Talk:Bernoulli's principle

Removed GF content in March
The equation v^2/2 +gz + p/row = constant

is in terms of energy per kgm, i.e. it has been divided-through by M - but the text refers to the term  gz  as quote  "force potential". This has no meaning, and serves only to confuse. It is really the potential energy in earth's gravity - per kg.

The term p/row, is same as (N/m^2 /kg) x m^3, which cancels to N-m/kg, so it is in fact, energy/kgm.

Since we were considering only INcompressible flow, row is constant and so dissappears to join the constant on the other side, to give

V^2/2 + gh + p = K

where p= pressure (N/m^2), g= 9.8 m/s/s, h = relative height, V = velocity m/s

It is clearer to not divide by mass, so that the equation is directly in terms of energy, i.e.

0.5.M.V^2 + M.g.h + P.Volume = k

i.e. Volume = M/row What Bernoulli did was yet another example of the Conservation of Energy Principle.

He added k.e. (M.V^2/2) to Potential enerergy, (m.g.h) to P.Volume and states that the total will remain constant - in an isentropic, or streamlined, flow.

However, what does not so far seem to have been pointed-out, is one hideously "obvious" fact, which is - disastrously - often over looked. i.e. that in a duct of varying csa, the speed at any plane, z, along the the duct, is entirely determined by the csa at that plane. (INcompressible fluid)

An example of this is the guy who went to great effort to try to make a litre of water fall onto a fan on a vertical axis, to turn an alternator. He directed the water - or attempted-to! - with a parallel pipe, and, as I explained to him, the water cannot accelerate AND keep the same diameter - that is mathematically impossible. But I had no reply.

What happened was that air was drawn into the lower end of the pipe to effectively - but randomly - decrease its csa. This caused a drenching drowning kind of splatter onto the fan, rather than a streamlined flow, "wasting" most of the energy in oxygenating the water! Also, it is for this reason that a turbine which works very efficiently in its designed direction of flow, Cannot - In Principle - work efficiently with the flow reversed.

It will, however - in Principle - work as a compressor - or pump - if energy is supplied to the rotor, (reverse rotation), and a suitable exit nozzle fitted to slow the flow back to the inlet speed.

Bert Vaughan — Preceding unsigned comment added by Bert Vaughan (talk • contribs)

Possible error in reference [15]
Hi everyone, I think that I noticed an error in reference [15] about the use of incompressible flow bernouilli's equation. It gives the reference to page 602 of the book but it rather seems to be at page 610 as you can see here. This is the first time I suggest something on wikipedia so I don't know if I have to modify it by myself or signal it first. — Preceding unsigned comment added by 86.208.16.31 (talk) 19:04, 14 August 2020 (UTC)


 * Reference 15 quotes p.602 in the 6th edition of White’s book. Are you quoting from the 6th, or some other, edition? I haven’t been able to download the .pdf file you supplied. Dolphin ( t ) 03:40, 15 August 2020 (UTC)
 * Seems they linked the 7th edition so I think we can put this to rest. 35drake (talk) 16:45, 1 February 2023 (UTC)

Adjustment to the lead
Currently the second paragraph speaks about some fairly high-level concepts such as isentropic flows, irreversible processes, non-adiabatic processes, incompressible flows and compressible flows. In contrast, the third paragraph is confined to simpler concepts such as conservation of energy, kinetic energy, potential energy and internal energy. In the interests of WP:Make technical articles understandable, I feel these two paragraphs should be reversed - the third should become the second, and the second should become the third. I will make the change. Dolphin ( t ) 12:21, 11 May 2023 (UTC)


 * It appears this suggested edit has already been done. I would go further and move the current third paragraph (isentropic flows, irreversible processes, non-adiabatic processes, incompressible flows and compressible flows) to the end of the lead, moving current paragraphs four and five up to three and four. Mr. Swordfish (talk) 01:06, 28 May 2023 (UTC)


 * Good idea. I have no objection to the change you are proposing for the third paragraph. Dolphin ( t ) 07:50, 28 May 2023 (UTC)

Not how but why?
There is no explanation here of why an increase in flow velocity should decrease the dynamic pressure, as described by Euler's equation. That is to say, why does the conservation of energy manifest as a drop in dynamic pressure and not in, say, a change in temperature (as it does in some other circumstances, such as gas expansion), or simple flow disruption and back-pressure, similar to transonic choking? Bernoulli observed the effect, Euler figured out the equation, but has anybody explained why it happens this way in the first place? &mdash; Cheers, Steelpillow (Talk) 16:45, 12 June 2023 (UTC)


