Talk:Cauchy momentum equation

Lead Section

This highly technical article would benefit from some non-technical explanation, such as a paragraph or two at the top that explains the history, importance, and uses of the equation. The lead section would benefit from some elaboration on what makes the equation notable and a few sentences that summarize the significance of the extensive mathematics. Altair2013 (talk) 02:04, 28 June 2020 (UTC)

There is an obvious relation between the Cauchy momentum equation and Euler's equation that should be mentioned: The latter is a special case of the former and considers the case of pressure and not the general form of a stress tensor, but it also predates Cauchy's equation by many decades: Cauchy was born 35 years after Euler wrote his version of the equation. [Peter Schupp, 5 December 2023] — Preceding unsigned comment added by 2003:CB:727:B300:C62B:44FF:FE44:451F (talk) 10:06, 5 December 2023 (UTC)

Sloppy Notation

Perhaps I've made a mistake in reading this article, but it seems as if the flux term should be $\mathbf {F} =\rho \mathbf {u} \otimes \mathbf {u} -\mathbf {\sigma }$, which comes from the more general form of the linear momentum equation $\nabla \cdot \sigma +\rho \mathbf {g} ={\frac {\partial (\rho \mathbf {u} )}{\partial t}}+\nabla \cdot (\rho \mathbf {u} \otimes \mathbf {u} )$ that can be rearranged to be $\mathbf {s} =\rho \mathbf {g} ={\frac {\partial (\rho \mathbf {u} )}{\partial t}}+\nabla \cdot (\rho \mathbf {u} \otimes \mathbf {u} -\sigma )={\frac {\partial (\mathbf {j} )}{\partial t}}+\nabla \cdot (\mathbf {F} )$. AndrewWinter (talk) 03:21, 21 October 2016 (UTC)

The tensor derivative needs a direction. (no signature. Who posted this?)

Furthermore, it is unclear what $\mathbf {v} \cdot \nabla \mathbf {v}$ and $\nabla \cdot \sigma$ mean. My interpretations are $\partial _{j}v^{i}v^{j}$ and $\partial _{j}\sigma _{j}^{i}$, respectively.

Correct. I put this in. 89.217.22.3 (talk) 16:14, 8 November 2014 (UTC)

Etoombs (talk) 09:12, 4 February 2009 (UTC)

It's true, this equation is technically correct only in the Cartesian coordinate system. I'll edit it into something more plausible, but this article needs an expert's attention. — Kallikanzarid (talk) 10:09, 30 July 2009 (UTC)

I edited the Derivation section, it should make sense now, mathematically. I replaced partial derivatives of the tension tensor with the covariant ones. I decided to use mixed components so that later stress tensor components could be decomposed in a straightforward way:

$\sigma =-p\mathbb {I} +\mathbb {T}$

$\sigma _{i}^{j}=-p\delta _{i}^{j}+\mathbb {T} _{i}^{j}$

There is, of course, no difficulty in using components of any other type, but in this particular case we can cheat and avoid explaining to the reader what a metric tensor is and why covariant derivatives of its components are zero. — Kallikanzarid (talk) 12:52, 30 July 2009 (UTC)

I have been unable to find any definition of the "del" of a vector or the "divergence" of a tensor -- or whatever you call the juxtaposition of the nabla symbol and a vector, and the "dot product" of the nabla symbol and a tensor.

I think this article is unhelpful for most readers. Does everybody learn tensor algebra nowadays? I am quite certain Cauchy did not know about tensors, and it should be possible to write something useful for readers that only know the stress tensor as a matrix.

I agree that the article is bad. I inserted the component definitions of the index-less nabla expressions. The index-less ones are useful once you get used to them, but not before. (I face these things constantly and I prefer to express everything both ways.) 89.217.22.3 (talk) 16:11, 8 November 2014 (UTC)

I have by chance read about tensors and general relativity, but not yet found the nabla symbol used as here. (There seems to be no end to the variants of the tensor language, in how many ways can you write the directional derivative of a scalar field in the direction of a coordinate basis vector? I found six notations the other day!) I may be able to reconstruct a meaning for the nabla combinations used in this article, but that is not how it should be. At very least, if the Wikipedia is to be taken over by GR enthusiasts (but thank you for a formidable work anyway), there should be an easy way for the reader to know which link to click to find an explanation of the notation used. How do you google the "juxtaposition operator which sometimes is used to express multiplication of real and complex numbers, but often means functional application" to learn what it means in the case of a nabla next to a vector? Many readers don't even know the name "nabla", even if they know the three (or four) standard uses of nabla to indicate gradient, divergence, and curl (and Laplace).

