for the next week, so not able to respond to any questions for a while.

(End Zapletal) 101.170.255.230 (talk) 11:43, 8 December 2014 (UTC)


 * Your rant about ideology closes this discussion as far as I am concerned. It is a breach of WP:GOODFAITH, it demonstrates that you do not have a neutral point of view (WP:NPOV) and are therefore WP:NOTHERE to build an encyclopedia and that this discussion remains as utterly pointless as I have suggested all along. &mdash; Cheers, Steelpillow (Talk) 15:06, 8 December 2014 (UTC)

Does this event really cause the other event, according to accepted theory?
This question is in regards to the text, "the flow following the upper surface contributes strongly to the downward-turning action."

To paraphrase, one event - "the flow follows the upper surface" - causes (or more precisely, is a contributing cause of) a second event - "the air turns downward".

Is this in accordance with accepted science? If not, I think that the text should be reworded to convey what was meant.

I think I may understand the important point that the author was trying to convey. I think the author meant to refer to a baseline of 'atmospheric pressure on all subsurfaces'--which applies when there is no flow--and compare that to the conditions of flight, where the deviation from the baseline is greater on the upper surface than on the lower (so the "contribution" relative to the baseline is mostly from the changed conditions on the upper surface, not the lower as intuition would suggest). But that is very different I think than what was said. The fact is that under conditions of lift, the upper flow is following the surface of the wing, more or less, "because of" (if we must allow a causal interpretation) the excess of pressure from the air above over that applied by the wing. And the pressure on the upper surface of the wing under conditions of lift is opposing, not enhancing lift.

Mark.camp (talk) 23:44, 1 December 2014 (UTC)


 * At some point in the article's history, the idea expressed by this passage was followed by some discussion of airflow on both sides of the wing and the point made that some explanations only involve air being deflected by the bottom of the wing. (sometimes called "bullet theroy" or "skipping stone theory"). This was accompanied by a graph showing the pressure distribution along the top and bottom of a wing where the pressure difference along the top was substantially larger than along the bottom.  At some point this was excised, whether by accident or not.


 * Which is to say that your interpretation of the intent is correct. I can see where what is actually written may be interpreted differently than what was meant.  A suggestion for how to reword would be welcome.  I'll take a look at earlier versions and see if we edited something out that shouldn't have been.  Personally, I don't read the passage as a statement of cause and effect but I can see how someone might interpret it that way.  Mr. Swordfish (talk) 00:24, 2 December 2014 (UTC)


 * Sorry, I didn't say why I thought the current text suggested a causal relationship, so I will provide some examples that clarify why I inferred one. (I address this point of yours before answering your good request for proposed wording, since if I'm off a bit in my reading then no change is needed.)

The addition of fertilizer to the poor soil common to the region contributes strongly to the higher corn yields. (The sentence implies, for me, that "the addition of fertilizer" is a contributing cause of "higher corn yields")

Having three unusually strong candidates for the lower house contributed strongly to the party's gains in that election. (The sentence implies for me that "having three unusually strong candidates for  the lower house" was a contributing cause of "the party's gains in that election."

Adding two extra cylinders to the V6 engine contributed strongly to the increased torque in the new engine. (The sentence implies, in my reading, that "adding two extra   cylinders to the V6 engine" was a contributing cause of "the increased torque in the new engine".

In these cases, which are I hope analogous to the text under discussion, it seems that the first event ("addition of fertilizer") is implicitly claimed as being a contributing cause of the second ("increased corn yields").

Does my reading of the original text seem reasonable to you?

Mark.camp (talk) 23:38, 7 December 2014 (UTC)


 * I am not sure there is a significant problem here:

The higher yields from fertilizer are predicated on it being poor soil. (this is specifically mentioned in the  example sentence)

The strong election gains are predicated on the election being for the lower house. (this is not exactly  mentioned in the example, although the description "that election" might suggest it)

The increased torque is predicated on the existing cylinder capacities remaining constant. (this is not mentioned  at all in the example sentence)

The contribution of the upper surface flow is predicted on the downward turning of the upper surface itself. (this  is not mentioned in the example sentence)


 * Whether these assumptions are made explicit in the given sentence is neither here nor there since they are, one way or another, trivially evident from the overall discussion. One might write, "the flow following the downward-turning upper surface contributes strongly to the downward-turning action of the flow," but I personally think that this would be unnecessarily pedantic. There comes a point where the sheer amount of word salad begins to confuse. Whatever the present version says in terms of phrasing, the physical mechanism it is explaining is obvious enough, and at the end of the day that is what language is for. Steelpillow 10:54, 8 December 2014 (UTC)

Why I now find "The Statment" to be problematic
Since I don't know quite where to insert this into the wall of text above I'm starting fresh with a new section. I'm also starting more or less from the beginning. Please bear with me, and thanks in advance for your patience.

Let's take a look at a simple model, i.e. a plane flying straight and level at a constant speed. In this model, the only things are the plane, the atmosphere, and the ground. After decomposing the total aerodynamic lift into it's components we have four forces on the plane: lift, drag, weight, and thrust. Lift is opposed by weight and thrust is opposed by drag. Since the plane is flying straight and level L+W=0 and D+T=0. With no net force in either the vertical or horizontal direction, there is no acceleration and consequently no momentum change. For the plane, dp/dt = 0.

The ground, of course, is stationary, so for the ground dp/dt is also zero.

The only other thing in the model is the air, and if its momentum is not constant we have a violation of conservation of momentum - how can the air experience a change in momentum without anything else experiencing an equal but opposite change in momentum? So the air (ie the atmosphere as a whole) must have constant momentum, that is dp/dt for the atmosphere as a whole is zero.
 * NB. That is not to say that there is no transfer of vertical momentum from the atmosphere as a whole to the foil. On the contrary, if there is a vertical force between wing and air, then there must be a momentum transfer, unless one supposes that there is a non-mechanical force in play, such as gravitation, electrostatic, etc.  It is only to say that, since the momentum of the air as a whole is constant, then if the air is transferring momentum to the wing at rate x, then it must be receiving momentum at an equal rate from some other interaction, such as an interaction with the earth)Mark.camp (talk) 03:23, 10 February 2015 (UTC)

We don't need to compute integrals of momentum flux and take the limit as the boundary goes to infinity to arrive at this, although making those calculations gives the same result.

However, there are subsets of the air that experience non-zero dp/dt. And if we take just the right subset, it can be shown that for that subset of the atmosphere dp/dt = -L. It is reasonable to interpret the lift physically as the result of the momentum change of that subset of the atmosphere. (not everyone agrees with or likes that interpretation, but it is supported by many reliable sources)

There's nothing special about the vertical direction in the argument above. Looking at the horizontal component of momentum change, thrust is equal in magnitude to drag so the net force in the horizontal direction is zero. For the plane, the horizontal component of dp/dt is zero as well. The ground is just as stationary as it was a couple of paragraphs ago, so we can conclude that for the air the horizontal component of dp/dt is zero.

This does not contradict the fact that the propeller or jet is pushing a fairly large amount of air backwards. Unlike with lift, there seems to be no controversy over whether air is pushed backwards by the engine. ( NASA has a concise treatment of thrust and momentum change at http://www.grc.nasa.gov/WWW/k-12/airplane/thrsteq.html). The physics in the horizontal and vertical directions is basically the same - if you take a close look at the propeller blades they're just small airfoils generating lift. One can say that the "real" reason the propeller drives the plane forward is the imbalance of pressure between the front and back of the propeller blades, but most folks are satisfied with the 'push air backwards' idea, so much so that for many simple explanations thrust isn't even explained - most people just get it without any explanation. In any case, looking at the Talk page of the Thrust article I don't see any long drawn out discussions about how to explain thrust, the article simply says it's a reaction force and this unremarkable statement has collected few remarks on the talk page.

So what does all this have to do with the statement? Well, one point of the exercise is to show in an easy-to-follow manner that when you consider "the air" as the entire atmosphere then the rate of momentum change of "the air" is zero, not -L. So, if a reader gets the idea that "the air" means the whole atmosphere rather than just a subset then including the statement as previously worded will give the wrong impression and mislead the reader. However, if we provide context then some version of the statement would be appropriate.

Another point is to show that dp/dt=0 is not inconsistent with some subset of air being accelerated - either downward as occurs with lift or backwards as occurs with thrust. Some subset of the air is accelerated downward with dp/dt = -L; some subset is being accelerated backwards with dp/dt = -T. I don't know how important it is to specify exactly what subset(s) meets this criteria, but I do think it's important to specify that it is a subset and not the entire atmosphere, that is if we are going to include it at all.

Moving forward, I'll try to collect in one place the cites that relate lift to momentum change of air. then we can put on our wikipedia editor hats and evaluate the material provided by the various reliable sources and perhaps craft a subsection that reflects that material. Mr. Swordfish (talk) 20:08, 8 December 2014 (UTC)


 * The analogy with thrust is a good one. Since a propeller is a rotating aerofoil, there's little difference in the physics, except for geometry.


 * I can see why someone familiar with Lanchester's "Principle of No Momentum" might initially see The Statement as a claim that the atmosphere as a whole is given downwards momentum. The trouble with that interpretation is that for it to be true, the atmosphere would have to fall through the ground. That is a nonsensical assumption, which highlights the straw man in the argument that The Statement is false.


 * It's true that momentum can only be calculated for a specified portion of air. I think you've expressed the point well, that that subset of air (and the momentum that is communicated* to the rest of the air) exists whether the location of that subset it is specified or not.


 * I think we've all agreed from the outset that talking about control volumes at this early stage of the article is more likely to confuse the reader than help them understand how the physical principles expressed in Newton's laws contribute to lift.


 * I think once some text for the proposed new section is drafted it will be much easier to agree the facts, and decide what changes (if any) should be made to the section on Newton's laws to improve the coherence of the article as a whole.


 * [* Lanchester's choice of word: When a loaded aerofoil is dynamically supported by a fluid, we know that its weight is eventually sustained by the surface of the earth, and that the transmission of the stress is effected by the communication of momentum from part to part p.148]
 * Burninthruthesky (talk) 10:15, 9 December 2014 (UTC)


 * I'd second that gathering and correlating sources is far the best way forward. And do you know, I have clean forgotten what "the statement" said, it was so long ago. Perhaps it is better not to remind ourselves but to just stick with the sources and a clean sheet? &mdash; Cheers, Steelpillow (Talk) 11:06, 9 December 2014 (UTC)


 * "The atmosphere as a whole" isn't the only definition of "the air" for which a reader is likely to mistakenly think The Statement is true. Another potential trap is the idea that The Statement is true for "the local airflow" in the neighborhood of the foil and that it is only when you look farther afield that other effects creep in.  This is a misconception that's been expressed by several participants in this discussion and may still be held by some of them.


 * Burninthruthesky seems to have a strange definition of a "straw man". It's obvious to him that The Statement is false for the atmosphere as a whole, so the mentioning of that as an example is a "straw man"?  The Statement is also false for "the local airflow", which is perhaps not so obvious.  So I disagree with the idea that it's not important to discuss the "location" where this "subset" with dp/dt = -L "exists".  I think it's important to know that "the local airflow" isn't that location.


 * In terms of basic physics, a propeller is different from a wing in important ways, and the analogy between thrust and lift is a poor one for the following reasons:


 * Even in a reference frame moving with the airplane, the propeller blades are moving and doing work on the fluid, leaving behind a slipstream with higher total-pressure than in the rest of the field. In even the simplest model of propeller flow, like Rankine's actuator-disc model from 1859, the boundary of the slipstream is a vortex sheet that has no counterpart in the flow around a 2D lifting foil.


 * A propeller has a substantial effect on the part of the flow that passes through the disc and becomes the slipstream, and relatively little effect on the rest of the flow. Thus the kind of model used by Clancy and by Chris Waltham, in which only a limited part of the stream is assumed to be affected, applies reasonably well to propeller flow.  It doesn't apply so well to the flow around a lifting foil.


 * Cross-stream pressure gradients play only a minor role in propeller flows but play a key role in the momentum balance in airfoil flows. This is to be expected, given that thrust is a streamwise force, and lift is a cross-stream force.


 * So I don't think analyses of thrust are much help in understanding lift.


 * I also second Mr. Swordfish's initiative to gather the sources.


 * That said, I have some reservations about including direct quotes in the article's list of sources. I understand that it makes things easier for the reader in a way.  But it also has a downside.  It isn't generally practical to provide a long enough quote to avoid the "out-of-context" problem, and the result can be an incomplete and possibly misleading view of what the quote really represents.  In a quote of reasonable length you often get only the author's conclusion, not the assumptions he made or the analysis he used to reach that conclusion.  I'd propose that instead of a direct quote we should consider paraphrasing the findings, including a brief summary of the assumptions and methods of analysis.


 * Now that we're collecting sources, I propose that we also collect comments by those who take the time to read the sources, e.g. brief summaries of the models and assumptions that were used, and resulting criticisms etc. My preference would be to append comments directly under each source to make things easy to correlate and to facilitate later discussion.


 * J Doug McLean (talk) 01:57, 10 December 2014 (UTC)


 * Yes, I know that there is more than one way to misinterpret The Statement in a way which makes it false. What I'm saying is that the language used is not specific enough to imply anything whatsoever about how to calculate the rate it describes. I still believe the control volume analyses which account partly for momentum and partly for pressure do not answer the question posed by The Statement. I see that you now describe it as "misleading" rather than "false". The former is an opinion to which you are entitled and I thank you for moderating your language.


 * Personally, I have no problem with the suggestion made above to keep quantitative statements out of the Newtonian section and introduce them later if required. I noticed myself that Langeweische doesn't make any quantitative claims.


 * I look forward to seeing your suggestions for the structure and focus of the new section you propose. Burninthruthesky (talk) 17:00, 10 December 2014 (UTC)


 * Doug, I think a big reason why you've received so much pushback here on the talk page about your objection to the statement is due to the lack of direct quotes from the sources you cite. Citations and verifiability are what makes the entire wikipedia party happen.  While I agree that we need to be careful to not take a single sentence or two out of context, we need more than just a work and a page number to back-up contentious claims.


 * Wikipedia can be a funny place sometimes; I'm sure you are quite capable of looking at a page of equations and concluding in words that "these equations clearly show that the time integral of foo is equal to bar". But unless we have a reliable source that draws the same conclusion putting that statement into the article is synthesis or interpretation which is prohibited.  If you publish the same statement in a book or a paper we can use it, but if you just say so here on the talk page we can't.  Like I say, Wikipedia can be a funny place, but those are the rules of evidence.


 * So, I'd prefer to see some direct quotes rather than editors' interpretations.  Right now, we have an excerpt from your book and the clarifying aside from Waltham about "doing it right"  that we can weigh against the imprecise assertions of Smith and the AAPT committee. Another one or two and we're probably there.


 * Sorry you didn't like the thrust analogy - like any analogy it only goes so far, but I think it serves to illustrate that dp/dt = 0 for the atmosphere as a whole is consistent with non-zero mass flow is some subsets of the atmosphere.


 * Finally (for now) I'd prefer that the raw material thread not be interrupted by a lot of back-and-forth exchanges amongst the editors. Once that starts happening the thread will become difficult to follow.  Instead, I'd suggest a sub-thread for each source below the initial post.  I'll start with an example.  Mr. Swordfish (talk) 20:45, 10 December 2014 (UTC)

Two examples that may help clarify the discussion

Consider an iceboat oriented so that the wind is perpendicular to its centerline with its sail trimmed in and the brake engaged. The sail generates lift which is parallel to the centerline of the boat and the lift force (L) is opposed by the friction force of the brake (B). We'll assume in this idealized model that the brake is strong enough to hold the boat motionless. L + B = 0, the total force on the boat is zero and therefore it's acceleration and dp/dt are zero.