 * The third paragraph contains the following two sentences:
 * If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.
 * Newton's second law says F=ma, and humans tend to think that the force comes first and causes the acceleration, not the other way around. It's possible to get way into the weeds arguing over whether the laws of physics describe causes and effects or whether they just quantify relationships between things like force and acceleration, but intuitively most of us think of acceleration as being caused by a force. For instance when you kick a football you apply a force on the ball and that "causes" the ball to accelerate; the ball accelerating does not somehow magically cause your foot to kick it.
 * This (in my opinion) is why so many people have trouble understanding Bernoulli's principle: it is often explained that the speed change happens "first" and this somehow "causes" the pressure to change. This reverses our usual intuitive notion of forces causing acceleration. But if you think about BP as the pressure differences exerting a force, and that force "causing" an acceleration it's much easier to understand.
 * Perhaps the article could be more clear about this. Mr. Swordfish (talk) 15:15, 13 June 2023 (UTC)


 * Your question is entirely valid. I doubt I have the skill to answer it to the satisfaction of a physicist, but I can think about it and record my thoughts.
 * Firstly, an important technicality: an increase in flow velocity is accompanied by a decrease in static pressure, not a decrease in dynamic pressure. (Dynamic pressure is defined to be one half rho times the square of the flow speed, so there is no mystery as to why an increase in velocity is accompanied by an increase in dynamic pressure - it is a consequence of the definition of dynamic pressure.)
 * Early scientists observed that providing the various forms of energy were defined consistently, energy always seemed to be conserved. They used these observations to formulate the law of conservation of energy. Those observations included Bernoulli’s principle which was the inescapable conclusion that in the flow of an incompressible fluid the sum of the static pressure and dynamic pressure is the same along a streamline, and in many situations, it is the same throughout the flowfield. So rather than say a fluid flow is constrained to conform to the law of conservation of energy, it is more accurate to say that the observations made by Bernoulli contributed to the formulation of the law we now know as conservation of energy.
 * Another example that might have helped formulate the law of conservation of energy is the motion of a pendulum - as the speed of the bob of the pendulum increases, so the gravitational potential energy of the bob decreases. Your question regarding fluid flow is analogous to asking “why does the potential energy of a pendulum bob decrease when the speed of the bob increases?”
 * The bob of a pendulum is incompressible so as its kinetic energy changes its temperature does not. Similarly Bernoulli’s principle talks about incompressible fluids so any change in kinetic energy won’t be accompanied by a change in temperature.
 * Energy can be identified in an incompressible fluid in two ways - its kinetic energy per unit of volume, and its potential energy per unit of volume. The kinetic energy per unit of volume is what is called the dynamic pressure. (The pressure unit pascal is equivalent to joules per cubic metre where the joule is the unit of energy.) The potential energy per unit of volume is the sum of the static pressure and the height above the datum multiplied by density. Along the datum the gravitational potential energy is arbitrarily zero.
 * The universe behaves in a manner that, in many ways, is uniform. We describe this uniformity using the law of conservation of energy. Fluid flow is included in this. The law of conservation of energy predicts that kinetic energy plus potential energy will always remain constant throughout a flowfield. Bernoulli’s observations confirmed that it had always been so. Dolphin ( t ) 15:24, 13 June 2023 (UTC)
 * Thank you both for your thoughts. However they pretty much address the problem rather than the solution, the how and not the why. The question remains; why is the flow incompressible, with both the total pressure and the temperature remaining constant? Why does it accelerate and squeeze down into a venturi and not slow down and bunch up, increasing the total pressure under its kinetic impact, like a crowd queueing to get out the door? For example it is easy to see that if a pressure gradient develops then the speed will increase, but not why the gradient develops in the first place, when there is no compressive or thermal buildup at the mouth (arguing that it is caused by, or inseparable from, the acceleration is just begging the original question as to why this circular cause-and-effect spirals up out of nothing). I should add that I am not alone in this concern; see for example Ed Regis; "No One Can Explain Why Planes Stay in the Air," Space & Physics, Scientific American website, 1 February 2020. &mdash; Cheers, Steelpillow (Talk) 16:18, 13 June 2023 (UTC)
 * why is the flow incompressible, with both the total pressure and the temperature remaining constant? 
