Talk:Lift (force)/Archive 12

Parts of the article should clarify that some of the claims are made are from the reference frame of the wing
For example, in what is reference #10: at the time of this posting: "... The cause for the flow turning is the simultaneous conservation of mass, momentum (both linear and angular), and energy by the fluid. ...", this is only true from the wing's frame of reference and ignores friction effects that convert mechanical energy into heat. From the air's frame of reference, momentum and energy are increased from zero to some non-zero value when a wing passes through a volume of air. Most of the increase in energy is a pressure jump that occurs as air crosses the plane swept out by a wing passing through, similar to this NASA explanation for propellers: propeller analysis. Rcgldr (talk) 20:13, 28 April 2018 (UTC)


 * You raise an interesting issue here that seems to confuse a lot of people. "Conservation" in this context doesn't mean that a quantity remains unchanged.  It just means a quantity is allowed to change only in keeping with the relevant "conservation law".  The laws allow mass, momentum or energy to change locally by transfer within the field or by exchange with the surrounding environment, and they allow energy to be exchanged between different forms (e.g. kinetic and heat).  In their most general forms, the laws are the same and are applicable in any reference frame, inertial or not.


 * You seem to be asserting that conservation of energy is somehow different in the reference frame of the wing from what it is in the reference frame of the air mass. Actually, it's not different in any fundamental sense.  The conservation law is the same in both frames.  What's tricky about energy in particular is that although the law is the same in both frames, the transfer of energy between forms can look different in the two frames.  In the reference frame of the air mass, the initial kinetic energy of the air is zero, and the heating of the air in the boundary layer comes from the work done by the body against the surface shear stress.  In the reference frame of the wing, the wing can't do any work, and the heat energy comes from the kinetic energy of the air.  There's no contradiction here:  The heating of the air in the boundary layer is the same phenomenon regardless of what frame you view it in, and if you do the analysis correctly in both frames, you should get the same answer for the temperature rise.  But it's perfectly consistent for the mathematical description to look different in different frames.  It doesn't mean there's a fundamental difference if the physics.  Work, power, and kinetic energy take on different values in different frames even though the conservation law that relates them is the same.


 * The propeller analysis isn't relevant to flows around wings because it involves an "actuator disc", a dividing surface across which mass is conserved but energy and pressure are actively added.


 * So I don't think reference frame needs to be specified in connection with reference 10, or most of the other "claims" in the article. About the only place reference frame is crucial is with regard to Bernoulli's equation in its usual form.  Bernoulli can be derived from Newton's second law, which holds in any inertial frame.  But the derivation assumes the special case of steady flow, which for the flow around a wing can apply only in the frame of the wing.  J Doug McLean (talk) 05:50, 4 May 2018 (UTC)


 * The conservation of energy from a wing's frame of reference only needs to include the air, since the wing doesn't perform any work on the air from the wing's frame of reference. From the air's frame of reference, energy isn't conserved unless you extend whats included so that it's a closed system. For example consider a high end 1500 lb glider with a 60 to 1 glide ratio at 60 mph (like a Nimbus 4T). The power involved is 4 hp and corresponds to the decrease of gravitational potential energy of the glider, and the increase of kinetic energy of the air, most of that being an increase from zero velocity to a mostly downwards (lift) and somewhat forwards (drag) increase in velocity. The point here is that the kinetic energy of the air is increased as a wing passes through a volume of air. Rcgldr (talk) 06:52, 4 May 2018 (UTC)


 * You still seem to be assuming that "conservation of energy" applies only to situations in which the total energy of a system remains constant, as it must for a thermodynamically closed system. My point was that that's not what "conservation of energy" generally refers to in fluid mechanics.  When reference 10 says the flow around a wing satisfies "conservation of energy" it means the flow "obeys" the thermal/mechanical energy equation locally everywhere in the field.  In its usual form that equation applies to local Eulerian fluid parcels, which are not closed systems and needn't maintain constant total energy.  Energy is "conserved" in the sense that it's neither created nor destroyed, but it can move from  parcel to parcel, from one form to another, and in and out of the flow domain, which needn't be a closed system, either.  The usual form of the energy equation, valid for steady or  unsteady flow, applies in all inertial reference frames (additional terms would have to be added for non-inertial frames).  As far as the equation is concerned, the only difference between the reference of the wing and the frame of the air mass is in the boundary conditions.  So the statement in reference 10 doesn't need to be clarified by specifying a particular reference frame.  It's valid regardless of what inertial frame the flow is viewed in. J Doug McLean (talk) 00:19, 5 May 2018 (UTC)


 * Yes, I was thinking that "conversation of energy" meant that the total energy remained constant, as that seems the be the usual meaning of "conservation of energy", such a elliptical orbits, or Bernoulli's equation where in an idealized situation, total mechanical energy remains constant, and as pressure energy (pressure times unit volume) decreases, kinetic energy increases, such that the total energy remains constant. I was also under the impression that there was some similarity between a propeller and a wing, using the air as a frame of reference, as air flows down and through the "plane" swept out by a wing as it passes through a volume of air, there's a pressure jump within that "plane" and that the air continues to accelerate downwards (and forwards) as it's pressure returns to ambient and it's velocity reaches what would be called "exit velocity" for a propeller. I'm not sure that the usage of "conservation of energy" as used in fluid mechanics would be clear to the average reader. Rcgldr (talk) 02:25, 5 May 2018 (UTC)


 * I think your understanding of what "conservation of energy" usually means in connection with general physics problems is correct, and you're right that the usual fluid-mechanics usage of the term is likely to confuse the general reader. So I think there's one change to the article that should come from all this.  We need to make it clear what it means to enforce "conservation of energy" in the fluids equations of motion.  What we have now sends a mixed message.  Under "Mathematical theories of lift" the bullet item "Conservation of energy" defines it as saying that energy is neither created nor destroyed, which is correct. but it links to the Wikipedia article "Conservation of energy", which deals only with the case of closed systems.  This link should be changed to something that describes the actual "local" energy equation used in fluid mechanics, for example https://www.engineeringtoolbox.com/fluid-mechanics-equations-d_204.html.


