Uniform asymptotic solutions for small and large sessile drops

The problem of determining the equilibrium shape of either a small or large sessile drop is studied. By the correspondence principle, the problem of approximating the static meniscus in a vertical right circular cylinder is also solved. A parameter ε is introduced which is ratio of the physical length scale to the capillary length scale. Perturbation solutions, which are uniformly valid over the entire surface of the drop, are obtained for either small or large values ε using the method of successive substitutions. Comparisons of the results with solutions obtained by other approximate methods are presented and discussed. The solutions for small and large values of ε are then combined using an ad hoc, but straightforward, technique, resulting in an approximate solution which is valid for all positive values of the parameter ε.