Entropy growth and entropy production rate in binary mixture shock waves

Entropy growth and entropy production rate are analyzed in the stationary shock waves in a multitemperature model of a binary mixture of gases. The model is developed within the framework of extended thermodynamics. It is shown that the entropy density of the mixture increases monotonically through the shock profile, whereas the entropy densities of the constituents may not be monotonic. The analysis of entropy growth rate distinguished the contributions of momentum exchange (diffusion) and internal energy exchange between the constituents, and showed a substantially larger contribution of diffusion. At the same time, it is shown that the entropy production rates of the constituents may not be non-negative throughout the whole shock structure.

Figure 1

Stationary shock structure: shock wave [red (dashed)] and shock structure [black (solid)]; $U$ represents the generic state variable; $U_{-}$ and $U_{+}$ are upstream and downstream equilibrium values; $u=v_x-s$ is the relative speed of the gas with respect to the stationary shock.

Figure 2

Profiles of the velocities: (a) $M_0=1.20,c_0=0.35,\mu =0.05$; (b) $M_0=1.60,c_0=0.21,\mu =0.10$; (c) $M_0=1.30,c_0=0.11,\mu =0.10$; (d) $M_0=2.00,c_0=0.09,\mu =0.02$.

Figure 3

Profiles of the temperatures: (a) $M_0=1.20,c_0=0.35,\mu =0.05$; (b) $M_0=1.60,c_0=0.21,\mu =0.10$; (c) $M_0=1.30,c_0=0.11,\mu =0.10$; (d) $M_0=2.00,c_0=0.09,\mu =0.02$.

Figure 4

Profiles of the diffusion flux and diffusion temperature: (a) $M_0=1.20,c_0=0.35,\mu =0.05$; (b) $M_0=1.60,c_0=0.21,\mu =0.10$; (c) $M_0=1.30,c_0=0.11,\mu =0.10$; (d) $M_0=2.00,c_0=0.09,\mu =0.02$.

Figure 5

Profiles of the entropy density: (a) $M_0=1.20,c_0=0.35,\mu =0.05$; (b) $M_0=1.60,c_0=0.21,\mu =0.10$; (c) $M_0=1.30,c_0=0.11,\mu =0.10$; (d) $M_0=2.00,c_0=0.09,\mu =0.02$.

Figure 6

Profiles of the entropy production—dissipation mechanism: (a) $M_0=1.20,c_0=0.35,\mu =0.05$; (b) $M_0=1.60,c_0=0.21,\mu =0.10$; (c) $M_0=1.30,c_0=0.11,\mu =0.10$; (d) $M_0=2.00,c_0=0.09,\mu =0.02$.

Figure 7

Profiles of the entropy production—components: (a) $M_0=1.20,c_0=0.35,\mu =0.05$; (b) $M_0=1.60,c_0=0.21,\mu =0.10$; (c) $M_0=1.30,c_0=0.11,\mu =0.10$; (d) $M_0=2.00,c_0=0.09,\mu =0.02$.