Asymptomatics of pinhole

It is well known that, in a thin liquid film, corresponding to any particular thickness there is precisely one axisymmetric equilibrium solution to the Laplace-Young capillary equation; i.e., corresponding to each film thickness there is a unique pinhole radius which has been shown to be unstable( 1973, Taylor and Michael, J. Fluid Mech. 58, 625-639). We deduce an asymptotic relationship between film thickness and pinhole height in the limit of very thin films.