The formation of rotational regimes in the thermocapillary flow of a non-uniform fluid in a layer

Abstract

The bifurcation of solutions describing thermocapillary flows of a non-uniform fluid in a horizontal layer of finite thickness, under the influence of a temperature gradient at the free boundary is studied. On the assumption that the flow is axially symmetric and has no peripheral component, the velocities of the points of the free boundary are numerically determined as functions of the layer thickness and the temperature gradient. Regions of the parameters are determined in which there are either no solutions, or one or more solutions differing from each other in the form of their velocity profile and the number of flow and counter-flow zones. It is shown for the solutions obtained that at every bifurcation point a pair of new symmetric solutions arises differing from the fundamental solutions by the presence of rotation about the axis of symmetry. Computation of the coefficients of the bifurcation equation reveals the existence of three types of bifurcation point for which the bifurcation equation in the principal approximation contains just two non-zero coefficients. The two-dimensional case of bifurcation is investigated. The bifurcating solutions are constructed asymptotically in the neighbourhood of the bifurcation points and numerically outside such neighbourhoods.