Effects of surfactants on the motion and deformation of a droplet in thermocapillary migration

We examine the effects of surfactants on the terminal velocity and the shape of a droplet translating in an unbounded, otherwise quiescent liquid, under the influence of an externally imposed constant temperature gradient. It is assumed that the droplet surface tension depends much more weakly on the surfactant concentration than on the temperature. In such a circumstance and assuming no bulk transport, the surface concentration of the surfactant can be found exactly for all Peclet numbers, confirming the intuitively expected result that the surfactant tends to accumulate near the low temperature side of the droplet. This, in turn, partially lowers the surface tension on that side, thereby decreasing the terminal migration velocity of the droplet as a whole. At the same time, to first order in capillary number, the droplet assumes a prolate spheroidal shape.