Properties of Solutions for the Problem of a Joint Slow Motion of a Liquid and a Binary Mixture in a Two-Dimensional Channel

Abstract

Under study is a conjugate boundary value problemdescribing a joint motion of a binary mixture and a viscous heat-conducting liquid in a two-dimensional channel, where the horizontal component of the velocity vector depends linearly on one of the coordinates. The problemis nonlinear and inverse because the systems of equations contain the unknown time functions—the pressure gradients in the layers. In the case of small Marangoni numbers (the so-called creeping flow) the problem becomes linear. For its solutions the two different integral identities are valid which allow us to obtain a priori estimates of the solution in the uniform metric. It is proved that if the temperature on the channel walls stabilizes with time then, as time increases, the solution of the nonstationary problem tends to a stationary solution by an exponential law.