Inertial coalescence with a little viscosity

We consider analytically the inertial regime of coalescence of two fluid drops, when inertial forces balance surface tension, at early times when the drops are joined by a small fluid bridge of radius r_b propto (gamma a/ rho)^{1/4} t^{1/2}. This regime applies when Oh ≡ mu/(rho gamma a)^{1/2} << r_b/a << 1. Remarkably, despite the small viscosity (Oh<<1), viscosity has a leading-order effect! Momentum created by surface tension at the edge of the fluid bridge is left behind in a thin viscous wake as the bridge grows. Fluid is entrained into the wake, driving an inertial flow on the scale of the fluid bridge. This flow opens up the gap between the drops, thus reducing the driving capillary pressure and resulting in an O(1) decrease in the rate of coalescence. Our matched asymptotic solution for the flow shows excellent qualitative agreement with previous results by Anthony et al. (2020) for the streamlines and gap profile in full Navier-Stokes simulations for Oh=0.001.