Talk:D'Alembert's paradox/Archive 1

Old discussions
What about the Superfluidity experiments on 4He? While "no drag" has not been observed, most other non-viscous elements have been observed, including no vortexes and no temperature hotspots.

This article is more or less incomprehensible to the layman, unfortunately. I think what's needed is an opening section that explains in non-mathematical terms what the paradox is, and what it means in practice, and what its consequences for the field of fluid dynamics were. While what's there is no doubt of value for the more enthusiastic or technical reader, at present it is not terribly useful in a general purpose encyclopedia. Graham 02:20, 18 August 2005 (UTC)

Yes, I agree. I have no idea what the bits of mathematical logic have to do with d'Alembert's paradox. I thought it had to do with zero drag in inviscid flow.

I agree too on the need for an opening section on what the paradox is before getting into an in-depth discussion over how it's been resolved, whether it's been resolved, and whether there's a controversy over whether it's been resolved or not.

I went to Wikipedia after reading that D'Alembert's paradox had "proven" that airplanes were impossible. Is that true? What does it mean? Whether or not the "parodox" still seems paradoxical, and no matter to what extent it's been satisfactorily resolved, the history of the effect it had in the years following its discovery seems the most important thing this page should describe.

Also, the article isn't neutral: apparently some people think there's no remaining controversy and the current author thinks there is. Somehow there has to be a neutral way to describe the situation rather than one guy laying out long technical arguments and documentation to bolster one position.

Also, if someone has resolved the paradox, that person should not be writing the article about it, it's a Wikipedia rule as I understand it. SteveWitham (talk) 23:40, 16 May 2008 (UTC)

---

You have correctly stated the crux of the paradox. Inviscid fluids do exhibit drag but the bits of mathematical logic in the peice show that this set of very useful equations predict that they will not. Yet the equations are properly derived and make useful predictions about real fluids behavior.

I found this page very very helpful - In fact I believe I may have derived a solution to the paradox. If I am correct it will be an important acheivement. I have written up my thoughts in a small article and I will be seeking to have this work published in an appropriate journal. In all events I will soon publish my thoughts here as well. Many thanks to the author of this article.

Sincerly,

Tony Gallistel A. Gallistel Innovation tgallist@aol.com

Sorry to break your bubble, but the paradox has already been "solved". It is even a matter of opinion whether this is a paradox at all. It is thought to be one because we expect to have drag when an object is in a moving fluid from our everyday life experience, yet the theory predicts that there is none. However, we can't blame a set of equations which are based on big assumptions (inviscid) for not representing the exact physical model and call the result paradoxal! it's like assuming that if gravity is neglected, a ball which is thrown accross a room will travel in a straight line, then carrying the experiment and finding out that it does not (although part of the result is still correct: the x velocity will be the same in both cases. Just like part of the result of assuming inviscid can still be correct)

New version
I am sorry to having implemented such a major change of the page without discussing it here first, but it appears that there has been some debate over the content of this paradox-page and I would like to help to make things more clear. I have made a complete rewrite (apart from the introduction) that hopefully will be helpful for anyone interested. I realize there are some mathematical terms that are not fully explained here or elsewhere in Wikipedia, but I intend to add material when neccessary. I also plan to add some pictures illustrating the article. The content in this article now also conforms with the German version of Wikipedia. Visitor22 09:37, 16 August 2007 (UTC)


 * The new version seems to be very biased towards the resolution of Hoffman and Johnson. The paradox is well known in fluid dynamics and I thought that Prandtl's resolution of the paradox is universally excepted. Apparently, Hoffman and Johnson put out a preprint last year in which they proposed a different resolution. This preprint has apparently been rejected by three journals . I doubt that Hoffman and Johnson's theory can be included in Wikipedia in the light of the verifiability and no-original-research polices. Even if it is to be included, the "undue weight" provision implies that it should be treated very briefly, in one sentence or so. -- Jitse Niesen (talk) 05:04, 18 August 2007 (UTC)

I read the policies and I think you are right; it seems that the idea is that the amount of text should somehow reflect the acceptance of the different views. I will edit and rearrange things the coming days to better follow this policy. As for the verifiability, the reference in the new version of the article is to the book (not the preprint), in which the underlying research (several articles referenced in the book) is published in well established journals in the field (Journal of Fluid Mechanics, Computational Mechanics etc.). So the underlying arguments (turbulent Euler solutions, computational method, etc.) are published and thus accepted by the scientific community, even though the consequences for the d'Alembert paradox is still under debate (as is clearly indicated in the new version of the article). Visitor22 07:52, 20 August 2007 (UTC)
 * I should add that apart from that, it is an immense improvement. What article on the German Wikipedia are you referring to? -- Jitse Niesen (talk) 05:13, 18 August 2007 (UTC)

Thanks! The german link (which I have nothing to do with) is: http://de.wikipedia.org/wiki/D%27Alembertsches_Paradoxon Visitor22 07:52, 20 August 2007 (UTC)

Also, I should add that I am the first author of the book referenced in the article (Hoffman), so I may of course be considered biased as a person. But my Wikipedia-article is based on (published) research that I have encoutered in my work, and I would be very happy to discuss the content of the present article with you or anyone else interested on this discussion page (or elsewhere). My main motives for updating this page is not to market my own research, but to carry out my duty as a researcher to communicate the present state of research in areas that I am familiar with, including my own findings, to the public. Visitor22 08:11, 20 August 2007 (UTC)

I have now minimized the material on the new resolution. Hopefully this gives a better balance to the article. Visitor22 10:02, 20 August 2007 (UTC)


 * First, despite the name, I don't really consider this a paradox at all. The "no drag with steady inviscid irrotational flow" is an provable mathematical result. The reason this doesn't agree with experiments is that the real flow isn't everywhere steady inviscid and irrotational.
 * It looks to me that this article is still lacking a full account of the accepted resolution: namely how even at high Reynolds numbers there's still a thin viscous boundary layer and this can lead to separation, with a low pressure region behind the body. It would also be good to include details of the agreement between boundary-layer theory and experiments in the case of slender bodies (where separation is not an issue). There must be lot of citable work out there to support the boundary-layer separation view -- lets see some more of that referenced. And some images would be really good too.
 * I haven't seen what's in the referenced book, but if it's anything like the preprint it leaves a lot to be desired in terms of good scientific argument. In the pre-print:
 * The numerical scheme is not fully described, nor is the effect of numerical diffusion considered. Therefore no weight can be given to the numerical results.
 * There is no satisfactory explanation for how vorticity is generated computationally in the inviscid flow -- something that is not permitted by the equations that are claimed to be being solved. There's some suggestion of unbounded velocity gradients, but how you expect to capture these numerically I'm not sure.
 * There's no evidence offered to suggest that the traditional explanation is unsatisfactory at explaining any experimental results, and hence no reason for a new explanation to be needed.
 * Since the Reynolds number doesn't appear in the system solved, the results must be independent of Re, if (as it is argued) viscous boundary layers are unimportant for $$Re \gg 1$$. However, experiments show a clear transition in behaviour between $$10^5$$ and $$10^6$$, which the results cannot be used to explain.
 * In conclusion, the article still seems very biased in favour of the Hoffman ideas, considering that they have yet to be accepted for publication in a peer-reviewed journal.
 * -- Rjw62 13:35, 23 August 2007 (UTC)

What would seem to be needed to support the commonly accepted resolution attributed to Prandtl, is original scientific work claiming to resolve the paradox, since Prandtl does not do so in his 1904 article. Otherwise, Prandtl´s resolution would not be suitable to present on Wikipedia because of its verifiability and no-original-research policies. The review article of Stewartson seems to indicate that no such original work is available up to 1981, and so far I have been unable to find any such work later either.

Regarding your concerns about the book; it is available for download for anyone to inspect. The points your are listing (numerical diffusion, drag crisis, vorticity generation etc.) are all clearly presented in the book as well as published by leading peer-reviewed journals of the field. Visitor22 07:18, 24 August 2007 (UTC)

About references to boundary layer theory: I have added an internal link to "Boundary_layer", and among the references in the current article are Stewartson and Schlichting. Visitor22 07:22, 24 August 2007 (UTC)


 * I'm not sure what you consider the actual 'paradox' that needs to be resolved to be, but in my mind it's the discrepancy between the mathematically rigorous steady Euler solution and what is observed in reality at large Reynolds numbers. The theory for a resolution is provided by Prandtl in his 1904 paper -- to claim otherwise is simply incorrect. He develops boundary layer theory, showing boundary layers can exist for arbitrarily large Ra, and that adverse pressure gradients can cause these layers to separate. As a result the limit as $$Ra \to \infty$$ does not have to correspond with the Euler solution. Separation can break fore-aft symmetry and a major way and allow a new pressure force on the body to be responsible for the observed drag. Even if Prandtl didn't feel in necessary to spell out all the details, there are numerous respected text books that do.
 * I've had a look through the book chapter and it appears quite similar to the pre-print. I stand by what I said above, and see nothing there (or in the referenced material) to address those issues (execpt a description of the numerical scheme). I'm also rather concerned about the claims in the book that steady potential solutions can not generate lift, and that fact that you've missed the whole class of lift-generating solutions for the model problem you consider in 10.6.
 * I'm also rather confused by your analysis of Stewarton's paper. I can't access the full text, but the abstract clearly states This three-pronged attack has achieved considerable success, especially during the last ten years, so that now the paradox may be regarded as largely resolved. I find it hard to believe that he would contradict this inside the paper. Moreover, I believe what he is referring to here is not just the essential theory of boundary layers and separation, but also a full understanding of the details of real flows. And even that he finds largely resolved.
 * Unless or until there is a peer-reviewed paper directly advancing your claims, I feel any suggestion in the wikipedia article that the drag can be accounted for without viscous boundary layers should be flagged as 'unproven' and confined to a couple of sentences at the most.
 * -- Rjw62 10:09, 24 August 2007 (UTC)

It seems that we agree on the definition of the paradox. I agree that Prandtl in his 1904 paper introduces boundary layer theory, for 2d steady laminar flow. Prandtl also presents a scenario for separation (p.6): with an adverse pressure gradient the flow may separate due to loss of kinetic energy in the boundary layers. This appears reasonable for laminar boundary layers up to Reynolds number Re of about 105 for e.g. a circular cylinder. Although for higher Re the boundary layers undergo transition to turbulence, which results in delayed separation and reduced drag, so called drag crisis. To get to Re=infinity one has to pass through turbulent boundary layers and drag crisis, so it appears that the boundary layer theory of Prandtl's 1904 paper (2d steady flow) is an over simplification for Re beyond drag crisis. And this theory also seems unable to explain the subsequent rise in drag for Re beyond drag crisis, reported in experiments.

