Thermocapillary migration of long bubbles in cylindrical capillary tubes

Abstract

We consider the problem of the migration of a long bubble in a tube with a prescribed axial temperature gradient. The resulting thermocapillary stress moves the bubble toward hotter regions and we are interested in determining the speed of the bubble. Assuming small Peclet, Reynolds, Bond, and capillary numbers, Ca allows the uncoupling of the temperature field from the flow field, the use of creeping flow and lubrication theory, the assumption of axisymmetry, and the use of matched expansions in Ca, respectively. The structure of the solution is that of a constant thickness film bounded by constant curvature cap regions, with transition layers in between. A modified Landau–Levich equation governing the film profile in the transition regions is solved numerically, establishing the relationship between the unknown film thickness and the unknown bubble speed. A global mass conservation relation is then used to complete the solution and relate the bubble speed to the thermophysical properties. The solution is a function of a single dimensionless parameter, Δσ*=γTβa/σ where β is the temperature gradient, a the tube radius, σ the mean surface tension, and γT the temperature coefficient of surface tension. It is found that the speed increases with the temperature gradient as expected, and has a nonlinear dependence on Δσ* that is ultimately connected with the nonlinearity of the fluid dynamics in the transition regions.