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

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. INTRODUCTION Half-space problem of evaporation and condensation is one of the most fundamental boundary-value problems in kinetic theory and has been intensively studied (see, for example, [1, 2] and the references therein). One of the important aspects of the problem is that the information about the conditions for the steady evaporation and condensation flows provides the fluid-dynamic equations with the appropriate boundary conditions; that is, it allows us to complete the macroscopic description of vapor flows with the phase change at ordinary pressure. According to the studies so far, the conditions for the steady flows depend qualitatively on whether evaporation or condensation takes place for a pure vapor and for a mixture of a vapor and a noncondensable gas. (There is also a qualitative difference between subsonic and supersonic condensations.) In the present paper, we will first show the corresponding results for a binary mixture of vapors in the case of weak evaporation and condensation. By limiting ourselves to weak evaporation and condensation, we can take an analytical approach [3] to obtain the conditions explicitly for both evaporation and condensation cases and to clarify a qualitative difference between them. The solution discussed here is closely related to the suction boundary layer [4] in the conventional gasdynamics. Next, we will study the two-surface problem of a binary mixture of vapors for small Knudsen numbers by the Euler set of equations with the conditions above being the boundary conditions. We show that the Euler system has two branches of solutions in general, so that two different solutions may occur for the same physical situation in the continuum limit. This result is discussed in connection with the ghost effect [1, 5] at the end of the paper. PROBLEM AND FORMULATION We consider a binary mixture of vapors, say vapor A and B, in a half-space in contact with their plane condensed phase. The condensed phase is kept at a uniform temperature Tw, and its surface (or the interface) is located at X1 = 0, where Xi is the rectangular coordinate system. The mixture of vapors occupies the region X1 > 0 and is in the equilibrium state characterized by the pressure p∞, temperature T∞, concentration (molecular number fraction) XA ∞ of species A, and flow velocity (v∞,0,0) at a far distance from the interface. The Mach number M∞ of the flow at a far distance, which is defined by M∞ = |v∞|( 5 3 kT∞/m∞)−1/2, is supposed to be small. Here k is the Boltzmann constant, mα (α = A,B) is the mass of a molecule of species α , and m∞ = mAXA ∞ + mBXB ∞ with XB ∞ = 1−XA ∞. We will investigate the steady behavior of the vapors on the basis of kinetic theory, mainly aiming at deriving the relation among the parameters that characterize the state of the condensed phase and that of the vapors at a far distance in order that the steady