Two-dimensional stationary thermocapillary flow of two liquids in a plane channel

The problem of two-dimensional stationary flow of two immiscible liquids in a plane channel with rigid walls is studied. On the one of walls a temperature distribution is imposed and the another wall is heat-insulated. On the common interface the interfacial energy change is taken into account. The temperature in the liquids is distributed according to a quadratic law. It agrees with velocities field of the Hiemenz type. The conjugate boundary value problem is nonlinear and inverse for pressure gradients along the channel. The tau-method is used for the solution of problem. Three different solutions are obtained in results. It is established numerically that the obtained solutions converge to the solutions of the slowly flow problem with a decrease the Marangoni number. For each of the solutions the characteristic flow structures are constructed.