The stability of pendent liquid drops. Part 1. Drops formed in a narrow gap

Abstract

We consider a drop of liquid hanging from a horizontal support and sandwiched between two vertical plates separated by a very narrow gap. Equilibrium profiles of such ‘two-dimensional’ drops were calculated by Neumann (1894) for the case when the angle of contact between the liquid and the horizontal support is zero. This paper gives the equilibrium profiles for other contact angles and the criterion for their stability. Neumann showed that, as the drop height increases, its cross-sectional area increases until a maximum is reached. Thereafter, as the height increases, the equilibrium area decreases. This behaviour is shown to be typical of all contact angles. When the maximum area is reached, the total energy is a minimum. It is shown that the drops are stable as long as the height and the area increase together.

The possible equilibrium shapes of static pendant drops

“…Also, selfcrossing drops are not real solutions and should be excluded from consideration, which has often been overlooked. 7,14 In cases ͑ii͒ and ͑iii͒, a zero would occur at z Յ 8, but the selfcrossing condition occurs at a lower height, as seen below. Equation ͑3͒ may be integrated 7 to give…”

“…7,10 Whereas those studies specified the liquid-gas interfacial contact length 2 to be held constant during the minimization procedure, thus making the contact line pinned in effect, we do not ͑as we should not for our problem!͒ impose this additional constraint. In a typical problem of this class, with the end point held fixed, one would need to specify an additional boundary condition at this point.…”

“…The shape of the liquid-gas interface is described by x͑z͒ or r͑z͒, where x for a two-dimensional drop is the horizontal distance of the liquidgas interface from the z-axis and r is the corresponding radial distance for an axisymmetric drop. Following Pitts,7,8 we write…”

“…III A to adopt a numerical approach. This approach is more elegant than looking for minimum energy shapes by making arbitrary perturbations which may not satisfy force balance, such as that of Pitts,7 where the curvature at the bottom was not disturbed. Pitts's procedure would result in a minimum energy for any trial shape, but a class of correct shapes was obtained by imposing Young's contact angle at the surface.…”

Existence and regularity for the problem of a pendent liquid drop

“…The behavior of a solution of (0.4), in its dependence on the initial value u(0) -u 0 < 0, has been studied extensively by P. Concus and R. Finn in a series of papers ( [12,2,3,4]; see also [28,29]). We refer to [4] for a recent detailed exposition on this argument.…”