Talk:Derivation of the Navier–Stokes equations
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Notation?
editDoes it change from Q_i to b_i for a source/sink of momentum? —Preceding unsigned comment added by 140.184.21.115 (talk) 13:40, 19 September 2007 (UTC)
Q is used for the generic conservation, b is used in the context of momentum. —Preceding unsigned comment added by Ben pcc (talk • contribs) 23:11, 24 October 2007 (UTC)
The bounding surface of the control volume is denoted $\partial\Omega$, but later the boundaries are defined as $\Gamma$ within the section ‘Continuity Equation’. Seems to be a mistake? — Preceding unsigned comment added by S. M. Peyres (talk • contribs) 19:04, 7 December 2023 (UTC)
Reynolds Transport Theorem
editI'm surprised that, in a derivation of the Navier-Stokes equations, that there is no mention of the Reynolds Transport Theorem.
--71.98.78.28 04:12, 11 June 2007 (UTC)
Good point. -Ben pcc 02:23, 28 June 2007 (UTC)
There is something fishy about how Leibniz's rule is applied just after Reynold's transport theorem is mentioned. In addition, the sign on Q looks wrong in the latter portions of that section. Worth looking into. 128.83.68.26 (talk) 18:01, 6 October 2008 (UTC)
Shouldn't it be extensive, instead of intensive property? —Preceding unsigned comment added by 85.24.185.34 (talk) 14:55, 2 September 2009 (UTC)
It defenitely should if these examples are correct Intensive_and_extensive_properties#Examples_2 — Preceding unsigned comment added by 46.242.70.122 (talk) 20:29, 3 May 2016 (UTC)
Outer Product?
editWould the relations for in the discussion on Newtonian fluids be equivalent to saying , where is the identity matrix? --Zemylat 17:57, 25 October 2007 (UTC)
I think so. I just added something that looks a lot like that minus the outer products, I think they're equivalent. I'm not using the outer product notation because my sources (MIT OCW "Surface tension module" and my fluid mechanics teacher) don't either. — Ben pcc 02:30, 3 November 2007 (UTC)
Scratch that. Using the outer product is more mathematically sound and there is zero ambiguity. Thank you! — Ben pcc 17:30, 4 November 2007 (UTC)
- I think this section is needlessly confusing and there is no reason to introduce the notation at all. Please correct me if my algebra is wrong but why not have the large expansion and then "and, more compactly, in vector form"
- This clearly shows how the incompressability reduces the stress contribution to and also why there is no contribution from bulk viscocity. — Jon —Preceding unsigned comment added by 131.227.66.187 (talk) 13:27, 8 September 2010 (UTC)
- See Volume viscosity for an example of this expansion already in use. — Jon —Preceding unsigned comment added by 131.227.66.187 (talk) 11:17, 9 September 2010 (UTC)
Are they correct?
editIn conservation of momentum, why is the following derivation correct: 1) ; 2) . The is not equal to —Preceding unsigned comment added by Run Jiang (talk • contribs) 10:23, 14 July 2008 (UTC)
- This is most easily seen in Cartesian coordinates, by using the summation convention to write the momentum equation with body forces as
- summing the second term over all coordinate directions j, so j from 1 to 2 in two spatial dimensions and j from 1 to 3 in 3D. Here vj are the components of the vector v in the respective coordinate directions associated with xj. Then, by using the chain rule:
- Which is equal to the last equation in the expansion. The third of the three rules, as mentioned by you above, has to be read as (∇v)•v=v•(∇v) and is also an identity (with ∇v the tensor derivative of vector v). The same holds in curvilinear coordinate systems, provided care is taken with respect to using contravariant or covariant vector component representations, and by using the covariant derivative instead of a simple partial derivative. In vector notation, as in the article, it is independent of the coordinate system used. -- Crowsnest (talk) 21:48, 15 July 2008 (UTC)
- While the simple method shown in the above response is correct, it does not answer the original question posted by Run Jiang. The answer to that question is that the first line of the derivation on the main page is, in fact, incorrect and so is the second one. It is only by magic that the third line (which we could have arrived at using Crowsnest's choice, namely the derivative of products) is correct again.
