Coalescence of liquid drops

When two drops of radius R touch, surface tension drives an initially singular motion which joins them into a bigger drop with smaller surface area. This motion is always viscously dominated at early times. We focus on the early-time behaviour of the radius r_m of the small bridge between the two drops. The flow is driven by a highly curved meniscus of length 2πr_m and width Δ ≪ r_m around the bridge, from which we conclude that the leading-order problem is asymptotically equivalent to its two-dimensional counterpart. For the case of inviscid surroundings, an exact two-dimensional solution (Hopper 1990) shows that Δ ∝ r_m³ and r_m ∼ (tγ/πη) ln [tγ/(ηR)]; and thus the same is true in three dimensions. We also study the case of coalescence with an external viscous fluid analytically and, for the case of equal viscosities, in detail numerically. A significantly different structure is found in which the outer-fluid forms a toroidal bubble of radius Δ ∝ r_m^3/2 at the meniscus and r_m ∼ (tγ/4πη) ln [tγ/(ηR)]. This basic difference is due to the presence of the outer-fluid viscosity, however small. With lengths scaled by R a full description of the asymptotic flow for r_m(t) ≪ 1 involves matching of lengthscales of order r_m² r_m^3/2, r_m, 1 and probably r_m^7/4 .