Entropy in self-similar shock profiles

In this paper, we study the structure of a gaseous shock, and in particular the distribution of entropy within, in both a thermodynamics and a statistical mechanics context. The problem of shock structure has a long and distinguished history that we review. We employ the Navier-Stokes equations to construct a self-similar version of Becker's solution for a shock assuming a particular (physically plausible) Prandtl number; and that solution reproduces the well-known result of Morduchow & Libby that features a maximum of the equilibrium entropy inside the shock profile. We then construct an entropy profile, based on gas kinetic theory, that is smooth and monotonically increasing. The extension of equilibrium thermodynamics to irreversible processes is based in part on the assumption of local thermodynamic equilibrium. We show that this assumption is not valid except for the weakest shocks. We conclude by hypothesizing a thermodynamic nonequilibrium entropy and demonstrating that it closely estimates the gas kinetic nonequilibrium entropy within a shock.