Interactions and collisions of bubbles in thermocapillary motion

Abstract

A calculation of the motion of two bubbles in a constant temperature gradient is presented for the limit of creeping flow and vanishing Péclet number. The results are valid for undeforming, spherical bubbles of arbitrary size and orientation with respect to the temperature gradient, and for separations up to near touching. Trajectories of two bubbles, as they approach one another in an infinite fluid, are then determined from these instantaneous velocity calculations. The limiting trajectories, where the bubbles ‘‘just miss’’ each other, are used to tabulate collision efficiencies over a wide range of bubble size ratios. On the basis of the two-bubble results, a model for the evolution of a statistically homogeneous cloud of bubbles with a given initial size distribution is formulated in terms of a discrete stochastic collection equation. Using this simple model, the dependence of the rate of coalescence in a bubble cloud upon material properties, the bubble volume fraction, the average temperature gradient, and other characteristics of the system, is demonstrated. A significant conclusion is that the bubble collision rate increases with the standard deviation of the bubble size distribution.