Thermocapillary migration of a slightly deformed spherical fluid drop in an immiscible fluid

In this work, we report on theoretical investigation of thermocapillary migration of a slightly deformed spherical fluid drop in a viscous incompressible fluid with a uniformly prescribed but arbitrarily oriented temperature gradient in the steady limit of vanishing Péclet and Reynolds numbers. The flow fields in the exterior and interior of the drop are governed by the Stokes equations whereas the temperature fields in both the regions are governed by Laplace's equation. The governing equations are solved asymptotically using a method of perturbed expansions under suitable boundary conditions. The deformation from spherical shape is characterized by a small parameter, and we have solved the problem up to the second order of this deformation parameter. The thermocapillary migration velocity of the slightly deformed spherical drop is obtained by applying force free condition. The explicit expressions for the migration velocity of the drop are obtained for the special cases of prolate and oblate spheroidal drops for various values of the internal-to-external viscosity ratio, axial-to-radial aspect ratio and relative thermal conductivity of the drop.