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. Commun., 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^2a if Oh≪1 due to a compressional boundary layer of length L∝OhrE, 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∼(32aγ/3ρ)^(1/4)t^(1/2), where γ is the surface tension and ρ is the density.