 * It isn't, and those two quantities don't remain constant. But these are useful simplifying assumptions that make the math easier, so they are commonly assumed. A more thorough model deals with compressibility and temperature variations etc.
 * Why does it accelerate and squeeze down into a venturi and not slow down and bunch up...
 * The narrow part of a venture tube acts as an obstruction which does cause the fluid to "slow down and bunch up" behind the obstruction, with an associated increase in pressure. Once the fluid moves past the obstruction, the higher pressure behind it pushes on the fluid and accelerates the fluid.
 * it is easy to see that if a pressure gradient develops then the speed will increase, but not why the gradient develops in the first place
 * For an airfoil, it's easy to see why the gradient develops: the streamline curvature theorem says that any time a fluid follows a path that is curved there is a pressure gradient perpendicular to the fluid flow. Lowered pressure on the top, higher pressure on the bottom. As the air flows from ambient pressure to the region along the "top" of the foil the pressure decreases.  As BP predicts, the air speeds up, but this is more of an interesting factoid than a reason why airfoils do what they do.
 * To understand the streamline curvature theorem, think of a tornado or a hurricane or simply a low pressure system in the atmosphere. Lower pressure on the inside of the curve. And if you want to derive it, just take the kinematics of circular motion and apply Newton's second law at the differental level and it pops out in one step, two or three if you want to be pedantic.
 * As for Regis's article, it's pure horseshit. The best advice I can give is to ignore it. Mr. Swordfish (talk) 17:51, 13 June 2023 (UTC)
 * Well, thank you, the editors of Scientific American, curator of aerodynamics at the National Air and Space Museum John Anderson, our sometime resident expert Doug McLean, and the other verifiable experts whom Regis gives voice to, will be glad to know they have wasted their careers. For the benefit of any subsequent readers, I should mention that your other points are no better placed. &mdash; Cheers, Steelpillow (Talk) 19:05, 13 June 2023 (UTC)
 * My reading of Doug McLean is that he shares my opinion of the Regis article. Perhaps not in the stark language that I would use, but his book is a hundreds pages long explanation of why Regis is off base.
 * Don't know about how Anderson feels about it, but my reading of his works is not that he thinks "nobody can explain" why planes stay in the air.
 * It very settled, well understood engineering and physics. Saying "nobody understands it" is sensationalist crap. I'm sure it gets lots of clicks, but... Mr. Swordfish (talk) 00:44, 14 June 2023 (UTC)
 * To the best of my knowledge, the current thread is the third time this theme has been aired on Wikipedia. The previous two were:
 * Talk:Lift (force)/Archive 8
 * Talk:Lift (force)/Archive 12. (User:Steelpillow contributed to this thread in three edits - 2 May, 3 May and (again) 3 May, all in 2020.)
 * Several Users made the point that the Scientific American article is technically sound and has high-quality artwork, but nothing therein supports the sensational title given to the article. I have challenged supporters of the SA article to identify some element of the article that supports, or is directly related to, the title but no-one has accepted my challenge. It looks like the person(s) who came up with the title was not the same person who wrote the body of the SA article. It is conceivable that the title is due to an editor or sales manager who wanted a sensational title to catch the eye of potential customers. Dolphin ( t ) 14:01, 14 June 2023 (UTC)
 * Yes. It is very common for magazine and newspaper articles to have their title written by the editor, not the author. It's even more common in the internet age where an article will be given several different titles to see which one gets the most engagement.  The SA article's title is effective clickbait, but as you observe it is at odds with the body of the article.
 * I re-read it last night, and it's not as bad as I recall, other than the title that is.
 * It's like the following hypothetical article:
 * We asked several famous chefs how to make tomato sauce, and their recipes varied widely. One of them said directly, "There is no one singular way to make tomato sauce." We then asked a chemist, and he said that while the chefs make tasty sauces, their recipes present an incomplete understanding since they don't reflect all the chemical reactions that occur when preparing the sauce.
 * And then some idiot editor comes along and gives it the title "Nobody knows how to make tomato sauce."
 * Anyway, the purpose of the talk page is to discuss how to improve the article, not navel gaze about epistemology.
 * There are two common ways to derive BP - apply conservation of energy or apply Newton's second law. The former is easier and only involves algebra so it is tractable for students at the grade school level. But it somewhat obscures the physics - the latter approach makes the physics clearer by starting with forces and acceleration, then applying a bit of calculus to compute the speed changes that occur because of the forces.  Many people are confused as to why the air should have reduced pressure just because it has sped up, but when presented with an analysis of the forces due to pressure and the ensuing acceleration due to those forces it becomes much easier to see why the relationship between speed and pressure happens.