On the other hand, in the book (Section 35.5-35.8) it is shown that it is possible to simulate drag crisis, including the rise in drag for very high Re, by parameterizing the boundary layer by simply a friction coefficient corresponding to the skin friction. This skin friction is then reduced towards zero corresponding to increasing Re, resulting in delayed separation (drag crisis) and the development of streamwise vorticity in the separation points (with increasing drag).

Discontinuous potential solutions is discussed in Section 3.3.

I have access to the full Stewartson paper, and it is clear to me that although he is satisfied with recent work within the area, he admits that it is still a long way to go to fully characterize boundary layer flow. In particular since unsteady flow and 3d flow is poorly understood. His statement that: "...the paradox may be regarded as largely resolved.", does not sound very convincing to me. Either the paradox is resolved or not. I also quote from his summary: "...Much remains to be done; in particular, the development of a rational theory for three-dimensional, and possibly also unsteady two-dimensional flows may be in its infancy....".

The book [4] is published with an established scientific publisher, and the results on the computational method, simulation of drag crisis etc. are published as peer-review papers in scientific journals, which is evident from the list of references in the book. Visitor22 14:01, 24 August 2007 (UTC)

Removed section: Boundary condition: slip or no-slip?
This section is original research and contradicts measurements, due to confusing skin friction coefficient with skin friction. See for instance:. The skin friction (or wall shear stress) τ is related to the far-field velocity U (or another characteristic velocity for the problem at hand), by:
 * $$\tau = C_f \frac{1}{2} \rho U^2,$$

with Cf the skin friction coefficient and ρ the mass density of the fluid. From experiments, the skin friction coefficient Cf decays with the Reynolds number as:
 * $$C_f \sim R_\text{e}^{-0.2},$$  with   $$R_\text{e} = \frac{U\,L}{\nu}$$

the Reynolds number depending on a characteristic length scale L and kinematic viscosity ν.

As a result, for a given object and fluid, i.e. L, ρ and ν constant, with increasing velocity U : as expected. — Crowsnest (talk) 10:06, 21 May 2008 (UTC)
 * the Reynolds number Re will increase: $$R_\text{e} \sim U,\,$$
 * the skin friction coefficient Cf will decrease: $$C_f \sim U^{-0.2}\,$$ and
 * the skin friction τ will increase: $$\tau \sim U^{1.8}\,$$,

I do not see that this section contradicts measurements: it should be clear that what is referred to is the skin friction coefficient, if the "coefficient" was missing that was a typo. What is relevant in terms of the d'Alembert paradox is the relative importance of the drag force from the pressure drop over the body compared to the drag force from skin friction. Thus you have 2 options: (i) either compare the skin friction coefficient Cf with the normalized drag coefficient Cd connected to the pressure drop, where Cd is about 0.5-1 and $$C_f \sim U^{-0.2}\,$$, or (ii) compare the skin friction "τ" with the actual drag force from the pressure drop that will increase as $$\sim U^{2}\,$$.

That is, I do not see that the section on slip vs no slip boundary conditions is either confusing nor misleading.

Visitor22 (talk) 15:08, 11 June 2008 (UTC)


 * This is the removed section "Boundary condition: slip or no-slip?", with the lines numbered by me:
 * Experiments show that the skin friction from a turbulent boundary layer decreases towards zero as Re-0.2 as the Reynolds number Re increases.
 * This indicates that for large Reynolds numbers (small viscosity) a slip boundary condition (or more generally a friction boundary condition with small friction), is a better model than a no-slip boundary condition.
 * Computational simulation of drag crisis supports this approach,
 * which opens entirely new possibilities for simulation of high Reynolds number flow without resolving very thin boundary layers.
 * This is contrary to Prandtl's claim that even for very high Reynolds numbers thin boundary layers need to be resolved (which is impossible) to get a correct drag.
 * This indicates that a correct resolution of d'Alembert's paradox can have important consequences also in applications.
 * References:
 * On which I have the following comments, refering to the above line numbers:
 * This has been addressed. If it should read "skin friction coefficient" instead of "skin friction", the form drag divided by the skin friction increases as Re0.2. This means that for very large to huge Reynolds numbers Re, say in the range 108–1010, skin friction is of the order of 1% of the form drag. Which is small, but may at all not be negligible with respect to the current claims regarding the d'Alembert paradox.
 * The conclusion that a slip (zero friction) or partial-slip (friction proportional to velocity to some power at the wall) boundary condition is better, is speculative. Especially with respect to the claims regarding the d'Alembert paradox.
 * I do not see how a complex computational model, which may contain bugs, numerical diffusion, round-off errors, etc., can support these claims with regard to mathematical-physical modelling.
 * Only in case of a slip boundary condition there is no boundary layer. Partial slip conditions also have a boundary layer to be resolved.
 * Prandtl did not claim that thin boundary layers need to be resolved (in a numerical model), only that "''In the thin transition layer, the great velocity differences will then produce noticeable effects in spite of the small viscosity." How these effects are brought into account, will depend on the solution method used (so also if there are problems with "resolving" the boundary layer).
 * This is also an unverifiable claim.
 * So in my opinion there is much more wrong with this removed section than the omission of the word coefficient in the first sentence. -- Crowsnest (talk) 17:18, 5 July 2008 (UTC)
 * So in my opinion there is much more wrong with this removed section than the omission of the word coefficient in the first sentence. -- Crowsnest (talk) 17:18, 5 July 2008 (UTC)

Good, this is a very important discussion. Let me address your points:
 * We agree that skin friction in problems of practical interest may be of the order 1%. For many applications this may be considered to be negligible, for example it is typically less than the experimental margin of error of the total drag (e.g. for the well known circular cylinder with experimental accuracy of a few percent). The most important cause for the dominating form drag is the separation of the flow, which determines the size of the wake, which gives the pressure drop and drag. A key question is then if it is possible to capture correct separation with slip boundary conditions. This is shown to be possible in the work cited in the article, with also in particular the following article to be published in Journal of Mathematical Fluid Mechanics this year: http://www.csc.kth.se/~jhoffman/archive/papers/dal-jmfm.pdf
 * That slip/friction bc is better is in the sense of computational efficiency: with no slip bc and full resolution of the boundary layer, computation of high Re flow is simply not possible whereas with slip/friction bc it is, as is shown in the references.
 * This is fully described in the references: basically it comes down to well-posedness of mean value output such as drag, using mathematical techniques of weak solutions and duality. That is, drag can be computed with reliable a posteriori error control (that is the result can be verified after having been computed from the computational model).
 * Yes, but this boundary layer does not attach to the wall with the same velocity as the wall (typically zero for a stationary wall), but instead just a few percent for high Re. This means that the corresponding boundary layer is not thin, and thus cheaply computationally resolvable.
 * But still Prandtl claims that there are "noticeable effects". Computing accurate drag using slip bc (without any boundary layer effects), in particular showing non-zero drag for inviscid flow past a cylinder without any boundary layer effects, shows that Prandtl's claim is wrong.
 * It is obvious that if you can use slip/friction bc to compute high Re flow in advanced applications, compared to the inability to solve high Re flow with resolved boundary layers in no slip bc, this has dramatic practical consequences.

Visitor222 (talk) 08:52, 10 July 2008 (UTC)


 * I am not going to comment on computational efficiency since this article is on d'Alembert's paradox: inviscid potential flow solutions to the Euler equations predict zero drag, which contradicts measurements finding substantial (form) drag — not reducing with increasing Reynolds number — while, in non-dimensional formulations, the Navier-Stokes equations converge asymptotically to the Euler equations with increasing Reynolds number.
 * Using the same numbering as above:
 * Although a skin friction of the order of one percent of the total drag is often not interesting from a practical point of view, it is important with respect to the solution of d'Alembert's paradox: the energy dissipation in the boundary layer attributes to the start of separation of the flow, which – after the separation points move upstream – leads to the much larger form drag. The article, cited by you, uses a numerical approach not solving the Euler equations but a "regularized" form including a heuristic non-isotropic dissipation. How such a numerical approach may lead to a solution of d'Alembert's paradox is unclear to me:
 * The viscous dissipation of the Navier-Stokes equations (whose formulation is based on measurements, physical and symmetry considerations) has been replaced by some other, heuristic (non-symmetrical tensor) form.
 * As is well known, the addition of dissipation in non-linear systems may have a destabilizing effect and lead to chaotic behaviour.
 * Perhaps the Navier-Stokes equations with slip boundaries will give similar numerical results. But this is all speculative and ignores the experimental fact of the occurrence of viscous boundary layers.
 * Better in the sense of computational efficiency is no argument with respect to the resolution of d'Alembert's paradox. There is no proof that a slip boundary condition is "better".
 * This only says that the code converges to some — reasonable looking or not — solution, not that it proofs the solution of d'Alembert's paradox, nor that a slip boundary condition is better than a no-slip.
 * Yes.
 * The numerical solutions are not inviscid, since explicitly some dissipation has been added. An option is, that any form of dissipation may trigger separation and lead to a reasonable prediction of form drag. Prandtl gave a theory, backed up by experimental evidence, that there is a viscous and turbulent, dissipative boundary layer at high Reynolds numbers. This becomes thinner with increasing Reynolds numbers, but does not vanish.
 * I agree. But how does this relate to the claim that "...a correct resolution of d'Alembert's paradox can have important consequences also in applications...". All kinds of CFD codes exist giving in many instances reasonable predictions of the flow characteristics.
 * Crowsnest (talk) 15:03, 29 July 2008 (UTC)