 * A wing in the frame of the air mass and a propeller are similar in that work is done, and changes in total-pressure are introduced into the flow. But the analogy isn't close because there are key differences.  In the idealized actuator-disc model of propeller flow, the flow is steady in the frame of the disc, and total-pressure changes only across the disc.  Flow with increased total-pressure is confined to the slipstream downstream of the disc, bounded by a vortex sheet surrounding it.  Flow outside the slipstream retains its freestream total-pressure.  In the case of a wing in the frame of the air mass, the entire flow is unstready, and total-pressure changes associated with the unsteady effects are spread diffusely throughout the field.  Wing flow is easier to analyze and understand in the frame of the wing, where the flow is steady, and total-pressure changes are limited to the boundary layer and wake.  There is a "jump" in static pressure only across the airfoil itself.  Away from the airfoil, the pressure field is continuous and smooth. J Doug McLean (talk) 00:24, 9 May 2018 (UTC)

Article organization again
I think that the current overall structure of the article is fine, so I'm not trying to reopen the debate from last year, but I would like to have a discussion of other organization details. While working on my proposed changes to "Momentum balance" I noticed that some of the subsections don't seem that consistent with the headings of the major sections they've been put in. I'm going to suggest moving some subsections around.

For example, the section "A more comprehensive physical explanation" was originally intended to include only the first three subsections. The subsequent subsections "Boundary layer", "Stalling", and "Bluff bodies" it seems to me aren't really parts of the explanation of lift and would fit better in the previous section "Basic attributes of lift". I think the subsection now titled "Lift reacted by overpressure on the ground under an airplane" belongs farther down in the article, as I'll indicate below.

Similarly, in the section "Mathematical theories of lift" the last three subheads starting with "Lift coefficient" aren't really theories of lift and thus don't really fit in this section. I would give "Lift coefficient" and "Pressure integration" their own major heading "Quantifying lift" and put it ahead of "Mathematical theories of lift". I'd put "Lift reacted by overpressure on the ground under an airplane" together with "Integrated force/momentum balance in lifting flows" under their own major heading "Manifestations of lift in the farfield", below "Three dimensional flow".

Making these changes, which don't affect anything until late in section 3, would result in the outline appearing as follows:

1 Overview

2 Simplified physical explanations of lift on an airfoil

2.1 Flow deflection and Newton's laws 2.2 Increased flow speed and Bernoulli's principle 2.2.1	Conservation of mass 2.2.2	Limitations of explanations based on Bernoulli's principle 3 Basic attributes of lift 3.1 Pressure differences 3.2 Angle of attack 3.3 Airfoil shape 3.4 Flow conditions 3.5 Air speed and density 3.6 Boundary layer 3.7 Stalling 3.8 Bluff bodies 4 A more comprehensive physical explanation 4.1 Lift at the airfoil surface 4.2 The wider flow around the airfoil 4.3 Mutual interaction of pressure and velocity 5 Quantifying lift 5.1 Pressure integration 5.2 Lift coefficient 6 Mathematical theories of lift 6.1 Navier-Stokes (NS) equations 6.2 Reynolds-Averaged Navier-Stokes (RANS) equations 6.3 Inviscid-flow equations (Euler or potential) 6.4 Linearized potential flow 6.5 Circulation and Kutta-Joukowski 7 Three-dimensional flow 7.1 Wing tips and spanwise distribution 7.2 Horseshoe vortex system 8 Manifestations of lift in the farfield 8.1 Integrated force/momentum balance in lifting flows 8.2 Lift reacted by overpressure on the ground under an airplane 9 Alternative explanations, misconceptions, and controversies 9.1 False explanation based on equal transit-time 9.2 Controversy regarding the Coandă effect 10 See also

11 Footnotes

12 References

13 Further reading

14 External links

I think changes along these lines would preserve the overall structure but put some of the subsections into more understandable context. Does this make sense?

J Doug McLean (talk) 04:15, 20 April 2018 (UTC)


 * Hello Doug, and welcome back. Your contributions are greatly appreciated.


 * What you have proposed makes a lot of sense. I would suggest making a copy of the current article in your personal space, implementing the proposed edits there and presenting it here fro review.  Looking forward to seeing it.  Thanks again for your efforts.  Mr. Swordfish (talk) 17:11, 25 April 2018 (UTC)


 * Thanks. I've  put a draft in my sandbox that implements the shuffling of subsections.   It also fixes the sourcing issue under "Integrated force/momentum balance in lifting flows", and tries out a better graphic under "Lift reacted as overpressure on the ground under an airplane".


 * I'm also considering recommending some expansion of "Limitations of explanations based on Bernoulli's principle". For one thing, the current version implies that among the simplified explanations only the Bernoulli-based ones have limitations and that the flow-turning-based ones don't have any.  The "more comprehensive physical explanation" farther down makes a good case (citable) that the simplified Bernoulli and flow-turning approaches are roughly equally deficient (incomplete).   I'm considering renaming "Limitations" as "Limitations of the simplified explanations", making it shorter and more general, and putting all the specific shorcomings under "How simpler explanations fall short" as a new subsection in the "comprehensive" section.  Just a thought so far.  Does it sound reasonable? J Doug McLean (talk) 05:29, 4 May 2018 (UTC)


 * I will eagerly take a look at the draft. It may take me a few days to comment.


 * As for the comments in the second paragraph, I have to say I'm a bit confused: on one hand you're considering some expansion of "Limitations of explanations based on Bernoulli's principle"., but then you say you're considering combining all the limitations, making it shorter. So, I'm not sure what to expect; I guess I'll have to wait and see the draft.


 * Agree that the article as it currently stands may imply that the flow-turning explanations don't have any limitations. Several years ago the article contained a subsection on the limitations of deflection/turning, to parallel the similar subsection re Bernoulli. I would support restoring that or something like it:


 * ====Limitations of deflection/turning====
 * While the theory correctly reasons that deflection implies that there must be a force on the wing, it does not explain why the air is deflected. Intuitively, one can say that the air follows the curve of the foil, but this is not very rigorous or precise.
 * The theory, while correct in as far as it goes, is not sufficiently detailed to support the precise calculations required for engineering.  Thus, textbooks on aerodynamics use more complex models to provide a full description of lift.


 * My organizational preference would be to have the limitations/shortcomings in the sections themselves rather than coming back to it later. Many of our readers only read the first few paragraphs.  Mr. Swordfish (talk) 21:23, 6 May 2018 (UTC)


 * Sorry for the confusion. I'm advocating expanding "Limitations" to make it cover both flow turning and Bernoulli.  I'm not advocating keeping the title, but changing it to "Limitations of the simplified explanations" and bumping it up a level in the outline.