 * We present the conservation of energy approach first, I think because that's the order it is usually presented to students, but it tends to confuse people. Historically, the law of conservation of energy was not discovered for many decades after Bernoulli and Euler did their work, so perhaps that should come first? Mr. Swordfish (talk) 14:39, 14 June 2023 (UTC)
 * Thank you for the reminder. My understanding has moved on a bit since then. I still dislike the "Woo!" aspect, such as Regis's headline, which is why I only mentioned it here in passing. What a dangerous thing to do on Wikipedia! The problem I have with the conservation-of-energy argument as an "explanation" is that it does not rule out other energy-conserving phenomena, such as flow stalling or temperature change: it is as incomplete as the model it is trying to explain. But what does come out of this thread is that there is evidently still no clear answer available. &mdash; Cheers, Steelpillow (Talk) 15:41, 14 June 2023 (UTC)
 * You refer to “no clear answer available”. Are you suggesting that there is, or should be, one truly correct explanation for the Bernoulli effect or fluid dynamic lift? My view is that there is no “one true explanation of lift” (and similarly no “one true explanation of the Bernoulli effect”.) I believe these things can be explained satisfactorily in two or more ways. Unfortunately this is sometimes interpreted incorrectly as disharmony within the scientific community and therefore evidence that “No-one really knows why ... ...” Dolphin ( t ) 15:55, 14 June 2023 (UTC)
 * It is out of place to speculate here, but I do think that this is the billion-dollar question. The maths works, no question. But is the reason why that is the right math something waiting to be understood, or is the reason an irreducible complexity? It would be nice to be able to cite an answer to that. &mdash; Cheers, Steelpillow (Talk) 17:22, 14 June 2023 (UTC)
 * There is definitely a philosophical question there, worthy of a philosophical discussion. The answer and the discussion won’t be unique to Bernoulli’s principle. They will be equally applicable to conservation of energy, conservation of linear momentum, conservation of angular momentum; in fact all the conservation laws. These discussions have likely already taken place somewhere like Philosophy of science. Dolphin ( t ) 14:42, 15 June 2023 (UTC)
 * That is not quite the point being considered. There is the narrower question as to why, in the particular case of Bernoulli/Venturi, the conservation laws manifest as a reduction in pressure, and not as a change of say temperature and/or density. &mdash; Cheers, Steelpillow (Talk) 16:11, 15 June 2023 (UTC)
 * The simple reason is that when deriving the BP, temperature and density are assumed to be constant. If the model includes temperature and density changes then a more complex formula that relates speed, pressure, temperature, and density will be produced, i.e. the Euler equations.
 * Of course, this begs the question of why ignoring temperature and density changes is a good approximation for many scenarios. I don't have a simple answer to that, and even if I did we couldn't put it in the article unless we could find a source for it. Mr. Swordfish (talk) 23:26, 15 June 2023 (UTC)
 * Indeed. This is exactly the question I asked at the beginning: is there any such source? &mdash; Cheers, Steelpillow (Talk) 06:51, 16 June 2023 (UTC)
 * If your question can be presented as “when a gas is compressed its volume decreases pressure increases a little and its temperature increases a little. Why is the temperature change not more, or less? Why is there any change at all, in the temperature?” This type of question puzzled many scientists in the 19th century and it was eventually solved by formulation of the Second law of thermodynamics which, among other things, says entropy can increase or remain zero, but it never decreases. It may be that what you are questioning is why, in a Venturi or other example of the Bernoulli principle, does entropy not decrease? The Second law tells us that a reduction in entropy has never been observed so we assume it never will.
 * I acknowledge that Bernoulli’s principle is confined to incompressible liquids but my point is still valid. In a Venturi or other example of the Bernoulli principle the resulting pressure, temperature and velocity are always the same. What determines these resultant parameters? The answer is that these parameters are those that involve no change in entropy. Why does entropy remain unchanging in the absence of irreversibilities? See the second law of thermodynamics. Dolphin ( t ) 03:03, 17 June 2023 (UTC)
 * Of course, the second law of thermodynamics applies only to closed systems, while a body moving relative to a fluid is an open system, but something like that and/or the principle of least action may be at work here. If it is, then it is surprising that none of the smart people who have studied this question have ever figured it out. As Wikipedians, it is the sourcing that matters to us, not the explanation per se. &mdash; Cheers, Steelpillow (Talk) 08:05, 17 June 2023 (UTC)