In this model the boat can't move, the ice doesn't move, and the only other thing is the air. Since momentum is conserved, dpboat/dt + dpair/dt = 0 and the net momentum change of the air is zero.

Now, let's release the brake. If we ignore the negligible friction force of the runners (skates), the net force acting on the boat is L. (drag is opposed by the runners that don't move sideways). If F is the total force on the boat we have F = L = ma = dpboat/dt.

Again, conservation of momentum says dpboat/dt + dpair/dt = 0, and in this case dpair/dt = -L. So in this special case we can say that the lift is equal to the time rate of change of momentum of the air.

When we had the brake engaged, the situation was very similar to a plane in steady level flight, with the brake playing the role of gravity in opposing the lift force. When we release the brake, the situation is similar to the idealized model of a wing in an infinite atmosphere in the absence of gravity. In this second situation, the statement is true even when "the air" is the entire atmosphere. But the usual example when explaining lift is steady level un-accelerated flight in a gravitational field, not a plane flying in the absence of gravity or an iceboat accelerating from a stand still.

To be a bit nit-picky, I should add that in the case of the iceboat, dpair/dt = -L is only strictly true for a moment -  once it begins accelerating the apparent will will move forward and L will not be parallel to the centerline of the boat anymore so L != dpboat/dt in general. Further, since it is accelerating our usual frame of reference will no longer be an inertial reference frame. And eventually the boat will reach a steady speed at which point dpair/dt = 0. So I don't think an iceboat accelerating from rest is a good example to place in the article.

Hopefully these two examples (a foil constrained to have no acceleration and a foil that is allowed to accelerate with L being the total net force) will cast a bit of light on how momentum is (or is not) transferred to the air. This thought experiment helped clarify things for me anyway. Mr. Swordfish (talk) 20:52, 15 December 2014 (UTC)
 * You have changed the statement from the version originally stated (and which I copied above before archiving). This change makes the phrase "the air" more ambiguous and easier to attribute an unintended meaning. Have I missed something or is this effectively a new discussion about a new problematic statement? &mdash; Cheers, Steelpillow (Talk) 22:01, 15 December 2014 (UTC)


 * Apologies. I didn't recall whether the word "downward" appeared in the original, so I went to the talk page archive to find the earliest example.  On July 27th, Doug wrote The more specific statement "lift is equal to the time rate of change of momentum of the air" is not correct in general. I recommend deleting this sentence. leaving out the word "downward".  I mistakenly took this to be the definitive version of the statement but looking back at the actual draft and subsequent versions of the statement on the Talk page indicates that the word "downward" was in the original. Thank you for restoring it.  Mr. Swordfish (talk) 22:15, 15 December 2014 (UTC)
 * The version I copied also included the word "deflected". This makes a big difference to what one assumes is the air in question. Without it the Statement is ambiguous and requires a suitable preceding remark to give it context. We have lost that context so, if the Statement as now presented is to mean anything at all that can be discussed, it needs that context restoring. &mdash; Cheers, Steelpillow (Talk) 11:23, 16 December 2014 (UTC)


 * Agree that "downward" needs to be part of the statement, otherwise the air deflected in the horizontal direction by the thrust and drag would be included. However, even with "downward" included the statement is still ambiguous.  What is meant by "the air"?  If it's the entire atmosphere then the statement is false.     If parsed narrowly, the statement says that the total momentum change of all the air with a negative vertical deflection is -L and this does not agree with the results of control volume analysis. If "the air" is a carefully defined subset of the atmosphere then the statement is true, but it's not true for arbitrarily chosen subsets and it's not true for most subsets that one would intuitively choose as representative.


 * Until recently, I was mislead by the statement. Only after reading up a bit on control volume analysis did I realize my intuitive ideas were not supported by rigorous quantitative analysis. I think something like the statement can go into the article if we provide sufficient context, but as it stands it's likely to give the reader the wrong idea. Mr. Swordfish (talk) 16:06, 16 December 2014 (UTC)


 * Are the sources which convinced you that your ideas were "not supported by rigorous quantitative analysis" listed below? Burninthruthesky (talk) 17:02, 16 December 2014 (UTC)


 * Yes. Zero net momentum change for the atmosphere as a whole is a fairly standard result. Chapter 8.5 of McLain's book lays it out in words fairly clearly.  Google books has a generous excerpt at http://books.google.com/books?id=_DJuEgpmdr8C&q=Manifestations+of+Lift+in+the+Atmosphere+at+Large#v=snippet&q=%22Manifestations%20of%20Lift%20in%20the%20Atmosphere%20at%20Large%22&f=false Here are some quotes:


 * "...there is no net downward momentum imparted to the atmosphere as a whole and that the lift is reacted by pressure differences on horizontal planes above and below the wing, or on the ground plane, if there is one. We'll also consider how conservation of momentum applies to control volumes that don't encompass the entire atmosphere. ... the lift can show up at the boundaries either as pressure differences on the horizontal surfaces or as fluxes of vertical momentum mainly through the vertical surfaces, or as combinations of the two, depending on the proportions of the control volume."


 * "Prandtl and Tietjens (1934) showed how in steady level flight the lift is balanced by an overpressure on the ground under the airplane, so that of course there is no need for net momentum transfer."


 * I've been trying to find the actual passage in Prandtl that supports this, but I haven't yet. In any case, I thought that "the air" in the statement meant the entire atmosphere, or some arbitrary box around the airfoil, but the only interpretation of "the air' that makes the statement true is a tall thin subset of the atmosphere - this is not intuitively obvious. Mr. Swordfish (talk) 21:11, 16 December 2014 (UTC)


 * Thank you for clarifying. Yes, the net change of momentum of the whole atmosphere is zero, but momentum is necessarily imparted to air within the atmosphere. See my comments on the AAPT paper below. Burninthruthesky (talk) — Preceding undated comment added 10:06, 17 December 2014 (UTC)


 * Agreed. The net change of momentum of the whole atmosphere is zero. And within the atmosphere there are subsets  which experience non-zero momentum change. If one chooses the subset carefully, the rate of momentum change for that subset is equal to -L.  But if one chooses a different subset then in general dp/dt != -L.  I think the main point of disagreement is over the lack of precision in referring to that particular carefully chosen subset for which dp/dt = -L as simply "the air" or "the air deflected downward".  I'm ok with a bit of imprecision in a qualitative introductory section aimed at the lay person as long as it leaves a basically correct impression. Here, I don't think it leaves a basically correct impression. Mr. Swordfish (talk) 16:50, 17 December 2014 (UTC)


 * It is imprecise to refer to the net momentum of a body of air without defining the body, but we don't do that. What is imprecise about saying that momentum is imparted to air within the atmosphere at a rate equal to lift? Burninthruthesky (talk) 17:48, 17 December 2014 (UTC)


 * For information, the whole of the disputed section containing what I understand to be "The Statement" can be seen here. Burninthruthesky (talk) 17:24, 16 December 2014 (UTC)


 * @User:Mr swordfish, I think there is some discussion at cross-purposes here. In saying that "downward" should be included in the Statement, are you implying by omission that "deflected" should not be? That is the word which you removed and which I am more concerned about. To me, the phrase "the air deflected downward" has a very clear meaning which is garbled when any one word is removed. I cannot help but wonder if it is the removal of "deflected" which might have caused the ambiguity which originally confused you. &mdash; Cheers, Steelpillow (Talk) 19:30, 16 December 2014 (UTC)


 * The word "deflected" does not appear in the AAPT version of The Statement or in any other of the sources, as far as I know. Thus adding "deflected" to it is something that has no citable source.  And for what it's worth, it's easy to show (though I know of no citable source for this) that adding "deflected" doesn't make The Statement more correct anyway.  There is much more negative dp/dt in the region of "the air deflected downward" than just -L.  J Doug McLean (talk) 21:15, 16 December 2014 (UTC)


 * No, I'm not implying anything by omission. And removing whatever word I did was an error on my part for which I apologize (I think I removed "downward' but it doesn't really matter).
 * I disagree that "the air deflected downward" has a clear meaning. If you take it literally then the statement is false.  Mr. Swordfish (talk) 21:24, 16 December 2014 (UTC)
 * In Clancy's cylinder model (see below) the air which is deflected downwards is described with mathematical precision. If you take "the affected air" to be the air deflected downwards (and what other interpretation is even remotely plausible?) then the Statement is demonstrably true. So I find your assertions to the contrary to be utterly baffling, an absolute failure between us to establish any common use of language. I get the feeling that we agree on the maths but just not on how to describe it. Still, I don't see how we can take this forward between us, so I guess I will have to withdraw from this conversation. &mdash; Cheers, Steelpillow (Talk) 23:00, 16 December 2014 (UTC) [Updated 10:01, 17 December 2014 (UTC)]


 * I have not read Clancy, but I have the book on order. Unfortunately, I have to get it via inter-library loan so probably won't see it until after the new year.  It may well be true that in his model the integral of all the air experiencing downward deflection is equal to -L.  But from what I've seen in the excerpt posted here, his cylindrical model is not as accurate as the potential flow model.  In the potential flow model, a subset of the air experiences dp/dt = -L, but if you add up all the air being deflected downward the magnitude of dp/dt is larger than |L|.


 * So, perhaps we are just looking at different models? Mr. Swordfish (talk) 16:27, 17 December 2014 (UTC)


 * Yes, I am sure that we are referencing different models. Some models give dp/dt=0, some 0.5L, some L, some >L. Each model is applicable under different assumptions or conditions, i.e. they are modelling different aspects of the problem, and - crucially to this debate - all are well attested in the literature. Taking a result (or assumption) from one of these aspects of the problem and then complaining that it doesn't match the results derived (or assumed) for a different aspect is at best futile. Asserting that it is therefore "wrong" is nonsensical. If the Statement disturbs you, it is because you and it are addressing different aspects of the problem. &mdash; Cheers, Steelpillow (Talk) 17:49, 17 December 2014 (UTC)


 * I agree. The idea that since these analyses give a numerical value for dp/dt, they all answer exactly the same question is tempting, but wrong.


 * Lanchester §112 asks, in effect, how much momentum is transferred per unit time between the foil and the ground, via the atmosphere.
 * Prandtl and Tietjens ask how much overpressure is exerted on the ground due to lift.
 * The control volume analyses of various other shapes ask how lift can be accounted for by a mixture of pressure and momentum.
 * The Statement is only concerned with momentum.


 * Just to avoid any confusion, I haven't seen any sources saying |dp/dt| > |L|, [//en.wikipedia.org/w/index.php?title=Talk:Lift_%28force%29&diff=636945452&oldid=636890166 neither] has J Doug McLean. Burninthruthesky (talk) 10:56, 18 December 2014 (UTC)


 * I think Steelpillow's and Burninthruthesky's line of argument above mischaracterizes the issue.


 * What we're discussing here is the question of how much integrated dp/dt is imparted to the flow by a lifting foil, and as part of that question we're concerned with how dp/dt is distributed in the field.


 * Both Clancy's cylinder model and the classical uniform-flow-plus-vortex model address these questions. They are not modeling "different aspects of the problem"; they just model the same flow in different ways.  Clancy's model for the velocity field is much more crude than the classical model, and Clancy's model ignores the pressure field, while the classical model models it realistically.  Clancy's model assumes dp/dt = -L in the near field (within the cylinder), while finding dp/dt = -L in the classical model requires looking much farther afield (the tall sliver control volume).  In this regard, the classical model is realistic, and Clancy's model is not.  Comparing the realism of different models in this way is not "futile".


 * Lanchester's analysis and the other classical analyses don't deal with momentum exclusively, but they do address the question of dp/dt. In that sense they all address the same question, i.e. the value of integrated dp/dt in the flowfield.


 * True, I have not seen any source saying that control volumes exist for which |dp/dt| > |L|, but it's easy to show that it's true. It's original research and can't be used in the article, but I think it's fair to use as a counterargument against other original research arguments on this page.  J Doug McLean (talk) 22:36, 19 December 2014 (UTC)

sources relating momentum transfer and lift
In this section I'm trying to collect source material for a proposed section on momentum transfer and lift. Additions cheerfully accepted, but let's try to keep extensive discussions out of this thread so we can see what raw material we have to work with.

"What supports an airplane aloft? ... Newton has given us the needed principle in his third law: if the air is to produce an upward force on the wing, the wing must produce a downward force on the air. Because under these circumstances air cannot sustain a force, it is deflected, or accelerated, downward.

Newton's second law gives us the means for quantifying the lift force:


 * Flift = m∆v/∆t = ∆(mv)/∆t.

The lift force is equal to the time rate of change of the momentum of the air."

Bernoulli and Newton in Fluid Mechanics Norman F. Smith The Physics Teacher 10, 451 (1972); doi: 10.1119/1.2352317 http://dx.doi.org/10.1119/1.2352317

"Most of the texts present the Bernoulli formula without derivation, but also with very little explanation. When applied to the lift of an airfoil, the explanation and diagrams are almost always wrong. At least for an introductory course, lift on an airfoil should be explained simply in terms of Newton’s Third Law, with the thrust up being equal to the time rate of change of momentum of the air downwards. See C. Waltham, “Flight without Bernoulli,” Phys. Teach. 36, 457 (Nov. 1998)."

Quibbles, misunderstandings, and egregious mistakes AAPT Physics Textbook Review Committee Citation: The Physics Teacher 37, 297 (1999); doi: 10.1119/1.880292 http://dx.doi.org/10.1119/1.880292

"Birds and aircraft fly because they are constantly pushing air downwards:


 * L=dp/dt    (3)

Here L is the lift force and dp/dt is the rate at which downward momentum is imparted to the airflow...

If we were to do this more correctly, we would box in the wing with a control volume of infinite vertical thickness. "

Flight without Bernoulli C. Waltham Phys. Teach. 36,457 (Nov. 1998). http://users.df.uba.ar/sgil/physics_paper_doc/papers_phys/fluids/fly_no_bernoulli.pdf

"Now let’s move on to conservation of momentum: the force exerted on a fluid equals the time rate of change (derivative with respect to time) of its linear momentum. If you exert a force on something, you change its momentum. If you don’t exert a force on something, its momentum stays unchanged or is conserved. This is Newton’s laws, if you choose to call it that. When an airfoil is producing lift, that force does in fact change the vertical component of the airflow’s linear momentum, and the drag force changes the horizontal component of the airflow’s linear momentum. ...Measuring lift by measuring the increase in downward vertical velocity in the flow coming off the trailing edge of the airfoil is conceptually possible. This downward velocity is definitely there and is known as downwash. I have never heard of anyone actually measuring it with sufficient precision to calculate lift, not because it is physically unsound but because it is not a practical experiment."

An Aerodynamicist’s View of Lift, Bernoulli, and Newton Charles N. Eastlake THE PHYSICS TEACHER Vol. 40, March 2002 http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf

"There is a widespread notion that an airplane in steady level flight continuously imparts net downward momentum to the atmosphere. ... Thinking in intuitive physical terms, we might also expect the impulse imposed by the airplane on the air (the product Lt) to produce a net vertical momentum in the atmosphere that grows with time. This expectation is not satisfied by the mathematics, however. ... If we expected to see a net downward momentum equal to Lt, the result comes as a surprise: The value of the integral over the semi-infinite space above the ground is zero, which means that the airplane imparts no net downward momentum to the atmosphere in steady level flight over a ground plane, regardless of height above the ground."

Understanding aerodynamics arguing from the real physics  sec 8.5 McLean, Doug Chichester, West Sussex, U.K. : Wiley, 2013. http://mirlyn.lib.umich.edu/Record/012482734

On basic control-volume analysis of the rate of change of momentum in a moving fluid:

The Dynamics and Thermodynamics of Compressible Fluid Flow Section 1.5 Shapiro, A. H. 1953. New York: The Ronald Press Company.

