Occurrence of rotational regimes of thermo-capillary fluid flows near the free surface

Abstract

The nonlinear boundary layers caused by the non-uniform heating of the free surface, were studied intensively at the end of the last century in connection with the experiments in space. For the Marangoni boundary layers self-similar solutions were found, the conditions for the existence and uniqueness of solutions were studied and numerous numerical calculations were conducted. It was shown for the layers of finite thickness that the non-uniform heating of the free surface can lead to the rotation of an inhomogeneous fluid over the entire thickness of the layer. This paper shows that under the positive temperature gradient there can occur a rotation of the fluid as a result of bifurcation in a thin boundary layer near the free surface only, the fluid out of this layer does not rotate. In the case of the negative temperature gradient rotational modes are not detected. For the stationary axisymmetric thermocapillary flows of liquid there were obtained self-similar solutions of the Marangoni boundary layer equations with the power-law dependence on the temperature of the free surface of the radial coordinate. The speed of the outer non-viscous non-swirling flow is of the same order as the velocity in the boundary layer. With a positive temperature gradient along the free surface, the equations of the boundary layer have two 'basic' solutions describing thermocapillary non-swirling fluid flows. These solutions exist if the speed of the external inviscid flow on the free surface exceeds a critical value. The bifurcation diagram was calculated and two secondary modes, which differ from the 'basic' modes by the presence of rotation around the axis of symmetry, were found. Under a negative surface temperature gradient there was found only one 'main' mode, bifurcation points and secondary modes were not detected.