Inviscid coalescence in the presence of a surrounding fluid

Coalescence of fluid droplets is initially controlled by surface tension forces and viscosity. For low viscosity fluids such as water, the viscous lengthscale is quickly reached, yielding a new balance between surface tension and inertia. Numerical and asymptotic calculations have shown that there is no simply connected solution for the coalescence of inviscid fluid drops surrounded by a void, as large amplitude capillary waves cause the free surface to pinch off. However, the flow induced in a low density surrounding fluid such as air may prevent pinch-off from ever occurring. We consider asymptotic solutions to a small-displacement version of this free boundary problem, and show that simply connected solutions are again not available for zero density surrounding fluid, but that for a distinguished density ratio in which the surrounding fluid has much lower density than the coalescing drops, we can obtain suitable asymptotic solutions.