Shock Structure in a Macroscopic Model of Binary Mixtures

In this chapter, we study the shock wave structure in a mixture on the basis of the model of multi-temperature (MT) mixtures explained in Chap. 28 . For simplicity, the study is restricted to shocks in a binary mixture of ideal gases (component-1 and 2) that have no viscosity and no heat conductivity. In the first part, we focus our attention on subshock formation. We prove that there exist, for a prescribed value of the mass ratio of the constituent molecules, four regions in the plane of the concentration and Mach number: no subshock region, the region where a subshock appears only in the component-1 gas, the region where a subshock appears only in the component-2 gas, and the region where variables of both component-1 and 2 gases have jumps. This is one of few examples for which subshock arises also for the shock velocity less than the maximum characteristic eigenvalue evaluated at the unperturbed state. The model predicts also the existence of the temperature overshoot of the component gas with heavier constituent molecules. This fact was predicted also by other sophisticated approaches. The overshoot is a consequence of weak energy exchange between the gas components due to either large molecular mass difference or large rarefaction of the mixture. In the range of small Mach number, it is also shown that the shock thickness (or equivalently, the inverse of the Knudsen number) decreases with the increase of the Mach number: a behavior similar to that of a single fluid.