 * What do you mean by "the Navier-Stokes equations converge asymptotically to the Euler equations with increasing Reynolds number"? In what sense? Do you mean that solutions of NS converge to solutions of Euler? What type of solutions; smooth solutions has not been proved to exist for NS or Euler, and weak solutions have only been shown to exist for NS? And what about boundary conditions, since you argue on behalf of boundary layer theory I guess you cannot mean that the boundary conditions of NS (no slip) goes to Euler (slip)? In fact, in mathematics a leading opinion is that as Re goes to infinity (vanishing viscosity) a limit (weak) solution (if it exist: which has not been proved) is dissipative (http://www.iop.org/EJ/abstract/0951-7715/13/1/312).
 * Since smoothness of solutions of Euler cannot be proved, we cannot rule out that an initially smooth solution (like the potential solution) will develop into a non-smooth weak dissipative solution. The numerical method is constructed to allow computation of such weak dissipative Euler solutions, but also smooth solutions to Euler if they exist. Thus the breakdown of a smooth Euler solution (here the potential solution) into a weak (turbulent) Euler solution can be studied in a computer simulation with a posteriori error control. In these simulations for the cylinder you can find the mechanism for breakdown being the 3d vortex instability which trigger separation and generate drag. You can also find this mechanism of exponentially unstable streamwise vorticity by a standard linear stability analysis of the vorticity equations. This as mentioned connects to the idea of Birkhoff, that instability of the potential solution could be responsible for the paradox.
 * You seem very confident when you say that the 1 percent skin friction observed in experiments "attributes to separation", whereas my impression is that separation from a turbulent boundary layer is not very well understood in the community?
 * This is a fact for all non-DNS CFD codes solving NS for high Re (through turbulence modeling or numerical methods used). And since DNS is impossible for high (>10^4) Re this is always the case. There seem to be an agreement (as you also point out) that still important flow characteristics can be computed.
 * Yes. But here it is shown analytically (linear stability analysis) that even the inviscid system (Euler equations) is unstable.
 * There are many things that are ignored in the model (geometric deficiencies, non-Newtonian effects,...) but the question is what is critical to include in the model to capture the phenomena you are interested in. You argue the no-slip boundary conditions is vital, whereas this work claim it is not.
 * Ok. I just mean that this resolution have positive consequences for CFD simulation.
 * It shows that the computed solution is a well defined weak solution to Euler, which means that we can draw mathematical conclusions based on this object.
 * Not really, the added viscosity is such that it is identically zero for an exact stationary solution to the Euler equations (as the potential solution), and thus this dissipative term does not trigger any instability. The instability comes from inherit instabilit of the inviscid Euler equations.
 * I think that the statement "all kinds of CFD codes...give reasonable results..." is way too optimistic. In RANS and LES type codes there are modeling parameters that needs to be tuned (for the subgrid/turbulence model and the boundary layer model) to obtain results matching experiments, in particular for a new application without any experimental reference data I would say that CFD codes are not very reliable.
 * I think that the statement "all kinds of CFD codes...give reasonable results..." is way too optimistic. In RANS and LES type codes there are modeling parameters that needs to be tuned (for the subgrid/turbulence model and the boundary layer model) to obtain results matching experiments, in particular for a new application without any experimental reference data I would say that CFD codes are not very reliable.

Visitor22 (talk) 10:21, 13 August 2008 (UTC)


 * At the moment the discussion has shifted towards whether the Hoffman & Johnson material should or should not be included in the article, see . At the moment there are no convincing arguments to include this material in the article. Discussing the text of the removed section, being based on the research of Hoffman & Johnson, can wait until the discussion below is settled. -- Crowsnest (talk) 14:11, 13 August 2008 (UTC)

Garrett Birkhoff
Why the variant spelling of his name? Richard Pinch (talk) 11:06, 13 July 2008 (UTC)


 * I can see no valid reason to depart from the spelling used in the Wiki article of the same name - Garrett Birkhoff. I have rectified the error.  Dolphin51 (talk) 11:57, 13 July 2008 (UTC)

Vague reference?
Jitse Niesen removes a reference to a closely related published article claiming that the article is vague. The criticism of the article should be made more explicit to motivate the removal. —Preceding unsigned comment added by Egbertus (talk • contribs) 12:34, 5 August 2008 (UTC)


 * Well, there is already some discussion, see above. Further, a relation between the solution of d'Alemberts paradox and the solution of the Clay millenium prize problem on the existence and smoothness of solutions to the Navier-Stokes equations seems to be speculative, at best. So I support the removal of this off-topic reference, and removed it again. -- Crowsnest (talk) 13:20, 5 August 2008 (UTC)

Speculative? In what sense? In fact, showing blowup of vanishing viscosity solutions of the Euler equations is precisely what solves d'Alembert's paradox: It is shown that initially inviscid incompressible irrotational (smooth potential laminar) flow (with zero drag) over time breaks down into a non-smooth turbulent Euler solution (with substantial drag). So there is a very strong connection between the two problems, which motivates cross-reference, —Preceding unsigned comment added by Egbertus (talk • contribs) 13:55, 5 August 2008 (UTC)


 * What are "...vanishing viscosity solutions of the Euler equations..."?. The Euler equations, according to all renowned textbooks on the subject, do not have viscosity by definition.
 * Further, why do you stay re-reverting this edit, which was reverted by two editors; instead of discussing this on the talk page. That is where talk pages are for. Also be aware that it is not done to stay reverting edits, see WP:3RR. -- Crowsnest (talk) 14:24, 5 August 2008 (UTC)
 * The talk page is also meant to discuss controversial edits, in order to reach consensus, see WP:CONS. -- Crowsnest (talk) 14:30, 5 August 2008 (UTC)


 * (in response to the implicit question at the beginning of this section) My reason for thinking that the resolution in the paper is vague because it doesn't explain how non-existence for Euler (what the paper claims to give evidence for) implies non-existence of Navier-Stokes. There is a difference between zero viscosity and finite viscosity. There are probably other points where the paper is too vague to be called a proposed solution; for instance, as far as I can see, all simulations discussed in the paper are flow past an object while the Clay Institute problem has no boundary conditions. -- Jitse Niesen (talk) 14:34, 5 August 2008 (UTC)

I wellcome the discussion. More people should tak part. Euler solutions are defined as Navier-Stokes solutions as the viscosity tends to zero (vanishing viscosity), and turbulent Euler solutions as solutions with substantial dissipation. Now, initially smooth non-turbulent Euler solutions show breakdown to non-smooth turbulent Euler solutions, and thus show breakdown of Navier-Stokes solutions under vanishing viscosity. Of course the considered case with boundary is more relevant than without, but this is not essential for the breakdown. Smooth forcing can have the same effect. —Preceding unsigned comment added by Egbertus (talk • contribs) 15:31, 5 August 2008 (UTC)


 * Please explain why "breakdown of Navier-Stokes solutions under vanishing viscosity" implies breakdown of Navier-Stokes solution with finite viscosity. The equation ax = 1 does not have a solution when a vanishes but it does have a solution for all a > 0. Also, explain that smooth forcing can have the same effect as a boundary (i.e., give a reference to a theorem). -- Jitse Niesen (talk) 16:08, 5 August 2008 (UTC)

The question is relevant: First, the Euler equations with zero viscosity do not seem to admit any wellposed solutions. (A potential solution is a Euler solution with zero viscosity, but it is shown to be unstable and not wellposed in any sense.) This is like the equation ax=1 with a=0. Instead, viscosity solutions are sought, as limits of Navier-Stokes solutions with viscosity tending to zero. This is like solving ax=1 with a tending to zero. Now, NS solutions with small viscosity are observed to become turbulent (by computation), so what is sought is a limit of turbulent solutions. This is like the solution x=1/a tending to infinity. Pointwise limits of turbulent solutions cannot exist, but limits of certain mean-value outputs such as drag an lift are shown to exist (up to some tolerance), by duality based a posteriori error estimation. Mean-value outputs are thus shown to be wellposed under vanishing viscosity. It is like showing that the limit of the output 1/x exists and equals 0. As concerns boundaries or not, the real problem (both mathematical and physical) has boundaries, and the problem in the whole space or with periodic bc is only for convenience to eliminate possible effects of boundary layers. An essential aspect of viscosity solutions of Euler with slip bc is that boundary layers do not form, and thus is similar to a problem without boundaries. Egbertus (talk) 06:13, 6 August 2008 (UTC)


 * This is exactly why the paper cannot be called a proposed resolution of the Millennium Problem (which the paper doesn't claim to be). There are no proofs, only computational evidence and non-rigorous arguments. Even if we accept that the paper shows that the Euler equations with slip BC do not have a smooth solution, we still need a proof that it follows that Navier-Stokes on R^3 or R^3/Z^3 has no smooth solution. There is no proof in the paper, thus it can't be a resolution. -- Jitse Niesen (talk) 12:50, 6 August 2008 (UTC)

Dear Jitse Niesen: Fine, you express your opinion, but you are not the judge for the prize, I think. Isn't it reasonable to let the scientific community evaluate the evidence and come to a decision? Maybe you are right, or maybe not. If there is some evidence missing maybe it can be supplied, once it is clear what is missing. I am not aware of any other proposed resolution and instead of just waiting, why not scrutinize the arguments of the article and clearly express what is missing?Egbertus (talk) 14:09, 6 August 2008 (UTC)


 * Well, you certainly try to push your opinion. Let me focus on the paper:
 * The paper does not claim to be a proposed resolution to the Millenium Problem, while Hoffman and Johnson seem to be well aware of the conditions for the Clay prize and that they do not comply with these conditions.
 * Further, they solve in this paper neither the Euler equations nor the Navier-Stokes equations, but what they call the "regularized Euler equations" with some dissipation formulation involving a grid-size dependent and coordinate-system dependent shear stress (see e.g. Batchelor, 1967, An introduction to fluid dynamics, §3.3, pp. 141-144 on the basic requirements regarding the formulations of shear stress in an isotropic fluid in order to be independent of for instance a pure rotation with constant angular velocity).
 * Reference to this paper as solving either the Euler or Navier-Stokes equations is in conflict with the common nomenclature of fluid dynamics, as used in the scientific community.
 * The paper uses a homogeneous Neumann boundary condition for the tangential velocity $$u_\tau\,$$, i.e. $$\frac{\partial u_\tau}{\partial r} = 0$$. The no-slip boundary condition $$u_\tau=0\,$$ would have been appropriate for the Navier-Stokes equations, while for the Euler equations the curvature of the cylinder wall has to be taken into account leading to $$\frac{\partial(r u_\tau)}{\partial r} = 0$$ for the velocity component $$u_\tau\,$$ in the circumferential direction of the cylinder (see the potential flow solution, or Batchelor, 1967, p. 602), which is different. The used boundary condition generated vorticity, invalidating the numerical results with respect to all claims on vorticity instability of the potential flow solution.
 * The stability analysis in the paper, of the vorticity blow-up at the rear stagnation point, is not for the Euler equations, nor for the Navier-Stokes equations, but for a heuristic approximate formulation, also with assumptions on boundary conditions. In the incompressible Euler equations, the pressure enforces incompressibility, giving a delicate coupling between the components of the velocity vector. Further there are quite some delicacies involved in the interactions between the components of the vorticity field in vortex stretching. So, this stability analysis does not proof that the potential flow solution for a cylinder in uniform flow is unstable. Further the Euler equations have symmetries and conservation laws, e.g. energy conservation, which may easily be lost (possibly in the direction of instability) in such approximations.
 * The introduction of a small amount of dissipation in a non-linear dissipationless system may have a de-stabilizing effect, as is well known from chaos theory. The paper does not discuss this vital aspect with respect to the d'Alembert paradox at all.
 * Crowsnest (talk) 21:18, 6 August 2008 (UTC)

