Thermocapillary migration of slightly deformed droplets

A small viscous droplet which is suspended in an immiscible liquid in a zero-gravity environment can be made to migrate by subjecting the system to an external temperature gradient. This so-called thermocapillary migration occurs since the surface tension of the interface is a function of temperature. If inertial effects are negligible and the interface is clean, the shape of the migrating droplet remains spherical. However, when inertial effects are significant or when surfactants are present at the interface, the droplet assumes a prolate or oblate spheroidal shape. In this work we calculate the change in the net migration velocity of the droplet which results from such a deformation. This is done by solving the requisite viscous flow problem in the deformed geometry by perturbation, in conjunction with a generalization of the so-called Lorentz reciprocal theorem which simplifies the calculations. It is found that the migration velocity could increase, decrease or remain unchanged depending on the values of certain controlling parameters.