Thermocapillary Bubble Migration - An Oseen-Like Analysis of the Energy Equation

The thermocapillary migration of a bubble in a liquid possessing a temperature gradient is analyzed in the limit of large Reynolds and Marangoni numbers. Crespo and Manuel (1983) performed an analysis in this limit wherein energy conduction is completely neglected and obtained the bubble migration velocity using energy dissipation arguments. In the present analysis, performed in a coordinate system moving with the bubble, the velocity field in the convection term in the energy equation is approximated in an Oseen-like manner by replacing it with the velocity field far away from the bubble (i.e., the migration velocity of the bubble). Conduction is retained to satisfy the zero conductive heat flux boundary condition on the bubble surface. An approximate solution has been obtained for the Oseen-like energy equation. The bubble velocity obtained using energy dissipation considerations is in agreement with the result of Crespo and Manuel. The solution shows the thermal boundary layer and wake structure in the vicinity of the bubble. The Oseen-like analysis, however, has inherent limitations, as the flow penetrates the bubble surface. These issues are discussed and the result are compared to those in the literature.