Half-space problem of weak evaporation and condensation of a binary mixture of vapors

Abstract

Abstract. Half-space problem of weak evaporation and condensation of a binary mixture of vapors is investigated on the basis of the BGK-type Boltzmann model. By a systematic asymptotic analysis, it is shown that the steady evaporation and condensation takes place only if the parameters that characterize the state of the condensed phase and that of the vapors at a far distance satisfy one relation for the condensation case and three relations for the evaporation case. As an application, the resulting relations are used as the boundary conditions of the Euler system in the study of the two-surface problem of a vapor mixture for small Knudsen numbers. This system has two branches of solutions, which causes two different solutions for the same physical situation in the continuum limit. This result is discussed in connection with the ghost effect.

Evaporation and condensation of a binary mixture of vapors on a plane condensed phase: Numerical analysis of the linearized Boltzmann equation

“…Incidentally, the qualitative difference of the jump condition between the evaporation and condensation was studied recently in Ref. 24 in the case of a binary mixture of vapors by an asymptotic analysis of the nonlinear problem for small Mach numbers.…”

“…The same is true in the case of a mixture of vapors. 24 Because of this failure, the solution of the linearized problem is meaningless for condensation as a certain approximation to that of the nonlinear one. Nevertheless, we should stress that, as mentioned above in the third paragraph, the linearized problem plays an essential role to determine the jump conditions for evaporation and condensation for the Navier-Stokes or Stokes equations.…”

Half-space problem of the nonlinear Boltzmann equation for weak evaporation and condensation of a binary mixture of vapors

“…Such monotonic property is also true in the case of a single-species vapor for the Boltzmann equation for general molecular models; and it was fully used in Refs. [12,7,1] in deriving the conclusion. However, the macroscopic quantities do not monotonically vary in general in the case of mixtures.…”

“…Preceding the present work, the first author studied the same problem by the use of the BGK-type model Boltzmann equation [9][10][11] in the case where there is no flow in the direction tangential to the surface [12]. In this case the fluid-dynamic system can be (formally) directly solved to show the monotonic behavior of the macroscopic quantities.…”

Global existence and stability of mild solutions to the Boltzmann system for gas mixtures

“…The mathematical analysis of (1.1) with suitable boundary and initial conditions is a very formidable job; hence several simplified kinetic models [8,11,15] similar in spirit to the BGK model for the Boltzmann equation have been studied. Recently K. Aoki and S. Takata and their group have systematically studied multi-component gases on the evaporation, condensation problem, hydrodynamic limit and gas separations using asymptotic analysis and numerical experiments in a series of papers [1,2,16,17,19,20,23,24].…”