Two-layer stationary creeping thermocapillary flow in a three-dimensional channel

We study the problem of three-dimensional stationary creeping flow of two immiscible liquids in a channel with solid parallel walls, one of which a given temperature distribution is maintained and the other is hear-insulated. Thermocapillary forces act on the flat interface. Temperature in the liquids depends quadratically on the horizontal coordinates, and the velocity field has a special form. The resulting conjugate problem for the Oberbeck– Boussinesq model is inverse and reduces to the system of ten integro-differential equations. The total energy condition is taken into account on the interface. The problem has up to two solutions and if the heat fluxes are equal, then it has one solution. Characteristic flow structures are constructed for each of the solutions. The influence of dimensionless physical and geometric parameters on the flows is investigated.