Layered motion of two immiscible liquids with a free boundary

The unidirectional motion of two viscous immiscible incompressible liquids in a flat channel is studied. An unsteady temperature gradient is set on the bottom solid wall, and the upper wall is a free boundary. Liquids contact on a flat interface. The motion is caused by the combined action of thermogravitational and thermocapillary forces and a given total unsteady flow rate in the layers. The corresponding initial boundary value problem is conjugate and inverse, since the pressure gradient along the channel is determined together with the velocity and temperature field. An exact stationary solution was found for it. In Laplace images, the solution of the non-stationary problem is found in the quadrature forms. It was established that if the temperature on the bottom wall and the flow rate stabilize with time, then the motion goes to a stationary state with time. This fact indicates the stability of the stationary solution with respect to unidirectional unsteady perturbations. The calculation results showing various methods of controlling motion by setting the temperature on the wall are given.