McLean: "This analysis shows that for a steady flow the integrated time rate of change of momentum of fluid parcels passing through the interior of a control volume is equal to the integrated (net) flux of momentum through the boundary. This is a basic ingredient in the other analyses cited below."

For the atmosphere as a whole, including a ground plane:

Applied Hydro- and Aeromechanics Prandtl, L., and O. G. Tietjens. 1934. . New York: Dover Publications. Derivation in connection with figure 150.

McLean: I don't have a copy at hand, so I can't provide a quote, but this is the classic analysis showing that the pressure pattern on the ground constitutes a downward force on the ground, and thus an upward force on the atmosphere, equal to L. The net force on the atmosphere due to the lift, (i.e. the vector sum of the forces exerted by the wing and the ground) is therefore zero, so that the integrated rate of change of vertical momentum for the atmosphere as a whole must be zero.

The Prandtl and Tietjens analysis is for the 3D case. It is easy to show that the same overall conclusion applies in 2D. A citable source for the 2D analysis probably exists, but I don't know of one offhand.

For a circular region centered on the airfoil:

Aerodynamic Theory, vol. 1. Sections B. V. 6 and B. V. 7. Durand, W. F., ed. 1932. New York: Dover Publications.

McLean: This is a control-volume analysis of the flow around a 2D lifting body of arbitrary cross-section in an infinite atmosphere, using a circle of large radius as the outer boundary of the volume. It shows that in the far field the flow is independent of the details of the body, and that significant contributions to the pressure and the momentum fluxes at the outer boundary come only from the combination of the uniform flow and the bound vortex. It arrives at a derivation of the Kutta-Joukowski theorem in equation 7.3. Equation 5.6 shows that the flux of vertical momentum across the outer boundary, and thus the time rate of change of vertical momentum in the air in the interior, is equal to only half the lift. Equation 6.6 shows that the integrated vertical pressure force on the outer boundary is upward and equal to half the lift. The net force on the air due to the lift is therefore downward and equal to half the lift, and Newton's second law is satisfied. It is explicitly stated that this result holds regardless of how large the radius of the circle is made.

Reference added by J Doug McLean (talk) 21:08, 16 December 2014 (UTC):

An Introduction to Fluid Dynamics Batchelor, G. K. 1967 Cambridge University Press

Applying the momentum theorem to incompressible inviscid flow around a 2D body of arbitrary cross-section (a general "cylinder") with circulation, and using a control volume bounded by a circle of large radius, Batchelor finds on p. 407:

"It follows from the calculation of the integral that the side-force exerted by the cylinder appears in the fluid far from the body half as a momentum flux and half in the form of a pressure distribution."

For rectangular control volumes:

"For a large rectangular control surface, part of the lift is attributable to pressure and part to momentum, depending on the aspect ratio of the surface. For a square control surface the contributions on the surface due to momentum and pressure are equal; for a tall, long vertical surface the contributions are mainly momentum, while for a streamwise long, flat, horizontal surface the lift is primarily due to pressure. This illustrates that it doesn't make much sense to attribute the lift on an airfoil to either pressure or momentum effect, unless one takes a control surface on the actual airfoil surface, when the lift is indisputably due only to pressure!"

The facts of lift. Section titled "Lift in thin slices: the two dimensional case". Lissaman, P. B. S. 1996. AIAA 1996-161. http://arc.aiaa.org/doi/pdf/10.2514/6.1996-161

For a tall column of air:

"When a loaded aerofoil is dynamically supported by a fluid, we know that its weight is eventually sustained by the surface of the earth, and that the transmission of the stress is effected by the communication of momentum from part to part, and is thereby distributed over a considerable area as a region of increased pressure"

Fig. 62 illustrates the forces on a narrow column of air where W is the weight of the foil acting downwards and the pressure at the base of the column is w. "Consequently the downward momentum imparted per second to the fluid leaving the prism plus the upward momentum received per second from that entering must be equal to W – w."

"When the height at which the aerofoil is sustained is great in comparison with its own dimensions, the area over which the weight is distributed on the earth's surface is obviously also great, and the quantity w becomes negligible. Under ordinary conditions this would usually be the case, so that the weight may be regarded as in no part statically supported."

Aerodynamics. §112 – Aerodynamic support Lanchester F.W. (1907) Archibald Constable & Co. Ltd. https://archive.org/stream/aerodynamicscons00lanc#page/146/mode/2up

"The downwash also varies in the streamwise direction. It reaches its ultimate value little more than a chord length behind the trailing edge; and its mean value at the wing itself can be shown to be one half of this ultimate value." (Page 75)

...

"The lift produced by a wing is imparted to it through the variations in pressure over its surface. This lift force has its reaction in the downward momentum which is imparted to the air as it flows over the wing. Thus the lift of the wing is equal to the rate of transport of downward momentum of the air.
 * 5.15 Lift and downwash

"This downward momentum is measured in terms of the induced downwash described above." (Page 76)

...

"Consider, then, this cylinder of air, as illustrated in Fig. 5.21.[longitudinal axis with respect to the foil, bends down to form the downwash] The area of cross-section of the cylinder of affected air is 1/4 πb2. The rate of mass flow of affected air past the wing is therefore 1/4 πρVb2. The rate of transport of downward momentum is therefore 1/4 πρVb2w, and this must equal the lift, L." (Page 76)

...

"If we consider unit span of an infinite wing, however, the air above this unit span forms part of a cylinder of infinite radius, and its mass is therefore infinite. Since the downward momentum imparted to the air in unit time is finite, and since the mass of the air is infinite the induced downwash velocity must be zero." (page 77)

Aerodynamics, Clancy, L.J. Pitman Publishing (1973)

"All attempts to fly in heavier-than-air machines must embody some means of forcing the air downwards so as to provide the equal and opposite reaction which is to lift the weight of the machine."

"...if we reject the idea of flapping wings, we must replace it by some other device which will deflect the air downwards."

Mechanics of flight Kermode, A.C. Eighth (metric) edition, 1972. Pitman Publishing

Reference added by J Doug McLean (talk) 21:08, 16 December 2014 (UTC):

Elements of Practical Aerodynamics Jones, B. 1939 John Wiley and Sons, Inc.

In the context of an analysis of induced drag on p. 82 he makes the following statement about the downward momentum imparted to "the mass of air affected by the wing":

"It has been proven mathematically that the downward momentum in unit time is equal to one-half the lift".

Section added by J Doug McLean (talk) 21:08, 16 December 2014 (UTC):

On the popular qualitative flow-deflection explanation based on Newton's laws:

"The main fact of heavier-than-air flight is this: the wing keeps the airplane up by pushing the air down.

Stick and Rudder - An Explanation of the Art of Flying. Langewiesche, W. McGraw-Hill Education

"In momentum-based explanations, it is generally argued that the airfoil produces a flowfield in which some of the air is "deflected" downward and thus has downward momentum imparted to it. To acquire downward momentum, the air must have a downward force exerted on it by the airfoil, and thus, by Newton's third law, the airfoil must have an upward force exerted on it by the air."

Understanding Aerodynamics -- Arguing from the Real Physics sec 7.3.1.7 McLean, D. Chichester, West Sussex, U.K. : Wiley, 2013.

Section added by J Doug McLean

On the Newtonian theory of lift

"The fluid itself is postulated as a collection of individual particles that impact directly on the surface of the body, subsequently giving up their components of momentum normal to the surface, and then traveling downstream tangentially along the body surface. That fluid model was simply a hypothesis on the part of Newton; it did not accurately model the action of a real fluid, as Newton readily acknowledged.  However, consistent with that mathematical model, buried deep in the proof of Proposition 34 is the result that the force exerted by the fluid on a segment of a curved surface is proportional to sin^2 (theta), where theta is the angle between the tangent to the surface and the free-stream direction.  That result, when applied to a flat surface (e.g. a flat plate) oriented at an angle of attack alpha to the free stream, gives the resultant aerodynamic force on the plate:

R = rho*V^2*sin^2(alpha)

This equation is called Newton's sine squared law...."

A History of Aerodynamics Anderson, J. D., Jr. Cambridge University Press

(work in progress - to be continued) I'll try to track down the cites provided by Doug McLean and see if I can pull out the relevant direct quotes instead of relying on his summaries. Not that I don't believe his summaries, it's just that we'd be remiss as editors if we just took his word for it. Mr. Swordfish (talk) 21:33, 8 December 2014 (UTC)

Comments on the sources above
Please make any comments below here, so that we can keep the listing of sources clean and uncluttered.

"...the integrated rate of change of vertical momentum for the atmosphere as a whole must be zero." Thus for the most obvious assumption a reader is likely to make regarding what is meant by "the air" (i.e. the atmosphere as a whole), The Original Statement is false. - DougMcLean (from earlier)
 * Comments on Prandtl and Tietjens

It is explicitly stated that this result holds regardless of how large the radius of the circle is made. Thus a large circle is another example of a region of "the air" for which a reader might reasonably expect The Original Statement to apply, but for which it is in fact false. - DougMcLean (from earlier)
 * Comments on Durand

According to Lissaman's results, if "the air" is taken to be the air in a rectangular box surrounding the airfoil, The Original Statement isn't even close to being true unless the box is a tall, slender sliver, and even then it isn't strictly true until the vertical dimension of the box is taken to infinity. Steelpillow quotes the section of my book that describes the result for the infinitely tall, slender sliver, the only control-volume shape for which The Original Statement has been shown to be true, and interprets it as being "in support of The Statement". A balanced recounting of what my book says would also quote the discussion in connection with figure 8.5.4, which deals with other control-volume shapes for which The Original Statement isn't true. - DougMcLean (from earlier)
 * Comments on Lissaman


 * Comments on the AAPT paper

The quote from the AAPT textbook committee given above is the entire discussion of lift in that paper. The paper itself is a 25 pager devoted to reviewing seven high school physics textbooks, and that's all the space they had to address lift. As such it is quite cursory and lacks context. They suggest seeing Waltham's paper for more details and context. Waltham in turn cautions that to do it "correctly" requires a control volume of infinite thickness.

The AAPT does not explicitly say dp/dt of the entire atmosphere (or the "local flow" for that matter) is equal to -L. By not defining what they mean by "the air" they leave an imprecise statement that can be interpreted in different ways, one of which is correct and most of which are incorrect. If we repeat it with the same imprecision it is likely that our readers will latch on to one of the incorrect interpretations.

In sum, I would not give it the same weight as other more in-depth treatments of momentum transfer. Mr. Swordfish (talk) 21:09, 10 December 2014 (UTC)

The AAPT describe how much momentum is imparted to air within the atmosphere per unit time, consistent with results obtained by Lanchester, Lissaman and others.

I find it hard to think of a better form of words to describe this rate simply and concisely without going into details of a mathematical proof. Burninthruthesky (talk) 09:20, 16 December 2014 (UTC)

I agree with Mr. Swordfish that the AAPT paper doesn't deserve "the same weight as other more in-depth treatments of momentum transfer", and that "to describe this rate simply and concisely without going into details" would be misleading. J Doug McLean (talk) 21:08, 16 December 2014 (UTC)

Lanchester §112 says that it is generally possible to show that dp/dt = -L. He goes on to clarify the exception commonly known as ground effect. Burninthruthesky (talk) 10:13, 11 December 2014 (UTC)
 * Comments on Lanchester

Without the use of calculus, he derives the total directly from Newton's laws.

The pressure w is entirely accounted for on the surface of the earth. Burninthruthesky (talk) 07:20, 13 December 2014 (UTC)


 * Lanchester's §112 does not say "that it is generally possible to show that dp/dt = -L." His analysis only shows that it is possible if you make one particular assumption for the shape of the control volume.


 * Lanchester's §112 amounts to a verbal version of a control-volume analysis for a tall-sliver control volume in contact with the ground.  Lissaman (1996) published the corresponding analysis for the free-air case.  The result is the same whether there is a ground plane or not, i.e. that as the height of the sliver goes to infinity the pressure force at the bottom (and top in the free-air case) vanishes, and thus dp/dt = -L for a control volume that is infinitely tall compared to its width.


 * Lanchester's result is essentially the same as Lissaman's and doesn't support an unapologetic version of The Statement any more than Lissaman did. The pressure is "entirely accounted for", but it's effect vanishes only because the height of the sliver is taken to infinity and the width is kept finite.  J Doug McLean (talk) 21:08, 16 December 2014 (UTC)




 * Yes it does. Providing the right physical conditions exist, it is always possible to make the assumptions which show dp/dt is practically equal to -L. Lanchester suggests that when an aerofoil is in ground effect, w may become significant, in which case it would not be possible to show dp/dt = -L. He characterises the former situation as "ordinary conditions" which would "ususally be the case".


 * I only mentioned the fact that Lanchester accounts for pressure on the surface of the earth in response to your previous [//en.wikipedia.org/w/index.php?title=Talk:Lift_%28force%29&diff=636794728&oldid=636312368 comment] suggesting Lanchester supports your argument against The Statement.


 * It is clear from previous discussion and your obvious expertise in the subject that you need no explanation of the physics. So it is a mystery to me why you repeatedly misrepresent what has been written by others. You jest that "chuckling at the mistakes of others" is "part of the fun of being an aerodynamicist". I have not made a mistake here. We are WP:NOTHERE for an exercise in schadenfreude.


 * I've no doubt that that Lanchester's §112 is essentially the same as Lissaman's result. Both show that momentum is imparted to air within the atmosphere at a rate equal to lift. I don't see anything misleading about describing this rate as a simple fact in an introductory text, as advocated by the AAPT. You may disagree, but they are experts at educating people. Burninthruthesky (talk) 12:07, 17 December 2014 (UTC)


 * What I've said repeatedly is that the only published sources that use a realistic flow model and find dp/dt = -L to be true are those using a very-tall-sliver control volume. These sources make no claim that dp/dt = -L is true in any more general sense than that.  So what, specifically, have I misrepresented?


 * You still seem to have concluded that The Statement is somehow true in a more general way. To me, this doesn't seem consistent with the physics or the published evidence.  Just because dp/dt = -L applies in one particular control volume doesn't mean it applies to "the atmosphere" in any general sense.  Yes, the tall sliver control volume "exists within" the atmosphere, but so do other control volumes for which dp/dt is different from -L.  I see no basis for thinking one control volume is the "correct" one and the others are not.  In general, the force exerted on the air by the foil is manifested as a combination of momentum changes and pressure differences in the flowfield.  The one control volume for which the integrated pressure differences happen to vanish isn't in any fundamental way more "correct" than the others.


 * I know of no published source that uses a realistic flow model and shows that dp/dt = -L is true in the general sense you're proposing. The AAPT paper presents no analysis, and Chris Waltham's paper doesn't count on this score because he uses a simplified flow model that omits the crucial effect of the pressure field, as discussed in my comments on Clancy.


 * No, the Statement is not a "simple fact". It is a statement that is misleading unless it is accompanied by a somewhat arcane caveat.  And I'm not making these arguments for fun.  I'm here to improve the article by avoiding the inclusion of a misleading statement.  J Doug McLean (talk) 22:52, 19 December 2014 (UTC)


 * In answer to your question, I have already clarified why I said, "Lanchester §112 says that it is generally possible to show that dp/dt = -L. He goes on to clarify the exception commonly known as ground effect." My comment is supported by the citation above. Your contradiction takes my words out of context.


 * The earlier comment I referred to said, This is a rebuttal for a proposition that was never made. The point of discussion was whether or not The Statement is true.


 * Several sources clearly explain how to calculate the result dp/dt = -L. As I have explained repeatedly, all of them account only for momentum. There are other calculations which account partly for pressure differences and for a smaller proportion of momentum. Your repeated assertion that other results for dp/dt are different, yet somehow answer the same question defies common sense. Again it is a rebuttal for a proposition that the cited authors did not make.


