Thermocapillary convection due to a stationary bubble

Abstract

We analyze the velocity and temperature fields due to thermocapillary convection around a gas bubble that is stationary in a liquid. A linear temperature field is imposed in the undisturbed liquid. Our interest is in the effect of convective transport of momentum and energy on the velocity and temperature fields. We assume that buoyant convection is negligible. The relevant Reynolds and Marangoni numbers are assumed to be small compared with unity. When the Reynolds and Marangoni numbers are set equal to zero, the steady velocity field far from the bubble decays inversely with distance from it. We show that this behavior of the velocity field, coupled with the linear variation of the imposed temperature far away from the bubble, leads to an ill-posedness in the problem for the perturbation temperature field at steady state for small Marangoni numbers, when the Reynolds number is zero. We also consider the effect of convective transport of momentum for small Reynolds number, when the Marangoni number is zero. In this case, the velocity field far away from the bubble decays inversely as the distance from the bubble even in the presence of inertia. It follows that, for a fixed Prandtl number, when both the Marangoni and Reynolds number are nonzero, the behavior of the velocity field far from the bubble continues to yield an ill-posed problem for the steady perturbation temperature field. Therefore, inertia cannot relieve the ill-posedness of the perturbation problem for the steady temperature field. In order to understand the origin of this ill-posedness, we analyze a related problem involving the unsteady development of the temperature field in the liquid from the moment that a point force is applied to it, and held constant subsequently. We show that including the effect of convective transport of energy by a perturbation expansion leads to a temperature field in the liquid that is always time dependent. The temperature field never achieves a steady state.