Explanation on the molecular level
There is a lot of misinformation on Bernoulli, allowing many people to get an incorrect understanding of how it works. And the correct explanations typically use abstract terms like internal energy and dynamic pressure. While correct, these explanations are unintelligible to most people.

If we were able to explain it on the molecular level, as balls bouncing around, I would think that readers would have an easier time understanding the concept. There are few articles and videos explaining how it works on the molecular level. And the one’s I’ve seen are not as clear as I would like.

So my question - what does the community think about having a section in the article explaining what is going on at the molecular level, in a way that is fairly easy for average readers to understand? Showing that the molecules can only have so much velocity, and if that velocity is in the direction of flow, less velocity is available to make pressure. I’m happy to write it and make illustrations. But I’m also wary that there may be significant resistance from some editors… I’m not interested in an edit war. Thoughts? Thanks! --Zojj tc 21:04, 30 July 2023 (UTC)


 * Bernoulli's Hydrodynamica was the seminal work in Statistical Mechanics, and that's probably worth mentioning in the article, but I'm skeptical that a statistical approach to the topic will provide much in the way of an intuitive understanding of the BP. Of course, I'd be willing to look at a draft. That's what our sandboxes are for.
 * I'd be careful with the molecules can only have so much velocity, and if that velocity is in the direction of flow, less velocity is available to make pressure. argument - it's been somewhat refuted in this article: []
 * The reason so many people get an incorrect understanding of BP is that so many presentations get it backwards - they say the fluid speeds up for some reason (call it reason X) and this causes the pressure to drop. The problem with all these reason X explanations is that they are always wrong. If you start with pressure differences, it is easy to see that more pressure behind will exert a force that accelerates the fluid (and more pressure in front will decelerate it).  The pressure differences cause the speed changes, not the other way around.
 * Granted, the equations don't talk about cause and effect, but for any intuitive understanding of force and acceleration, we humans are predisposed to think of forces causing acceleration, not the other way around. When you try to present the phenomena as "acceleration causes the force to appear" it's counter-intuitive. Mr. Swordfish (talk) 14:06, 31 July 2023 (UTC)
 * Good call on the sandbox. And thank you for link and your thoughts on how to better help people understand. At first glance the paper makes a leap on the perpendicular speed concept... I’ll look into it more. --Zojj tc 12:24, 2 August 2023 (UTC)


 * A word of caution is appropriate here. On all scientific topics, Bernoulli included, newcomers appreciate intuitive explanations and newcomers often identify analogies and simple explanations that help them climb the ladder of understanding. It is easy to find such helpful explanations on the internet but they don't constitute reliable sources. Wikipedia works on the principle that everything that is likely to be challenged should be verifiable using a reliable published source (WP:VERIFY). It is not sufficient for the source to be published on the internet - it must be a reliable source. Given the state of knowledge of Bernoulli's principle, these popular, helpful, easy-to-understand are incompatible and peripheral to that state of knowledge. I have not seen the "molecular level" explanation of Bernoulli so I think it might be impossible to find a reliable published source to support it. It should not be added to Wikipedia unless it can be supported by a genuinely reliable published source, regardless of how helpful you might find it in helping you assimilate the information about Bernoulli.


 * With all due respect to Mr Swordfish's considerable expertise on these topics, ideas about "cause and effect", and whether pressure causes acceleration or vice versa, are outside the mainstream of Bernoulli's principle, and outside the state of knowledge of most topics in physics. Daniel Bernoulli correctly avoided writing anything about cause and effect, or acceleration causing pressure change. Similarly, Newton correctly avoided writing anything about these things. The cause and effect approach to physical laws is something developed since these laws were published; something developed by well-meaning people looking for a simple and intuitive way to understand or explain these profound physical laws. In the case of Bernoulli's principle, pressure and speed change simultaneously - it is incorrect to imagine that one causes the other; the thing that is the original cause of both is the change in cross-section area of the stream tube or the pipe in which the fluid is flowing.


