Thermocapillary migration of a compound drop inside a spherical cavity

Abstract

This study investigates the thermocapillary migration of a compound drop placed concentrically within a spherical cavity under the limit of vanishing Péclet and Reynolds number. The imposed temperature gradient, which is constant along the line connecting the centers of the drop and cavity, is the driving force for the migration of compound drop. The compound drop is assumed to translate with an unknown velocity to be determined using force-free conditions. The flow field in each phase of the drop and the continuous phase is governed by the Stokes equations, whereas the thermal problem in each phase is governed by the heat conduction equation. The hydrodynamic problem and the thermal problem are coupled through specific boundary conditions. A complete general solution of the Stokes equation is used to solve the hydrodynamic problem in each phase. The migration velocity of a compound drop inside a spherical cavity is presented for various values of the physical parameters involved such as viscosity ratio, thermal conductivity ratio, Marangoni number. It has been observed that the migration velocity which represents the rate of movement of compound drop due to thermocapillary effects, decreases as the ratio of the compound drop’s radius to the cavity radius increases. On the other hand, this velocity decreases with an increase in relative conductivity of the cavity wall and increases with Marangoni number at the interface of the compound drop. The analytical solution provides a closed-form expression for the migration velocity of the confined compound drop, and it is seen that the boundary effects play significant role in thermocapilary migration.