On a nonlinear differential equation of electrohydrodynamics

A nonlinear differential equation which governs the equilibrium of two closely spaced drops, suspended from two circular rings, and maintained at electric potentials $\pm$ V is considered. When the potential reaches a critical value V$_m$, experiments show that the drops coalesce and the equation has, in fact, no solutions for V > V$_m$. Two limiting cases are studied. At first we consider drops which at zero potential difference, are films of zero curvature. In this case it is shown that the equation may not have a unique solution for V < V$_m$. In the second case we study drops which are nearly touching when there is no potential difference, with non-zero curvature. Using the method of matched asymptotic expansions, it is shown that when the drops are about to coalesce, the original separation distance is reduced by one-half. Thus the drops are not drawn out as might have been expected from experiments on isolated drops.