A joint creeping motion of three fluids in a flat layer: a priori estimates and convergence to a stationary regime

We study a partially invariant solution of rank 2 and defect 3e to the equations of a viscous heat-conducting fluid. It is interpreted as a two-dimensional motion of three immiscible fluids in a flat channel bounded by solid walls for which the distribution of temperature is known. From a mathematical point of view, the resulting initial boundary value problem is nonlinear and inverse. Under some assumptions (often fulfilled in practical applications), the problem is replaced by a linear one. We obtain a priori estimates as well as the exact stationary solution and prove that, the solution tends to a stationary regime if the temperatures of the walls stabilize with time.