Thermocapillary Deposition of a Fluid Droplet Normal to a Planar Surface

Abstract

An analytical study is presented for the thermocapillary migration of a fluid sphere in a constant applied temperature gradient perpendicular to a planar surface. The Peclet and Reynolds numbers are assumed to be small, so that the appropriate energy and momentum equations of the fluids inside and outside the droplet are governed by the Laplace and Stokes equations, respectively. The asymptotic formulas for the temperature and velocity fields in the quasisteady situation are obtained by using a method of reflections. The plane surface may be a solid wall or a free surface. When the droplet is migrating normal to a solid plane, the boundary effect of the planar surface retards the droplet motion, reducing the thermocapillary velocity of the droplet. In the situation of droplet migration toward a free surface due to thermocapillarity, the droplet velocity can be either greater or smaller than that which would exist in the absence of the plane surface. In general, the boundary effect on the thermocapillary migration is found to be weaker than that on the motion driven by a gravitational force. However, the interaction between the plane and the droplet can be very strong when the gap thickness approaches zero. Considering thermocapillary mobility, the deposition time for a droplet translating across the thermocapillary boundary layer is integrated. Also, it is predicted that the deposition time will be postponed if the fluid sphere is migrating normal to a solid wall. However, the deposition time for a droplet moving normal to a free surface may be shorter than predicted if there is no boundary influence. Generally speaking, a free surface exerts less influence on the droplet movement than does a solid surface.