Bifurcations of rotation in a fluid near free surface

Abstract

Based on the Navier-Stokes equations, the thermocapillary fluid flow in a semi-infinite space bounded from above by a free surface is calculated. At the free surface, the temperature is distributed locally according to a power law from the radial coordinate. It is shown that when the boundary is cooled, rotational regimes of fluid flows appear in the boundary layer near the free surface. There is no rotation outside the boundary layer. Both untwisted and rotational modes are calculated numerically. Non-swirling modes exist only if the external flow velocity is greater than its limit value. Rotational regimes arise as a result of bifurcation of untwisted regimes. The bifurcation values of the external flow velocity are found numerically by solving the linearized eigenvalue problem. A bifurcation diagram is constructed. In a small neighborhood of bifurcation points, asymptotic expansions of the velocity and pressure fields are constructed. Two small parameters are introduced in the construction of asymptotic formulas. It is shown that two rotational regimes branch off from the bifurcation points, which differ from each other only in the direction of rotation. Outside small neighborhoods of the bifurcation points, the rotational regimes are constructed numerically.