Solution of the Boltzmann Equations for Strong Shock Waves by the Two-Fluid Model

Abstract

The structure of a plane shock wave is studied by an extension to the gas dynamics of the two-fluid model already introduced in the superfluidity theory. The classical form is used for the linearization of two Boltzmann equations, and restricted to the first approximation. This is different from Mott-Smith's solutions which cannot be derived by linearization from a single Boltzmann equation, and as a result has to be introduced arbitrarily. Six equations of change for conservative quantities (mass, momentum, energy) are obtained which makes possible the solution of the wave problem without having to introduce other equations of change. Calculations are made using the elastic-sphere model. Comparison with observations made on argon gives very satisfactory results. The solutions tend asymptotically to a stationary solution corresponding to a Mach number equal to infinity, and to a shock wave thickness 11.4 times the mean free path ahead of the shock for all monatomic gases. This establishes the possibility of coexistence of two fluids in nonequilibrium, even thermal, in shock waves with large gradients.