 * I'm afraid this is going round in circles again. Hopefully in the New Year there will be some progress on improving the article. Burninthruthesky (talk) 12:26, 20 December 2014 (UTC)

It is interesting that Clancy sees no contradiction between his Newtonian description and the fact that at the trailing edge, only half the momentum has yet been transferred. &mdash; Cheers, Steelpillow (Talk) 09:51, 15 December 2014 (UTC)
 * Comments on Clancy

As far as I can tell, in the idealised infinite situation he goes on to describe Clancy is arguing that although the downward change of momentum in any finite portion of air is now infinitesimal, the sum of infinitely many such portions remains finite and is in fact still equal to the reaction to the lift. &mdash; Cheers, Steelpillow (Talk) 16:11, 11 December 2014 (UTC)

Clancy refers to "the affected air". This compares to the phrase "the air" which has caused so much supposed scope for misunderstanding here. Simply inserting the word "affected" as Clancy does would clear that side issue up. &mdash; Cheers, Steelpillow (Talk) 09:36, 12 December 2014 (UTC)


 * I don't think inserting the word "affected" will solve much of anything. According to the model, the entire atmosphere is affected by the presence of the moving airfoil so the "affected air" is all of it.  And we have seen that for the entire atmosphere dp/dt =0.    A few months ago we tried inserting "the air deflected by the foil" and that didn't do it either.  There is some subset of the air for which dp/dt=-L, but I'm at a loss for how to state this in layman's terms without it being so awkward and convoluted that it distracts from the flow of the simple introductory section. Mr. Swordfish (talk) 13:10, 15 December 2014 (UTC)
 * It is not true that, as you suggest, "According to the model, the entire atmosphere is affected by the presence of the moving airfoil." Nowhere does Clancy's model address the entire atmosphere. It addresses a certain cylinder of air which he first makes finite and then expands to infinite size. I wonder if you are confusing this infinite cylinder with the whole atmosphere? We can be sure that they are not the same thing because within the interior of Clancy's model infinite cylinder, dp/dt = F while we know that within the even more infinite atmosphere which contains his model, dp/dt = 0. "The affected air" in his model is just the air within the rear part of the cylinder, aka "the air deflected downwards". &mdash; Cheers, Steelpillow (Talk) 16:52, 15 December 2014 (UTC)


 * Apologies for not being clear, but I wasn't referring to Clancy's model - I was referring to solutions to the Navier-Stokes equations, Euler equations, potential flow, etc.


 * In the very simple case of potential flow, the resulting flow field is the superposition of a steady uniform flow (i.e. what the flow would be like in the absence of the foil) and a vortex flow. While the vortex flow field diminishes as one gets farther away from the foil, at least in theory it is non-zero everywhere.  So in the potential flow model the entire atmosphere is affected by the foil.  More rigorous models (N-S, CFD, etc) give similar non-zero deviations from uniform flow throughout the entire atmosphere.  These mathematical treatments are what I meant by "the model".


 * It appears that when Clancy says "the air" he means some specific subset of the atmosphere which he has defined beforehand and not the entire atmosphere. Fair enough, but unless we provide such a definition to our readers it is likely that they will interpret "the air" as the entire atmosphere. I think we are in agreement that for a non-accelerated foil dp/dt of the entire atmosphere is zero.  If so, we're just arguing over what is meant by "the air".  If not, then our disagreement is more fundamental.  Which is it? Mr. Swordfish (talk) 19:55, 15 December 2014 (UTC)
 * Ah, in an item headed "Comments on Clancy", I hope you will forgive my misunderstanding which model you were discussing. The sources certainly bear out that the overall dp/dt is zero, I have no problem with that. Perhaps the best approach to "the air" is simply to write the new section intelligibly and not worry about our past usage. My comment on Clancy's usage was really just a flag to that end, in case it came in useful. &mdash; Cheers, Steelpillow (Talk) 20:50, 15 December 2014 (UTC)


 * Clancy himself admits that his flowfield model is "very crude". It assumes that "the air affected" by the wing in 3D is limited to a stream of circular cross-section with diameter equal to the wingspan (fig 5.21) and that this air is uniformly deflected downward by its interaction with the wing.  There is no upward turning ahead of the wing or behind as there is in more-realistic flow models.


 * His model is unrealistic in another respect, and that is that it completely neglects the pressure field. It is assumed that the only force acting on the affected stream is that exerted on it by the wing, and this force is assumed to show up entirely as a rate of change of momentum of the steam of affected air.  Thus dp/dt = -L is not really a result of this analysis; it is more of an a priori assumption.


 * His model becomes a little less unrealistic in 2D, where the wingspan and the diameter of the affected cylinder have gone to infinity, and the entire flow is thus affected, but it is still unrealistic in assuming the flow deflection is uniform, and in neglecting the pressure field.


 * Both Durand and Batchelor rigorously show that a uniform flow plus a vortex is a good approximation for the far-field flow in 2D, regardless of the details of the airfoil shape. The classical control-volume analyses I've cited use this model and show that the pressure field exerts significant forces on "the air" except in the case of the infinitely tall sliver.


 * I would not give Clancy the same weight as I would to the classical analyses I've cited, which use a much more realistic model. And I agree with Mr. Swordfish that adding "affected" doesn't clear up what is meant by "the air".  J Doug McLean (talk) 21:08, 16 December 2014 (UTC)
 * I am quite sure that you would give your own work more weight than you give to those whom you criticise. Yawn. Worse, you cherry-pick from Clancy's full text to support your PoV. It also notes that his simplified model is consistent with a more complex analysis he gives later - and when we turn to that later analysis we find that it embraces the very flow-plus-vortex model you approve of. To selectively quote and then claim that Clancy's simplified model is at odds with the flow-plus-vortex model is somewhat invidious. Turning to the classical analyses you have quoted from, we may note that Clancy also writes that the downwash at the trailing edge is only half of its ultimate value, which it achieves a little over a chord-length further downstream. This is entirely supported by the classical analyses you quote. If I had had a different textbook on my shelf introducing these standard results using the standard introductory model, I am sure you would have laid into them equally. To claim that a standard textbook such as Clancy is at odds with the mainstream is to seriously undermine your own position. &mdash; Cheers, Steelpillow (Talk) 11:02, 17 December 2014 (UTC)


 * All of the quotes attributed to Clancy in the "sources" section above are from chapter 5, and all are based on the simplified flow model he uses there. In criticizing this model, I am not "cherry-picking".  I'm sticking to the topic under discussion, i.e. the weight these different sources should be given regarding the question of dp/dt in the flow around a lifting foil.


 * You say that his results from chapter 5 are "consistent" with his results from the more realistic model in chapter 8. That's simply not true with regard to the question we're discussing here.  The issue of dp/dt isn't addressed at all in chapter 8.  The only part of the book that deals with dp/dt due to lift is chapter 5.  So to criticize the model he uses in chapter 5 is not to "selectively quote" him.


 * The only thing for which Clancy claims the two models yield equivalent results is in the variation of the downwash from the near field to the far field, which does not address the question we're discussing here. The downwash velocity is a local quantity, while the dp/dt we're discussing is an integrated quantity.


 * In chapter 8 he uses the horseshoe-vortex model for a 3D wing, for which it is said that both the bound vortex and the trailing vortices contribute to the downwash field. But when he says that the downwash in the near field is only half the value in the far field, he's changed gears and he's referring only to the "3D" part of the downwash, the part associated with the trailing vortex system, and omitting the part of the downwash associated with the bound vorticity.  I mention these details only to rebut your claim that "This is entirely supported by the classical analyses you quote."  No, the analyses I quote deal with integrated dp/dt in a control volume, not with the variation of the downwash velocity downstream.  And all but one of them deal with the 2D case, where the downwash velocity doesn't behave at all as Clancy describes for the 3D case anyway.


 * I'm not criticizing Clancy's book as a whole. I'm criticizing the applicability of the model he uses in chapter 5, and the statements he derives from it, to the question of dp/dt.  And yes, I do find that in neglecting the pressure field he's at odds with the mainstream analyses on this particular issue.  If you have any actual specific counterarguments to make in this regard, please let us know what they are.  J Doug McLean (talk) 22:52, 19 December 2014 (UTC)


 * Comments on Chris Waltham's Flight without Bernoulli 

In the section titled "A Simple Model", Waltham uses the same kind ot model Clancy uses in sec 5.15, i.e. the model in which the air is affected only within a stream of limited cross-section. He starts with the assumption that the cross-section is rectangular, but he later says that it could just as well be circular. This model is unrealistic for the same reasons I give in my comments on Clancy. Because the model does not represent the flow realistically, it is wrong on some important details, such as dp/dt in the local flow around the wing. Regarding the question of dp/dt, I would not give Waltham the same weight as I would to the classical sources. J Doug McLean (talk) 21:08, 16 December 2014 (UTC)


 * Comments on the popular qualitative flow-deflection explanation based on Newton's laws

I have added new section to the sources list: "On the popular qualitative flow-deflection explanation based on Newton's laws". It lists only two sources so far, but more can be found on NASA websites and elsewhere.

I think this kind of thing is the appropriate level of detail for the "Flow deflection and Newton's laws" section, and that we should not include a quantitative dp/dt statement there. J Doug McLean (talk) 21:08, 16 December 2014 (UTC)


 * Comments on the "Newtonian" approach

In the "Sources" section I've added a quote from J. D. Anderson describing Newton's theory of lift, based on modeling the flow as a hail of bullets. The flow models used by Clancy and by Chris Waltham are similar to Newton's model in the sense that they assume that only a limited portion of the flow interacts with the foil, and they take no account of a continuum pressure field. Like Newton's theory, these "fire-hose" models are wrong about important details of the flow, such as dp/dt in the near field. As I argue above, Clancy and Waltham should be given less weight as sources on the topic of dp/dt than the classical sources that use a more realistic model. J Doug McLean (talk) 20:03, 20 December 2014 (UTC)

Some diagrams to help clarify the discussion of 'The Statement'
We've created quite a wall of text arguing about the statement, and don't seem to be getting anywhere. Perhaps some pictures can help clarify things.

Here's a diagram of an airfoil generating lift, with the airflow coming in from the left. Ahead of the airfoil in region A the air is accelerated upwards (upwash), in regions B and C the air is accelerated downward, and in region D the air is accelerated upwards as it returns to horizontal flow.



Let's say we define the regions accordingly, where A is the region in front of the foil where dp/dt>0 (upward acceleration) B and C are the regions above and below the foil where dp/dt<0, and D is the region behind where dp/dt>0.

When I encounter the phrase "the air deflected downwards" I parse that to mean the union of regions B and C. Perhaps there's some other interpretation that can be used, but that's what "the air deflected downwards" means to me. If there are other interpretations floating about I'd be happy to hear them.

What can we say about the net momentum transfer of B ∪ C? I do not know of any source that gives this value dpB∪C/dt. But we can infer how it relates to L by looking at the following sub-region of B ∪ C:



Region E is a tall thin sliver and if we take the limit of dpE/dt as the height goes to infinity and the width goes to zero we get dpE/dt = -L. Since this is a proper subset of B ∪ C and all of B ∪ C has negative dp/dt, E must have a lower magnitude of momentum change than all of B ∪ C. That is, for B ∪ C  |dp/dt| > |L|. Earlier, Doug claimed to have calculated it to be -1.6L, which sounds plausible.

Now, I'm not suggesting that the above analysis (along with my crude diagrams) go into the article - it's certainly too arcane for the intro section and it's not ready for "prime time" as a later section. But I hope that my fellow editors can read it and understand why I have a problem with saying "the time rate of change of momentum of the air deflected downwards" is equal to the lift. By my interpretation of 'the air deflected downwards' the statement is incorrect. It took me a while to come around to this view - recall that I actually wrote the statement and included it in my re-draft last summer - but I now see that it's problematic. Comments? Mr. Swordfish (talk) 17:21, 22 December 2014 (UTC)
 * I am having trouble following your summary:
 * The "width" which goes to zero is not defined - the width of what? Is it of the aerofoil chord or just of the column E, or perhaps the span orthogonal to the screen?
 * Whatever the "width" is, I find it hard to justify the statement that "as ... the width goes to zero we get dpE/dt = −L" (unless you are also implying that as the width goes to zero, L also tends to zero, which is not very helpful).
 * If you believe what you wrote, then I can indeed understand why you have a problem with the Statement. But more fundamentally, I think you are still confusing a broad statement of the Newtonian principle with an attempt at detailed analysis. The Statement is expressing the Newtonian principle in a few words, while you are seeking to interpret some of those words ab initio, i.e. without prior acceptance of the principle. That is bound to end in tears. &mdash; Cheers, Steelpillow (Talk) 14:51, 23 December 2014 (UTC)


 * Mr. Swordfish presents a clear, detailed analysis of the problem with The Statement. There is one substantive problem with the analysis as presented, which Steelpillow has picked up on, and that is that it doesn't make sense for the width of column E to go to zero.  But this problem doesn't invalidate Mr. Swordfish's conclusion.


 * In his analysis of the tall sliver control volume, Lissaman took the limit as the height goes to infinity, but not as the width goes to zero. With an actual foil in the picture, a proper control-volume analysis of the lift requires that the control volume be wide enough to bracket the projected chord of the foil.  I'm sorry if the word "sliver" confused the issue here.  I used it to refer to the control volume's proportions, meaning to convey that the control-volume width is small compared to the height, not that it is small compared to the chord.  Anyway, even though the width doesn't go to zero in Lissaman's analysis, the pressure difference between the top and bottom vanishes as the height goes to infinity, and the integrated vertical pressure force vanishes, with the result that dp/dt = -L, independent of the width, as long as the width is kept finite.


 * For E to be a proper subset of B ∪ C, and to not include part of A ∪ B, the width of E must be restricted compared to what Lissaman assumed, with the vertical boundaries pushed up against the leading and trailing edges of the foil. But Lissaman's analysis still applies, and dp/dt = -L for E.


 * Steelpillow's point 3 is mistaken. He says "E has a smaller magnitude which means that it is closer to zero and is therefore 'greater than' −L."  No, E is the subset with dp/dt =-L, so its dp/dt cannot be "'greater than' −L.".  Mr. Swordfish's analysis is correct.
 * Ah, yes, thank you for pointing that out. Too much Christmas spirit in my glass, I fear. I have now deleted that item. &mdash; Cheers, Steelpillow (Talk) 22:59, 23 December 2014 (UTC)
 * Steelpillow is also mistaken when he says "But more fundamentally, I think you are still confusing a broad statement of the Newtonian principle with an attempt at detailed analysis." No, dp/dt = -L for "the air" can be established as a valid "statement of the Newtonian principle" only after a detailed analysis has shown that -L is indeed the resultant force acting on "the air".  And analysis has only shown this to be the case for the tall sliver control volume.


 * Mr. Swordfish's analysis above clearly shows that dp/dt = -L is false for "the air deflected downward". However, we have no citable source for this argument.  On the other hand, even if we thought dp/dt = -L was true for "the air deflected downward", we have no citable source that supports including the word "deflected".  To stay within what our sources support, our only choice is The Original Statement from the AAPT paper, without the word "deflected".


 * I agree that The Original Statement in unapologetic form is unacceptably vague. If we include it in the intro section, we must, at a minimum, also specify that the only definition of "the air" for which it's been shown to be true is "a region that is very tall compared to its width", citing Lissaman.  As I said long ago, I think the best option is not to make any quantitative statement in the intro section and to discuss the quantitative momentum balance in a new later section.


 * I'd like to see something like Mr. Swordfish's ABCDE discussion in this new section, but I know of no citable source for it. From the sources I know of, I think the best we can do is to describe the results for the circular and rectangular (tall, square, and flat) control volumes in free air, and the atmosphere with a ground plane.  Here are two possible diagrams I've made for the purpose, and I've started drafting candidate text to go with them.






