Heavy gas relaxation in a light gas shock wave at small Prandtl number

The shock wave structure of a binary mixture with very different molecular weights is analyzed using a two-fluid theory in the limit of small Prandtl ($\text{Pr}$) number of the light gas. In particular, the case in which both the density ratio and $\mathrm{\Lambda }=\text{Pr}/\delta \text{Sc}$ are order unity or below is considered, where $\delta \ll 1$ is the molecular mass ratio and $\text{Sc}$ is the Schmidt number. Approximate analytical solutions are given for weak, and for moderately strong, shock waves. In the former case the solution is continuous, while in the latter case the light gas shock exhibits a discontinuity and the heavy gas relaxes towards the final equilibrium state.

Figure 1

Regions in the $\epsilon -M_0$ plane where the numerical integration of (6, 7, 8) is straightforward from “0” to “$f$” or from “$f$” to “0”, as indicated, for $\gamma =5/3$.

Figure 2

Comparison between the approximate analytical solution (thick line) and the numerical one (thin line) on the $\xi -\eta$ (a) and $\xi -\theta$ (b) phase planes for $\gamma =5/3, \epsilon =0.25, M_0=1$ (small circle in Fig. 1), and $\mathrm{\Lambda }=1$. The dashed lines in (a) are the slopes corresponding to the two eigenvalues at each singular point.

Figure 3

Structure of the shock wave in the physical coordinate $s$ obtained from the approximate analytical solution for the case plotted in Fig. 2.

Figure 4

(a) Solution (12, 13) for $\gamma =5/3,\epsilon =0.25$, and $M_0=1.3>M_0^{*}$ with $\mathrm{\Lambda }=0.5$ plotted vs $\eta$. (b) Shock structure vs $s$ given by the composite approximate solution.