 * I share your preference to keep the limitations/shortcomings in one place, but I see a problem with it. Though the limitations part fits well where it is, and I'm going to suggest keeping it there, the items on the shortcomings list are really shortcomings relative to "A more comprehensive physical explanation" farther down.  Putting them in with the "Limitations" part hangs them out there before the needed background has been covered.  So I'm still going to suggest splitting them as I suggested above.  I've incorporated a draft of this arrangement into my sandbox.  It shouldn't interfere with your review of the earlier organization changes.J Doug McLean (talk) 01:04, 9 May 2018 (UTC)


 * I've now had a chance to give your draft the attention it deserves. (see https://en.wikipedia.org/wiki/User:J_Doug_McLean/sandbox) I think the overall re-org is an improvement.  As it currently stands, your draft is a cogent, logical exposition that presents the material in a much more logical order that the current article.


 * I do not feel as positive about the change to the "limitations". From an organizational perspective, waiting to discuss the shortcomings until after the background material has been presented is very logical and cogent and would be the right approach for a textbook or an academic paper.  The problem I have with waiting is that  most readers use wikipedia the way they read newspapers - look at the headlines, read the first couple paragraphs ("above the fold"), and when it becomes too "inside baseball" move on to the next article.  We need to think carefully about what goes "above the fold" and what can wait until later in the article.
 * I don't think I actually disagree with anything in your draft content-wise, but I think we do have a disagreement about our audience and how best to serve it. Your presentation is perfect for the eager young student of aerodynamics who will be reading it end-to-end.  For the casual user (or almost all of the readers - apologies for not having numbers to back this up) they're coming to the article to get a basic gist of how lift works, see the section on "Simplified explanations", read that, maybe stick around to skim the "Basic attributes" section, and then head for the exits once they get to the "More comprehensive" section.
 * In particular, waiting until the 21st subsection to address the equal transit time explanation (the most common, if erroneous, explanation) means that few readers will still be reading when we get to it.  And I think one of the most important objectives of this article is to correct that misguided notion.  At minimum, I'd need to see that discussion moved up.


 * What makes the most sense to me is to begin with a brief synopsis of the 2nd-3rd Law / Flow-turning explanation, with a short discussion of its shortcomings. Then, since the literature is full of claims that Bernoulli can be used to explain lift and we repeat that claim early in the article, we're kinda forced to provide one. I've yet to see one that's comprehensible to the lay reader; what's there is not really an explanation at all since it doesn't explain why the streamtubes change size, but at least it's an accurate discussion of (part of) the real physical phenomenon.  Since it's obvious to us that this is a non-explanation we should state that clearly rather than leave the readers wondering.  And since Bernoulli and Equal Transit Time go hand in glove in popular explanations, this seems to be the right place to briefly address that misconception.  The discussion on geometrical arguments and "squeezing" can probably be cut or moved to the end beside the longer ETT section.


 * So, my strong preference would be to address the shortcomings of each explanation immediately after each explanation is given. I don't think the shortcomings need to be terribly detailed, at least at this point in the article. More detailed discussion can come later, after the "needed background" has been covered.


 * An exercise I've heard newspaper writers use is to imagine someone reading only the first paragraph. What info do they come away with?  Is it the most important thing for them to know?  Does it misrepresent things by omission?  Then iterate for the first two paragraphs, the first three, etc. Expect most readers to stop reading at a certain point, so each "initial word segment" should allow them to come away with a reasonable impression.  It's not like a book or a movie  where it's expected that most of the audience reads/watches from beginning to end.  That's my assessment anyway.
 * I sum, thanks for coming back and helping whip this article further into shape. The overall re-org is definitely an improvement and I'd suggest if there are no disagreements in the next couple days to implement those changes.  Mr. Swordfish (talk) 22:32, 15 May 2018 (UTC)

However, the first step, kinematics of the flow field, is not so easily explained. There is no simple physical law such as Bernoulli or conservation of energy that neatly explains the shape of the streamlines. Continuity (or conservation of mass) is helpful in explaining that as streamlines get closer together the speed must increase; but continuity doesn't explain why streamlines converge across the top of the airfoil but don't do so across the bottom. Hence we see amateurish attempts to explain this first step such as the equal transit time theory. Part of the reason so many occasional visitors to this article try to dismiss Bernoulli as a legitimate explanation of aerodynamic lift is because they imagine Bernoulli and the equal transit time theory are intertwined when in fact the two are entirely separate: one is used in an attempt to take the first step, and the other takes the second step. Equal transit time theory is an attempt to explain the kinematics of the flow field to the layman; whereas Bernoulli says nothing about the kinematics but it does provide the link between the kinematics and the pressure field in the immediate vicinity of the airfoil. In my view, the best quantitative explanation of the kinematics of the flow field is given by the Kutta condition and the horseshoe vortex. The difficult task is translating these two into layman's language. An alternative approach is the qualitative one taken by Holger Babinsky in which the explanation is a one-step process rather than two-step. Dolphin ( t ) 14:19, 17 May 2018 (UTC)
 * I don't have any problem with your proposal, or that of . However, here is a quick comment regarding an explanation of aerodynamic lift. Many of the popular explanations rely, correctly, on a two-step process. The first step is to explain the affect the airfoil has on the flow field; the shape of the streamlines and the distance apart of streamlines at any point around the airfoil; this can be called the kinematics of the flow field. The second step is to relate the kinematics of the flow field to the pressure distribution over the surface of the airfoil. The second step is very simple - Bernoulli's principle directly relates the dynamic pressure at any point to the static pressure at that point - the sum of the two is the total pressure, and total pressure is constant throughout the flow field (providing we avoid the boundary layer and ignore variations in elevation around the airfoil.)


 * Dolphin, you raise an important point. We need to be careful that any critique that we make of the popular Bernoulli-based descriptions do not give the readers the impression that there's something wrong with Bernoulli's principle or that Bernoulli's principle is unrelated or irrelevant to explaining lift.  As you say, the problems with the popular explanations arise in the first step (kinematics) not the second step where Bernoulli's principle is applied. Likewise, if we are to add discussion of the limitations of  the flow turning/Newtonian explanation we also need to be careful to not create the impression that there's something incorrect about that approach.


 * My point above is that I've yet to see a Bernoulli-based presentation that's both correct and easy to understand for the layman. Equal transit time is easy to understand but dead wrong.  Streamtube constriction and conservation of mass is a bit more complicated, but not beyond the ken of most people; unfortunately it begs the question of why the streamtubes change size. Many popular treatments have been edited to remove equal transit time, leaving basically nothing in it's place.  You see things like "the airplane wing is designed to make the air go faster over the top than the bottom", which while technically true doesn't really explain anything. That's why I refer to it as a "non-explanation". All that said, I'm not advocating any particular change to that section.


