Thermocapillary migration of bubbles: convective effects at low Péclet number

The effect of a weak convective heat transfer on the thermocapillary interaction of two bubbles migrating in an externally imposed temperature gradient is examined. It is shown that, for short and moderate separation distances, the corrections to the individual migration velocities of the bubbles are of O(Pe), where Pe is Péclet number. For separation distances larger than O(Pe−1/2) the correction is of O(Pe2) as previously found for an isolated drop. The perturbations to the bubble velocities have opposite signs: the motion of the leading bubble is enhanced while the motion of the trailing one is retarded. A newly found feature is that equal-sized bubbles, which otherwise would move with equal velocities, acquire a relative motion apart from each other under the influence of convection. For slightly unequal bubbles there are three different regimes of large-time asymptotic behaviour: attraction up to the collision, infinite growth of the separation distance, and a steady migration with equal velocities, the steady motion separation distance being a function of the parameters of the problem. Sufficient conditions for the realization of each regime are given in terms of the Péclet number, initial separation and radii ratio.