Thin static drops with a free attachment boundary

Abstract

A thin fluid drop is at rest on a plane vertical surface, supported against gravity by surface tension. The perimeter of the drop is required to lie on a given closed curve, upon which the contact angle is arbitrary. If the drop is of sufficient volume, it can wet the whole area interior to this curve. However, for any given curve, there is a certain critical volume below which this fully wetted configuration is not physically acceptable, the formal solution having negative thickness. It is suggested here as an alternative that the upper portion of the drop, above a free boundary to be determined, must drain completely. Some time-dependent computations in two dimensions are presented to illustrate this draining property. In three dimensions, the static free boundary has zero contact angle, and must be determined as part of the solution. An example solved here is that where the original boundary is a circle, and the free boundary is a non-trivial curve lying inside it, whose shape is found by numerical methods. This problem also has relevance to the shape of a raindrop on a windowpane where surface contamination prevents contact-line motion, and the drop may again be considered to be confined within a prescribed boundary.