Dear Crowsnest: Yes, HJ push their opinions by publishing scientific papers in scientific journals of high standard. Maybe you should express your criticism similarly, instead of using wikipedia as some kind of blog for your opinions? Your points: HJ solve the Euler equations with slip bc augumented by residual dependent viscosity effectively enforcing a weak Neumann condition on tangential stresses. With slip as indicated the vorticity does not originate from the boundary condition. The computation shows the actual development of the instability at rear separation, and the analysis is for conceptual understanding of the phenomenon. You claim that introducing viscosity has a destabilizing effect. Where does this occur?Egbertus (talk) 06:34, 7 August 2008 (UTC)


 * Thank you for your advice on what I should do.
 * The destabilizing effects of small dissipation directly occur in many fluid dynamics problems and elsewhere, see e.g, and the discussion and references therein (also available here). The given references contain many examples of fluid flow where the inclusion of a small amount of dissipation changes the flow behaviour from marginally stable to unstable. Several of the given references are on interfacial instabilities, as also relevant for the present case (i.e. the plane through the cylinder axis where the flows along either side of the cylinder meet again). A model for this destabilizing behaviour of dissipation can also be found in the Duffing equation.
 * All points in my previous post of 21:18, 6 August 2008, can be backed up by reliable sources, like the one above. While there is only one source, i.e. the book and one published paper by Hoffman and Johnson (H&J), contradicting the consent in the scientific world that the reason for the occurrence of the d'Alembert's paradox lies in the the dissipation and vorticity production in the thin viscous boundary layers, as explained and measured by Prandtl. See e.g., Vol. 2, §41–5: The limit of zero viscosity, pp. 41–9 — 41–10.
 * The threshold for inclusion in Wikipedia is "...verifiability, not truth...", and "...Exceptional claims require exceptional sources...". See also Fringe theories. My opinion is shifting towards altogether removing these references to this work by H&J from here and in Navier–Stokes existence and smoothness, since their extravagant claims, especially in their book and the (draft) papers on their website/wiki, are not backed up by any other work in the field, on the contrary. The Navier-Stokes equations are well established by theory and experiments, and they solve numerically another set of equations containing an unproven sort of viscous dissipation which does not fullfill the basic requirements regarding coordinate-invariance (see my previous post; Batchelor, 1967, An introduction to fluid dynamics, §3.3, pp. 141-144). Further they apply the wrong slip boundary conditions at the cylinder, see also my previous post and the reference there.
 * The point raised by H&J, that the potential flow solution around a cylinder has 3D instabilities, may or may not be true. They give no rigorous proof of this in the referred (draft) papers and book. The only indication is the instabilities seen in the results obtained with there code, solving their own proposed model equations for fluid flow (not supported by theory or experiments). These instabilities may as well have a very different cause (code bugs, the used fluid flow model, the wrong boundary conditions).
 * I propose to minimize the amount of text and focus spent on these claims by H&J, or perhaps remove them alltogether, given the unbalance between the claims made — contradicting the general opinion on this subject that from a practical point of view d'Alembert's paradox became irrelevant by Prantl's theory and measurements on viscous boundary layers — and the lack of proof for these claims (refinement added later on) , as well as the doubts with respect to the applicability of the used flow models for solving these kind of fundamental problems.
 * Crowsnest (talk) 09:34, 7 August 2008 (UTC)

The work by HJ is new and challenges established truths. But it is published in refereed journals of high standard, an it is not contested or shown to be wrong in any published scientific work. It has been presented at several scientfic meetings with acceptance. It is natural that new ideas meet opposition, and science is about letting different viewpoints meet in open discusssion. It is natural to give a reference to the work by HJ on wikipedia, since it is the only published work claiming a resolution of d'Alembert's paradox.Egbertus (talk) 10:08, 7 August 2008 (UTC)


 * Like d'Alembert's paradox, i.e.:
 * "the dimensionless Navier-Stokes equations (having viscous dissipation) converge asymptotically to the Euler equations for increasing Reynolds number, yet the potential flow solutions to the Euler equations predict zero drag, contradicting experiments",
 * Hoffman & Johnson just introduce a new paradox, equally difficult to resolve, but not interesting from a physics point of view:
 * "the regularized Euler equations (having a grid-dependent dissipation) converge asymptotically to the classical inviscid Euler equations for finer grid size, yet they do not converge towards potential flow solutions of the inviscid Euler equations".
 * This does not solve anything, it just poses new and as difficult to solve problems. So there is no reason to include this material in WP at length. -- Crowsnest (talk) 11:09, 7 August 2008 (UTC)


 * This would be interesting material if it contained rigorous proof that the "real" inviscid incompressible Euler equations are unstable at the rear side of the cylinder. But such a proof is lacking in this reference, making it hardly relevant for the subject of this article. -- Crowsnest (talk) 11:13, 7 August 2008 (UTC)


 * I posted a request for contributions by other editors on Wikipedia Talk:WikiProject Mathematics and Wikipedia Talk:WikiProject Physics. -- Crowsnest (talk) 12:13, 10 August 2008 (UTC)

We should wait for general recognition that the HJ work solves D'Alembert's paradox
Egbertus wrote (on 7 August):"It is natural to give a reference to the work by HJ on wikipedia, since it is the only published work claiming a resolution of d'Alembert's paradox"We need more than just Hoffman and Johnson *asserting* that in their opinion, they have solved d'Alembert's paradox. We need publications by independent third parties in refereed journals agreeing that H & J have solved the problem. Lacking such confirmation, early-stage work like the HJ work probably does not belong in a general article like this one. EdJohnston (talk) 02:49, 11 August 2008 (UTC)


 * Not only that, it is particularly important to follow this standard Wikipedia policy (of independent third party confirmation), since one of the authors has acknowledged that they have engaged in self promotion of the work on this article. Incidentally, it's clear (and even more so if you read through the correspondence between Terrence Tao and Johnson) that all they have done is give some numerical evidence for a resolution.  This is standard practice in applied mathematics, and indeed, people can go and give many talks all over the work on the basis of such a work.  But it takes more than that before the entire applied community will acknowledge that the work is indeed a "resolution".  Unfortunately, it is much too common (and even expected) in the applied fields, that people will take meager evidence, drum it up as the end all solution, and then spend years arguing with other people before admitting they are wrong.  Thus it is imperative that there be independent confirmation by reliable sources.  The simple fact that they've published some articles and given some talks is definitely not enough.  By that criteria, we would be able to include a lot of bogus stuff into Wikipedia.  --C S (talk) 03:37, 11 August 2008 (UTC)

Dear CS: Of course one could wait until general consensus is reached, but I wonder if that really is the policy of WP. The article as it now reads without the HJ references does not make sense at all, since then there is no resolution whatsoever. The article should then just be two lines. Is this better? What science is about is to make progress, not to block progress. This can be made if different viewpoints and arguments can meet in an open discussion. WP can here play an important role by allowing controverisal subjects, as long as different viewpoints are included. So the choice is between (i) a one-line article stating that potential flow has zero drag (which is trivial), and (ii) an article giving information about both the resolution suggested by Prandtl and proposed by HJ. Which is best? It is notable that the article with HJ has been on WP for a year, without any notable criticism. Why remove HJ now without factual criticism of what they say? Are you sure that HJ are wrong? If so why? Do you know of anybody claiming that they are wrong? If not, why remove them from WP?Egbertus (talk) 06:07, 11 August 2008 (UTC)


 * Please see WP:NPOV (particularly WP:UNDUE), and WP:RS.  The other material does in fact satisfy these policies.  Wikipedia is not here to allow an "open discussion".  It is here to explain what discussion has taken place.  What discussion has taken place about H&J's work by independent parties, published in reliable sources?  Your choice reflects complete misunderstanding of policy.  I hope after reading the relevant pages, you will see there is a 3rd choice, which is the current state of the article.  It is not notable that the deleted material was here for a year.  Why would you think it is?  As for your last questions see WP:UNDUE.  There are several false proofs of famous conjectures on the arXiv.  No one has published criticisms of them, but we do not include these "proofs" into Wikipedia.  Simply put, they are so crappy it was not worth anyone's time to publish refutations.  The "due weight" policy is designed to take care of this by not including minor viewpoints that are so minor nobody has chosen to comment on them.  As far as I can see, the H&J work seems to fall into this category.  You are welcome to prove me wrong.  I look forward to seeing the significant published academic commentary on the H&J work by independent 3rd parties, so that we can choose to include their work into this article.  --C S (talk) 06:16, 11 August 2008 (UTC)

Dear CS: Note that the HJ resolution is accepted for publication in Journal of Mathematical Fluid Mechanics, which is the highest authority in the subject of fluid mechanics based on mathematics. It is thus accepted by the experts of the field. Does this count for nothing? How could it get accepted if it is all wrong? You description of the work by HJ shows that you have not understood/read the work by HJ. Is this a reason for removal? Egbertus (talk) 06:28, 11 August 2008 (UTC)


 * I didn't say it was wrong or even "all wrong". I said it gives "evidence" of a resolution.  Indeed, their language is more circumspect in the paper and abstract than in the title, isn't it?  They "propose a resolution" and give "analytical and computational evidence".  I also said people give "evidence" all the time in applied fields.  That doesn't mean everybody agrees that it is correct or the resolution.  Find some sources written by other people that say Hoffman and Johnson have resolved D'Alembert's paradox, or heck, even find any sources by other people that discuss their proposed resolution in a way which indicates they think it's important.


