Thermocapillary migration of a gas bubble in an arbitrary direction with respect to a plane surface

Abstract

The problem of the thermocapillary migration of a gas bubble in an unbounded fluid in the presence of a neighboring rigid plane surface is analyzed in the quasistatic limit of negligible Reynolds and Marangoni numbers. The temperature gradient in the fluid is uniform in the undisturbed state and is oriented at an arbitrary angle relative to the plane surface. The solution is constructed by superposing solutions of the corresponding problems of motion normal to the surface and motion parallel to it. Results are reported in the form of a scalar interaction parameter defined as the ratio of the speed of the bubble in the presence of the plane surface to the speed in its absence. The direction of motion of the bubble is, in general, different from that of the temperature gradient, and is given. The corresponding problem of gravity-driven migration of a gas bubble normal to a plane surface is already solved in the literature. The solution of the parallel migration problem, not reported in the past, is given here. Comparisons of the results in this case show that the plane surface exerts a weaker influence in the thermocapillary migration case. This is due to the more rapid decay, away from the bubble, of the disturbance velocity and temperature gradient fields in this case. Results presented herein show that the surface exerts the greatest influence in the case of motion normal to it, and the weakest in the case of parallel motion.