Two-dimensional plane thermocapillary flow of two immiscible liquids

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