# Validity of the Multi-Species Navier-Stokes Equations for real gases

I'm wondering what are the validity limits of Multi-Species Navier-Stokes equations. I'm aware of the limit for rarefied gases. But is there any new limit that arises in the context of real gases? I hope that an answer could be found by making use of the kinetic theory.

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## 1 Answer

There are several more validity limits; I know of two:
1. Flow at relativistic speeds needs to be described by more complex equations (Israel-Stewart or Mueller-Ruggeri, say).
2. At sufficiently high temperature, one must reckon with chemical reactions or dissociation effects, and needs reaction-diffusion equations.
In both cases, one can derive validity limits form the corresponding more complex theories.

On the other hand, I wonder whether there ever has been a derivation of the multi-species Navier-Stokes equations for real gases from statistical mechanics (Boltzmann equation or Liouville equation) - I don't even know a derivation for the single fluid case, say for a common substance such as pure water. If someone knows better, I'd be highly interested to see appropriate references.

The usual derivation is given for simple cases, and the general case is then established on the basis of the principles derived from these cases - conservation laws, positive entropy production, Maxwell relations, and low order derivatives. Of course these do not tell you the limits of validity.

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