 * The idea of elevating Babinsky's approach to earlier in the article is intriguing, but since it isn't found in many (if any) other sources we'd risk mis-representing the body of reliable sources. We do present something very close in the next section under "Pressure Differences" where the streamline curvature theorem is cited.  To my eyes, that's the simplest, shortest way to see how pressure gradients arise from an airfoil generating lift but the reader needs to know a bit of calculus to follow it. I think that material is fine where it is. Mr. Swordfish (talk) 21:26, 18 May 2018 (UTC)


 * This is some good discussion. I agree that pointing out that equal transit time is false should have a more prominent place.  A solution that looks good to me is to make "Alternative explanations, misconceptions, and controversies" a subsection of "Simplified physical explanations of lift on an airfoil", just under "Limitations of the simplified explanations".  I still like keeping the "Limitations" of turning and Bernoulli together because the main points are common to both.  I've implemented this move in my sandbox, keeping the detailed critiques of the simplified explanations down under "How simpler explanations fall short".  This ordering looks pretty good to me.  My only misgiving is that "Simplified physical explanations" is now a long section that some readers won't get past.  On the other hand, it debunks equal transit time early, where it should, and it alerts the reader who wants a better explanation that he'll find it under "A more comprehensive physical explanation".


 * I agree that streamtube-pinching/Bernoulli is a non-explanation. But the simple flow-turning explanation also has serious flaws as a physical explanation in the cause-and-effect sense.  It establishes a logical necessity (If there is downward turning, there must be a downward force on the air and an upward force on the airfoil), but to me it doesn't make the cause-and-effect relationship clear.  The cause-and-effect relationship between the downward force and the flow turning is actually reciprocal.  A close reading of the first two paragraphs of the explanation suggests this is the case (It starts with the downward force and comes back around to the downward force), but the mutuality of the interaction isn't clearly pointed out.  The simple downward-turning explanation also doesn't explain how turning is imparted to a deeper swath of flow than is touched by the airfoil.  These are things that are pointed out later under "A more comprehensive physical explanation".


 * Anyway, does moving "Alternative explanations, misconceptions, and controversies" make sense? J Doug McLean (talk) 01:26, 19 May 2018 (UTC)


 * I would support moving that section to earlier in the article. (I'm not so sure that giving that level of prominence to the Coanda controversy is best, but I also don't know where else to put that material, so I'm fine with leaving them together.)  Equal transit time is perhaps the most common explanation of lift (or at least it was a few  years ago) so it's appropriate to address that earlier than we do now.
 * Regarding the limitations sub-section, the draft now contains two: wiki/User:J_Doug_McLean/sandbox early in the arrticle and wiki/User:J_Doug_McLean/sandbox much later under the "More comprehensive" section.  My reading is that the second is the stronger of the two and covers a point that I'd like to see addressed early (i.e. that no reason is given why the streamtubes change size).  I'd support swapping the two.  I'm going to take the liberty of implementing that in the draft to see how it looks.  If there's pushback it's easy to undo.
 * I think we're getting close to consensus, at least among the three participants. Mr. Swordfish (talk) 17:02, 23 May 2018 (UTC)


 * I agree we're getting close, but I see a couple of problems that shouldn't be hard to fix.


 * First, in the "Limitations" section as you've proposed it, I see that the first paragraph no longer refers the reader to "A more comprehensive physical explanation", where the supporting background is covered. This leaves the paragraph making some strong claims with no support.  I think the reference should be restored.


 * Second, "How simpler explanations fall short" was meant to cover the shortcomings of the simplified explanations that are remedied by the more comprehensive one. So putting the paragraph about the simplified explanations' lack of quantitative predictive capability in this section seems problematic to me because the more comprehensive explanation isn't quantitatively predictive either.  The easy way to fix this would be to move this paragraph back to the top of "Limitations".


 * I've implemented these in my sandbox. "Limitations" is now longer, and "How simpler explanations fall short" is quite short, but that looks OK to me.  The references in the last two paragraphs of "Limitations" don't seem to be translating correctly, but that can be fixed later.  J Doug McLean (talk) 00:42, 25 May 2018 (UTC)

, The current draft is acceptable and I am ok with publishing it as-is (after fixing the refs). But I'm still a bit concerned about how the casual reader will absorb the material, in particular as to how the Bernoulli-based treatments are really non-explanations. The current published version, Lift_(force), clearly and concisely states the main issues with Bernoulli-based descriptions immediately afterwards; the current draft version postpones that by several paragraphs, so many readers will miss it.

BTW, it is acceptable to present "strong claims" without supporting material having been previously presented - any claim does need to be supported by cites or references, but it's not necessary to provide proof or some convincing argument in the body of the article, or that such material must be presented first. That is to say, a wikipedia article is more like an executive summary than a mathematical proof. The mission is to present established facts, not necessarily to provide a logical argument why they are so. That job is for the source material; it's fine to recapitulate some of that here but it's not a requirement.

Another concern is that the three sentences treating the geometrical arguments for streamtube pinching has been removed. Given that there still widely distributed misinformation about this (see this video, for instance http://howthingsfly.si.edu/media/lift-bernoulli%E2%80%99s-principle). Agree that the limitations section is growing and may be too large already, but let's think twice before cutting this. Mr. Swordfish (talk) 16:33, 25 May 2018 (UTC)


 * Regarding the "geometrical arguments", I assume you're referring to the first two of the three bullet items under "Limitations" in the current article. I agree with your concern about these, and I now think my removing them was a mistake.  I also think my new first paragraph under the new "Limitations" went too far.  That's the one that points out that the simple explanations can't make quantitative predictions, citing some of the same sources as in the deleted bullet items.  Of course they can't make quantitative predictions, as pointed out in some sources, but a qualitative physical explanation shouldn't be expected to.  So I propose deleting that paragraph and restoring the bullet items.


