Generalized hydrodynamics and shock waves

In this paper, the generalized hydrodynamic equations are applied to calculate the shock profiles, shock widths, and calortropy production (energy dissipation) for a Maxwell and variable hard sphere gas. Shuck solutions are shown to exist for all Mach numbers (N-M) studied, ranging up to N-M = 10, but this upper Mach number can be in principle extended to infinity. This is in contrast to the Grad moment equation method [H. Grad, Commun. Pure Appl. Math, 5, 257 (1952)] which does not admit shock solutions for N-M greater than or equal to 1.65 and to the method of Anile and Majorana [A. M. Anile and A. Majorana, Meccanica 16, 149 (1982)] and Weiss [W. Weiss, Phys. Rev. E 52, R5760 (1995)] who also used moment equations and found the shock solutions do not exist for N-M greater than or equal to 2.09 and N-M greater than or equal to 1.887, respectively. The difference of the present theory from the aforementioned theories lies in the closure relations used for higher-order moments. The nonlinear factor in the dissipation terms in the flux evolution equations of generalized hydrodynamics significantly contributes to producing the shock width increasing with the Mach number. The results calculated are comparable with the Monte Carlo simulation results and the results by various closures of the Mott-Smith method. The present method is also applied to calculate the experimental shock widths for argon and found to give results in good agreement with experiments. The energy dissipation is shown to increase with N-M as (N-M-a)(alpha), where a and alpha are positive constants.