 * I find your claim as to the prestige of the journal surprising. It was started in 1999.  Is a publication in that journal the pinnacle of the subject of fluid mechanics?  I don't think so.  Perhaps to give me an idea...what is its impact factor?  When I see mathematicians like Chorin or Sethian publish, generally it's in something like Annual Review of Fluid Mechanics or PNAS.  These have extremely high impact factors.  But I'm getting distracted by your nonsequitor.  It really isn't important that there is a single publication by them in even a very prestigious journal.  In a field with high citation numbers, I would expect some citations and relevant discussions in other prestigious journals.  Where are they?  --C S (talk) 06:51, 11 August 2008 (UTC)


 * Also, see WP:REDFLAG: "...Exceptional claims require exceptional sources...". The paper has been rejected by Science and the Journal of Fluid Mechanics, which are both of higher prestige than the Journal of Mathematical Fluid Mechanics, see the authors wiki. -- Crowsnest (talk) 08:04, 11 August 2008 (UTC)

There is fluid mechanics and there is mathematical fluid mechanics, and in the second field I see no more prestigious journal. Do you? D'Alembert's paradox concerns mathematical fluid mechanics, right? If scientific publcation in refereed journals is to count at all in science, and I think it should, what more is required to merit a short reference in WP? You understand of course that opposition to the work by HJ is to be expected since it challenges established "truths". Instead of seeking various formal reasons to block the work by HJ from discussion, I suggest that you scrutinize it on scientific grounds and express factual criticism.Egbertus (talk) 07:23, 11 August 2008 (UTC)

As concerns the merits of HJ in mathematical fluid mechanics, you should consult experts in the field, since you seem to be unaware. Remember that you have responsibility by changing the WP article. Are you sure that you have enough expertize to do so?Egbertus (talk) 07:35, 11 August 2008 (UTC)


 * It is your responsibility to familiarize yourself with Wikipedia policies. I have explained their relevance and the appropriate links.  If you are unable to understand that we as editors are not supposed to individually verify facts, but only rely on using reliable sources, then I suggest you think things over until you do.  Whether I "scrutinize [the material] on scientific grounds" and "express factual criticism" is utterly irrelevant, as you would know if you had bothered reading the policy pages.  --C S (talk) 07:42, 11 August 2008 (UTC)


 * You forget that mathematical fluid mechanics is a sub-field of fluid mechanics. So most prestigious in the sub-class will in most instances not mean most prestigious in the wider class of journals on fluid mechanics. -- Crowsnest (talk) 08:26, 11 August 2008 (UTC)

To CS: A reliable source is an article in a refereed journal of high standard, such as Journal of Mathematical Fluid Mechanics. I cannot see how you can deny the existence of such as source for the HJ reference. Of course you do not have to read nor evaluate the work by HJ, but if you remove a reference from WP to a reliable source, you must express a reason to do so. What is your reason in clear words? Unless you express a clear reason the removed material will be reinserted.Egbertus (talk) 08:31, 11 August 2008 (UTC)

To Crowsnest: You must have expertize in a scientific discussion, right? I take it that your expertize is fluid mechanics, but not mahematical fluid mechanics, right?Egbertus (talk) 08:31, 11 August 2008 (UTC)


 * Yes and no. To be more precise: both on subjects of mathematical fluid mechanics as well as on other topics in fluid mechanics. Crowsnest (talk) 08:59, 11 August 2008 (UTC)

To CS: WP states: "neutral point of view (NPOV), representing fairly, and as far as possible without bias, all significant views that have been published by reliable sources." To suppress reference to HJ is thus a violation of NPOV, right?Egbertus (talk) 09:04, 11 August 2008 (UTC)

To Crowsnest: Congratulations.Egbertus (talk) 09:04, 11 August 2008 (UTC)

To CS: You should carefully read about Information Suppression on NOPV. Have you done that? Egbertus (talk) 10:49, 11 August 2008 (UTC)


 * I guess H&J is not (yet) a significant view. There must be hundreds of books that talk about d'Alembert's paradox and only the one of H&J themselves mentions their view.
 * I had a look at the home page of the Journal of Mathematical Fluid Mechanics and I couldn't find the article. Where can we find confirmation that it has been accepted for publication? -- Jitse Niesen (talk) 11:12, 11 August 2008 (UTC)

Dear Jitse: You guess that the reference is not (yet) significant. How do you judge its significance? Are you the judge? What about violation of WP rules about Suppression of Information? Would you remove the reference? The article has been accepted by chief editor Rolf Rannacher. Ask him for a confirmation. The discussion preceeding the acceptance is available on the book home page.Egbertus (talk) 12:08, 11 August 2008 (UTC)


 * "How do you judge its significance?" By looking at how many people refer to it and what they say about it. "Are you the judge?" Yes; all Wikipedia editors decide by consensus which views are significant enough to be included. "What about violation of WP rules about Suppression of Information?" These rules are irrelevant if a view is not significant enough, as explained by C S below. "Would you remove the reference?" I don't know; if there had only be one sentence on it and the article gave a fairer view of what people actually believe then I might not have bothered, but your relentless pushing invites a reaction. "The article has been accepted by chief editor Rolf Rannacher. Ask him for a confirmation." Private communication is not allowed under the Wikipedia rules. "The discussion preceeding the acceptance is available on the book home page." But the discussion to accept does not seem to be there; all I can see is acceptance subject to revisions which implies that changes have been made in the manuscript. -- Jitse Niesen (talk) 14:16, 11 August 2008 (UTC)

To Ed Johnston: Note that WP does not seek to present the truth, whatever that is, but a neutral point of view with possibly different opinions being represented according to WP:NPOV. Thus WP does not wait until everbody agrees on a topic, or even that a majority does. Science is not developed by majority decision, but by arguments based on reliable sources and scientific expertize, right?Egbertus (talk) 12:45, 11 August 2008 (UTC)

To CS: I undid your complete removal of any reference to HJ work because it violates NPOV. Please motivate the removal.Egbertus (talk) 13:14, 11 August 2008 (UTC)

To CS: You quickly repeated you violation of NPOV, without any motivation. This is getting serious. Please motivate why you suppress all ref to HJ work!Egbertus (talk) 13:20, 11 August 2008 (UTC)


 * It's natural that some passages may just jump out at you while you ignore other parts of the NPOV policy. That's quite common for people who have looked at the policy for the very first time.  For example, you choose to see any effort to remove what you want to include as "information suppression" so if you see a section labeled thus in the policy, naturally, you see our actions as being contrary to the policy.  Nonetheless, I think some patience on your part should be exercised.  Everyone else has, in order to educate you on the policies.  Indeed, all of us, have contributed for a while to Wikipedia (several years average), and have seen this NPOV policy grow, evolve, and be refined.  Common sense should tell you that we may know something you don't about contributing to Wikipedia.  Indeed, since you apparently have some personal stake in this, while we do not, I wonder why you think we are acting in concert to suppress information from Wikipedia.


 * The next section after "information suppression" is called "expertise". In it, it is written, "Coverage should also be roughly in proportion to the number of experts holding each view...Views held by only a tiny minority of experts may not merit inclusion."  Note it doesn't say that because these are experts that there views must be included.  Indeed, the acknowledgement that experts may think something to the contrary of other experts is implicit in the sentence.  So in other words, H&J can be experts, but if their view of their work is in the minority, and a tiny one at that, their views may not in fact "merit inclusion".   Both sections are in a larger section called ""space and balance".  They are meant to explain the principle of how we allocate appropriate space to balance out the different views in a way to reflect the larger body of existent knowledge.  With that in mind, it should be clear that extraordinary claims of having resolved a famous paradox, or having refuted the common establishment belief, requires more than just one set of author's claims.


 * With that, I have reverted your latest attempt to add back the information. Since you are in the tiny minority here of one, further attempts of this type will only be liable to bring sanctions upon you.  Do not misunderstand me: that's not a threat; there is no need to threaten, as I am in the majority.  I only say that as a warning that reversion wars in order to push your minority view will not get you very far.  Reflect on the policies and explain your reasoning in terms of it and you may sway someone.  --C S (talk) 13:28, 11 August 2008 (UTC)

Well CS, do you really believe that because you are in the majority of people without a solution of d'Alembert's paradox, that you can dictate over a minority which has a solution? Do you? If so, why is this not a violation of NPOV? Finally it comes down to scientific expertize: Who would you say have more expertize in mathematical fluid mechanics than HJ? I would like to ask these persons about their opinions about d'Alembert' s paradox and if they support the idea of removing the work by HJ from WP. Do you really believe that HJ represent a tiny insignificant minority of "one"? Are you sure that you are well informed?.Egbertus (talk) 14:07, 11 August 2008 (UTC)


 * I do not see a proven solution of d'Alembert's paradox, as already discussed above in . H&J just solve their own set of equations, named the regularized Euler equations, which are neither the Euler equations as used in the fluid dynamics literature (which are inviscid), nor the Navier-Stokes equations. I repeat my comment above, from 11:09, 7 August 2008 (UTC):


 * Like d'Alembert's paradox, i.e.:
 * "the dimensionless Navier-Stokes equations (having viscous dissipation) converge asymptotically to the Euler equations for increasing Reynolds number, yet the potential flow solutions to the Euler equations predict zero drag, contradicting experiments",
 * Hoffman & Johnson just introduce a new paradox, equally difficult to resolve, but not interesting from a physics point of view:
 * "the regularized Euler equations (having a grid-dependent dissipation) converge asymptotically to the classical inviscid Euler equations for finer grid size, yet they do not converge towards potential flow solutions of the inviscid Euler equations".
 * This does not solve anything, it just poses new and as difficult to solve problems. So there is no reason to include this material in WP at length.


