Axisymmetric thermocapillary migration of a fluid sphere in a spherical cavity

The quasi-steady thermocapillary migration of a spherical fluid drop situated at an arbitrary position in a second fluid within a spherical cavity is studied theoretically in the limit of negligible Marangoni and Reynolds numbers. The imposed temperature gradient is constant and along the line connecting the centers of the drop and cavity. To solve the thermal and hydrodynamic governing equations, the general solutions are constructed from the fundamental solutions in the two spherical coordinate systems based on the drop and cavity. The boundary conditions at the drop surface and cavity wall are satisfied by a collocation technique. Numerical results for the thermocapillary migration velocity of the drop normalized by its value in an unbounded medium are presented for various values of the relative viscosity and thermal conductivity of the drop, the relative conductivity of the cavity phase, the drop-to-cavity radius ratio, and the relative distance between the drop and cavity centers. In the particular case of the migration of a spherical drop in a concentric cavity, these results agree excellently with the exact solution derived analytically. The normalized thermocapillary migration velocity of the confined drop decreases monotonically with an increase in the drop-to-cavity radius ratio or its relative distance from the cavity center and vanishes in the touching limit of the drop and cavity surfaces. On the other hand, this velocity increases with an increase in the relative thermal conductivity of the cavity phase, but can increase or decrease with an increase in the relative viscosity or thermal conductivity of the drop for a given configuration. The boundary effect on thermocapillary migration can be significant, but is weaker than that on sedimentation.