Talk:Euler equations (fluid dynamics)

message: I would assume that the first equation expresses the conservation of mass and is the equation of continuity, the second one is the conservation of momentum and is the true Euler equation. The third is the conservation of enthalpy, and is the enthalpy or energy equation. None of them incorporates outer force field effects, wich would be nice tough.
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should I put stuff about Rankine-Hugoniot conditions here or under shock waves?
Mo ena ene test lor eulers la. Ki pou fou????
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Is it possible for someone to define all the variables in this article?  Rtdrury 10:59, 19 December 2005 (UTC)
replies: text: Should the note about the other equation not being true dropped? Also, the appended comment of both being "correct"? (what does that exactly mean? I can see both are correct only if the fluid is incompressible) --Daniel (talk) 12:51, 27 April 2008 (UTC)
thread_title: Old discussions and comments

message: I think there should a gravity term rho * g be included in the momentum equation since dm * Dv/Dt = dF = -dP*dA + dm*g  ...
rho * Dv/Dt = -grad(P) + rho * g.  —Preceding unsigned comment added by 193.190.253.144 (talk) 22:33, 5 October 2010 (UTC)
replies: text: Body forces, of which gravity is an important one, are not included in the presented form of the Euler equations. -- Crowsnest (talk) 11:10, 6 October 2010 (UTC)
thread_title:  the gravity term (!) 

message: I removed these comments from the conservation and component section:
in particular, it is not intuitively clear why this equation is correct and \left(\partial/{\partial t}+{\mathbf u}\cdot\nabla\right)(\rho{\mathbf u})+\nabla p=0 is incorrect) -- Both forms are "correct"
FR - This note represents the author's opinion of what he/she believes is more intuitive. As opposed the this author, I would say that it is the non-conservation form that is more directly linked to the classic representation of Newton's second law F = m du/dt when a fixed mass is followed in time. For Fluids the non-conservation expression is just: mass/volume du/dt = F/Volume, where du/dt is the derivative of the velocity of the fluid particles as they pass through a certain point in the flow field (= substantial derivative, typically written as Du/Dt to remind the reader that an Eulerian frame of reference is used). The non-conservation form maps directly to the interpretation of Newton's 2nd law for fixed mass particles. What is more, the conservation form can be easily reworked from the non-conservation form by means of the continuity equation. That is using additional information (conservation of mass) thereby obscuring Newton's 2nd law. Of course, the conservation form can also be directly derived from principles of momentum changes, representing an alternative interpretation of Newton's 2nd law. This is then just a limiting case of the Reynolds Transport Theorem for which the control volume size is shrunk to an infinitesimal one. But, whether this shows the "direct connection" to Newton's 2nd law can be argued as I have done. Therefore I would suggest to delete the opinions from this page. Best Regards, FR
PAR (talk) 06:27, 22 February 2012 (UTC)
thread_title:  Removed comment 

message: What about external forces, such as gravity? Are they not supposed to be included in these equations? Or maybe this fluid isn't affected by gravity??
At least, the section about the Bernoulli equation for unsteady potential flow states that the equation is obtained by integrating the momentum equations of the Euler equations, and that equation does contain gravity. —Kri (talk) 16:29, 9 March 2012 (UTC)
replies: text: External forces can off course be added to the momentum equations, and might also appear in the energy equation if they perform work. -- Crowsnest (talk) 14:01, 10 March 2012 (UTC)
replies: text: Well, you say that they can off course be added, but they are not present in the equations in their current form in the article, which has to be the most important thing. Why are they not there? They are obviously supposed to be there. Do you agree with me? —Kri (talk) 10:55, 12 March 2012 (UTC)
replies: text: Sure I agree with you. There are however not many textbooks of renowned fluid-dynamics scientists including them in the equations. Emphasis is often on how to present/derive them in a conservative form. And further, in aerodynamics (where they are mostly used) gravity is often of minor importance. So a reliable source including the external (body) forces, like gravity, has to be found. -- Crowsnest (talk) 18:04, 12 March 2012 (UTC)
replies: text: What do you mean by that a reliable source including the external (body) forces has to be found? —Kri (talk) 23:15, 15 March 2012 (UTC)
replies: text: Ah, never mind. I realized that you where talking about a literary source and not a source that should be included in the equations. —Kri (talk) 23:17, 15 March 2012 (UTC)
thread_title:  What about external forces? 