 * So, to my opinion, H&J work does not contribute to the resolution of the d'Alembert paradox. They only suggest a mechanism through which the (inviscid) Euler equations might become unstable (if they are at all unstable for this flow), but proof is lacking. Crowsnest (talk) 14:40, 11 August 2008 (UTC)

The page has some correspondence between J and Tao (an expert in this area). Tao appears to be rather skeptical of the claims made by J. The work by H&J should not be included on wikipedia unless it is confirmed by multiple reliable sources. The account Egbertus has no edits except for attempts to promote the work of H&J. R.e.b. (talk) 14:52, 11 August 2008 (UTC)

To Crowsnest: Your opinion of the HJ work is of no interest as long as it does not have a reliable source. This is a basic principle of WP. What is your reliable source? To R.e.b: How many reliable sources are required? I refer to certain reliable sources and it is not my responsibility to cover all possible sources. This is a basic idea of WP as a joint project of many editors. Egbertus (talk) 15:37, 11 August 2008 (UTC)
 * More than zero. R.e.b. (talk) 16:00, 11 August 2008 (UTC)

To R.e.b: Clever answer. Is the HJ work more than zero as reliable source? If not, what is your reliable source for this conclusion?Egbertus (talk) 16:25, 11 August 2008 (UTC)
 * More than zero reliable sources, that are independent of H & J, must express the opinion in a refereed article that H & J have solved d'Alembert's paradox. EdJohnston (talk) 19:17, 11 August 2008 (UTC)

To Ed Johnston: Is this your own interpretation or can you give source in WP policy?Egbertus (talk) 06:13, 12 August 2008 (UTC)

To Ed Johnston stating that "It is not enough that the author decides to award himself the victory. We require outside confirmation". Maybe for a famous open problem such as d'Alembert's paradox, one could make a distinction between (i) published proposed/conjectured solutions and (ii) commonly accepted solutions. The HJ work would then fall into category (i). The alternative is to mention nothing on WP until (ii) is available, but most references should then be deleted, including those to Prandtl. Would that be better? A very short article stating the paradox. What do you think?Egbertus (talk) 08:50, 12 August 2008 (UTC)


 * One could also make a distinction between (a) work that has seen a lot of discussion in the literature, with many agreeing that it provides at least a partial resolution and (b) work that has seen no independent discussion in the literature. I'd argue that that is the relevant distinction here. -- Jitse Niesen (talk) 12:37, 12 August 2008 (UTC)

To Crowsnest: You seem insist that the paradox has been resolved. What is the reliable source for the resolution? You say that it is not Prandtl. Who is it then? You must understand that this utterly important and requires a response, not just simple deletion. Where is a resolution published?Egbertus (talk) 13:57, 12 August 2008 (UTC)


 * From a practical and physics point of view, the paradox has been resolved with Prandtl's work on boundary layers. After that, not much work on this was done any more. From a mathematical point of view: there is no resolution available. -- Crowsnest (talk) 07:04, 13 August 2008 (UTC)

I shortened the presentation of Prandtl, because he does not claim to have resolved the paradox.Egbertus (talk) 14:49, 12 August 2008 (UTC)

To Jitse: What you are asking for seems more relate to experimental science, than mathematical. To repeat a published proof or computation is not considered to be publishable. Or is it? To make a publication relating to another publication, you must either make an improvement or prove that something is wrong. Right?Egbertus (talk) 15:10, 12 August 2008 (UTC)


 * There are journals where such discussion would appear, such as Notices of the American Mathematical Society. They might also occur in a survey article. And the original paper will eventually have a review. But until such references appear the original article is not notable. Ozob (talk) 22:03, 12 August 2008 (UTC)

To Ozob: You do not seem to understand the crucial role of refereed journals in independent checking of research before publishing. Discussions or review articles is something different which reilies on refereed publications.Egbertus (talk) 06:06, 13 August 2008 (UTC)

To CS, Crowsnest, Jitse et al: By suppressing/deleting any ref. to HJ work, even the most minute, you violate the WP rule of NPOV information suppression: "Editing as if one given opinion is "right" and therefore other opinions have little substance: Entirely omitting significant citable information in support of a minority view, with the argument that it is claimed to be not credible. Ignoring or deleting significant views, research or information from notable sources that would usually be considered credible and verifiable in Wikipedia terms (this could be done on spurious grounds)". Don't you see that? To delete as you are doing you have to give evidence that HJ work published in articles in refereed journals and books is insignificant and not credible. What is your positive evidence that it is insignificant and not credible? Just ignorance is not sufficient as evidence. What reliable source says that HJ are wrong? The reader of the present version of the d'Alembert article must get the impression that nothing of significance has happened in fluid mechanics for 250 years, since its most basic paradox has not been resolved at all. Is this good, and what you seek to display in your energetic deletion on WP? Egbertus (talk) 06:06, 13 August 2008 (UTC)


 * Again you seem to highlight only passages in a narrow way, while ignoring certain key terms, such as "significant". It is not up to us to demonstrate something is insignificant.  As you must be aware (and I mentioned it before), insignificant works are often not described in other sources, by virtue of being so insignificant they are not worth commenting on.  Thus for insignificant works, often the only evidence they are insignificant is lack of independent sources discussing them.  If you think H&J's work is significant, as demonstrated by independent 3rd party sources (as explained to you over and over), then you need to demonstrate that.  You have not even been able to produce one, which is the very minimal standard of what would "usually be considered credible and verifiable in Wikipedia terms".  By the way, since Hoffman has apparently terminated his part of the discussion above and you coincidentally appeared only days later, is he aware of your aggressive promotion efforts on his behalf?  This entire talk page discussion thus far doesn't make him look particularly distinguished.  If you are truly concerned on his behalf, consider dropping your promotional efforts.  --C S (talk) 07:18, 13 August 2008 (UTC)


 * Personally, it is not my intention to suppress all references to H&J's work. With regard to the N-S smoothness and uniqueness, I think it is off-topic (see there). With regard to d'Alembert's paradox: in my opinion this does not resolve the paradox; but what may be interesting is the proposed possible 3D mechanism of instability (which lacks rigorous proof, either of instability or stability). So when there appear peer-reviewed articles on this, or other independent peer-reviewed articles supporting the findings of H&J, these H&J references can be re-inserted. -- Crowsnest (talk) 06:57, 13 August 2008 (UTC)

Very good Crowsnest, but what you do is effectively deleting any reference to published refereed work by HJ. This is violation of NPOV. Don't you see that? Again: your opinion on the HJ work is irrelevant as long as it is not based on a reliable source. It is good that you see something of interest. Maybe you fill find more....Egbertus (talk) 07:08, 13 August 2008 (UTC)


 * As stated before: exceptional claims require exceptional sources. The H&J book and paper(s) boost claims which are not supported by strong evidence. On the contrary, for instance for the cylinder they find a drag coefficient of 1.0, while experiments find 0.7 (at Re=1·107) (see e.g. Roshko, J. Fluid Mech. 1961, p.345; Batchelor, Introduction to fluid dynamics, 1967, p.341), with no indication that the drag coefficient should increase towards 1.0 for higher Reynolds numbers). So, again, the H&J results do not seem to be in agreement with measurements. And they do not discuss these discrepancies, they just claim a: "...drag coefficient ≈ 1.0 in accordance with experiments for high Reynolds number flow (beyond the drag crisis)..." in the BIT paper. Without references or further comment.
 * Until there is evidence in accordance with the claims made, I oppose inclusion. Or if there was a big discussion going on (not here on WP) in the scientific community on this work, making it notable enough to be encyclopedic. -- Crowsnest (talk) 07:47, 13 August 2008 (UTC)


 * So again, like in the d'Alembert paradox on the discrepancy between potential theory giving zero drag and on the other side experiments giving drag, the H&J work is also in discrepancy with experiment. It does not solve the paradox, nor are the numerical code results in good agreement with experiments (despite the claims made). -- Crowsnest (talk) 07:56, 13 August 2008 (UTC)

Again Crowsnest: You are not the judge for the scientific discussion and decides what is good or bad. OK? HJ have reliable sources and are well respected world leading experts and cannot be defined as insignificant. Are you credentials really superior to those of HJ, since you act as if they are?Egbertus (talk) 08:10, 13 August 2008 (UTC)


 * Well, if you think George Batchelor is not a leading expert on fluid dynamics, who is? -- Crowsnest (talk) 08:17, 13 August 2008 (UTC)
 * Before, you did not even think a reference of Richard Feynman on the subject relevant. -- Crowsnest (talk) 08:19, 13 August 2008 (UTC)

Oh, so you are Batchelor? Did Feynman resolve the paradox?Egbertus (talk) 08:30, 13 August 2008 (UTC)


 * No, but the H&J book and BIT paper are in contradiction with experimental facts, while they claim to be in agreement with these facts, without giving references or further comment on the discrepancies. That makes them questionable sources, WP:QS, in this respect. -- Crowsnest (talk) 12:17, 13 August 2008 (UTC)

To EdJohnston: I ask you to look into NPOV suppression of information as concerns the consistent deletion of any reference, however minute, to the work by Hoffman and Johnson published in refereed journals. Is this not violation of NPOV according to your experience of how WP should work? I would appreciate your assistance very much .Egbertus (talk) 08:45, 13 August 2008 (UTC) epxer


 * WP:NPOV, 2nd sentence:
 * "...All Wikipedia articles and other encyclopedic content must be written from a neutral point of view (NPOV), representing fairly, and as far as possible without bias, all significant views that have been published by reliable sources..."
 * So if sources can not be qualified as being significant and reliable, it does not violate NPOV to not include them, on the contrary. -- Crowsnest (talk) 14:42, 13 August 2008 (UTC)

Very good Crownest: Now it is up to you give evidence that articles published in the refereed high-quality journals JMFM and BIT are not reliable sources. What is the evidence? Egbertus (talk) 15:03, 13 August 2008 (UTC)


 * You turn things around: the proof is to show that the BIT paper (the JMFM article is not published yet) is significant and verifiable. And verifiability with respect to exceptional claims requires exceptional sources. So it is up to you to proof this material should be included. -- Crowsnest (talk) 15:18, 13 August 2008 (UTC)
 * See WP:PROVEIT. -- Crowsnest (talk) 15:20, 13 August 2008 (UTC)

What is so exceptional? The JMFM article is accepted. Two articles on the same theme in two different high-quality journals. What more can you ask. That you like it?Egbertus (talk) 15:37, 13 August 2008 (UTC)


 * The JMFM article is not published, it is not on the JMFM website.
 * The BIT article refers to the JMFM article as having the title: "Resolution of d'Alembert's paradox", also the title of the draft manuscript for JFMF on the web site of J&H. It is an exceptional claim, to say you have solved a 250 year old paradox.
 * -- Crowsnest (talk) 15:53, 13 August 2008 (UTC)
 * The H&J findings are in conflict with long-established experimental facts, i.e. the drag coefficient (see before), while stating to be in a agreement (without giving any references or comment). That questions the reliability of this work on the subject of the d'Alembert paradox, which is about drag. In such a case, reliable secondary sources are required, see WP:PSTS. Which are lacking. -- Crowsnest (talk) 16:03, 13 August 2008 (UTC)

