Overshoot of the non-equilibrium temperature in the shock wave structure of a rarefied polyatomic gas subject to the dynamic pressure

Abstract

The shock wave structure in a rarefied polyatomic gas is analyzed on the basis of non-linear extended thermodynamics with 6 independent fields (ET6); the mass density, the velocity, the temperature and the dynamic pressure, which permits us to study the shock profile also for large Mach numbers. The first result of this paper is that the shock wave structure is substantially the same as that obtained previously from the linear theory for small or moderately large Mach numbers. Only for very large Mach numbers there exist some differences in the relaxation part of the profile between the model with a non-linear production term and the one with a linear production term. The mathematical reason of this behavior is due to the fact that the non-linear differential system has the same principal part of the linear one.The classical Meixner theory of relaxation processes with one internal variable is fully compatible with the ET6 theory and this fact gives us the explicit expressions of the internal variable and the non-equilibrium temperature in the Meixner theory in terms of the 6 fields, especially, of the dynamic pressure. By using the correspondence relation, the shock wave structure described by the ET6 theory is converted into the variables described by the Meixner theory. It is shown that the non-equilibrium Meixner temperature overshoots in a shock wave in contrast to the kinetic temperature. This implies that the temperature overshoot is a matter of definition of the non-equilibrium temperature.