Besides the use of clever nabla shorthands, there is a lack of context. Since the material derivative is used, I understand that the article is written from the perspective of fluid mechanics, where the system is a fixed volume through which some fluid flows. The f is not just body force, it is body force per unit volume.

Finally, there is an error in the derivation. The density belongs with the acceleration, not outside the integral where it would be multiplied with all the terms. The "$F_{i}$" that is mentioned in the very last statement has not been named or explained anywhere. What is it? Cacadril (talk) 00:40, 3 December 2009 (UTC)

I agree, the derivation is bad.

1)The MAIN point is to understand that the "divergence" of the stress tensor should yield a force. But this step is skipped and instead some wordy material about volume elements is put in that looks like high-school motivation of an integral. In other words, the already-known stuff is explained and the essential new part skipped.

2) The long-winded coordinate expressions are not properly part of the derivation. So I labelled them into a new section. As the article stands, these formulas take up way too much space, but if the article gets more filled out, maybe there is a place for them. 89.217.22.3 (talk) 16:11, 8 November 2014 (UTC)

Physical Background

To me the derivation is not very convenient from the physical point of view. Perhaps it is easy to cope with for the ones really understanding covariant notation and stuff like that, but I tried to figure out the essence of what

$\rho \int \limits _{\Omega }{\frac {\mathrm {d} u_{i}}{\mathrm {d} t}}\,\mathrm {d} V=\int \limits _{\Omega }\nabla _{j}\sigma _{i}^{j}\,\mathrm {d} V+\int \limits _{\Omega }f_{i}\,\mathrm {d} V$

should to tell me:

It seems to me the left hands side shall be expressed in a more feasible form with the right hand side. So the question is how do I end at the right hand side? It might be possible that it is mathematically triggered but I think it reminds me about the derivation of Maxwell equations. As a model one has this control volume comprising of N different $H_{2}0$ molecules (we are dealing with fluids right?) or units cells to speak more generally on which the forces can act individually. Here instead of dealing with finite sums we will already go for the limit using integrals, that's alright. So we might break the forces per unit volume:

$\rho \int \limits _{\Omega }{\frac {\mathrm {d} u_{i}}{\mathrm {d} t}}\,\mathrm {d} V$

down into a part which orginates form a global force field e.g. gravitational forces which acts on every subdivision $i$:

$\int \limits _{\Omega }f_{i}\,\mathrm {d} V$

and a second part which is a little bit more tricky (and actually I am a bit uncertain concerning this part and would appreciate some encouraging discussion) is due to some surface forces excerted on the control volume's surface $\Sigma$ which might be described by the stress tensor $\sigma _{ij}$ which describes the stress on the $i$th subunit in the $j$th direction (<- uncertain!) in the following way

$\int \limits _{\Sigma }\sigma _{ij}\,\mathrm {d} S$

and using Gauss–Ostrogradsky theorem one finds

$\int \limits _{\Omega }\nabla \sigma _{ij}\,\mathrm {d} V$

The following conclusion made in the article that the control volume is arbitrary is fine, too, but I'm not sure how to end at the final formula for the whole system without subscript $i$ then.

But the very critical part is the stress tensor and its components. Can anybody shed a bit more light on that?

And can anyone explain me why one uses the partial derivatives instead of ${\tfrac {\mathrm {d} v}{\mathrm {d} t}}\,$ ? I think I figured out the mathematics behind that since the total derivative of $v$ which is a function of $r,t$ is

$\mathrm {d} v={\frac {\partial v}{\partial t}}\mathrm {d} t+{\frac {\partial v}{\partial r}}\mathrm {d} r$

Division through $\mathrm {d} t$ yields

${\frac {\mathrm {d} v}{\mathrm {d} t}}={\frac {\partial v}{\partial t}}+{\frac {\partial v}{\partial r}}{\frac {\partial r}{\partial t}}$

since ${\tfrac {\partial r}{\partial t}}={\tfrac {\mathrm {d} r}{\mathrm {d} t}}$ because $r$ is just a function of $t$

(MArras (talk) 19:30, 10 June 2010 (UTC))

Misleading link

The link on the word "continuum" misleadingly sends the reader to the Wikipedia article about the continuum as the set of real numbers. This is clearly not the right interpretation of the word in this context. — Preceding unsigned comment added by 66.207.95.54 (talk) 19:53, 24 June 2011 (UTC)

Lamb form section

In this section after using assuming stress tensor as isotropic, it is mentioned that in the steady incompressible case the mass equation is simply $$\mathbf {u} \cdot \nabla \rho =0\,$$ however for a steady incompressible case the mass equation is $$\mathbf {\nabla } \cdot u=0\,$$ It should be written that for a incompressible case gradient of density is zero. Navin123explorer (talk) 11:57, 28 July 2023 (UTC)