Thermocapillary convection in a spherical container due to a stationary bubble Original Research Article

We analyze the velocity and temperature fields at steady state due to thermocapillary convection around a gas bubble that is stationary in a liquid present in a spherical container. The bubble is assumed to be located concentrically within the container. The temperature distribution on the container wall is such that the temperature field in the liquid in the absence of the bubble is linear. Our interest is in the effect of convective transport of 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 size of the container is large compared with the size of the bubble, the situation may be considered an approximation to the case of a stationary bubble in an unbounded liquid. In this case, when the Reynolds and Marangoni numbers are set equal to zero, the velocity field far from the bubble decays inversely with distance from it. It can be shown 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 for small Marangoni numbers, when the Reynolds number is zero. When the size of the container is finite, we show that the ill-posedness disappears. We obtain a perturbation temperature field that satisfies all the boundary conditions, when the Marangoni number is small compared with unity, in contrast with the case in which the bubble is present in a liquid of unbounded extent.