Thermocapillary motion of a fluid droplet parallel to two plane walls

Abstract

The steady thermocapillary migration of a fluid droplet located between two infinite parallel plane walls is studied theoretically in the absence of fluid inertia and thermal convection. The imposed temperature gradient is constant and parallel to the two plates, and the droplet is assumed to retain a spherical shape. The plane walls may be either insulated or prescribed with the far-field temperature distribution. The presence of the neighboring walls causes two basic effects on the droplet velocity: first, the local temperature gradient on the droplet surface is enhanced or reduced by the walls, thereby speeding up or slowing down the droplet; secondly, the walls increase viscous retardation of the moving droplet. To solve the thermal and hydrodynamic governing equations, the general solutions are constructed from the fundamental solutions in both the rectangular and spherical coordinate systems. The boundary conditions are enforced first at the plane walls by the Fourier transforms and then on the droplet surface by a collocation technique. Numerical results for the thermocapillary migration velocity of the droplet relative to that under identical conditions in an unbounded medium are presented for various values of the relative viscosity and thermal conductivity of the droplet as well as the relative separation distances between the droplet and the two plates. For the special cases of thermocapillary motions of a spherical droplet parallel to a single plate and on the central plane of a slit, the collocation results agree well with the approximate analytical solutions obtained by using a method of reflections. The presence of the lateral walls can reduce or enhance the droplet velocity, depending upon the relative transport properties of the droplet, the relative droplet-wall separation distances, and the thermal boundary condition at the walls. In general, the boundary effect on thermocapillary migration is quite complicated and relatively weak in comparison with that on sedimentation.