Self-similar solutions describing unsteady thermocapillary fluid flows

Unsteady thermocapillary flows in thin layers and layers of infinite thickness with non-uniform heating of the free boundary are investigated at high Marangoni numbers. In the plane and axially symmetric cases, self-similar solutions of the non-linear boundary-layer equations are constructed and asymptotic formulae are presented. It is shown that the self-similar solutions may be non-unique for certain values of the parameters of the problem. The branching points are calculated numerically and the branched solutions are investigated.