Coalescence of liquid drops

When two drops of radius R touch, surface tension drives an initially singu lar motion which joins them into a bigger drop with smaller surface area. T his motion is always viscously dominated at early times. We focus on the ea rly-time behaviour of the radius r(m) of the small bridge between the two d rops. The flow is driven by a highly curved meniscus of length 2 pi r(m) an d width Delta << r(m) around the bridge, from which we conclude that the le ading-order problem is asymptotically equivalent to its two-dimensional cou nterpart. For the case of inviscid surroundings, an exact two-dimensional s olution (Hopper 1990) shows that Delta proportional to r(m)(3) and r(m) sim ilar to (t gamma/pi eta)ln [t gamma/(eta/R)]; and thus the same is true in three dimensions. We also study the case of coalescence with an external vi scous fluid analytically and, for the case of equal viscosities, in detail numerically. A significantly different structure is found in which the oute r-fluid forms a toroidal bubble of radius Delta proportional to r(m)(3/2) a t the meniscus and r(m) similar to (t gamma/4 pi eta)ln [t gamma/(eta R)]. 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 asymptoti c flow for r(m)(t) << 1 involves matching of lengthscales of order r(m)(2), r(m)(3/2), r(m),1 and probably r(m)(7/4).