Thermal-concentration convection in a system of viscous liquid and binary mixture in a plane channel with small Marangoni numbers

A conjugated initial-boundary-value problem occurring in the movement of a binary mixture and viscous heat-conductive liquid with a common interface surface under the action of thermal-concentration forces is under consideration. A solution describing a stationary flow in layers, temperature distribution, and concentration distribution is determined. The Laplace transform method is used to obtain a nonstationary solution for the problem in images, which makes it possible to describe the evolution of the movement using the numerical inversion of images.