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Can you derive a conservation law for entropy per unit mass? I know that there is a proof for this conservation.

My assumptions are:

  • $(\nabla p)/n = D \vec{v} + \vec{v} \cdot \nabla\vec{v}$ (Euler equation)
  • $\nabla \cdot (n\vec{v}) + D n = 0$ (continuity equation)
  • $p/n + u = Ts$ (chemical potential is zero)

Here I define:

  • $p$: pressure
  • $n$: mass density
  • $u$: internal energy per unit mass
  • $s$: entropy per unit mass
  • $D$: time derivative
  • $T$: temperature
user109107
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  • You are trying to write a continuum mechanics type of conservation law for specific entropy variation in the flow of an ideal (inviscid) fluid. Certainly, in such a case, the contribution of viscous dissipation to entropy generation in the fluid is zero. But, are you also assuming that heat conduction in the fluid is negligible, such that the contribution of conductive generation of entropy in the fluid is also zero? – Chet Miller Feb 22 '16 at 12:08
  • Possible dupe of http://physics.stackexchange.com/q/116779/ (restricting oneself to an ideal gas) – Kyle Kanos Feb 22 '16 at 12:42

1 Answers1

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Yes, you can. You have given the equations in a typical set of primitive variables. If you transform the equations into another set of primitive variables, ie.

$$\vec{Q} = \lbrace p/(nc), u, v, w, s\rbrace$$

where $c$ is the speed of sound and $n$ is the density, you can end up with an conservation equation for entropy. If you take your equations and write them as a PDE of a vector $\vec{W} = \lbrace n, u, v, w, T\rbrace$ you can then transform from one set to another by computing the Jacobian matrix $\partial \vec{W}/\partial \vec{Q}$ and you will get your new conservation laws.

tpg2114
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