 * I've implemented this in my sandbox. The restored bullet items are now in the third paragraph in the section, which is maybe not as prominent as we'd like, but the first two paragraphs are short and fit best where they are, and I think the bullets will still grab readers' attention.  I shortened the third bullet item because it was partly redundant with "False explanation based on equal transit-time".  I hope this eases your concerns.   I also fixed the misbehaving refs.  I think everything is in pretty good shape now. J Doug McLean (talk) 19:01, 26 May 2018 (UTC)
 * Looks good. I made a couple of minor tweaks, but I think it's ready. It's a holiday weekend in the US, so maybe wait until Tuesday for publication to allow for other editors to weigh in.  Mr. Swordfish (talk) 22:05, 26 May 2018 (UTC)
 * I think the latest draft is an improvement on the currently published version. You have both done very well. I have no objection to Doug's version going live. Dolphin ( t ) 12:42, 27 May 2018 (UTC)
 * I have published this version. Before doing so I made one minor text change which can be viewed in the sandbox history.  Thanks to everyone for their help and cooperation.  Mr. Swordfish (talk) 13:20, 30 May 2018 (UTC)

Animation Error Regarding Flow
Under "The wider flow around the airfoil" there is an animation which implies that an entire column of air ss bisected by the airfoil and the top half is shifted to the right. This is certainly not the case. Local flow/speed changes do not propagate all the way to the top of a given column of otherwise static air.

The top most and bottom most black dots should remain in alignment.Myndex (talk) 05:15, 14 March 2020 (UTC)


 * I agree that at a great distance from the airfoil the vertical lines of black dots remain in alignment. This diagram only shows one chord-length above the airfoil; and one chord-length below. At this scale the lines of black dots are not in alignment, but they are sloping in the required direction to enable them to align beyond the limits of the diagram. Dolphin ( t ) 07:57, 14 March 2020 (UTC)


 * Look more closely: the top column of black dots moves TWICE as fast as the bottom column, while transvering the chord. After passing the trailing edge, the lower column is correct in being deflected forward closer to the wing, BUT the top column should be deflected BACK closer to the wing, and the upper part of the column should not increase in speed but it does in the animation. Myndex (talk) 11:17, 31 March 2020 (UTC)


 * The animation was created many years ago by . I will invite him to join the discussion and comment on your criticism. I think your comments are applicable to the whole of the flow field but the diagram shows only one-chord length above and below the airfoil. Dolphin ( t ) 11:27, 1 April 2020 (UTC)


 * I just did a crude analysis of the animation that I think supports what Dolphin ( t ) is saying.


 * First, what flow speeds should we expect to see at the top and bottom of the graphic? According to the file notes, the graphic represents the theoretical potential flow around a Karman-Trefftz airfoil at 8 degrees angle of attack (alpha).  For these airfoils, the same conformal-mapping transformation that defines the airfoil shape also yields the potential-flow solution.  So it's probably a good assumption that the graphic was plotted using the analytic expressions for the transformation and that it represents the actual potential-flow solution.


 * A moderately cambered airfoil like this one at 8 degrees alpha should produce a lift coefficient in the neighborhood of 1.5 in potential flow, about 10% higher than it would see in a real viscous flow. The top and bottom of the graphic frame are about one chord away from the airfoil.  We can get a decent first estimate of the flow speed increments at these locations by assuming the bound vorticity associated with the airfoil is concentrated in a single potential vortex, with strength proportional to lift coefficient, located near mid-chord.  The result, for a lift coefficient of 1.5, is that the flow speed at the top of the frame should be about 9/8 of freestream, and the speed at the bottom should be about 7/8 of freestream.


 * How does this expectation compare with the animation? Local flow speeds are proportional to distances, measured along streamlines, between dots in the vertical columns (timelines).  For a freestream baseline, I measured this spacing at the right edge of the frame, midway between top and bottom, using pencil marks on a piece pf paper held against my computer screen.  Then I measured the spacings directly above and below the airfoil at the at the top and bottom of the frame.  The resulting speed ratios were about 9/8 and 7/8, as expected.


 * So, within the margin of error of a crude analysis, I'd say the flow speeds (dot spacings) at the top and bottom of the frame are just as they should be. J Doug McLean (talk) 18:56, 8 April 2020 (UTC)


 * Please see the comments above by Doug McLean. Doug is the author of Understanding Aerodynamics: Arguing from the Real Physics. For example, see Footnote No. 102 in the article.
 * I am hopeful that User:Crowsnest, creator of the animation, will respond to your comments in a short time. Dolphin ( t ) 00:38, 10 April 2020 (UTC)
 * Thanks for the very interesting question and discussion. The first half of this video – a flow visualization for another airfoil shape in a wind tunnel – illustrates further the effects analyzed by J Doug McLean and Dolphin51. According to Doug's far-field approximation, the deviation of the flow velocities far above and below the airfoil from the freestream velocity will be proportional to $$c/y.$$ Where $$c$$ is the chord length and $$y$$ the vertical distance from mid-chord. So for an animation of an area extending five times as far in each direction, the speed ratios on top and bottom would be about 41/40 and 39/40. Which is still about 2.5% deviating from the freestream velocity.
 * For the same flow situation, File:Streamlines_relative_to_airfoil.png and File:Streamlines_relative_to_ground.png show the streamlines in frames of reference relative to the airfoil and the ground. In the former, as well as in the animation, it can be seen that streamlines are not straight near the top and bottom. This is another indication that flow velocities there will deviate from the freestream velocity. -- Crowsnest (talk) 22:27, 10 April 2020 (UTC)

Intuitive Simplified Explanation of an Airfoil
Anybody have a comment on the value of this explanation? It's what I say nowadays, but I don't see it anywhere, and don't really know if it's correct. It's very intuitive to me, and I expect it resolve questions quickly.

An airfoil has an inherent angle of attack, and this functions to redirect the force of the air striking it. An angled surface when struck obliquely will experience a force perpendicular to the direction of strike. This is the lift to the airfoil provided from below. The air below is slowed and turned downward, because it is roughly physically striking the wing that is in its way.

Similarly above the airfoil, the air in front of the wing is being pushed downward, but the wing is moving forward, leaving an empty space behind it that air must rush in to fill. This empty space has decreased pressure because the airfoil that used to be in it has just left it. This decreased pressure contributes to pulling the wing upwards, and the air travels faster above the wing because the decreased pressure is pulling it into this space.

