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In this paper, the problem of a two-dimensional stationary flow of two immiscible viscous heat-conducting fluids in a cylinder is solved. The fluids have a common movable non-deformable interface. The cylinder has a solid outer wall. At the same time the mass forces are absent. The total energy condition at the interface is taken into account. The temperature in liquids is distributed in a quadratic law, which is consistent with the velocity field of the Himenz type. From a mathematical point of view, this initial-boundary value problem is nonlinear and inverse with respect to pressure gradients along the cylinder axis. The modified Galerkin method is used to solve the problem. The effect of the Marangoni number on the fluids flow is investigated.

Two-dimensional stationary flow of two immiscible fluids in a cylinder taking into account the internal energy of the interface