Motion of a drop in a vertical temperature gradient at small Marangoni number – the critical role of inertia

When a drop moves in a uniform vertical temperature gradient under the combined action of gravity and thermocapillarity at small values of the thermal Péclet number, it is shown that inclusion of inertia is crucial in the development of an asymptotic solution for the temperature field. If inertia is completely ignored, use of the method of matched asymptotic expansions, employing the Péclet number (known as the Marangoni number) as the small parameter, leads to singular behaviour of the outer temperature field. The origin of this behaviour can be traced to the interaction of the slowly decaying Stokeslet, arising from the gravitational contribution to the motion of the drop, with the temperature gradient field far from the drop. When inertia is included, and the method of matched asymptotic expansions is used, employing the Reynolds number as a small parameter, the singular behaviour of the temperature field is eliminated. A result is obtained for the migration velocity of the drop that is correct to O(Re2 log Re).