 * If we mention the "fire-hose" models at all (Clancy, Waltham), it would only be to point out their deficiencies relative to the more rigorous analyses. I'm proposing that it be a new subsection titled "Analyses of the integrated momentum balance in lifting flows", under "Mathematical theories of lift", just after "Circulation and the Kutta-Joukowski theorem".  J Doug McLean (talk) 22:36, 23 December 2014 (UTC)
 * This is just the same old same old. The article is not going to be amended to support a view unsubstantiated by reliable sources. &mdash; Cheers, Steelpillow (Talk) 22:59, 23 December 2014 (UTC)


 * There is one substantive problem with the analysis as presented ... it doesn't make sense for the width of column E to go to zero.  Thanks.  I have struck that language.


 * I look forward to seeing the draft section. Agree that anything we put into the article must have a citable source. Mr. Swordfish (talk) 00:46, 25 December 2014 (UTC)


 * Mr. Swordfish, I suggest you think about how dt should be handled in these calculations. You may find Lanchester §3 helpful.


 * When making content decisions, we should be guided more by RS than by OR. The novel, unpublished analysis above will not help anyone understand the relevant physics. If you haven't already, I recommend reading Lanchester §112 for a straightforward explanation. Burninthruthesky (talk) 14:58, 27 December 2014 (UTC)


 * At this point I would like to ask both Burninthruthesky and Steelpillow to elaborate on what they think "the air deflected downward" means.  I think we are in agreement that it doesn't mean the entire atmosphere since dp/dt of the entire atmosphere is zero. So, what does "the air" mean?  I've given my interpretation and Doug has given several possible reasonable interpretations (of which only one makes the statement true).  What's yours? And is it likely that our readers will have the same interpretation?


 * Agree that we should value RS over OR discussion on the talk page, and I'll keep trying to chase down a copy of Lanchester and read sec 112. Mr. Swordfish (talk) 21:28, 27 December 2014 (UTC)
 * Since the Statement is just the application of Newton's laws to lift in words, in this context it means, "the air deflected downwards in reaction to the lift force." Not any other air deflected downwards because of some vortex or some distant pressure distribution or some clever analysis or whatnot. It is simply affirming that if we apply F=dp/dt to L, then there is a mass of air whose dp/dt is in reaction to L. Our clever analysis can then identify the location of this mass for us, and different analytical models will identify different locations (e.g. Clancy's firehose). Crucially, it is not saying that this is the only approach, nor even the best approach, just that it is an approach. Other models based on other approaches, say on pressure, will not even yield dp/dt=L because they have already accounted for much or all of L some other way. The sources can then indicate the due weight that each deserves. This is how pretty much every textbook treats it, and I am not aware of widespread misinterpretation among engineering students. I am confident that our readers will be no different. One can really only misinterpret it once one has gained a good deal of detailed knowledge, well beyond the introductory stage at which it is appropriate. &mdash; Cheers, Steelpillow (Talk) 22:10, 27 December 2014 (UTC)


 * Apologies for taking so long to reply.
 * I don't have any fundamental disagreement with your position. I do think Doug has a point (made elsewhere) that defining "the air" as the region that makes the statement true is somewhat circular, but the fact remains that there is such a region and I'm comfortable eliding over some of the details in the introductory section.  To that end, I've prepared a draft in my user space that may help move us towards consensus.  I'll introduce that in a new thread.  Mr. Swordfish (talk) 19:56, 7 January 2015 (UTC)

Suggested revisions
I have posted a proposed revised version of the article in my sandbox User:J_Doug_McLean/sandbox. Changes from the current version are limited to two places:


 * 1) Under "Flow deflection and Newton's laws" I have removed the quantitative statement about momentum and done a bit of rewording of what remains to be sure that both the second and third laws still get their due.


 * 2) At the end of "Mathematical theories of lift" I have added two new subsections:  "Analyses of the integrated momentum balance in lifting flows" and "Newtonian theories of lift".

Everything I've included has a citable source and is presented, I think, from a neutral point of view.

I think removal of The Statement is best for the following reasons:


 * 1) In this part of the article and at this level ("Simplified physical explanations..."), the flow deflection explanation is better off without it.  This explanation is usually presented in its qualitative form anyway.


 * 2) Without the qualification that it's only been found to be true for the tall sliver, The Statement is open to misinterpretation.  Steelpillow and Burninthruthesky still seem to think that it isn't, but I think their confidence that the typical reader will know how to interpret it correctly is unjustified.  I think it's just too easy for an uninitiated reader to assume that "the air deflected downward" could refer to the atmosphere as a whole or at least to some sufficiently large subset of it.  The atmosphere as a whole may be a "nonsensical" assumption as Burninthruthesky called it, but I wouldn't expect the typical reader to know that unless we tell him.  And I certainly don't expect the typical reader to realize that it's true only for a particular shape of region.


 * 3) A general, unapologetic "dp/dt = -L" isn't consistent with what the mainstream literature says about dp/dt.

The two new sections are pretty self-explanatory. I included a paragraph on Newton's bullet model because I think it's interesting in its own right, it had an impact on early assessments of the practicality of heavier-than-air flight, and it helps put the "fire hose" models in perspective, which I also included under "Newtonian theories".

Looking forward to comments and suggestions.

J Doug McLean (talk) 23:53, 30 December 2014 (UTC)


 * First of all I would like to thank for all the hard work that has gone into this. The only real issue I have with any of it is the well-worn one we are all familiar with.
 * Given all the quotations we so carefully collected above, quite how anyone can maintain that the mainstream sources do not support the momentum statement - and right at the introductory stage at that - is beyond me. Mainstream sources do include it, and even mandate it there. Ours is not to reason why or to sanitise it out of the article. It is there in the article, it is reliably cited, and there it must stay. Doug McLean makes some other minor textual changes to this section which are generally good.
 * The bulk of the new section on "Analyses of the integrated momentum balance in lifting flows" is good, though there is a certain residual defensiveness in the last few sentences which can probably simply be omitted. Also, I would shorten the heading to just "Integrated momentum balance in lifting flows".
 * The next section - the critique of the momentum model - is useful as far as it goes. However I think it needs a balancing critique of the other simple model we introduce, viz. the pressure model. These critiques could better introduce the whole section on "Mathematical theories of lift" rather than conclude it. Alternatively, they could be confined to their respective introductory subsections on "Limitations of ..." &mdash; Cheers, Steelpillow (Talk) 10:52, 31 December 2014 (UTC)


 * I agree with Steelpillow there is no justification for removing the existing momentum statement. I'm also grateful for progress on improving the article.


 * One minor point - I haven't been able to find Shapiro to reference the momentum equation, but I have found this page which says (in abbreviated terms):


 * F = dM/dt = FB + FS


 * I understand that in steady flow, FB = 0, so F = FS.


 * In the proposed new section, I read the last two sentences of the first paragraph as FS = FB. Is this the intended meaning? Burninthruthesky (talk) 18:06, 1 January 2015 (UTC)


 * I have now incorporated what I believe to be the bulk of the proposal into the article, save for two parts: I retained the Statement along with its citations, and I have not copied across the subsection on Newtonian theories of lift because I am not yet sure where to put it. &mdash; Cheers, Steelpillow (Talk) 14:01, 4 January 2015 (UTC)

Momentum theorem
I still don't have an answer to my question above. A Google search for "momentum theorem for a control volume" brings back mainly references to Reynolds transport theorem. Is this what is being described? Burninthruthesky (talk) 14:58, 4 January 2015 (UTC) Here is an actual quote from Shapiro:

Also, Shapiro's Eq 1.15 is identical to Eq 3.42 in the page I linked above. Burninthruthesky (talk) 17:05, 4 January 2015 (UTC); edited 17:17, 4 January 2015 (UTC) I've left "disputed" tags on these two sentences. I am disappointed they were added to the article despite my unanswered question. Please will someone with more specialist knowledge than myself correct them? Burninthruthesky (talk) 17:36, 4 January 2015 (UTC)


 * My apologies. I took the citation of Shapiro Section 1.5 at its word, as I was not able to check it and you had referred to it as a "minor point". I do think we need to be clear whether each case is allowing a dynamic flow where the net rate of momentum change within the control volume is non-zero, or restricted to a steady-state flow where the net rate of momentum change within the control volume must be zero. &mdash; Cheers, Steelpillow (Talk) 22:17, 4 January 2015 (UTC)


 * Thank you, and I'm sorry I didn't make myself clear. By "minor", I meant the content probably needed correcting before release, as opposed to not being inherently useful.


 * I think a link to Reynolds transport theorem may be useful, as I undertsand this momentum theorem directly follows from it.


 * In this case, I would be a little uncomfortable with adding material to the encyclopedia which I only learned myself yesterday, in the course of verifying another's work. Burninthruthesky (talk) 07:44, 5 January 2015 (UTC)


 * First to Burninthruthesky's question: Is FS = FB the intended meaning of the last two sentences of the first paragraph of the new section?  No.  That interpretation isn't consistent with what the variables represent.


 * FB is the body force acting throughout the volume of the fluid, which for an electrically neutral fluid is just gravity. In aerodynamics we usually neglect both gravity and the background hydrostatic pressure gradient that goes with it, under the assumption that they cancel each other.  Thus FB = 0 results from neglecting gravity.  Whether the flow is steady or unsteady has nothing to do with it.


 * FS is the sum of the surface forces (pressure and viscous shear stress) acting on the control volume boundaries. In the airfoil analyses I've cited it includes both the -L' imposed by the foil and the integrated pressure force on the outer boundary.  The volume integral I refer to in the sentences in question is the integral of the material rate of change (material derivative) of momentum in the interior, which is not at all the same thing as FB.  Note that the two integrals on the RHS of equation 3.42 of this page represent the volume- and surface-integral parts of dM/dt.  The fact that dM/dt has two parts has nothing to do with the decomposition of F into FB and FS, and the first term on the RHS is not equal to the first term on the LHS.


 * So yes, FB = 0, so that F = FS, but it's because we neglect gravity, and not because the flow is steady. And no, FS = FB does not follow, and is not the intended meaning of the last two sentences of the first paragraph of the new section.  These two sentences follow from the fact that the first integral on the RHS of Shapiro's Eq 1.15 is zero for steady flow, and that the entire RHS represents the same quantity as the RHS of the first equation (unnumbered) of section 1.5, i.e. the instantaneous time rate of change of the momentum of the material system that occupies the control volume at time t, commonly referred to as the material derivative.  The two sentences don't "need correcting".  They are consistent with Shapiro, and the "disputed" tags should be removed.


 * The Reynolds transport theorem as described in the linked article is similar to Shapiro's Eq 1.15, but it uses d/dt to represent a more general kind of time derivative, the total time derivative for some quantity contained within the volume as it evolves in time, including the effects of movement of the boundaries of the volume in the general case. The result is formally the same as Shapiro's only for the special case in which the boundaries of the volume move with the flow as described under "Form for a material element".  I think this article is unlikely to help anyone understand Shapiro because it provides less supporting detail than Shapiro does.


 * Steelpillow wrote:


 * "I do think we need to be clear whether each case is allowing a dynamic flow where the net rate of momentum change within the control volume is non-zero, or restricted to a steady-state flow where the net rate of momentum change within the control volume must be zero."


 * Are you saying that The Statement dp/dt = -L can be true only in unsteady flow? That would make no sense.  Actually, all of the sources we list that bear on the dp/dt issue analyze the flow in the frame of the foil and assume the flow is steady in that frame.  But the rate of change of momentum within a control volume that's stationary in that frame can still be non-zero in Shapiro's sense of the material derivative.  And although Chris Waltham and the AAPT don't say so explicitly, dp/dt in their versions of The Statement has to be referring to the material derivative.  Otherwise, if it referred to the conventional partial derivative with respect to time at a fixed location, it would be zero for steady flow, as you say, and The Statement would be false for any control volume.  And even I'm saying that there's one control volume for which The Statement is true.  J Doug McLean (talk) 06:40, 7 January 2015 (UTC)
 * I am certainly not saying that, as you say that would make no sense. Equations applicable to dynamic momentum distribution will presumably differ from the simpler equations required for steady state. Some of the remarks posted or referenced appeared to encompass the dynamic situation, and that concerned me. For example your new material contains this: "The momentum theorem states that the integrated force exerted at the boundaries of the control volume (a surface integral), is equal to the integrated time rate of change (material derivative) of the momentum of fluid parcels passing through the interior of the control volume (a volume integral).[under discussion] For a steady flow, the volume integral can be replaced by the net surface integral of the flux of momentum through the boundary." Grammatically, the addition of the caveat "For a steady flow" to the second sentence would suggest that the first sentence encompasses unsteady, aka dynamic, flow. Since the momentum theorem is not yet properly defined and cited, I cannot judge whether this is so. Some other materials left me similarly concerned. I have no knowledge of such dynamic situations, nor any references on my bookshelf, or I would be able to comment more sensibly. &mdash; Cheers, Steelpillow (Talk) 12:14, 7 January 2015 (UTC)


 * I see my assumption that each term on the LHS of (3.42) as discussed equates to the same term on the RHS doesn't necessarily follow from what is written, but that's how I read it. Anyway, this equation shows the total force is equal to a volume integral plus a surface integral and it requires some calculation to follow your argument. In shorthand (with the integral contents omitted), Shapiro's Eq 1.15 is ∑F = ∂/∂t ∫c.v. + ∮c.s.


 * As you say, In steady flow the first RHS term disappears, so ∑F = ∮c.s.


 * The first equation of 1.5 is a statement of Newton II in the x-direction. In general Newton II is ∑F = d/dt (mV)


 * So d/dt (mV) = ∮c.s.


 * I'm starting to think this wording may not be incorrect, so I have changed the tags. However, Shapiro 1.5 doesn't directly state any of the three equations above.


 * The wording in question is:


 * The idea that either of the relationships derived above for steady flow conditions are "The momentum theorem" is not consistent with any of the sources I have seen.


 * I don't see where Shapiro says that d/dt (mV) is the "material derivative" or that this term involves a volume integral.


 * Your assertion that in the discussed (3.42), "the first term on the RHS is not equal to the first term on the LHS" implies that neither term on the RHS is equal to the corresponding term on the LHS. Yet the quotation above seems to imply the opposite: i.e. the second term on the RHS (a surface integral) is equal to the surface forces, "the integrated force exerted at the boundaries of the control volume" and "the net surface integral of the flux of momentum through the boundary".


