On quasi-gasdynamic and quasi-hydrodynamic equtions for binary mixtures of gases
The quasigasdynamic (QGD) approach makes it possible to construct convenient and reliable difference schemes for the numerical solution of various gasdynamic problems. Its description can be found in several books. In particular, the Boltzmann kinetic equation for a mixture of monatomic gases is used to derive and test QGD equations for binary mixtures of nonreactive ideal polytropic gases. In this paper, we analyze and expand the capabilities of the QGD approach in this area. The original equations are rewritten as conservation laws, which are more conventional in viscous gas dynamics and convenient for discretization. Additionally, an external force and a heat source are taken into account. We briefly discuss the parabolicity of the system in the sense of Petrovskii, which ensures that the system is well defined. An entropy balance equation is derived, and the entropy production for a gas mixture is shown to be nonnegative, which ensures that the system is physically consistent (but does not hold in all available descriptions of gas mixtures). Importantly, to achieve the latter property, the expressions for the exchange terms in the total energy balance equation (initially derived only for monatomic gas mixtures) are properly generalized. Additionally, we introduce a simplification of the QGD system for binary mixtures, which is referred to as a quasihydrodynamic system and is used for the numerical simulation of weakly compressible sub and transonic flows. At the end of this paper, we present simplified barotropic versions of both systems and derive a corresponding energy balance equation with nonpositive energy production.