Thermocapillary migration of a droplet along a free surface

The proximity of a free surface can have a significant impact on the thermocapillary migration of droplets (or bubbles) submerged in a layer of fluid. For instance, since the thermocapillary effect produces forces on both the surface of the droplet and the surface of the fluid layer, the droplet can be forced to migrate in, or opposite to, the direction of the temperature gradient. To quantify the migration velocity we use the method of reflections to construct an explicit Lamb's series solution for the velocity field inside and outside the droplet in the presence of a uniform as well as a nonuniform temperature gradient.