Thermocapillary Motion of a Spherical Drop in a Spherical Cavity

A theoretical study of the thermocapillary migration of a fluid sphere located at an arbitrary position inside a spherical cavity is presented in the quasi-steady limit of small Reynolds and Marangoni numbers. The applied temperature gradient is perpendicular to the line through the drop and cavity centers. The general solutions to the energy and momentum equations governing the system are constructed from the superposition of their fundamental solutions in the spherical coordinates originating from the two centers, and the boundary conditions are satisfied by a multipole collocation method. Results for the thermocapillary migration velocity of the drop are obtained for various cases. When the fluid sphere is at the center of the cavity, the collocation result is in excellent agreement with the available exact solution. The normalized thermocapillari migration velocity decreases with increases in the drop-to-cavity radius ratio and in the relative distance between the drop and cavity centers, vanishing as the drop surface touches the cavity wall. For a given configuration, this velocity augments with increases in the relative viscosity of the drop and thermal conductivity of the cavity phase. The boundary effects on the thermocapillary motion perpendicular to the line connecting the drop and cavity centers is significant, but in general weaker than that parallel to this line.