Entropy is constant. How to express this equation in terms of pressure and density?

Question

In hydrodynamics of an ideal, non-compressive flow we use 5 variables: pressure $p$, density $\rho$ and velocity field $\mathbf{v}$. So we need 5 equations. Landau's "Hydrodynamics" states that the equations are:

1. The mass continuity equation $\frac{\partial \rho}{\partial t}+\nabla (\rho \mathbf{v}) = 0$

2. The Euler equation (3 components) $\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}\cdot \nabla)\mathbf{v} = -\frac{1}{\rho}\nabla p$

3. A statement of the fact, that there is no dissipation of energy $\frac{d s}{d t} = 0$ ($s$ is entropy per unit mass)

My question: how to express the last equation in terms of the actual variables we are using? We have to assume some form of the equation of state for the fluid in order to do it right? Landau in his typical fashion glance over it, assuming the readers' perfect understanding of thermodynamics, which is not my case unfortunately.

PS. The question is a little related to a more general one I posted yesterday: Explicit form of the entropy production in hydrodynamics

Answer 1

Generally the fifth equation is pressure conservation: $$ \frac{\partial p}{\partial t}+\mathbf v\cdot\nabla p+\gamma p\nabla\cdot\mathbf v=0 $$ which, depending on your particular subfield, is usually written in terms of the total energy: $$ \frac{\partial E}{\partial t}+\nabla\cdot\left[\left(E+p\right)\mathbf v\right]=0 $$ where $E=\frac12\rho v^2+p\left(\gamma-1\right)^{-1}$. However, if you want to do this in terms of the entropy and you are working with an ideal gas, then we can define a variable $$ S\equiv p\rho^{-\gamma} $$ as a measure for entropy (not entropy itself because it's missing certain constants, e.g. heat capacity at constant volume $C_v$). Then you can use the total derivative to evolve $S$: $$ \frac{DS}{Dt}=\frac{\partial S}{\partial t}+\mathbf v\cdot\nabla S=0 $$

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The connection between this measure of entropy, $S$, and the entropy density, $s$, is $$ s=C_v\ln(S)=C_v\ln\left(p\rho^{-\gamma}\right)=C_p\ln\left(p^{1/\gamma}\rho^{-1}\right) $$ where we used $\gamma=C_p/C_v$ between the latter two equalities. The above relation can be found in Section 83 of Landau's Fluid Dynamics text.