So why aren't airfoils just angled flat boards? This is because air doesn't quite behave like hitting an object with a ball, and bounces off of itself as well as what it hits, producing turbulence and eddies. The curves attempt to account for this turbulence. 184.14.134.15 (talk) —Preceding undated comment added 22:44, 10 April 2019 (UTC)


 * My apologies for the one-year delay in providing a response to your excellent question. One of the early scientists (and I think it was Isaac Newton) contemplated how the wing of a bird generated lift. His conclusion was an explanation very close to yours!
 * Your explanation would be accurate for the force on a flat plate that is generated when the plate is struck by a stream of solid objects such as sand or bullets. Fluids behave differently so Newton’s explanation is no longer considered accurate.
 * At every point on the surface of an airfoil a velocity vector can be assigned to the fluid flowing past that point. The local speed of the fluid varies significantly around the airfoil. The faster the speed of the fluid at a point, the lower the air pressure at that point. When an airfoil is experiencing lift, the total force on the top surface is less than that on the bottom surface because the speed of the air over the top is generally faster than over the bottom.
 * The challenge is to explain why the air moves faster over the top! To explore that topic I suggest you look at lifting-line theory. Dolphin ( t ) 01:08, 11 April 2020 (UTC)

Humility in the face of the unknown
I have the sense that this article could use a strong dose of humility. Similar to some other mysteries of the universe, we really don't know exactly what causes all the lift that is generated by an air foil. I strongly agree with the key points made in this article from this month's Scientific American https://www.scientificamerican.com/article/no-one-can-explain-why-planes-stay-in-the-air/ I think this Wikipedia article would be much better off if the introduction, overview, and explanation sections contained wording indicating that science does not currently understand what accounts for all the lift generated by an air foil. Put simply, the article should be much more humble in the face of the unknown. --Westwind273 (talk) 05:26, 12 February 2020 (UTC)


 * We have seen Scientific American’s line of argument before. It is aimed at a naive and uncritical audience, and on previous occasions Users watching this Talk page have had no difficulty dismissing it as misleading.
 * On a philosophical level it is self-evident that we humans don’t possess complete, total knowledge of anything. We can truthfully say things like “we know that energy is conserved but no-one knows exactly WHY energy is conserved.” Similarly, we can say “we know that all atoms contain protons but no-one knows exactly WHY all atoms contain protons.” So there is no surprise that some writers, even professional writers, are prepared to write “no-one knows exactly WHY an airfoil generates lift.” There is nothing unique or special about the phenomenon of lift, but some writers are prepared to write things that suggest the phenomenon of lift is unique and special.
 * Implied in the Scientific American article, and all previous attempts I have seen, is the notion that there must be one, true explanation of lift. The article points to the Bernoulli explanation, and separately to the Newton Third Law explanation, and then mischievously suggests that seeing there are two different explanations they must both be incomplete and inadequate. That approach is worse than misleading. There are many natural phenomena that can, and should, be explained using two or more different approaches. For example, there are phenomena in mechanics that can be explained using consideration of kinetic and potential energy, or alternatively explained using consideration of momentum. It would be incorrect to suggest only one of these approaches can be correct; or to suggest that the two simultaneous approaches show that no-one can fully explain these mechanical phenomena.
 * I would have some sympathy for the “more humility needed” theory if I could see that this “more humility” theory is being implemented throughout ALL scientific articles on Wikipedia. That won’t be happening, and there is no good reason why it should.
 * The Scientific American article acknowledges some very good work done by Doug McLean. Be aware that the same User:J Doug McLean has contributed significantly to the Wikipedia article on lift. <i style="color: green;">Dolphin</i> ( t ) 06:59, 12 February 2020 (UTC)


 * Users reading this thread should be aware that User:Westwind273 initiated a thread on this topic on this Talk page on 20 March 2013. The thread attracted a lot of interest and many very sound comments. The final comment was made 19 months later on 22 October 2014. The thread was then closed. The thread is still available for viewing - see “Limits of current human knowledge” at Talk:Lift (force)/Archive 8. <i style="color: green;">Dolphin</i> ( t ) 11:41, 12 February 2020 (UTC)


 * Thank you Dolphin. It was seven years ago, so my memory was a bit vague. I don't want to re-hash everything that was said then. I do find it interesting that the key editors of this article seem to find the Scientific American article misleading. I cannot remember a similar instance on another scientific Wikipedia article; I thought Scientific American was fairly well respected. Anyway, no need to rehash all that we worked through in 2013. I just thought the Scientific American article was timely and deserved mention here. Thank you. --Westwind273 (talk) 22:20, 12 February 2020 (UTC)


 * Timely? I am baffled by the article. I too always thought Scientific American was a leader in its field of serious scientific journalism but the title of this article - “No One Can Explain Why Planes Stay In The Air” - is quintessential pulp journalism. It is designed to catch the eye, and impress, the naive and gullible; those who are highly impressionable on matters of science. The artwork in the article is very good, and much of what is written is technically sound, but none of it supports the sensational title given to the article. I might be a lot more sympathetic if Scientific American published a series of articles with titles such as “No One Can Explain Why Water Flows Downhill” and “No One Can Explain Why Magnets Attract”.
 * The SA article is not serious science. It attempts to be sensational by presenting a dramatically alternative viewpoint, but I think it fails. It doesn’t deserve mention on Wikipedia. <i style="color: green;">Dolphin</i> ( t ) 23:27, 12 February 2020 (UTC)
 * What I think is missing is a sense of perspective. Whether something is explained or unexplained is not black and white. There are degrees to which things are or are not understood. I would argue that there is relatively more unexplained about lift than about water flowing downhill or magnetic attraction. This is what I think the Wikipedia article is missing, a sense of perspective. And I think that was the point of the Scientific American article. --Westwind273 (talk) 02:58, 13 February 2020 (UTC)
 * I do find the "woo, we don't understand this!" approach tiresomely sensationalist and provocative. It applies to all fundamental phenomena and thus loses significance in any given topic. What pleases me is to see our good giving a sound account of it all. My own intuitive understanding is a little extended, in that I also regard the fact that lift is not generated at very slow airspeeds as key to understanding how it is generated at higher speeds. That is to say, via circulation. This circulation is a consequence of the pressure differences and acts, in a way reminiscent of (but not analogous to) the rotation of an autogyro rotor, to significantly boost lift. But as the circulation theory seems absent from published intuitive explanations (hence, IMHO, their deep struggle for coherence) it should not be included here so early on. &mdash; Cheers, Steelpillow (Talk) 08:10, 2 May 2020 (UTC)
 * Wait, what? "...lift is not generated at very slow airspeeds..." ??? That's news to me. Granted, low airspeeds do not generate sufficient lift to keep a plane in the air, but they do generate lift, as any sailor who's ever managed to make their sailboat move in light air knows. I can't tell you how many sailboat races  I've  endured at 1 knot or less of windspeed.  If the force propelling the boat isn't lift then what is it? Mr. Swordfish (talk) 13:58, 3 May 2020 (UTC)
 * I should have clarified that full circulation lift is not present. At very low airspeeds the Kutta condition is not yet established as the rear stagnation point lies on the upper surface forward of the trailing edge. As the plane moves forward and circulation begins, a counter-circulating vortex - the starting vortex - is created above the rear section. The lifting circulation is sluggish and hence lift is greatly reduced. Once sufficient airspeed is reached the starting vortex detaches and the stagnation point progressively moves to the trailing edge. Once there, full circulation can build up and the full normal lift develop. The situation changes for high AoA and/or thin leading edges, and I cannot speak for thin sheet airfoils (perhaps form drag may be significant for certain wind directions), but the article on the Kutta-Joukowski theorem has a section on Lift forces for more complex situations. That article also cites a value of around half the full lift before the Kutta condition is established, but I am unsure of that statement's applicability or provenance. &mdash; Cheers, Steelpillow (Talk) 15:26, 3 May 2020 (UTC)
 * What is "full circulation lift"? I've never encountered the term before and Google strikes out here. https://www.google.com/search?q=%22full+circulation+lift%22 Mr. Swordfish (talk) 15:59, 3 May 2020 (UTC)
 * It is not a technical term per se, it is the full amount of lift predicted by circulation theory once the Kutta condition is met. &mdash; Cheers, Steelpillow (Talk) 16:59, 3 May 2020 (UTC)