message: There is hardly anything to find on "streamline curvature theorem": 4 papers in Japanese scientific journals. For a fundamental theorem in fluid dynamics this seems very unlikely, one would expect tons of search results. The same is the case for the term "Euler's normal equation".
What can be found is several references to the Euler equations in "streamline coordinates" or "path coordinates". These have a component normal to the streamline, the subject of the article proposed to merge here, and along the streamline one recovers Bernoulli's equation.
See:
*
*
* This book is the reference in the article, and talks about "Euler's equation in streamline coordinates".
See also this Google search for the combination "Euler", "equations", "streamline" & "coordinates".
So my suggestion would be to merge Streamline curvature theorem into a section in this article, e.g. "Steady flow in streamline coordinates". -- Crowsnest (talk) 14:15, 11 May 2012 (UTC)
replies: text: That looks really very good, くま兄やん, to my opinion. I changed Streamline curvature theorem into a redirect to this article in order to finish the merger. -- Crowsnest (talk) 12:54, 17 May 2012 (UTC)
replies: text:  Thank you, Crowsnest, for merging. くま兄やん (talk) 13:09, 17 May 2012 (UTC)
thread_title:  Merger of [[Streamline curvature theorem]] 

message: Energy equation in convective form is given as
{D e \over Dt} = \frac p \rho \nabla \cdot \mathbf u
Should not it be
{D e \over Dt} = -\frac p \rho \nabla \cdot \mathbf u
?
See, e.g., http://www.mathematik.uni-dortmund.de/~kuzmin/afc2.pdf , beginning of section 2, which has \nabla(\rho H \mathbf v), with \rho H = \rho E + p.
Also the section of "Enthalpy conservation" in the article seems to have lost a minus in 
{D e \over Dt} = {D h \over Dt} - \frac 1 \rho \left({D p \over Dt} + \frac p \rho {D \rho \over Dt} \right)
It should be 
{D e \over Dt} = {D h \over Dt} - \frac 1 \rho \left({D p \over Dt} - \frac p \rho {D \rho \over Dt} \right)
192.198.151.43 (talk) 13:56, 29 July 2015 (UTC)
thread_title:  Minus signs 

message: (I give up using wikipedia math expressions:) 
Section: "Incompressible Euler equations with constant and uniform density"
Contain the equation: du/dt + u . grad u = - grad w + g 
the u . grad u term is not very clear. Clearer to write (u.grad) u.
see  p.1-2
This is made worse by the equation:
du_j/dt+Sum_i u_j d(u_i + w e_i)/dr_i == 0
where to my eyes the second term is plainly wrong.
should be:
du_j/dt +sum_i u_i d u_j/dr_i + dw/dr_j == 0
see  p.1  — Preceding unsigned comment added by 194.47.168.94 (talk) 09:01, 10 May 2017 (UTC)
thread_title:  Error? 

message: Firstly, in my opinion, the second equation in the deduction of the sound speed for ideal gases in chapter Euler equations (fluid dynamics)#Ideal gas is wrong. There should be a minus sign instead of a plus sign. Can somebody confirm that?
Second, division of the second equation by ρ−γ dρ does not yield the third equation, because ρ−γ-1 divided by ρ−γ yields ρ−1 instead of ρ.
Belgariath (talk) 12:08, 12 November 2019 (UTC)
thread_title:  Deduction of the form of sound speed valid for ideal gases 

message: A preprint, posted online in October 2022, states that, under certain boundary conditions, the Euler equations fall in a mathematical singularity. It was the first time in which a singularity was discovered through a computer-aided demonstration. (source: Wired). 95.237.45.179 (talk) 20:59, 29 December 2022 (UTC)
thread_title:  singularity with a computer-aided demonstration 

message: This article notes that the eqns are ill-posed, without also specifying eqn of state. Additional refs for this are:
* C. De Lellis and L. Szekelyhidi, On admissibility criteria for weak solutions of the Euler equations, Archive for Rational Mechanics and analysis 195, 225 (2010).
* C. De Lellis and H. Kwon, On non-uniqueness of Holder continuous globally dissipative Euler flows, arXiv:2006.06482 (2020)
* S. Daneri, E. Runa, and L. Szekelyhidi, Non-uniqueness for the Euler equations up to Onsager’s critical exponent, Annals of PDE 7, 1 (2021)
At high Reynolds numbers, this apparently leads to "Eulerian spontaneous stochasticity" during turbulent fluid motion:
* A. A. Mailybaev, Spontaneously stochastic solutions in one-dimensional inviscid systems, Nonlinearity 29, 2238 (2016).
* S. Thalabard, J. Bec, and A. A. Mailybaev, From the butterfly effect to spontaneous stochasticity in singular shear flows, Communications Physics 3, 1 (2020).
I'd like to read more about how/why the eqns are ill-posed without the eqn of state, and a description of spontaneous stochasticity. 67.198.37.16 (talk) 21:11, 20 March 2024 (UTC)
thread_title:  Spontaneous stochasticity 

message: The equation is supposed to represent conservation of momentum for the general case where rho can be a function of space and time. In that case, the equation should be D/Dt (rho * u) = -grad p + g.  NotSqwerty (talk) 06:21, 5 May 2024 (UTC)
thread_title:  Error in Second Euler Equation For Compressible Fluid