As I said JMFM has accepted the article, see book page for acceptance letter by Rolf Rannacher. If you don't believe it ask him. Good to know that HJ have solved an exceptional problem. But maybe exceptional care should also be taken in suppressing it. You misunderstand the rules of WP. If you question the results of HJ, write an article about and publish, e.g. in JMFM.Egbertus (talk) 16:17, 13 August 2008 (UTC)


 * Is there an echo in here? If you are unable to understand the policies even after having it explained countless times, the discussion is at an end.  We have done all we could here.  Good luck in the real world getting your work accepted.  Don't bother trying to force Wikipedia articles and policies to work to your benefit though.  Unless you have anything new to add, don't be surprised if you get no more responses.  Expect, however, that people will still be keeping an eye on things here.  --C S (talk) 16:47, 13 August 2008 (UTC)

That's fine CS. Maybe the real world is what counts. Besides, are you happy with the article now, which I have written? What was your contribution?Egbertus (talk) 18:55, 13 August 2008 (UTC)

Stability of axi-symmetrical potential flow
I was wondering whether a reasoning along the following lines can be included used (changed later on) with respect to quotes from Birkhoff, or whether it is original research: -- Crowsnest (talk) 10:07, 18 August 2008 (UTC)
 * 1) Due to Kelvin's circulation theorem, the circulation around any closed material contour moving with the flow — satisfying the inviscid Euler equations — is a constant.
 * 2) For an object moving with uniform velocity through a fluid which is at rest at infinity, the initial circulation of any closed material contour is zero, corresponding with potential flow.
 * 3) Considering all possible closed material contours, for axi-symmetrical flow (for instance a sphere moving through the fluid) material contours can wrap around the sphere at a certain moment. As a result they also encircle the streamline emanating from the rear stagnation point.
 * 4) As a result, for this axi-symmetrical flow, the circulation has to stay zero everywhere: there are no ways vorticity can be generated. So it stays a potential flow.
 * 5) For a potential flow, a single-valued potential flow solution has minimum kinetic energy, due to Kelvin's minimum energy theorem (Batchelor, 1967, p. 384).
 * 6) In the considered flow, there is no external forcing and the energy in the system consists solely of kinetic energy.
 * 7) Then, starting from the steady potential flow solution at t=0, this has to stay a potential flow and has minimum energy, meaning it is a stable flow.
 * 8) From d'Alembert's paradox: the drag is zero.
 * 9) From measurements for a sphere: there is substantial drag (Batchelor, 1967, p. 341).
 * 10) So Birkhoff's proposal that instability of inviscid flow may be the cause of the paradox, is at least not valid for a translating sphere and many related (axi-symmetric) objects of finite 3D extent.
 * 11) For an object of infinite extent, such as a circular cylinder, material contours cannot completely wrap in every way the object. They cannot cross or encirle the stream surface emerging at the rear from the stagnation points. So this stream surface may in principle develop a jump in potential and tangential velocity (called a vortex sheet). In this case the stability of the Euler equations with respect to an initial potential flow solution has to be proven yet (if not already done so).


 * I'm afraid that that's a bit too OR (at least in my opinion), especially if you're using that in the article to criticize Birkhoff. On the other hand, how much emphasis to put on Birkhoff's remarks and whether it should be mentioned at all is an editorial decision, and we can use reasoning like this to decide to downplay his remarks, as C S suggested.
 * On the technical level, I am wondering whether you can really say there is no forcing if you drag an object through the flow. -- Jitse Niesen (talk) 12:00, 19 August 2008 (UTC)


 * I agree it is too OR to be included. With regard to external forcing: the only external forcing needed is to create the initial potential flow at t=0. Thereafter, there is no drag due to the paradox and no net force on the object. A noticeable effect of the passage of the object is the Darwin drift (closely related to added mass, see e.g. Brooke Benjamin, JFM, v. 169, pp. 251-256, 1986): after the passage of the object fluid particles are displaced with respect to their initial positions. But still, in such a potential flow there is no potential energy and only kinetic energy, which is minimum. -- Crowsnest (talk) 21:15, 19 August 2008 (UTC)

Of course you should suppress Birkhoff if you suppress HJ. Simple logic. Go ahead.Egbertus (talk) 13:20, 19 August 2008 (UTC)

The Knol article by CJ now lists 3rd on Google searching on "D'Alembert's paradox" with quotation marks. The reader can thus compare the Wikipedia article with the Knol article, and see that the WP is a very bleak copy of the Knol article, so bleak that it contains no information, just a hand-waving defense of an official resolution without substance. The Knol article will soon rank 1st, and I therefore put in a link at WP to Knol, since there is a link at Knol to WP. I hope that this time Crowsnest can keep fingers away and not again delete the link in a consistent removal of every trace of HJ on WP.130.237.251.226 (talk) 08:14, 9 September 2008 (UTC)

Birkhoff quotes
There are two quotes from the book ''Hydrodynamics. A study in logic, fact and similitude'' (1950) by Birkhoff in the article: Unfortunately, page numbers or section numbers are missing in these quotes.
 * 1) "...I think that to attribute them all to the neglect of viscosity is an unwarranted oversimplification The root lies deeper, in lack of precisely that deductive rigor whose importance is so commonly minimized by physicists and engineers...".
 * 2) "the concept of a "steady flow" is inconclusive; there is no rigorous justification for the elimination of time as an independent variable. Thus though Dirichlet flows (potential solutions) and other steady flows are mathematically possible, there is no reason to suppose that any steady flow is stable...".

Reading the second revised edition of 1960 (I do not have the first edition from 1950), I cannot find these quotes. Closest to the first quote comes, on page 5:
 * "...It is now usually claimed that such paradoxes are due to the differences between “real” fluids having small but finite viscosity, and “ideal” fluids having zero viscosity. Thus it is essentially implied that one can rectify Lagrange's claim, by substituting “Navier-Stokes” for “Euler”. This claim will be discussed critically in Ch. II; it may well be correct in principle for incompressible viscous flow. However, taken literally, I think it is still very misleading, unless explicit attention is paid to the plausible hypotheses listed above, and to the lack of rigor implied by their use. Though I do not know of any case when a deduction, both physically and mathematically rigorous, has led to a wrong conclusion, very few of the deductions of rational hydrodynamics can be established rigorously. The most interesting ones involve free use of Hypotheses (A)-(F)..."

The Lagrange claim is given by Birkhoff on page 3: "...One owes to Euler the first general formulas for fluid motion ... presented in the simple and luminous notation of partial differences ... By this discovery, all fluid mechanics was reduced to a single point of analysis, and if the equations involved were integrable, one could determine completely, in all cases, the motion of a fluid moved by any forces..."

The above is very different in its implications from the original one, as is in the article now. I cannot find the second quote at all. And the section on the d'Alembert paradox just states the paradox, without any criticisms on Prandtl or talking about possible instabilities in the Euler flow. Since this is the second revised edition of the book from 1960, I have to conclude that Birkhoff changed or removed them purposely (perhaps also influenced by criticism as given by e.g. Stoker, see article). Either because they were (in the 1st edition) in a form easily to be misinterpreted, or because he changed his opinion.

I intent to remove this section altogether. The Stewartson quote can go to a footnote, and be used in the lead to support the general view that the paradox is explained along the lines given by Prandtl. The Stoker critique can be removed, since it was on the 1st edition of the book. A reference to Birkhoff (his section on d'Alembert's paradox) can be included. -- Crowsnest (talk) 06:50, 21 September 2008 (UTC)


 * Good call. Haukur (talk) 11:49, 21 September 2008 (UTC)

As I said before, of course you must erase the criticism by Birkhoff, since you are suppressing the new resolution showing that Birkhoff was on the right track. But notice that the Knol article by Johnson ranks 2nd on Google. There is no reason that the WP article should rank 1st, since it contains no information of value, and Google will soon downgrade it. And what will then the WP value be? Also read the Knol article Wikipedia Inquisition appearing on wikipedia review.Egbertus (talk) 08:29, 23 September 2008 (UTC)

Found the Birkhoff quotes (in his 1950 edition)
I found the first edition of Hydrodynamics in a library, the one dated 1950. The first quote is on page 4, and it's just as stated in the article now, except it begins with 'Moreover':
 * "Moreover, I think that to attribute them all to the neglect of viscosity is an unwarranted oversimplification The root lies deeper, in lack of precisely that deductive rigor whose importance is so commonly minimized by physicists and engineers." [page 4, Birkhoff(1950)].

The second quote is from page 21, though the passage given above actually starts in the middle of a sentence. Here is the entire context:
 * "In any case, the preceding paragraphs make it clear that the theory of non-viscous flows is incomplete . Indeed, the reasoning leading to the concept of a "steady flow" is inconclusive; there is no rigorous justification for the elimination of time as an independent variable. . Thus, though Dirichlet flows and other steady flows are mathematically possible, there is no reason to suppose that any steady flow is stable. " [page 21, Birkhoff(1950)]. (I underlined the passages that were in italics in the original).

It's funny to suppose that Birkhoff changed his mind in the second edition (1960). The top quoted passage (above) seems like a good summary of his entire work, and it's not clear why he'd want to take that back later. And surely the feelings of the physicists won't be hurt by saying they lack deductive rigor. (Some physicists seem to pride themselves on it).