 * Anyway, this discussion has already taken up far more of my time than I wish. I am not going to embark on an undergraduate course in Fluid Mechanics just so I can continue to protect this article from dubious and misleading information, as I have done for many months now. Burninthruthesky (talk) 12:09, 7 January 2015 (UTC); last edited 06:54, 9 January 2015 (UTC)


 * Also, this page says,
 * Burninthruthesky (talk) 15:46, 7 January 2015 (UTC)


 * From a reader's perspective, all he needs to know in this context is that in steady flow, momentum changes to air passing through the control volume can be accounted for at its boundaries. I think a simpler form of words would be more understandable. Burninthruthesky (talk) 07:43, 8 January 2015 (UTC)
 * Looking at other sources, as far as I can tell the momentum theorem states that the integrated rate of change of momentum within the control volume equals the sum of the integrated forces acting on the internal volume plus the integrated forces acting on the surface. (e.g. George Emanuel; Analytical Fluid Dynamics, Second Edition, pp447-448 ) That is rather different from the definition currently given in the article. One can add that for a steady state flow where the force acting on the interior is zero, the integrated rate of change of momentum within the control volume equals the integrated forces acting on the surface, like this:
 * I would be happy for this to replace the "wording in question" quoted above. Any objections? &mdash; Cheers, Steelpillow (Talk) 16:35, 9 January 2015 (UTC) [Updated &mdash; Cheers, Steelpillow (Talk) 16:59, 9 January 2015 (UTC)]
 * I would be happy for this to replace the "wording in question" quoted above. Any objections? &mdash; Cheers, Steelpillow (Talk) 16:35, 9 January 2015 (UTC) [Updated &mdash; Cheers, Steelpillow (Talk) 16:59, 9 January 2015 (UTC)]


 * Thanks for suggesting a new wording and for the further reference. All the sources I've seen have presented the momentum theorem in the same form [Shapiro (1.15), Emanuel (14.3)]. I see this equates a force to the sum of volume integral and surface integral components of momentum, but I'm not sure how to express it accurately in words. Burninthruthesky (talk) 07:01, 10 January 2015 (UTC); edited 07:06, 10 January 2015 (UTC)
 * If the suggested wording can be reliably cited, I've no objection to its addition to the article. Burninthruthesky (talk) 11:55, 10 January 2015 (UTC)
 * Specific wording does not need to be cited verbatim (or we would fall foul of plagiarism), however their meaning needs to be clear and that meaning needs to be cited properly. This is what I have tried to do. &mdash; Cheers, Steelpillow (Talk) 12:35, 10 January 2015 (UTC)
 * Without plagiarising, we must somehow present the facts which are directly supported by the sources. Does Emanuel say the Momentum Theorem is:
 * the splitting of momentum into surface and volume components of force (14.1)
 * the splitting of momentum into surface and volume components of momentum (14.2)
 * the splitting a force into surface and volume components of momentum (14.3)
 * something else derived later?
 * I'm pretty certain some of those bullets are not the correct answer, but I'm not sure which is. Burninthruthesky (talk) 13:11, 10 January 2015 (UTC)
 * Emanuel gives the equation of integrals. I paraphrased that equation in words. The LHS is the momentum-change integral, the RHS is the sum of the two force integrals, effectively your 14.1. One could write the equation, which is not plagiarism but equally is not especially helpful unless one explains it in words as well. It would be better included in an article on fluid dynamics in general. &mdash; Cheers, Steelpillow (Talk) 13:31, 10 January 2015 (UTC)
 * I was attempting to describe Emanuel's 14.1, not make my own. I agree the suggested text above is also a verbal description of that equation. My question is whether that equation is in fact "The Momentum Theorem", since Emanuel describes it as "Newton's second law". Burninthruthesky (talk) 14:49, 10 January 2015 (UTC); edited 15:39, 10 January 2015 (UTC)
 * Ah, silly me. My apologies. I'll have to find time to digest Emanuel a bit more carefully. We also have 14.9 to consider. &mdash; Cheers, Steelpillow (Talk) 16:56, 10 January 2015 (UTC)


 * A lot of issues to deal with here. To begin:  Burninthruthesky wrote:


 * "I don't see where Shapiro says that d/dt (mV) is the "material derivative" or that this term involves a volume integral."


 * Good point. He doesn't explicitly use the term "material derivative", though that is the widely accepted term for the kind of time derivative he refers to as d/dt(mV).  He also doesn't explicitly say the term involves a volume integral, but he does define it as "the time rate of change of the total x-momentum of the system", which for non-uniform flow can be quantified only in terms of a volume integral.  Shapiro's notation is confusing in this regard, and Emanuel would be a better source to cite.  The RHS of his eq 14.2 is the same quantity as Shapiro's d/dt(mV).  It is clearly a material derivative (for which the capitalized D/Dt is the common notation), and it is clearly a volume integral.  So I think my wording is correct and well supported by Emanuel, just not well supported by Shapiro.  Thanks for pointing this out.


 * Burninthruthesky wrote:


 * " The idea that either of the relationships derived above for steady flow conditions are "The momentum theorem" is not consistent with any of the sources I have seen. "


 * I don't understand your basis for saying this. Your relationship ∑F = ∮c.s. is identical to Shapiro's eq 1.15, for the case of steady flow, where the first term on the RHS is zero.  And this is the final equation in a section headed "Working Form of Momentum Theorem".


 * Burninthruthesky wrote:


 * "Your assertion that in the discussed (3.42), "the first term on the RHS is not equal to the first term on the LHS" implies that neither term on the RHS is equal to the corresponding term on the LHS. Yet the quotation above seems to imply the opposite: i.e. the second term on the RHS (a surface integral) is equal to the surface forces. "


 * You're right about what is implied, but that doesn't mean it's contradictory. The two second terms are equal only in the special case in which the flow is steady, for which the first term on the RHS is zero, and the body force (the first term on the LHS) is neglected, as it usually is in aerodynamics.  This is the special case used in all of the airfoil analyses.


 * Burninthruthesky wrote:


 * "From a reader's perspective, all he needs to know in this context is that in steady flow, momentum changes to air passing through the control volume can be accounted for at its boundaries. I think a simpler form of words would be more understandable."


 * This is also a good point. I originally included the wording about the volume integral of the material derivative because that's the form of the momentum theorem that relates to the "time rate of change of momentum of the air" in The Statement in the intro section, and I think I had the editing community in mind more than the general reader.  I agree that the simpler form is better for the article, and I've made the change in my sandbox User:J_Doug_McLean/sandbox.


 * Steelpillow proposed replacing "the wording in question" with the following:


 * "The momentum theorem states that the integrated rate of change of momentum within the control volume equals the sum of the integrated forces acting on the internal volume plus the integrated forces acting on the surface.[cite] For a steady state flow where the force acting on the interior is zero, the integrated rate of change of momentum within the control volume simply equals the integrated forces acting at the boundary."


 * There are two reasons this isn't satisfactory:


 * 1) As I explained before, setting the body-force term ("the sum of the integrated forces acting on the internal volume") to zero comes from neglecting gravity and has nothing to do with whether the flow is steady or unsteady.


 * 2) This version doesn't mention the surface-integral form for the momentum term, which is the form used in the airfoil analyses that follow.


 * Burninthruthesky asks which of the equations is actually the "momentum theorem".  I think a reasonable reading of the sources is that the "momentum theorem" can be expressed in several forms and that it can be any of the equations that relate the integrated force to the integrated rate of change of momentum and/or momentum flux.  In Emanuel that would be eq 14.1, 14.3, 14.4, or 14.9.  It is not eq 14.2, 14.5, 14.6, 14.7, or 14.8 because they deal with forces or momentum changes separately, not with the second-law relationship between them.


 * In light of the above discussion I have changed the words in question to the simpler form suggested by Burninthruthesky, and cited both Shapiro and Emanuel. See in my sandbox User:J_Doug_McLean/sandbox.   J Doug McLean (talk) 00:19, 11 January 2015 (UTC)

I take on board much that has been said, and apologise again for looking at the wrong equation. But I am still puzzled by a couple of points in Doug McLean's latest proposal. It says, "For a steady flow, the momentum theorem states that the integrated force exerted at the boundaries of the control volume is equal to the integrated flux of momentum through the boundary." But in explaining his equation 14.3 Emanuel remarks, "the left side represents the sum of all the applied forces that act on the fluid inside the control volume." and later in explaining 14.9, "the force is therefore provided by a volumetric integral plus an integral over a stationary surface." i.e. Emanuel's generic analysis of fluid flows does not support two aspects of the new version: I am not saying that Doug's analysis is necessarily wrong, only that Emanuel does not appear to support it fully. Does Shapiro? &mdash; Cheers, Steelpillow (Talk) 13:28, 11 January 2015 (UTC)
 * Emanuel describes the force as acting within the control volume, McLean as acting at its boundary.
 * Emanuel does not observe that in air, gravity may be ignored and the momentum volume integral is zero.


 * That is a good question. There is also the question of whether it is correct to refer to a degenerate case of Shapiro's eq 1.15, with the first RHS term set to zero for steady flow, as "the momentum theorem". By way of comparison, if one described a relationship derived from Newton's second law where F = 0, it would look very much like Newton's first law.


 * If I was mistaken to say, 'The idea that either of the relationships derived above for steady flow conditions are "The momentum theorem" is not consistent with any of the sources I have seen', I ask J Doug McLean to specify where the evidence is. Failing that, he may wish to clarify or strike out his comment. Burninthruthesky (talk) 09:37, 13 January 2015 (UTC)


 * After looking again, I don't see anything in Shapiro 1.5 which says that the first RHS term of Eq 1.15 is zero in steady flow, although I've no doubt it's true. I don't see anything saying gravity may be excluded from the sum of body forces. In fact the paragraph headed BODY FORCES includes "gravitational attraction" in the description. I don't see Shapiro make any attempt to combine those terms in a single equation as Emanuel (14.4) does. I stand by the comment I quoted. Burninthruthesky (talk) 16:37, 13 January 2015 (UTC)


 * You're both questioning whether my summary of the momentum theorem is correct and uses the term correctly, and whether it is fully supported. First, let's look at the apparent contradiction between me and Emanuel, as raised by Steelpillow:


 * "Emanuel describes the force as acting within the control volume, McLean as acting at its boundary."


 * There's actually no contradiction here. Note that Emanuel defines the RHS of eq 14.1 as "the vector sum of the applied forces that act on the system, which here is the fluid inside V."  The force acting on the boundary (the second integral on the RHS) is part of that sum, along with the body force acting throughout the interior.  So Emanuel considers the force acting at the boundary to be part of the total force acting "on the fluid inside V".  Then if you neglect gravity, the first integral on the RHS (the volume integral) is zero, and  only the second integral (the force acting at the boundary) remains.  So my description is identical to his when gravity is neglected.


 * True, Emanuel doesn't say that gravity can be neglected, and neither does Shapiro's section 1.5, but gravity is neglected in every application of the momentum theorem to aerodynamics that I've ever seen, including the ones I cite in the article. Clancy's firehose analysis also neglects gravity.  And in the rest of Shapiro's book, when he uses Newton's second law, he leaves the gravity force out without explaining why.  So does every other aerodynamics book on my shelf, except Batchelor.  On p. 176, in a section headed "Modification of the pressure to allow for the effect of the body force", Batchelor shows that gravity does not affect the dynamics of a constant-density flow because the body force (the weight of a fluid parcel) is canceled by a background gravitational-hydrostatic pressure gradient that always accompanies it.  Thus neglecting both the body-force term and the gravitational-hydrostatic part of the pressure gradient is rigorously justified for constant-density flow.


 * So neglecting the effect of gravity on the flow is something everyone does in aerodynamics applications, and most take it for granted. In my quick check of the sources, Batchelor is the only one who takes the time to justify it rigorously.  Neither Shapiro nor Emanuel mentions the simplified form of the momentum theorem with the gravity terms omitted, so I see now that it's misleading for me to cite them when I state the simplified form.  I'll propose a fix for this problem below.


 * Now to the matter of steady flow. Burninthruthesky wrote:


 * "I don't see anything in Shapiro 1.5 which says that the first RHS term of Eq 1.15 is zero in steady flow".


 * True, he doesn't say it in 1.5. He defines steady flow in 1.2 as "a condition where at each point in space there is no variation of any property with respect to time", and writes it as the partial derivative of any property with respect to time equals zero, using density as an example.  So I'd say that Eq 1.15 with the first RHS term set to zero for the case of steady flow is fully supported by Shapiro.


 * On this same topic Burninthruthesky also wrote:


 * "If I was mistaken to say, 'The idea that either of the relationships derived above for steady flow conditions are "The momentum theorem" is not consistent with any of the sources I have seen', I ask J Doug McLean to specify where the evidence is."


 * I said that I didn't understand your basis for saying this because it seemed to me at the time, and it still does, that your relationship ∑F = ∮c.s. is identical to Shapiro's eq 1.15, for the case of steady flow, and is thus consistent with Shapiro. This version of the equation requires only steady flow; it doesn't require neglecting gravity.  So I think that in addition to being consistent with Shapiro, it is fully supported, though as I point out above, you have to look in section 1.2 in addition to 1.5.  The short answer to your question is that the evidence regarding the unsteady flow term is in Shapiro.


 * Whether it is correct to refer to it as "the momentum theorem" is the next question. Burninthruthesky wrote:


 * "There is also the question of whether it is correct to refer to a degenerate case of Shapiro's eq 1.15, with the first RHS term set to zero for steady flow, as "the momentum theorem". By way of comparison, if one described a relationship derived from Newton's second law where F = 0, it would look very much like Newton's first law."


 * In general I think it's fair to use the name of a theorem or law to refer to a special case, provided you also make it clear what special case you're talking about. To use your example, I see nothing wrong with saying "For F = 0, Newton's second law states that dp/dt = ma = 0".  Yes, it looks like the first law, but nothing logically forbids having a special case of one law look like another law.  And in the case of the "momentum theorem", the special cases don't have their own separate names anyway, as far as I know.  So I'd say that special cases of the momentum theorem are still examples of "the momentum theorem".


 * A proposed fix: I'm confident that my summary of the momentum theorem is correct for the case of steady flow when gravity is neglected.  And it's well supported, except for the problem, as I discussed above, that you need to look beyond Shapiro and Emanuel, e.g. to Batchelor, to find formal justification for neglecting gravity.    That said, however, the fact is that the sources I'm citing for the momentum analyses all use the form of the momentum theorem in which gravity is neglected.  So I'm not really on the hook to justify it; I really only have to observe that that's the form they use.  I've moved the Shapiro and Emanuel citations and changed the wording to reflect this in my sandbox User:J_Doug_McLean/sandbox.  I think this addresses the issues you've raised here.  Thank you for raising some interesting questions.   I also propose simplifying the introductory discussion of control volumes by eliminating the mention of the momentum volume integral and the material derivative (first paragraph), and providing further clarification of the lifting flow results (new fourth paragraph).  J Doug McLean (talk) 20:46, 23 January 2015 (UTC)

The Statement and Newtonian theories of lift
Regarding the proposed deletion of The Statement, I see nothing in Wikipedia policy that says that something "must" remain in an article just because it's already there and has a citable source. Nor do I see anything that says that removal of something that has a citable source requires a direct and citable refutation. No, it looks to me like we as editors are free to make changes based on weighing the sources available to us, even if those sources don't explicitly refer to each other. So it appears to me that Steelpillow's contention that The Statement "must stay" is unfounded. If I'm wrong, show me the specific policy wording.

When Steelpillow insists that "mainstream sources" support The Statement, he's being unduly selective. Yes, some of the sources support The Statement in unapologetic form, but all of the sources taken together, on balance and weighted according to the quality of their analyses, clearly show that The Statement is true only with qualifications. If we present The Statement without the qualifications, we're presenting a biased and misleading impression of what sources on this topic actually say.

The only published analyses we have that support The Statement in unapologetic form are based on the "firehose" model for the flow (Chris Waltham and Clancy). In my posts of 17and 19 December I presented detailed arguments as to why these sources should be accorded less weight than those that use the classical model based on uniform flow plus a vortex. I invited specific counterarguments and so far there have been none.

Likewise, Mr. Swordfish invited Burninthruthesky and Steelpillow to elaborate on what they think "the air deflected downward" means. Only Steelpillow responded, and his answer amounted to "It means whatever it has to mean to make The Statement true, depending on what flow model you prefer". That's an unsatisfactory answer in general, but especially when one of the models we're considering (the "firehose") assigns an unrealistic spatial distribution to dp/dt, as Clancy himself admits.

This is not a question of a choice between a "momentum model" and a "pressure model", as a matter of style or personal taste. No, a proper application of the momentum theorem requires that all of the forces exerted on the air be taken into account, including the pressure force. The "firehose" analyses ignore the pressure force, while the classical analyses properly include it. Thus in terms of the basic physics, these two types of analysis are not of equivalent quality. Steelpillow suggests that the new section on Newtonian models "needs a balancing critique of the other simple model we introduce, viz. the pressure model." No, there is no "balancing" of that kind to be done. The classical model is not just a "pressure model". It accounts for both momentum and pressure, and assigns them their realistic locations in the field. It has no faults that rise to the same level as the faults of the "firehose" model. If you disagree with this assessment, please state your specific counterarguments.

