Nonstationary laminar thermal and solutal Marangoni convection of a viscous fluid

We have determined and analyzed exact stationary and nonstationary solutions to the laminar Marangoni convection problem, which is an overdetermined boundary value problem. The numerical solution of this problem belongs to the class of Birikh solutions. The overdetermination of the resolving system of equations results from the zeroness of the velocity parallel to the applicate axis. Cases of thermal and solutal convection of a viscous incompressible fluid are considered. To make the boundary value problem solvable, the class of exact solutions is proposed for use, where velocities are one-dimensional in coordinates, the pressure and temperature fields are three-dimensional. Identical equality to zero of the convective derivative in the impulse conservation equation is typical of the class presented. The convective derivative remains in the caloric equation. The discussed boundary value problem is shown to be irreducible to the one-dimensional problem when the temperature gradient is specified on both boundaries of the fluid layer, as distinct from the classical Birikh solution and its numerous generalizations. The obtained stationary and nonstationary solutions have a stagnation point for velocities, thus suggesting the presence of counter flows in the moving fluid. It is demonstrated by localization of the polynomial roots of the stationary solutions that there exists such a value of layer thickness that the tangential stress can become zero on the lower boundary of the fluid layer only under thermal Marangoni convection. The nonstationary solutions obtained by the boundary element method, which can be treated as an exact method, tend to become stationary. The application of the boundary element method extends the class of exact nonstationary solutions considerably, since this method enables one to study not only invariant exact solutions.