Conservation of mass subheading
This subheading (under Increased flow speed and Bernoulli's principle) was recently removed.

I think the article is more readable with it than without. The reason is that the most common explanations involving increased flow speed and Bernoulli's principle do not address conservation of mass; conservation of mass is a separate but related concept best conveyed by the subheading structure.

I'm reverting it for now, but will cheerfully accept the consensus here. I would be interested in hearing the argument(s) in favor of removing it. Mr. Swordfish (talk) 22:56, 1 May 2020 (UTC)


 * Thanks for bringing this matter to the Talk page. I have read the sub-section “Conservation of mass” very carefully. What this sub-section is saying is true of incompressible flow but not generally true of fluid flow in general. It is only the second paragraph that states explicitly it is only referring to incompressible flow. The third paragraph is only true if the reader understands that it only applies to incompressible flow.


 * For example, there is the statement “Conservation of mass says that the flow speed must increase as the stream tube area decreases.” This statement is incorrect if applied to the flow downstream of the throat of a convergent-divergent nozzle (used to raise the flow of a gas to supersonic speed.) Downstream of the throat the flow speed increases even though the area of the stream tube is also increasing. (The explanation is that, downstream of the throat, the gas density is decreasing faster than the area of each stream tube is increasing.)


 * The point I am getting to is that this sub-section purports to explain the lift on an airfoil using the principle of conservation of mass but, in fact, it is not conservation of mass that is primarily at work; it is that in incompressible flows the fluid density is assumed to remain constant even as the pressure and the speed of the fluid vary considerably. The second paragraph should begin “For incompressible flow, the rate of volume flow (e.g. volume units per minute) must be constant within each stream tube since the fluid density remains constant.” It is true that conservation of mass applies to the fluid flowing around an airfoil but it applies equally to both compressible and incompressible flows. If this sentence is to end with “since matter is not created or destroyed” it is incorrect to restrict the sentence to incompressible flows only.


 * I am in favour of retaining this in its own sub-section but I think “Conservation of mass” is an inappropriate heading. It would be more appropriate to call it “Incompressible flow” or “Constant density”, and then tweak the wording to highlight that it is based on the assumption that the air density is not changing as the air pressure and airspeed change. <i style="color: green;">Dolphin</i> ( t ) 04:48, 2 May 2020 (UTC)


 * I removed it. Passing by this article after a long absence, it struck me as incongruous. There is a reason why most treatments do not go there. Pretty much every phenomenon in engineering would change if mass were not conserved, it is such a universal principle that localizing it under a subheading will more confuse than clarify. It should simply be taken for granted. &mdash; Cheers, Steelpillow (Talk) 07:26, 2 May 2020 (UTC)[updated 08:11, 2 May 2020 (UTC)]


 * Steelpillow has written "... would change if mass were conserved, …" Mass is conserved. Did you mean to write "... if mass were NOT conserved"? <i style="color: green;">Dolphin</i> ( t ) 07:32, 2 May 2020 (UTC)
 * Thank you, now corrected. &mdash; Cheers, Steelpillow (Talk) 08:11, 2 May 2020 (UTC)
 * RE: renaming it. Seems to me that "Streamtube pinching" would be the right alternative subheading. No quarrels with the other suggestions as long as we can do them without sacrificing readability. Or just remove it, as I suggest below. Mr. Swordfish (talk) 18:46, 2 May 2020 (UTC)

This sub-section is sourced to Anderson's Introduction to Flight(2004) sec 5.19, and my recollection is that it fairly explicitly invokes conservation of mass, thus it's usage in this article. Unfortunately, I don't have a copy of that edition and the only one I can find on line is an earlier edition (http://docshare04.docshare.tips/files/18502/185026212.pdf), in which sec 5.19 is titled "How Lift is Produced-Some Alternate Explanations " but there is no treatment of streamtube pinching. Moreover, a text search for "conservation of mass" produces no results. So we may have insufficient sourcing here. Anybody got a copy of the 2004 edition?