- If we wish to do this on a step-by-step basis, as in the main article, these lines should do as follows
- Which would greatly simplify the rest of the derivation, since in this form the terms of the continuity equation are already pulled together. Czigi (talk) 17:17, 13 May 2010 (UTC)
- Please correct the derivation on the main page, if you find mistakes (e.g. "first line of the derivation on the main page is, in fact, incorrect and so is the second one" by Czigi). Bkocsis (talk) 23:38, 4 June 2010 (UTC)
In Stream function formulation the derivation seems to assume that the body forces are conservative, but this is not stated. To fix this, I suggest to insert to the right hand side of the equation of motion of the stream function, and then note that this term drops out if . However, I am not sure what the analogous equation is for the 2D flow in orthogonal coordinates. Bkocsis (talk) 23:38, 4 June 2010 (UTC)
Messed up notation
editIn the beginning of the section "General form..." the very first equation incorrectly takes the divergence of the stress tensor components, not the tensor itself. I suggest replacing with —Preceding unsigned comment added by Kallikanzarid (talk • contribs) 19:42, 28 July 2009 (UTC)
Stress tensor for incompressible Newtonian fluid
editDoes the simplified stress tensor look like this or this . If i denotes rows and j denotes columns then I think the second one, right? Thanks. --kupirijo (talk) 16:02, 22 September 2010 (UTC)
Stress tensor for compressible Newtonian fluid
editI am fairly certain that the general expression for the stress tensor (i.e. compressible) is wrong. Instead
should read
See Landau and Lifshitz, Fluid Mechanics, Second Edition: Volume 6 (Course of Theoretical Physics) page 45. — Preceding unsigned comment added by Neeson.m (talk • contribs) 02:50, 5 July 2011 (UTC)
I second that opinion. While this has been corrected, further down the page (Navier–Stokes equations for a compressible Newtonian fluid) the term making the tensor T_ij traceless is still missing. 143.210.37.186 (talk) 09:26, 22 November 2012 (UTC)
It depends on whether is the volume viscosity or the second viscosity (Lame's first parameter). There is a lot of confusion over which is called bulk viscosity, volume viscosity, or second viscosity. If is the second viscosity, or Lame's first parameter, then the stress tensor is:
If is the volume viscosity, or bulk modulus, then the stress tensor is:
Consequently, the Stokes hypothesis is different for each. For the former one takes and for the latter . Lucky.v.tran (talk) 21:23, 28 November 2017 (UTC)
This article should be combined with the main article on the Navier Stokes equation
editIt is unclear why the authors have made this a separate article when the main article on the Navier Stokes equation covers almost all of the same material. Also this article was judged to be under WikiProject Physics whereas the main article is judged to be under WikiProject Mathematics... both (or more properly one article combined) should be under the former since it is about the physics and not the applied math (singular please).Danleywolfe (talk) 17:02, 8 September 2011 (UTC)
- Can't agree more, it seems an unnecessary duplication and source of confusion. Gpsanimator (talk) 07:52, 22 April 2024 (UTC)
NS is one of the very few mathematical models that is simultaneously insufferably difficult, both conceptually and mathematically, and widely applicable and even commonly used on a daily basis. similar examples would be derivations of the Boltzmann equation, Schrodinger equation, Kohn Sham system, Dirac Equation, etc. The usefulness of the derivation either laying in the examples of the historical ingenuity and/or a visual depiction of the amazingly complicated, and clunky, mathematics that lay at the heart of the various subsets of theoretical physics/math. Unrelated to the technical issue, I personally think that no wikipedia article should be excessively long. At a certain point, a very very long article detracts from some subset of its contents. In this case, I think the article might actually be in need of splitting it based upon the pure theoretical/mathematical, solutions (why? because this is a whole area of research in and of itself)/approximations/competing/equivalent theories, and maybe even an "applications" article.... or just a "lay article," to keep those people who never even saw a professor write the word "tensor" on a board, let alone personally handle them on paper, happy without too much loss of detail.184.189.220.114 (talk) 11:25, 28 March 2013 (UTC)
Stress tensors
editHi, I corrected an error in what I think is a confusion in notation between the deviatoric stress tensor and the viscosity stress tensor. The latter includes the volume viscosity, but is not required to be traceless. However, in the notation used in the article, has, by definition, to be traceless for all values of λ, and not just for the provided value of . I introduced π as the mechanical pressure, so as to preserve most of the notation in the latter parts of the article, where p corresponds to the thermodynamical pressure. Hope this improves the article coherence. Donvinzk (talk) 17:15, 24 May 2012 (UTC)
Momentum equations
editThe second line in the derivation of the momentum equation moves the source/sink term to the right hand side, but the sign is kept positive with no explanation why:
Why is not negative in the second line? — Preceding unsigned comment added by Voyvoyvoy (talk • contribs) 19:11, 26 August 2015 (UTC)
Energy equation/power balance
editA section on the derivation of the energy equation/power balance is missing from the article.
The section Compressible Newtonian fluid lists a formulation of the energy equation/power balance, but does not mention that is with the isotropic material model applied. 2.202.211.78 (talk) 10:09, 6 June 2022 (UTC)
LET'S ASK SIR ISAAC NEWTON ABOUT N-S
editRead if you wish.
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Sir, Newton if you could read this derivation of Navier-Stokes equations what would you say? MOTION OF THE FLUID IN THE CONTINUUM (Euler's approach): Sir Newton you are always right! Sorry about the ideal fluid. |
Continuity/conservation
editThis article appears to imply these two words are synonyms. Are they?