Thin-sheet flow between coalescing bubbles

When two spherical bubbles touch, a hole is formed in the fluid sheet between them, and capillary pressure acting on its tightly curved edge drives an outward radial flow which widens the hole joining the bubbles. Recent images of the early stages of this process (Paulsen et al., Nat. Common., vol. 5, 2014) show that the radius of the hole r(E), at time t grows proportional to t(1/2), and that the rate is dependent on the fluid viscosity. Here, we explain this behaviour in terms of similarity solutions to a third-order system of radial extensional flow equations for the thickness and velocity of the sheet of fluid between the bubbles, and determine the growth rate as a function of the Ohnesorge number Oh. The initially quadratic sheet profile allows the ratio of viscous and inertial effects to be independent of time We show that the sheet is slender for r(E) << a if Oh >> 1, where a is the bubble radius, but only slender for r(E) << Oh(2)a if Oh << 1 due to a compressional boundary layer of length L proportional to Oh r(E), after which there is a change in the structure but not the speed of the retracting sheet. For Oh << 1, the detailed analysis justifies a simple momentum-balance argument, which gives the analytic prediction r(E) similar to (32a gamma/3 rho)(1/4)t(1/2) where gamma is the surface tension and rho is the density.