 * Newton's second law of motion is another example. People commonly write that it is always the force that causes the acceleration. This may be true some of the time, but it isn't true all the time. An example of my point is a passenger vehicle accelerating. All passengers experience the same acceleration. The force acting on each passenger is this common value of acceleration multiplied by the mass of the passenger. It is the common acceleration which determines the magnitude and direction of the forces on the passengers. A more profound point to be taken from this is that, for the purpose of applying Newton, it is not necessary or meaningful to decide which one of force or acceleration is the primary cause; Newton's 2nd law simply says that the magnitude and direction of force and acceleration are related by the well-known equation, and they occur simultaneously. There is no time lag, and there is no cause and effect. It is often meaningful to talk about what causes what, but don't imagine that is a discussion about Newton's 2nd law. <i style="color: green;">Dolphin</i> ( t ) 13:14, 2 August 2023 (UTC)
 * ...ideas about "cause and effect", and whether pressure causes acceleration or vice versa, are outside the mainstream of Bernoulli's principle, and outside the state of knowledge of most topics in physics
 * Really? So all those allegedly reliable sources we cite in the reference for this page that clearly discuss cause and effect are outside the mainstream?
 * For instance Bauman is pretty clear about it:
 * Bernoulli’s principle is typically stated in the form that increasing the speed of a gas lowers the pressure. This illogical interpretation casts aspersions on Bernoulli’s equation, which is a direct application of Newton’s second law. Consequently, authors have sought alternative explanations, including an isoergic model and the “bounce” model, inconsistent with physics. The difficulties are removed by recognizing that Bernoulli’s equation tells us that a pressure difference causes a change in speed, and pressure differences are caused by curvature of flow, interpreted locally as producing a centrifugal force. (emphasis mine)
 * You'll also find discussions of "cause and effect" in the works by John D. Anderson, Doug McClean, John S. Denker, Holger Babinsky, Klaus Weltner etc. Just read the footnotes to this article, or the others that we work on together.  It's not an "outside the mainstream" view.
 * Granted, there are some physicists who take an agnostic view and say that there are only equations, not causes and effects. I don't know how mainstream this view is, but it doesn't help with an intuitive understanding of physics. Mr. Swordfish (talk) 19:29, 2 August 2023 (UTC)
 * Thanks for the pdf of the essay by Robert Bauman. I have read it closely and I am most disappointed that a Professor of Physics has written such a low-grade article on this topic. Bauman's essay is not mainstream science. It contains much that is misleading or incorrect. Two of the most egregious errors are as follows:
 * At Criticism 1, Bauman writes "The equation is derived for motion along a streamline, ... so it should only be applied along a single streamline." This nonsense deserves a place down there with the equal transit-time theory! See responses, including your own response dated 24 Aug 2021, at Talk:Lift (force).
 * "Pressure differences are caused by curvature of flow, ..." Bauman probably meant to write "curvature of flow is caused by a pressure gradient" but that is not what he wrote. Converging flow in a nozzle is an example of falling pressure and increasing speed, but all streamlines are straight - there is no curvature of flow. Bauman concludes this sentence by writing "... interpreted locally as producing a centrifugal force." This is incompatible with Wikipedia's view on centrifugal force which is that it is a pseudo force or fictitious force.
 * This article by Bauman offers much, considering his status as a Professor of Physics, but it delivers little of value. It is embarrassing. It should not be regarded as a reliable source on the subject. <i style="color: green;">Dolphin</i> ( t ) 13:17, 3 August 2023 (UTC)
 * So, Holfger Babinski gets it wrong too?
 * With these assumption we can now derive the rules governing fluid motion by considering the resultant pressure force acting on an individual fluid particle and applying Newton’s second law, which states that force causes acceleration.
 * http://www3.eng.cam.ac.uk/outreach/Project-resources/Wind-turbine/howwingswork.pdf
 * Or John D. Anderson who states in Introduction to Flight:
 * ...differences in pressure from one point to another in the flow create forces that act on the fluid elements and cause them to move. Hence, there must be some relation between pressure and veloc�ity, and that relation is derived in this section.
 * and
 * The pressure distribution acting over the surface of the wing is the fundamental cause of lift.
 * Or Klaus Weltner
 * ...it is shown that the high streaming velocity at the upper side of the aerofoil is not the reason for the low pressure. To the contrary, the low pressure generated by the aerofoil is the reason for the high streaming velocity. []
 * The conventional explanation of aerodynamical lift based on Bernoulli’s law and velocity differences mixes up cause and effect. The faster flow at the upper side of the wing is the consequence of low pressure and not its cause.[]
