Kinetic theory based nonlinear treatment for motions of a binary gas mixture involving slightly strong evaporation and condensation processes ? A system of macroscopic equations, the boundary conditions and the Knudsen-layer corrections

Abstract

An analysis for the steady behaviour of a binary mixture of a vapour and an inert gas in a general domain has been carried out based on the Boltzmann equation of BGK type subject to the diffusive boundary condition under the following situations: i) the amount of inert gas is dense enough so that the mean free path of the inert gas molecules is small compared with the characteristic length of the system. However, the collision between the molecules of the vapour and those of the inert gas is so frequent that the mean free path of the vapour molecules is comparable to that of the inert gas molecules; hence, the mean free path of the vapour molecules is adopted for the definition of the Knudsen number of the system, which is small compared with unity; and ii) the deviation of the system from a reference stationary equilibrium state is small, but its magnitude is of the order of the Knudsen number. In this case, the problem becomes nonlinear, and the kinetic equation and its boundary condition cannot be linearized. Since the Mach number of the system is a measure of the degree of the deviation of the system and is proportional to the Knudsen number times the Reynolds number, the Reynolds number in the present case is finite. By the singular perturbation method, we have derived, as an asymptotic solution to the Boltzmann equation of BGK type for small Knudsen numbers, the macroscopic equations governing the fluid dynamic quantities and the appropriate boundary conditions for them at the interface together with the Knudsen-layer corrections near the interface up to the second order of approximation in the analysis. The equations obtained are essentially of Navier-Stokes type at each order; the macroscopic boundary conditions are those of no slip and no jump at the first order of approximation but, at the second order, they are slip and jump conditions and comprise several terms, each of which is expressed by one of the set of solutions of the macroscopic equations at the previous approximation times aconstant; the Knudsen-layer corrections are of the same type as the boundary conditions with theconstant replaced by a rapidly decreasingfunction of the distance from the interface. Theseconstants andfunctions are universal in the sense that they are totally independent of the geometry of the problems. The derived system of the macroscopic equations and boundary conditions makes possible at the level of ordinary fluid dynamics the treatment of various flow problems of a binary gas mixture, giving an adequate description of the behaviour of the mixture and its component gases for moderate values of the concentration of inert gas. The vapour-mass transfer for these values of the concentration is quite small because it is governed by the diffusional ability of the vapour through the inert gas.