Occurrence of Fluid Rotation upon Cooling of the Free Boundary

Abstract

For the Navier-Stokes system and the heat equation with vanishing viscosity, the stationary thermocapillary flow of an incompressible fluid in a horizontal layer of infinite thickness is calculated. The layer is bounded above by a free non-deformable boundary on which an uneven temperature distribution is specified. In the approximation of the boundary layer, asymptotic expansions of the solution of the problem are constructed. The main terms of the asymptotics satisfy the Prandtl equations of the boundary layer. Two types of regimes in the boundary layer are calculated - non-swirling and rotational fluid flows.

Rotational regimes of fluid flows arise as a result of bifurcation of non-swirling regimes in the boundary layer. Bifurcation points are found by solving a boundary value problem for eigenvalues. It is shown that rotational regimes arise only when the free boundary is locally cooled. When the boundary is heated, fluid rotation does not occur.

Two types of rotational modes are numerically calculated. All modes exist only if the velocity of the external fluid flow does not exceed the bifurcation value. The modes of the first type have axial symmetry. There are only two such modes. Other modes do not have axial symmetry. For modes of the second type, an exact solution in cylindrical coordinates is obtained. These modes depend on two independent parameters that fill the circle of unit radius. So, at the bifurcation point, many rotational regimes arise, which are a two-parameter family.