Journal of Fluid Mechanics

Coalescence of liquid drops

a1 Universität Gesamthochschule Essen, Fachbereich Physik, 45117 Essen, Germany
a2 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver St, Cambridge CB3 9EW, UK
a3 Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA


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 rm of the small bridge between the two drops. The flow is driven by a highly curved meniscus of length 2πrm and width Δ[double less-than sign]rm 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[is proportional to]r3m and rm[similar](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[is proportional to]r3/2m at the meniscus and rm[similar](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 rm(t)[double less-than sign]1 involves matching of lengthscales of order r2m, r3/2m, rm, 1 and probably r7/4m.

(Received March 16 1999)
(Revised July 28 1999)