 * I could go on, but I'll stop here. You are entitled to think these reliable sources get it wrong, and say so here on the talk page, but what we put in the article is based on reliable sourcing. The notion that forces cause acceleration is hardly out of the mainstream.  Curious what you've been reading to think that it is. Mr. Swordfish (talk) 14:57, 3 August 2023 (UTC)
 * We appear to be at cross purposes. I'm not aware of what it is that I have written to prompt you to quote Babinski, Anderson and Weltner. I don't believe I have ever challenged the notion that forces cause acceleration (your words) because that is exactly what Newton's FIRST law of motion is all about. Perhaps I wrote that forces cause acceleration is not Bernoulli's principle and you interpreted my words as doubting that forces cause acceleration.
 * You and I are both aware of many instances of eminent authors and institutions that have published documents advocating the equal transit time theory of aerodynamic lift. Despite the number of these documents Wikipedia is not seduced by the credentials of their authors. We repudiate the equal transit time theory. Well meaning attempts to simplify complex topics by resort to intuition, analogy and misplaced theories don't begin and end with the equal transit time theory. We must be willing to be alert at all times that something we see published may, in fact, be unreliable. Red warning lights should definitely be flashing when we see an annoyingly complex topic reduced to refreshing simplicity; or when an isolated attempt to explain a topic omits some of the essential elements used by all the acknowledged specialists and all mainstream publications. We must avoid being seduced by such strategies as denigrating the efforts of a large number of others, followed by a claim by the author that he (or she) has found a simple, easy-to-understand, explanation that anyone can comprehend. For example: "The conventional explanation of aerodynamical lift based on Bernoulli’s law and velocity differences mixes up cause and effect." I recall that this strategy was common in the equal transit time documents.
 * The conventional explanations of aerodynamic lift using Bernoulli's law say nothing whatsoever about cause and effect, so how can they mix them up? Why does Weltner even make this claim? I think it is a claim that is likely to catch unsuspecting readers, and perhaps that is why it appealed to Weltner. <i style="color: green;">Dolphin</i> ( t ) 13:55, 4 August 2023 (UTC)
 * Ok I’ve digested Eastwell’s paper and agree with everything in it. The leap was conflating what I wrote (molecules can only have so much velocity, and if that velocity is in the direction of flow, less velocity is available to make pressure) with what Eastwell wrote (Since perpendicular vectors are independent, any net force that changes the motion of air particles in one direction (e.g., speeds up the air) will have no effect on the speed of these particles in a perpendicular direction. ). Both statements are true, the difference being that I am not applying any work/force/acceleration to the molecules, while Eastman is.  This is key and many people do not understand it.
 * As to Swordfish’s other notion that the pressure drop is essentially caused by pressure drop in the flow direction, I agree there is truth to this too. But to Dolphin’s point, it is also not as easy as it looks to say one aspect is the cause of the other. My goal is to solve this apparent paradox.
 * I see we all agree there is very little out there explaining what happens on the molecular level. And I agree I can’t just write what is happening, that would be taken as opinion. So I’ll try to find something before editing the article… FWIW here there is a video on youtube, “Bernoulli’s Principle on Atomic Scale”, with good intentions, but it is still not 100% right / clear.
 * --Zojj tc 12:53, 3 August 2023 (UTC)
 * So I just found this:
 * It has talk of molecules and simulations. Anyone with access able to see if any of those are helpful? --Zojj tc 04:57, 4 August 2023 (UTC)
 * This url seems to bring up the article. []
 * Quickly scanning it, I'm not seeing much about statistical mechanics. There's some discussion of the applicability of the  model of the fluid as a continuum (which the author mistakenly calls the continuum hypothesis), but the author basically concedes that it's almost universally accepted.
 * This article [] may be what you are looking for but I don't have access to the full text.
 * I'll take some time to read it more carefully, along with the cited articles - I thought I had read most of what's out there, but the article contains cites to many resources that I'm unfamiliar with. It's certainly an excellent compendium of links to aerodynamics and lift articles, even if I disagree with some of his conclusions. Mr. Swordfish (talk) 15:57, 5 August 2023 (UTC)
 * Mr. Swordfish states: The notion that forces cause acceleration is hardly out of the mainstream. However, this statement – interpreting Newton's second law in terms of force being the cause and acceleration the effect – is not correct. See e.g. this NASA page, or the section "Determinism (or, what causality is not)" of Causality (physics). The same is of course true for Bernoulli's principle relating changes in velocity to changes in pressure etc. in a fluid flow under certain assumptions/conditions. -- Crowsnest (talk) 20:50, 12 September 2023 (UTC)
