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In fluid mechanics, we derive the fact that the Cauchy stress tensor is symmetric by using a balance of angular momentum argument. We start the derivation by assuming that no body/surface couples act on fluid elements.

My question is, why is that a necessary assumption? To be exact, I don't think I get why body/surface couples are considered separately - why aren't they included in the integrals

$$\int_V \mathbb{x}\times \rho b \,dV + \int_S\mathbb{x}\times \mathbb{t}\,dS$$

where $b$ is body force per mass, $\mathbb{t}$ is surface force per area, and $x$ is the position vector.

My background is in math, not physics, so even a seemingly trivial physics description explanation would help.

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  • $\begingroup$ I think the sentence "no body/surface couples act on fluid elements." may be not too clear. $\endgroup$ – jaromrax Feb 20 '17 at 16:05
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Strictly speaking this is not a necessary assumption, but it is an assumption that does make sense in standard continuum mechanics. What we are saying is that the (infinitesimal) particles we are describing do not have angular momentum (or rotational energy). It would certainly be mathematically possible to construct a continuum mechanical model for particles that do have angular momentum, and I could even imagine that such a model would be useful to describe some exotic type of fluid where rotational degrees of freedom of the particles is important. You could then derive a continuum mechanics model for a fluid made up of such particles. However, most standard fluids do not behave in this way.

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