Thermocapillary convection of immiscible liquid in a three-dimensional layer at low Marangoni numbers

The joint convection of two viscous heat-conducting liquids in a three-dimensional layer bounded by solid flat walls is studied. The upper wall is thermally insulated, and a non-stationary temperature field is set on the lower wall. Liquids are assumed to be immiscible and complex conjugation conditions are set at the flat interface between them. The evolution of this system is described by the Oberbeck-Boussinesq equations in each fluid. The solution of this problem is sought in the class of velocity fields linear in two coordinates, and temperature fields are quadratic functions of the same coordinates. In this case, the problem is reduced to a system of 10 nonlinear integro-differential equations. It is conjugate and inverse with respect to 4 functions of time. To find them, integral redefinition conditions are set. The physical meaning of these conditions is the closeness of the flow. The inverse initial-boundary value problem describes convection in a two-layer system that occurs near the temperature extremum point on the lower solid wall. For small Marangoni numbers, the problem is approximated by a linear one (the Marangoni number plays the role of the Reynolds number for the Navier-Stokes equations). A stationary solution to this problem has been found. The linear nonstationary problem is solved by the Laplace transform method, and the temperature can have discontinuities of the 1st kind (change by a jump). In Laplace images, the solution is obtained in quadratures. It is proved that with increasing time, it tends to stationary mode if the temperature on the lower wall stabilizes over time. The evolution of the behavior of the velocity field in the transformer oil-water system has been studied using the numerical inversion of the Laplace transform.