The classical analyses (Durand, Batchelor, and Lissaman) clearly deserve more weight than the "firehose" analyses. And they clearly show that The Statement in unapologetic form is misleading. Thus it should either be deleted or properly qualified. This is not "a view unsubstantiated by reliable sources". It is a conclusion supported by an appropriately weighted and balanced consideration of all of the sources. Nor does it constitute synthesis, though I would also argue that the (WP:SYNTH) policy doesn't apply to a decision to omit something.

I agree with shortening the heading to "Integrated momentum balance in lifting flows". Regarding the overall organization of "Mathematical theories of lift", I think it's best just as it is, starting with the basic principles, followed by the predictive theories (the ones that actually predict lift starting with the airfoil shape). Those that merely relate one thing to another but don't make actual predictions, such as the Kutta-Joukowski theorem, the momentum-balance analyses, and the Newtonian theories, are not theories at the same level as the predictive theories and should not precede them. This is an ordering that befits an encyclopedia article as opposed to a textbook.

The "Newtonian theories of lift" belongs where I proposed putting it, at the end of Mathematical theories....". None of it should be moved to the introductory section.

J Doug McLean (talk) 06:40, 7 January 2015 (UTC)


 * Doubters of Wikipedia's policies and guidelines may find it useful to read WP:REMOVAL and, on the matter of neutrality, WP:NPOV. &mdash; Cheers, Steelpillow (Talk) 12:14, 7 January 2015 (UTC)


 * Re-reading the proposed section on Newtonian theories of lift, it is really two unconnected parts. The first is a summary of early theories and would go better as the start of a "Historical development" of theories of lift, the second is a critique of the firehose model and while that might be a useful topic to work in somewhere, frankly I find the treatment presented to be unbalanced. Neither part is fit to be moved into the present article as it stands, nor do I intend to engage with the author on their improvement. &mdash; Cheers, Steelpillow (Talk) 12:47, 7 January 2015 (UTC)


 * Your feedback that you don't see the connection between the two parts of "Newtonian theories...." is useful. Thank you.  I have revised the second part to make the connection clearer.  See User:J_Doug_McLean/sandbox.  Beyond that, I have already explained why I struck the balance I did in my criticism of the firehose model.  If you have specific suggestions for improving the balance, let's hear them.  Simply calling the treatment "unbalanced" isn't helpful.


 * You are the only one who has expressed opposition to including "Newtonian theories...." in the article, so at this point yours would seem to be a minority view. Does anyone else oppose my adding this new subsection?  J Doug McLean (talk) 00:26, 11 January 2015 (UTC)
 * Yes. Burninthruthesky (talk) 08:08, 12 January 2015 (UTC)
 * To explain the message a little more clearly. The first part is not current but historical. It has no place in an exposition of current theory. The second part discusses a model still to be found in the text books. No matter how one wraps words around them or applies other creative fixes, for encyclopedic purposes they are not linked. Rather, the flavour of your discussion - and of your response to my considered rejection - makes it clear that your purpose is to push your PoV that the Newtonian approach is flawed in comparison to your favoured approach, and that is your motivation for linking them. That PoV is not backed up by the sources. "Verifiability, not truth" is an old watchword of Wikipedia and no amount of yardage in filling this talk page is going to change that. I have absolutely no interest in conrtibuting any more than necessary to that yardage, in order to terminate it. If you wish for wider consultation on WP:VERIFICATION and its application here, then there are plenty of avenues to follow. Flogging a dead horse here is not one of them. Poor User:Burninthruthesky has suffered enough and I too have a life. If this endless charade persists I shall take it higher to seek a topic ban on this talk page. I trust that all is now clear. &mdash; Cheers, Steelpillow (Talk) 10:47, 12 January 2015 (UTC)


 * Steelpillow, I do not think this hostile tone and threats is conducive to building consensus.  I understand the frustration with a topic that has gone on for too long without resolution, but it is imperative that we keep cool heads and refrain from impugning the motives of fellow editors. Threatening a topic ban over a good-faith disagreement over emphasis is over the top. Please consider retracting those comments.


 * Regarding the section on "Newtonian theories..." My take is that it's not the place of this article to go into every historical failure. It is true that Newton tried to calculate lift using a flawed model and produced inaccurate results.  Unfortunately, some authors use this to confuse things - Newton's (flawed) model and using Newton's laws to explain lift are two distinct things with similar names.  If we are going to address Newton's (flawed) model we should be clear to make that distinction, and my take is that the best place for it would be to take a page from  NASA and treat it alongside the Equal Transit Time (see http://www.grc.nasa.gov/WWW/k-12/airplane/wrong2.html ) under the heading Alternative explanations, misconceptions, and controversies.


 * Although we shouldn't go into every historical failure, my feeling is that the ETT is sufficiently widespread that we would be remiss in not including it in the article. I'm not so sure about the "Skipping Stone" explanation, so I can go either way on whether we should include such a subsection.


 * Regarding the second paragraph, I've managed to borrow on a copy of Clancy and have read the short section 5.15 on page 76 introducing what is sometimes called the "firehose model". To my eyes, it just looks like an example of a spherical cow analysis. I don't think it is sufficiently noteworthy that we need to treat it in the article. Mr. Swordfish (talk) 19:18, 12 January 2015 (UTC)


 * My thoughts and feelings on the matter are very similar to Steelpillow and I thank him again for his support. Given the level of activity in this discussion, if there were any objections to his decision to omit that content from the article, there has been ample opportunity for them to be voiced.


 * I agree with Mr. Swordfish's comment that going into every historical failure is out of place in the article. It is very similar to what I wrote but decided not to post because I'm tired of repeatedly defending practically everything I have written since I joined this discussion in August. Burninthruthesky (talk) 07:49, 13 January 2015 (UTC)


 * I've said it before, but I would again like to provide a pointer to WP:LISTEN. Thank you. Burninthruthesky (talk) 11:04, 13 January 2015 (UTC)


 * In reply to Mr Swordfish, it is not in my power to threaten sanctions, only - as I said - to seek them. The disruption and distress are real and they need to be dealt with. I stand by all that I said. &mdash; Cheers, Steelpillow (Talk) 13:36, 13 January 2015 (UTC)


 * Regarding the firehose model, my "PoV", as Steelpillow calls it, is supported by the sources, but that's perhaps beside the point. Mr. Swordfish has a good point that the firehose model is an example of a "spherical cow" analysis, and as such isn't sufficiently noteworthy for inclusion in the article.  So I withdraw the second paragraph of my proposed subsection on "Newtonian theories....".


 * And I propose renaming the subsection "Newton's theory of lift" and moving it to "Alternative explanations, misconceptions, and controversies", after "Misconception regarding 'pulling down' of the flow". See User:J_Doug_McLean/sandbox.  Newton's theory isn't as prominent these days as EET, but it was quite influential in the early discussions of the practicality of heavier-than-air flight, and so I think is noteworthy for that reason.  J Doug McLean (talk) 23:21, 13 January 2015 (UTC)

Proposed re-draft of "Flow deflection and Newton's laws"
Doug McLean recently proposed a re-draft of this section, however it was not met with much support. (personally, I'd support it were there consensus to go in that direction) I've created another draft that will hopefully move us in the direction towards consensus: https://en.wikipedia.org/wiki/User:Mr_swordfish/Lift#Flow_deflection_and_Newton.27s_laws

The main ideas are:


 * I think we all agree that there is a region of air for which dp/dt = -L. The draft uses a instead of the; by using the indefinite article I hope to avoid the confusion that may result in referring to "the air" without specifying what is meant by "the air".
 * I've added a cite which I believe is sufficient to support the language that is in the draft:
 * "...if the air is to produce an upward force on the wing, the wing must produce a downward force on the air. Because under these circumstances air cannot sustain a force, it is deflected, or accelerated, downward. Newton's second law gives us the means for quantifying the lift force: Flift = m∆v/∆t = ∆(mv)/∆t. The lift force is equal to the time rate of change of momentum of the air." Norman F. Smith "Bernoulli and Newton in Fluid Mechanics" The Physics Teacher 10, 451 (1972); doi: 10.1119/1.2352317 http://dx.doi.org/10.1119/1.2352317


 * I've moved some of the material around so that the first paragraph deals with the third law and the second paragraph deals with the second law. That seems like a logical organization.
 * Some other minor tweaks such as referring to the 'air flow above the wing' rather than the 'air that follows the upper surface'

I invite the other editors to comment. I hope that I have crafted language that is technically correct, explains the momentum transfer idea as recommended by the AAPT, and is understandable by the lay reader. Suggestions for improvement cheerfully accepted. Mr. Swordfish (talk) 20:44, 7 January 2015 (UTC)


 * I did edit the article section to include some of Doug McLean's minor suggestions and, from that perspective, I am happy with it as it is. I do not think that Mr. Swordfish's version changes the substance in any way. No version can please everybody, so I have no strong opinion as to whether we stick with what we've got or go for Mr. Swordfish. Either way, the new citation is useful. &mdash; Cheers, Steelpillow (Talk) 21:36, 7 January 2015 (UTC)


 * I appreciate the work Mr. Swordfish has done here to try to bring us together. But although this proposal is less bad than the current version, it doesn't fix the problem.  True, "a volume of air" isn't as likely to be misinterpreted as referring to the atmosphere as a whole, but it still begs the question:  What volume of air does it refer to?


 * Couching The Statement in terms of "a volume of air" instead of "the air" or "the air deflected downward" changes the substance a bit, but not much. And it still gives prominence to the quote from Cliff Swartz, which makes The Statement in its most unapologetic form, a statement that I've said all along and Mr. Swordfish has recently argued is problematic.


 * We've been through this over and over, and I thought that recently Mr. Swordfish had come to agree with me on this: The only assumption for which authoritative published analyses have shown The Statement dp/dt = -L to be true, i.e. the tall sliver control volume, is so specific that making The Statement without spelling out the assumption is misleading.  The proposed new wording doesn't get us past this problem.


 * If we're going to make the quantitative statement dp/dt = -L, we must spell out what body of air it's true for. The wording wouldn't have to be overly technical or mathematical, but it would have to be specific.  Here's my idea of the minimum that would be required to replace Mr. Swordfish's second paragraph and to be technically correct:


 * In accordance with Newton's second law, some of the air surrounding the airfoil is accelerated downward. The rate at which downward momentum is imparted to the air can be calculated, but it depends on what portion of the air is included in the accounting.  Depending on the shape of the region considered, pressure differences acting on the outer boundary of the region offset some of the downward force exerted on the air by the airfoil, reducing the rate at which momentum is imparted, as described below under "Momentum balance in lifting flows".  For a column of air that extends to large distances above and below the airfoil and is relatively narrow, the pressure differences are not a factor, and the rate at which downward momentum is imparted is equal to the lift.[Lissaman]


 * This gets the momentum-transfer message across in a more accurate way, and it isn't that much longer than the current version or the proposed draft. It cites a different set of sources (Lissaman, Durand, Batchelor) from the proposed draft (Swartz, Clancy, Smith), but I've already made detailed arguments as to why that reflects an appropriate weighting.  I would still prefer to keep this section qualitative, an option that both Mr. Swordfish and Burninthruthesky have said they could live with.  But if we are to make a quantitative Statement, I'd urge you to consider a version with a technically correct qualification, as suggested above.


 * J Doug McLean (talk) 01:14, 11 January 2015 (UTC)


 * No, for reasons given here. Also, may I remind folks that WP:CONSENSUS is not just about a simple head count. &mdash; Cheers, Steelpillow (Talk) 11:10, 12 January 2015 (UTC)


 * Doug, I disagree that we must spell out what body of air it's true for at least in the introductory section.  I believe it is possible to be intentionally vague without prompting the question of "but which body of air?" in most reader's minds.  Something along the lines of:


 * In accordance with Newton's second law F=ma, some of the air surrounding the airfoil is accelerated downward. The rate at which downward momentum is imparted to a portion of the air is equal to the lift. (See "Momentum balance in lifting flows" for details)


 * It's not a question of length - the issue is whether or not it is a distraction for the reader and I think a detailed description of control volume analysis at this stage of the article would be a distraction for most readers. The more detailed it is, the more likely we are to lose our audience. I'd prefer to keep the intro sections simple. Those who want more details can follow the link. Mr. Swordfish (talk) 20:35, 12 January 2015 (UTC)
 * Mr Swordfish's suggestion is a good one. I like ... some of the air surrounding the airfoil is accelerated downward. The rate at which downward momentum is imparted to a portion of the air is equal to the lift.  Dolphin  ( t ) 00:35, 13 January 2015 (UTC)


 * Thank you for your support. I have integrated that language into the draft in my user space: https://en.wikipedia.org/wiki/User:Mr_swordfish/Lift#Flow_deflection_and_Newton.27s_laws . At this point both Dolphin and I are in favor of this language, Doug still doesn't like it but thinks it's an improvement over the current version, Steelpillow is neutral.  That may be as close to consensus as we are going to get at this point.  Absent any objections in the next 24 hours I'll make the change. I do not expect that this edit will be "permanent", but hopefully it will be an incremental step towards consensus. Mr. Swordfish (talk) 16:36, 13 January 2015 (UTC)


 * I think the further change in words and referring the reader to "Momentum balance in lifting flows" are a big step toward consensus, and I'd say we're almost there.


 * I'd like to suggest simplifying it by leaving out the "its mass times its acceleration" step and going straight to the imparting of downward momentum. My reason for favoring dp/dt over ma is that The Statement is rigorously shown to be true only for the infinitely tall sliver, for which m is infinite but dp/dt is finite.  So I'd suggest something like:


 * In accordance with Newton's second law, i.e. that a force produces a rate of change of momentum, some of the air surrounding the airfoil has downward momentum imparted to it at a rate equal to the lift. (See "Momentum balance in lifting flows" for details)


 * I'd also like to point out that the analysis in Lissaman's paper completely supports this language and that I'd prefer to cite him here (instead of Swartz, Clancy, and Smith) because his analysis is the more rigorous. J Doug McLean (talk) 22:45, 13 January 2015 (UTC)
 * Inserting "i.e." in the middle of a sentence, followed by words that properly belong in a separate sentence is done frequently but it makes for poor English expression. Also, beginning with "In accordance with Newton's second law" almost implies that the air surrounding the airfoil is being obedient to Sir Isaac Newton when in fact air has been diverted by airfoils for millennia prior to Newton's birth. To eliminate these problems, I suggest the following restructuring of Doug's proposed text:
 * Some of the air surrounding the airfoil has downward momentum imparted to it at a rate equal to the lift. This is consistent with Newton's second law of motion which states that the rate of change of momentum is equal to the resultant force.
 * Dolphin ( t ) 23:24, 13 January 2015 (UTC)
 * I've incorporated these suggestions and added a cite to Lissaman in the draft article. (I don't see why the cites should be either/or - we can cite them all) I'll give it another day for comment before making it live. Mr. Swordfish (talk) 15:50, 14 January 2015 (UTC)


 * I'm sorry to ask, but does this really clarify the text, when read in context? Currently the second paragraph starts:


 * The proposed version moves some of those words to the previous paragraph, and begins the relevant paragraph "some of the air".




 * Even though the implication we are talking about a tall sliver of air will be invisible to nearly all readers, I can't help wondering whether that implication is clearer in the existing version. To my mind "passing" carries that implication more strongly than "surrounding". Is it just me? Burninthruthesky (talk) 16:17, 14 January 2015 (UTC)


 * Agree that "passing" is a better word to describe the dynamic nature of the interaction than "surrounding", and using the same verb in both paragraphs helps readability. I've updated the draft.


