On the paradox of thermocapillary flow about a stationary bubble

When a stationary bubble is exposed to an external temperature gradient, Marangoni stresses at the bubble surface result in fluid motion. A straight-forward attempt to calculate the influence of this thermocapillary flow upon the temperature distribution fails to provide well-behaved solution [Balasubramaniam \& Subramanian, {\it Phys. Fluids} {\bf 16}, 3131 (2004)]. This paradox is resolved here using regularization procedure which exploits the qualitative disparity in the long-range flow fields generated by stationary bubble and moving one. The regularization parameter is an (exponentially small) artificial bubble velocity $U$, which reflects the inability of any asymptotic expansion to satisfy the condition of exact bubble equilibrium. The solution is obtained using asymptotic matching of two separate Reynolds-number expansions: an inner expansion, valid at the bubble neighborhood, and remote outer expansion, valid far beyond the familiar Oseen region. This procedure provides well-behaved solution, which is subsequently used to evaluate the convection-induced correction to the hydrodynamic force exerted on the bubble. The independence of that correction upon $U$ confirms the adequacy of the regularization procdure to descibe the stationary-bubble case. The ratio of the calculated force to that pertaining to the classical pure-conduction limit [Young, Goldstein Block, {\it J. Fluid Mech.} {\bf 6}, 350 (1959)] is given by $1 - Ma/8+ o(Ma)$, where Ma is radius-based Marangoni number.