Matched asymptotic expansion calculation of the equilibrium shape of a hole in a thin liquid film

We consider the occurrence of small axisymmetric pinholes in an otherwise uniform infinite thin liquid film. Corresponding to any particular undisturbed film thickness there exists precisely one (unstable) equilibrium solution reflecting a balance between surface tension and gravity effects. If a pinhole is smaller than this critical size the pinhole tends to close over and `heal'. If a pinhole is larger it tends to open out. So determination of this critical hole size is crucial. We examine this problem in the case of a `small' pinhole where the fundamental length-scale in the film is much smaller than the capillary length. Solutions are obtained using matched asymptotic expansions for which several different scalings are necessary.