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I have one question related to Hydrodynamics, using local equilibrium thermodynamics. The variables $\rho, s$, i.e. mass density and specific entropy, or $T,S$, temperature and entropy, are typically used variables. What is the physical law, or theorem the choice of the variables is based on? Is this expressed in Gibbs-Duhem relation?

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The general principle at work is the state postulate (which is itself a special case of the phase rule). Since hydrodynamics considers a simple compressible fluid, exactly two independent, intensive properties are required to define the thermodynamic state at each point. The "independent" constraint means that you can't choose a pair like $\rho$ and $v \equiv 1/\rho$ as your two properties, since fixing one would fix the other.

The choice of which independent, intensive properties should be used is completely arbitrary. Density is often a convenient choice in hydrodynamics because, for a liquid, it is essentially constant. I would expect $\rho$, $T$ to be a common choice because both are easy for the typical reader to visualize.

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