 * The motivation for moving the sentence was to have the first paragraph address Newton's third law and have the next paragraph address the second law.  That just seemed like a useful organizing principle, but I'm not wed to it.  To my eyes, both versions achieve roughly the same level of clarity, with the proposed version avoiding some of the issues raised here on the talk page. Mr. Swordfish (talk) 16:48, 14 January 2015 (UTC)


 * Thanks, I think that helps. Congratulations on finding a solution to what seemed an intractable problem. Burninthruthesky (talk) 17:10, 14 January 2015 (UTC)
 * indeed &mdash; Cheers, Steelpillow (Talk) 17:18, 14 January 2015 (UTC)


 * To me, this reordering of the sentences raises a clarity issue. The link "(See 'Momentum balance in lifting flows' for details)" now seems to me to be a bit disconnected from the sentence about dp/dt in the flow, which is what I think was the main thing it was meant to accompany.  I think it would be clearer if the citations and "(See 'Momentum balance in lifting flows' for details)" came immediately after the first sentence.


 * That's how it would work best for me. But in any case, it's a big improvement over the current version.  J Doug McLean (talk) 07:51, 15 January 2015 (UTC)


 * I've implemented this suggestion and will make the changes live. Thanks to everyone for their patience in this lengthy consensus-building process.  Mr. Swordfish (talk) 16:24, 15 January 2015 (UTC)

Why "The Statement, L = -dP/dt" is Throwing The Baby Out With The Bathwater.
(Zapletal writes ->). The main article, as at February 2015, under "Flow deflection and Newton's laws", still has "The Statement" ALIVE AND WELL. This is very disappointing.

As argued endlessly by Doug and others on these Talk pages, "TS" is VERY MISLEADING. Its prominence near the top of the article will drive yet another nail into the coffin of society's understanding of Fluid Dynamic Lift. Organisations such as the AAPT, and authors like Clancy and his "firehose model" (well described by Mr Swordfish as a "spherical cow" analysis) have already done enough damage. It is morally irresponsible for the editors of this widely read Wiki article to continue this degradation of society's knowledge base.

As should be obvious from the above paragraph, I am adding these notes to support Doug's stance. I am also addressing this mainly to Mr Swordfish and other readers who have the necessary mathematical skills to understand the Mechanical principles involved here. I note that one of the two pro-TS editors, "Burninthruthesky", has earlier said '' "I am not going to embark on an undergraduate course in Fluid Mechanics just so I can continue to protect this article from dubious and misleading information ..." ''. I strongly suggest that both the pro-TS editors DO, in fact, learn the Mechanics of FDL before imposing their currently limited understanding of this issue on the rest of society.

~o0o~

ENERGY CONSIDERATIONS - TS (ie. L = -dP/dt) has been extensively discussed here from the perspective of MOMENTUM in the fluid domain. To very briefly recap, the time-rate-of-change of the total momentum of a fluid domain in which is immersed a steadily moving Lifting Body, is ALWAYS ZERO.

This, of course, has been discussed ad nauseam on these pages. So, more recently, the validity of TS has been reduced to the question of "Which parts of the fluid should we focus on, to ensure, a priori, that L = -dP/dt?". This PoV currently appears in the main article as;

'' "SOME of the air passing the airfoil has downward momentum IMPARTED to it at a rate equal to the lift. ... consistent with Newton's second law ..." '' (My added emphasis.)

The hugely misleading aspect of this PoV is that it very strongly suggests that there is a CONSTANT ONGOING PROCESS whereby the aerofoil is intercepting new, "virgin", fluid, and then "IMPARTING" momentum to the fluid, namely throwing it downward, so as to develop the requisite Lift.

A misleading argument that goes with this PoV then suggests that a side-effect of this fundamental mechanism of "NII creates Lift", is that the now downwards moving fluid pushes other fluid out of the way, with some of the fluid ultimately moving upwards, for the unfortunate net result of zero dP/dt. To the pro-TS-ers, this final result of zero dP/dt is merely a sort of computational glitch, and NOT a genuine refutation of what they see (ideologically) as the "one true source of Lift", namely NII.

''' But what does this TS-PoV infer about the change in KINETIC ENERGY of the fluid domain? '''

Without being too rigorous (I will assume Mr Swordfish, et al, "get it"), because KE is a scalar, the more fluid that is set in motion by the ongoing "L = -dP/dt" mechanism, then the greater the total KE there will be in the whole fluid domain. So the fluid domain's total KE should be constantly increasing.

To repeat this for clarity, the above TS-PoV says that in order to generate Lift, the aerofoil has to constantly IMPART downward momentum (namely "m.V") to newly intercepted particles of fluid. It follows that because these massive fluid particles have now acquired finite velocity, they MUST also have acquired additional finite Kinetic Energy.

Put yet another way, the TS-PoV says that the longer the aerofoil is "in flight" through the fluid domain, then the greater the amount of fluid that it "stirs up" by the L = -dP/dt process, and the greater the measurable total KE in the whole fluid domian.

(Technical Note: For planar flow (ie. finite span wing between infinite walls) of an aerofoil+bound-vortex moving through an inertially stationary but UNBOUNDED fluid domain, the total KE is infinite, which can make calculation of "additional KE" difficult. But when a GROUND-PLANE is included (equivalent to a mirror vortex), the total KE in the flow becomes finite and easily calculable. Lanchester gives a particulary elegant explanation of this.)

~o0o~

THE BABY IN THE BATHWATER - The essence of FDL is the "Circulation Theory of Lift" (hereafter CToL, aka the "bound-vortex theory..."), developed by Lanchester, Kutta, Zhoukowski, Prandl, et al. The essence of CToL is that it requires NO WORK, OR ENERGY, to sustain it.

This is in COMPLETE CONTRADICTION to any reasonable interpretation of TS, as explained in the above section. TS demands that there MUST ALWAYS BE some fluid that is being given NEW momentum, or increased-velocity-per-massive-particle. Thus any reasonable interpretation of TS has the inevitable consequence that there MUST ALSO BE a time-rate-increase of total KE in the fluid domain.

''' Thus, any assertion that TS is true, is an assertion that the Circulation Theory of Lift is wrong. ''' So putting TS at the top of the article, starts-off the article by "throwing the baby (= CToL) out with the bathwater". This is, of course, a disgraceful thing to do.

Note that the purest version of CToL applies to planar flows of a (3-D) "wing between walls". It is only with this model that there is zero work required to sustain the lift (see also notes below). "Finite span wings" (ie. 3-D, but without the end-walls) DO leave stirred-up wakes behind them (ie. the wing-tip vortexes), so they do continously increase the total KE in the fluid domain. But it is well known, ever since Prandtl provided the quantitative maths in early 1900s, that increasing the wingspan for a given amount of Lift reduces the amount of this KE in the wake. In principle, this KE lost to the fluid can be reduced indefinitely by increasing span. Or, more simply, the KE can be reduced to zero by flying a real, 3-D, finite span wing between real end-walls. And, yes, there is in common fluids (but not "superfluids") always a small ongoing creation of KE that is a side-effect of viscosity.

But, and VERY IMPORTANTLY, none of the above KE increases of the fluid domain are a fundamental, or intrinisic, part of CToL. In CToL, once the aerofoil is moving at a steady velocity wrt the bulk fluid domain, there is NO CHANGE IN KINETIC ENERGY of the fluid domain, EVER.

~o0o~

SOME TECHNICAL NOTES - For reference, and in anticipation of objections.

1. When any Body is stationary wrt an inertially stationary fluid domain, the total KE in the fluid can be taken, arbitrarily, to be zero. If the Body starts to move through the fluid (ie. if it "accelerates"), then it must necessarily set some of the fluid in motion. This motion of the massive fluid implies an increase in KE of the total fluid domain, because KE is always positive (ie. 1/2.m.V-SQUARED). This increase of KE is a direct consequence of the WORK done by the Body as it first starts to push the fluid out of its way. That is, the Body must exert on the fluid a finite forwards force, which acts over a finite forwards distance (ie. F dot-product D).

Importantly, the equal-and-opposite (NIII) finite Drag experienced by the accelerating Body is felt even if the fluid is "Ideal" or "perfectly frictionless". All this was explained by Stokes and others in middle 1800s. The standard method of calculating this unsteady-flow Drag force is by equating it (dot-D) to the increase in KE of the fluid. In Potential Flow Theory these unsteady forces are calculated via the "dPhi/dt" term. In simple planar flows of such Ideal fluids (ie. not considering unsteady effects such as vortex shedding), whenever there is NO acceleration of the Body, there is NO net Drag force between Body and fluid (ie. "d'Alembert's Paradox"). So NO increase in KE of the fluid.

But there can most certainly be a net Lift force, because the dot-product of F.D ensures that NO work is done by this force, and so again NO increase of KE of the fluid.

2. When an asymmetric Body such as an aerofoil first starts to move through an unbounded fluid domain, which may be of Ideal fluid, it experiences both a Drag force as above (to get the fluid moving), and also a Lift force. Due to the inertia of the fluid, and other details (eg. see "Coffee-spoon" paper by Klein, 1910) a "Starting-Vortex" might be shed from the rear of the Body. This SV leaves a necessarily equal-and-opposite Bound-Vortex flow, or Circulation, around the Body, which henceforth generates Lift on the Body via CToL.

If the now moving Body settles down to a constant velocity, but the SV is still close to it, then the Body feels a positive Drag, so must do positive work to maintain its speed. Thus the Body continually increases the total KE of the fluid while it is in close proximity to the SV. A simplified quantitative analysis of this was given by Wagner in 1925. But this Drag force and the accompanying rate of increase of KE in the fluid diminishes asymptotically to zero as the Body moves away from the SV.

Importantly, in the big-picture of things, this SV/Wagner-Drag can be ignored as it is, again, NOT an essential feature of CToL. As someone said back on page 1 of these Talk pages, the gist if CToL is that the aerofoil is "always flying in ground-effect", so NO DRAG.

3. The Kinematics of a planar fluid domain that has a Lifting Body (eg. aerofoil) moving through it are interesting. Although this is more often described in a reference-frame fixed to the Body (because it makes the problem appear to be one of "steady flow"), it is more instructive to use a reference-frame fixed to the bulk, inertially stationary, fluid. Here the velocity field is that of a "potential vortex" (V~1/R) that moves with the Body, together with the localised "perturbation" velocity field that accounts for the fluid getting past the specific shape of the Body (this field is roughly that of a "doublet" that also moves with the Body).

The fluid a long way in front of the Body has asymptotically small upwards motion. That is, it is starting to move (ie. accelerating) upwards, but at a much lower rate the further in front of the Body. The fluid a long way behind the Body has asymptotically small downwards motion. So it is decelerating downwards and effectively coming to a halt a long way behind the Body. Similarly, the flow above the Body is moving rearwards, and the flow under the Body is moving forwards, but both asymptotically less at greater distances.

These motions of the massive fluid particles are CAUSED by the pressure field that travels UNCHANGED with the Body. The isobars of this pressure field are a series of circles tangent to the Body and with centres on a vertical line through the Body (assuming horizontal flight, and with localised distortions of the field near the Body). Highest pressure directly under the Body, lowest pressure directly above.

Importantly, the net result of all this is that NO Kinetic Energy is added to the fluid domain over time, as noted before. The above pressure field IMPARTS momentum and KE to the fluid particles in front of the Body. But then these very same fluid particles RETURN ALL THEIR MOMENTUM AND KE to the pressure field when they are behind the Body.

This is a very WAVE-LIKE process, as noted by Lanchester, with a finite and constant amount of KE propagating as an "entity" through the otherwise quiescent fluid domain. This wave-like nature is strikingly obvious when these motions are seen in this fluid-fixed reference frame.

However, it is interesting to note that NOT all is left unchanged after the Body passes by. Fluid particles that happen to pass above the Body finally come to rest rearwards of their initial starting points. Particles that pass below the Body are moved forwards, in the direction of the Body's motion. The resulting "Surface of Discontinuity" of fluid particle POSITIONS extends from the rear of the Body all the way back to the SV, where it curls up in a spiral to form the SV's core. But, as before, this transportation of the particles gives NO NET CHANGE OF MOMENTUM OR KINETIC ENERGY to the fluid domain.

4. Newton's calculation of Lift in his "Newtonian Medium" was NOT a "historical failure". It is perfectly VALID, and gives correct results, as long as the fluid matches Newton's assumption of "non-interacting particles". So, as long as the mean-free-path of the (gaseous) fluid's particles is in excess of a typical dimension of the aerofoil, such as the chord, then the model works just fine. This is typically the case in the upper atmosphere, or where there is very low air density, or with very small Bodies.

(End Zapletal)101.170.85.70 (talk) 04:24, 6 February 2015 (UTC)


 * Your post betrays an utter ignorance of Wikipedia's policies and guidelines. Wikipiedia is simply the wrong place for you to promote your message. For example our policies on verifiability and balance requires us to document well-sourced material, however much you might personally feel that spherical cows are sapping the intellectual fiber of a generation. You are utterly wasting your time repeating your endless message here. Also, your perjorative tone and use of block capitals, aka "shouting", do not fit well with our policy on civility. There is a whole wide world of web sites out there, your best bet is to find one more suited to your purposes. &mdash; Cheers, Steelpillow (Talk) 11:44, 6 February 2015 (UTC)

(Zapletal Writes ->) Steelpillow, regarding "verifiability", all that I have said above can be verified from "reliable sources". I briefly included some of those references in the above notes (and in more detail in earlier notes). None of the above is "original research". It is all well established knowledge from over 100 years ago.

Regarding "balance", most of the seminal work on FDL was done in the period roughly from middle 1800s to early 1900s. Very little of significance has been added since then. However, there have been countless textbooks and papers written in more recent times. Most certainly, NOT ALL of these are well-written. As Doug has pointed out at length, the TS-PoV is very much a minority view. It is a "dumbing-down" of the subject that is very misleading, and it only appears in some of the more superficially written works. It is perhaps best described by Von Karmann's quote, '' "When you are speaking to technically illiterate people you must resort to the Plausible Falsehood instead of the Difficult Truth." ''

Regarding "utter ignorance", on this Talk page under "Suggested Revisions - Momentum Theorem", both yourself and Burninthruthesky show, and also admit, that you lack a good understanding of how to use NII in Continuum Mechanics. Nothing wrong with that. But it strongly suggests that your PoV on this matter of TS is founded in "ignorance". Again, nothing wrong with that. But it is not a good base from which to preach, or insist on changes to the article.

Regarding "shouting", I have found that when writing explanations of difficult technical subjects, which for completeness necessarily requires many words, that occasionally adding some EMPHASIS helps with the communication. Many people reading my notes have told me that they appreciate this style. (End Zapletal)101.171.255.254 (talk) 04:36, 7 February 2015 (UTC)
 * To clarify. I do not preach, I do not tell you what to believe or why. Rather, I educate, I explain how Wikipedia works. It is the encyclopedia that "anyone can edit" - provided you obey the house rules. The issue you raise has just been settled by consensus debate and will not be re-opened for a good while, if ever. Your attempt to do so is misguided. If you are so blinded by the topic concerned that you cannot accept this, then you are clearly not here to build an encyclopedia but to evangelise your PoV. &mdash; Cheers, Steelpillow (Talk) 12:23, 7 February 2015 (UTC)

(Zapletal Writes ->) This section was originally addressed to Mr Swordfish and/or other readers who have both the mathematical skills to understand it, and enough historical background knowledge of the subject to appreciate it. I will wait to see what those people have to say before deciding whether there is a genuine "consensus" here. (End Zapletal)101.171.127.235 (talk) 03:14, 8 February 2015 (UTC)