It sounds like our WP article is taking the position criticized by Birkhoff in 1950. We say: The occurrence of the paradox is due to the neglect of the effects of viscosity. EdJohnston (talk) 01:25, 24 September 2008 (UTC)


 * Good work! A question I have is: is the 1st quote in agreement with the text in the WP article? Is he expressing "a clear doubt in their official resolutions"? And what is the "deeper root" pointing at? In the 1960 revised edition, Birkhoff gives in Chapter I (one) examples of paradoxes arising in inviscid theory (Euler equations and potential flow theory), and in Chapter II of paradoxes arising in viscous theory (Stokes flow and Navier-Stokes equations). The main statement he tries to make, as I read his book (from the start of his preface in "Hydrodynamics. A study in logic, fact, and similitude", 1960, 2nd revised ed.), is:
 * "The present book is largely devoted to two special aspects of fluid mechanics: the complicated logical relation between theory and experiment, and applications of symmetry concepts. The latter constitute “group theory” in the mathematical sense.
 * The relation between theory and experiment is introduced in Chapters I and II by numerous “paradoxes,” in which plausible reasoning has led to incorrect results..."
 * Showing that also in viscous flows paradoxes arise due to the use of seemingly "plausible intuitive hypotheses" added to the mathematical problem description in order to be able to obtain results. Birkhoff's focus (in the 1960 version) is not on the resolution of the mentioned paradoxes. He uses the paradoxes to show that you can get paradoxes starting from seemingly reasonable hypotheses, and that adding viscosity is not the panacea to prevent the occurrence of paradoxes. -- Crowsnest (talk) 20:29, 25 September 2008 (UTC)

You are right: The WP official resolution based on neglect of viscosity lacks evidence, as observed by Birkhoff. The lack of evidence is also admitted in the article. Why is WP presenting this form of desinformation?Egbertus (talk) 20:33, 24 September 2008 (UTC)
 * I'd be happy to replace the present bald statement with a cited one: 'Some authors attribute the paradox to neglecting the effects of viscosity..' but then we'd have to agree on who some authors would be. EdJohnston (talk) 21:22, 24 September 2008 (UTC)

These authors do not exist. Only a facade without body exists. Interesting that EdJ asks questions. But as long as Crowsnest controls WP fluid mech only the facade is allowed to be presented. Is this what WP readers want?Egbertus (talk) 05:41, 25 September 2008 (UTC)

Crowsnest enthusisastically enters Good work! in the middle of the discussion. But Crowsnest does not follow the suggestion of EdJ to name an author claiming that neglect of viscosity is the root of the paradox. If the "general view of the fluid dynamics community" is that this is the root, there must be several members of that community who are willing to make this claim in public. Who are they? The World reading WP demands an answer! And I do as well.Egbertus (talk) 12:44, 30 September 2008 (UTC)
 * I added some references. The Schlichting book is also in the references list of Johnson & Hoffman on their knol and in their draft d'Alembert paper, so Egbertus could have added that one himself. Here from the Introduction of the 2000 edition of Schlichting (and Gerben):
 * "...Prandtl (1904) showed how a theoretical treatment could be used on viscous flows in cases of great practical importance. Using theoretical considerations together with some simple experiments, Prandtl showed that the flow past a body can be divided into two regions: a very thin layer close to the body (boundary layer) where the viscosity is important, and a remaining region outside this layer where the viscosity can be neglected. With the help of this concept, not only was a physically convincing explanation of the importance of the viscosity in the drag problem given, but simultaneously, by hugely reducing the mathematical difficulty, a path was set for the theoretical treatment of viscous flows.
 * One particular property of the boundary layer is that, under certain conditions, a reverse flow can occur directly at the wall. A separation of the boundary from the body and the formation of large or small eddies at the back of the body can then occur. This results in a great change in the pressure distribution at the back of the body, leading to the form or pressure drag of the body.
 * --Crowsnest (talk) 15:20, 30 September 2008 (UTC)
 * --Crowsnest (talk) 15:20, 30 September 2008 (UTC)

Both Prandtl and Schlichting have passed away, as has Birkhoff. There must be some members of the fluid dyn community who are still alive.Or?Egbertus (talk) 15:52, 30 September 2008 (UTC)
 * According to which guideline or policy have references to be by people who are still alive? -- Crowsnest (talk) 20:19, 30 September 2008 (UTC)

If Prandtl had been alive, he could have stated if he had solved the paradox and on what grounds. But he did not say so in his 1904 paper, when he was alive. If there is today no prominent member of the fluid mech community, who can speak for himself and express that the there is strong scientific evidence that the paradox is solved as suggested by Prandtl, then neither WP can say so. But there is a new resolution presented by living scientists which should be presented to the readers of WP, and not be suppressed in the names of dead people.Egbertus (talk) 04:33, 1 October 2008 (UTC)
 * The discussion here is on the Birkhoff quotes. Regarding the inclusion of the H&J claims the discussion is closed, and are you not the proper person to start it again, due to your conflict of interest in this matter. -- Crowsnest (talk) 07:06, 1 October 2008 (UTC)

The discussion concerns if there is any living scientist of high stature claiming that the paradox is solved along the lines suggested Prandtl. If you cannot come up with any such person, you must understand that there is no scientific basis for the claims made in the WP written by you. If you are this person you you have to show your identity and scientific credentials. If you are not, then you have to point to someone else, who is alive and can speak for himself, and not through you from the other side. Prandtl did not claim to have solved the paradox. You claim that he has, but that is very different thing. This is serious: it concerns the essence of science and responsibility in scientific information. Don't you understand this? How come that you control WP?Egbertus (talk) 07:57, 1 October 2008 (UTC)


 * You are the only one over here asking for references by living scientists (of which one already is added to the article). I do not see that request being made by EdJohnston. Nor am I aware of WP guidelines or policies who demand that.
 * You still appear to be unable to grasp the nature of Wikipedia, which is an online encyclopedia, and not a peer-reviewed publisher of original research, see WP:NOR.
 * The only reason why older references would not suffice, is when they are outdated by new insights (which have to be notable and verifiable for inclusion in WP). I am not aware of scientists of high stature, claiming that Prandtl's explanation of the effects of small viscosity on drag is invalid. Only H&J claim so, and until now there are no reliable secondary sources backing up these claims, nor have they even appeared as primary sources in a peer-reviewed journal.
 * But the 1950 Birkhoff quotes - as they are now in the article - seem to be outdated by new insights, since Birkhoff himself changed or removed these quotes in the 2nd revised edition of 1960.
 * The funny thing is that the key issue of Birkhoff's book: that seemingly plausible hypotheses may result in paradoxes - both for inviscid and viscous flow descriptions - is just what you do in your book and (draft) papers. You assume that the boundary layer may be neglected, and assume that you may add some dissipation to "regularize" the Euler equations (without having effect on flow stability), as well as that you assume that the additional simplifications in your stability analysis still make the results transferable to the full Euler-flow problem. Then you end up with a paradox, i.e. results which are in contradiction with experiments, theory and computation by others:
 * two or more separation points on the circular cylinder in experiments and N-S based computations, only one in H&J;
 * a too-high drag coefficient for the circular cylinder as compared with experiment;
 * no results presented with respect to Strouhal number and oscillating lift for the cylinder.
 * But instead of looking critically at your assumptions and their effects, you try to push the publication of claims based loosely on these results (which are based on the assumptions you make). This is just an observation.
 * I added several references in high support of Prandtl's explanation of occurrence of friction and form drag at high Reynolds numbers. It is up to you to come with references which doubt the mechanism proposed by Prandtl, not me to come with references of "living scientists".
 * Neither I nor you control Wikipedia: this is a community project. -- Crowsnest (talk) 10:46, 1 October 2008 (UTC)

If you cannot find any living scientist claiming that what you sell on WP is true, then you should seriously reconsider your role. Science is carried by living people, not dead people who no longer can express themselves. I don't insist on putting up anything whatsoever on WP, including work by HJ, since I have giving up any hope about WP. But I have an obligation as scientist to react if I happen to see misleading information concerning a basic scientific problem of big importance in the top position of Google. And this what I see in the present case. So who is the leading living scientist you are relying on? Give me the name and I will contact this person and listen. If you have no name, then remove statements suggesting that the paradox has been resolved, or I will do it.Egbertus (talk) 11:38, 1 October 2008 (UTC) being


 * You do not seem to listen or read: there are living scientists (two in fact, Gersten and Veldman) in the given references. Just to please you, not because this is required.
 * Regarding your obligations with respect to misleading information I think a humbler position would better suit you: you misled WP-editors (and through WP also the scientific community, since the information you put there was biased by your views) for a long time by avoiding to state your conflict of interest (in the WP sense), despite being asked about this many times, and despite being pointed at the WP COI-guidelines. In fact, you never openly admitted that you have a COI. -- Crowsnest (talk) 23:03, 1 October 2008 (UTC)

Two things: Anybody, like you and Veldman, claiming that Prandtl resolved the paradox in the 1904 paper, should read the paper carefully and notice that Prandtl does not claim to have resolved the paradox. Prandtl only makes some vague suggestions. So you put your own words into a dead mans mouth and let him speak what you freely invent. You must understand that this is not allowed in science. Right? Second, a mathematical paradox, like d'Alembert's, can only be resolved mathematically. You simply put the paradox under the carpet, by stating that it is resolved "from a practical point of view". You must understand that to discard or forget a problem and to resolve a problem, is completely different. Right? I give you 24 hours to change the statements indicating that the paradox was resolved by Prandtl. Then I or someone else will change these statements.Egbertus (talk) 08:22, 2 October 2008 (UTC)


 * You're not in a position to make ultimatums here. Follow through and be prepared for a block (and maybe even something more permanent).  I'm puzzled by why a distinguished professor such as yourself is spending time trolling here.  Perhaps things aren't going so well off Wikipedia in regards to your paper?  --C S (talk) 10:27, 2 October 2008 (UTC)

The paper was accepted by Journal of Mathematical Fluid Mechanics on Sept 12, as you can read on the book home page, and will soon appear online. A distinguished professor, in particular, has an obligation to correct desinformation in top position on Google. If you cannot back your claims of resolution, then you propagate desinformation, and correction will be made. So again, which living scientist able to speak for himself, are you relying on?Egbertus (talk) 10:50, 2 October 2008 (UTC)


 * Congratulations! Now I noticed in your paper that you claim the Prandtl solution is the standard one, which has convinced several generations of fluid dynamicists.  --C S (talk) 11:00, 2 October 2008 (UTC)

Yes, but the point of the article is that the standard resolution is incorrect and no longer should be propagated, not even on WP.Egbertus (talk) 13:05, 2 October 2008 (UTC)


 * I understand that is your point. Nonetheless, why query Crow's nest repeatedly above for a statement that you yourself make in the paper?  Wikipedia's goal is to report on the established consensus with less reporting for the lesser established views.  In the case of your work, you don't seem to understand that this is not the place to promote it.  Wait until it becomes more established.  I'm sure then someone else will be including it with no push from you!  --C S (talk) 10:12, 6 October 2008 (UTC)

So am I. Maybe you will be that someone who will include the new resolution? It seems that only a WP editor can do it. The new resolution is being published in a leading journal and so can deserve at least a reference on WP. Right? Go ahead!213.113.216.240 (talk) 05:24, 7 October 2008 (UTC)

To be more explicit: To propagate incorrect information because of ignorance, can be understood even if it is not admirable. To suppress correct information for the same reason, has no excuse. So CS, since you show concern, proceed to action!Egbertus (talk) 08:13, 7 October 2008 (UTC)