I'm wondering: how widespread is the streamtube pinching explanation and is it sufficient to include it here in the article? I thought that since Anderson's Introduction to Flight was more or less the standard college text that that would be sufficient for inclusion, but if streamtube pinching is not there, what is our source? The other cite (http://users.df.uba.ar/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf) which does address streamtube pinching via conservation of mass, while reliable is probably not sufficient to pass the notoriety test. So maybe we just cut the whole subsection? The content seems to be problematic and the sourcing is suspect Mr. Swordfish (talk) 18:42, 2 May 2020 (UTC)
 * Having read the whole Bernoulli effect section more carefully, the bulk of it certainly needs to stay. But I remain unhappy about splitting it up with a subheading. The lead paragraph merely states the principle, it does not explain how the velocity difference between surfaces is obtained. You have to read right through. Thus, the lone subsection is not expanding on something summarised in the lead but takes the same logical flow forward. All the subheading does is force a hiccup in the flow. I really do think that removing it gives a clearer picture of the overall argument.
 * On conservation of mass, I see it as a rabbit-hole too far. It is sufficient to remark that the net mass flow along a streamtube is constant at all points; conservation of mass is thus implied in a simple, intuitive way but need not be stated. Stating it overtly immediately raises the need to explain its unusual significance, which frankly is just not there. All mention should be replaced with the idea of constant mass flow (as Clancy 1975 does).
 * On streamtube pinching, just as Anderson sees downward deflection of the air as an effect of lift, so too I see streamtube pinching (which is a consequence of Bernoulli's principle) as an effect of the increased velocity (which is integral to Bernoulli's principle). I have seen the pinching referred to elsewhere, but not with any authority. FWIW the Venturi tube reveres the cause-and-effect by forcing the streamtube to pinch in. With these complex inter-related effects, a perfectly good explanation of one situation will give a wholly wrong impression if taken out of context and we need to guard against that. So we should introduce the velocity increase first and use that to show why the streamtubes pinch.
 * &mdash; Cheers, Steelpillow (Talk) 19:55, 2 May 2020 (UTC)
 * Dealing with the Bernoulli-based explanations at the elementary level is always going to be something of a minefield. (c: Agree that the section doesn't explain how the velocity differences occur, and the idea of stream tube pinching just kicks the can down the road because it's not apparent why the stream tubes change size. Equal transit time, hump, half-venturi, and stream tube pinching seem to be the usual layman's explanation of why the velocity differences occur, but they all fall short because the "real reason" for the speed change is the pressure change. In quotes, because it's my opinion and I don't get to spout off my opinion in the article; the article needs to be based on what the reliable sources say. I'm not sure why we have chosen to present only one of the four usual non-reasons for the speed difference in this section, but I do think that relegating it to a sub-section helps imply that it's only one non-reason and not the most common. Maybe I'm reading too much into it.


 * I'm basically agnostic over whether we say it's "conservation of mass + incompressibility", "constant fluid density", or "constant mass flow". They are all equivalent in my mind. The best choice would be the one that is most readable to the lay audience and conforms to the terminology in the reliable sources.


 * BTW, searching for "streamtube pinching" via Google doesn't return many reliable sources. One is Doug Maclean's book (https://books.google.com/books?id=UE3sxu28R0wC&pg=PA272&lpg=PA272&dq=streamtube+pinching) which unfortunately leaves the details to Anderson (2008), so I would assume that this treatment is actually present in the later editions. If the libraries were open now I'd just wander over and check it out, but that's not an option. Seems to me that a good way forward here would be for one the editors here to obtain a copy, read the material carefully and make sure that we are using the terminology in a similar fashion as the source. Ideally, a snippet could be added to the ref tag. Mr. Swordfish (talk) 22:01, 2 May 2020 (UTC)

Since nobody took me up on my suggestion to go to the source (Anderson, Eighth Ed.) I procured a copy myself. A few relevant excerpts form chapter 5.19 (emphasis mine):


 * "...this raises the question of why the pressure is lower on the top of the airfoil and higher on the bottom. The answer is simply that the aerodynamic flow over the airfoil is obeying the laws of nature: mass continuity and Newton’s second law."


 * "... stream tube A is squashed to a smaller crosssectional area as it flows over the nose of the airfoil. In turn, because of mass continuity (ρ AV = constant), the velocity of the flow in the stream tube must increase in the region where the stream tube is being squashed."
 * "Because of the law of mass continuity —that is, the continuity equation—the flow velocity increases over the top surface of the airfoil more than it does over the bottom surface."


 * "The sequence of preceding items 1 through 3 are the fundamental laws of nature that result in lift being produced on an airplane wing. You cannot get more fundamental than this — mass conservation and Newton’s second law. "

Since he uses both "mass continuity" and "mass conservation" I think we are on solid ground using either. My preference is mass conservation (or equivalently "conservation of mass") since this is an easier concept for the lay reader to understand. I'd rather not have to stop and explain the continuity equation at this juncture. Also, we have an article on conservation of mass and no corresponding article on mass continuity. (The article on the continuity equation is a more general treatment of any quantity that is subject to a conservation law.)

I'm still unconvinced that the idea of using streamtube pinching to explain lift at the elementary level is sufficiently widespread to include it here. It's clearly Anderson's pet explanation but few other authors address it. Seems to me that if it wasn't in "Introduction to Flight" we'd probably dismiss it as WP:FRINGE. Mr. Swordfish (talk) 17:04, 13 May 2020 (UTC)


 * @Mr Swordfish: Thank you for getting a copy of Anderson's "Introduction to Flight" and informing us of relevant contents. On the question of "continuity" versus "mass conservation", George Batchelor says the following in his An Introduction to Fluid Dynamics (1967 - Section 2.2 Conservation of mass):
 * "The differential equation (2.2.2) is one of the fundamental equations of fluid mechanics. A common name for it in the past has been the 'equation of continuity', in which the word continuity is evidently being used in the sense of constancy (of matter), but in this text we shall adopt the more descriptive term 'mass-conservation equation'."
 * Batchelor (in Section 2.1 Specification of the flow field) defines the meaning of a stream-tube as follows:
 * "A related concept is a stream-tube, which is the surface formed instantaneously by all the streamlines that pass through a given closed curve in the field."
 * However, I haven't found any topic in which he uses the concept of the stream-tube to explain a fluid dynamic phenomenon. He certainly doesn't use it to explain the lift on a 2-dimensional airfoil or a 3-dimensional wing. <i style="color: green;">Dolphin</i> ( t ) 12:41, 14 May 2020 (UTC)
 * My understanding is that continuity is a stronger criteria than conservation - a quantity may be conserved as long as it pops up somewhere else when it disappears from the area in question; continuity requires conservation at the local level. For mass at the macro level (as opposed to the quantum level) I think the two are the same. I do not think it is appropriate to make that subtle distinction at this point in the article. Moreover, the other cite we have (Eastlake) uses the term "conservation of mass" so my take is to stick with that.
 * Batchelor is not alone in not using this explanation - I'm failing to find it anywhere other than Anderson and Eastlake (restricting the search to reliable sources per Wikipedia's definition.) Despite Eastlake's plea that "This HAS TO BE INCLUDED with Bernoulli’s law when explaining lift for it to really make sense." I remain unconvinced that it should be in the article. Mr. Swordfish (talk) 19:15, 14 May 2020 (UTC)