Interactions between two bubbles on a hot or cold wall

A temperature gradient normal to a planar wall produces two-dimensional motion and aggregation or separation of bubbles on the hot or cold wall, respectively. The origin of the motion is fluid convection driven by the thermal Marangoni stress on the surface of the bubbles. Previous theories for the dynamics of two or more bubbles have been based on an analysis of flow about a single bubble and the resulting convection that entrains its neighbors. Here we extend the theory by solving the quasi-steady equations for the temperature and velocity fields for two bubbles. The result is a quantitative model for the relative velocity between two bubbles as a function of both the distance between them and the gap between each bubble and the surface. Interactions between the bubbles strongly increase the approach velocity, which is counter-intuitive because the hydrodynamic resistance increases as the bubbles approach each other. An asymptotic analysis indicates the thermocapillary force bringing them together or pushing them apart is singular in the separation when the bubbles are close to each other. The two-bubble theory agrees reasonably well with the experimentally measured velocities of pairs of bubbles on hot or cold surfaces, though it slightly overestimates the velocities.