In this paper, a gas kinetic scheme for the compressible multicomponent flows is presented by making use of two-species GK model in [A. D. Kotelnikov and D. C. Montgomery, A Kinetic Method for Computing Inhomogeneous Fluid Behavior, J. Comput. Phys. 134 (1997) 364-388]. Different from the conventional BGK model, the collisions between different species are taken into consideration. Based on the Chapman-Enskog expansion, the corresponding macroscopic equations are derived from this two-species model. Because of the relaxation terms in the governing equations, the method of operator splitting is applied. In the hyperbolic part, the integral solutions of the BGK equations are used to construct the numerical fluxes at the cell interface in the framework of finite volume method. Numerical tests are presented in this paper to validate the current approach for the compressible multicomponent flows. The theoretical analysis on the spurious oscillations at the interface is also presented.

A computer-aided method for accurately carrying out the Chapman-Enskog expansion of the Boltzmann equation, including its inelastic variant, is presented and employed to derive a hydrodynamic description of a dilute binary mixture of smooth inelastic spheres. Constitutive relations, formally valid for all physical values of the coefficients of restitution, are calculated by carrying out the pertinent Chapman-Enskog expansion to sufficient high orders in the Sonine polynomials to ensure numerical convergence. The resulting hydrodynamic description is applied to the analysis of a vertically vibrated binary mixture of particles (under gravity) differing only in their respective coefficients of restitution. It is shown that even with this “minor”difference the mixture partly segregates, its steady state exhibiting a sandwich-like configuration.