 * The NASA page: "Considering the momentum equation, a force causes a change in velocity; and likewise, a change in velocity generates a force. The equation works both ways." -- Crowsnest (talk) 06:14, 13 September 2023 (UTC)
 * I have no argument with the statement that "the equation works both ways". But the NASA article also states:
 * Example of force involving aerodynamics:
 * An aircraft’s motion resulting from aerodynamic forces, aircraft weight, and thrust.
 * IOW, the aircraft's motion is a result of the forces on it. The notion that forces result in acceleration is hardly "outside the mainstream of physics." Or perhaps you think NASA is outside the mainstream?
 * I am familiar with the idea that there are no causes and effects in physics, only mathematical equations that express the relationship among certain quantities. And sometimes that may be the best way to think about things.  But I don't know how common that school of thought is among physicists, and many mainstream reliable sources describe forces as causing acceleration, and that acceleration is an effect of the applied force.  See the quotes above from Anderson and Babinsky for instance.
 * Doug McLean offers criticism of the Weltner paper on page 280 of his book. It's an interesting critique, and I'm inclined to agree with McLean, but note that he doesn't claim that Weltner is wrong, only that the idea is "...not entirely correct...".  If Weltner was "outside the mainstream" I doubt McLean would bother addressing the paper; certainly nothing McLean has written could be interpreted as claiming Weltner was "outside the mainstream".
 * If there's some source that claims "The notion that forces cause acceleration is out of the mainstream" I haven't seen it yet. I'd be curious to read it, assuming one exists. Mr. Swordfish (talk) 22:00, 13 September 2023 (UTC)
 * Accelerations are always caused by unbalanced forces. That knowledge is found in Newton’s FIRST law of motion. It is not found in Newton’s second law, and it is not found in Bernoulli’s principle or any other principle of physics. We all know of many reliable published sources that associate this concept with Newton’s FIRST law, but I don’t recall any reliable source that associates it with Bernoulli’s principle. <i style="color: green;">Dolphin</i> ( t ) 22:51, 13 September 2023 (UTC)
 * The classic derivation of Bernoulli's equation (i.e. not using conservation of energy) is simply a direct application of Newton's second law to the unbalanced forces encountered in a flow field with changing pressure.
 * Holger Babinsky clearly explains this in English (and provides the math in the appendix) in his article "How do Wings Work"
 * If the particle is in a region of varying pressure (a non-vanishing pressure gradient in the $x$-direction) and if the particle has a finite size $l$, then the front of the particle will be 'seeing' a different pressure from the rear. More precisely, if the pressure drops in the $x$-direction ($dp⁄dx < 0$) the pressure at the rear is higher than at the front and the particle experiences a (positive) net force. According to Newton's second law, this force causes an acceleration and the particle's velocity increases as it moves along the streamline... Bernoulli's equation describes this mathematically (see the complete derivation in the appendix).
 * Mr. Swordfish (talk) 12:50, 14 September 2023 (UTC)
 * If we have a vector equation R = P x Q where R and Q are vectors, does R cause Q, or does Q cause R? I think our vector equation provides no information to assist with this question; if there is a causal relationship between R and Q, we must look elsewhere to find it. <i style="color: green;">Dolphin</i> ( t ) 13:50, 14 September 2023 (UTC)
 * If the particle is in a region of varying pressure ...
 * The important premise is already in this If, which assumes a non-interaction between the velocity field and the (independent) pressure field.
 * There are discussions going on for a very long time whether cause-and-effect are a subject of physics or of philosophy/metaphysics. Personally, I would prefer to stay away from this tricky subject regarding this article on Bernoulli's principle. The more since the dominant case is for application in steady flow, where time does not play a role in the dynamics (Bernoulli equation) – only in the kinematics through the relationships between the position, velocity and acceleration of a fluid parcel. Without time being a variable in the system dynamics, how can there be cause and effect? And to my opinion this is also in full agreement with Dolphin51's above argument. -- Crowsnest (talk) 22:01, 15 September 2023 (UTC)
 * From "Understanding Aerodynamics", Doug McLean (2012), p. 3:
 * "Another is that the basic equations define implicit relationships between flow variables, not one-way cause-and-effect relationships. Because of these difficulties, misconceptions have often arisen, and many of the physical explanations that have been put forward over the years have flaws ranging from subtle to fatal." -- Crowsnest (talk) 06:49, 13 September 2023 (UTC)