Shape of sessile drops at small contact angles

Abstract

The shape of a sessile drop on a horizontal substrate depends upon the Bond number $Bo$ and the contact angle $\alpha$ . Inspired by puddle approximations at large $Bo$ (Quéré, Rep. Prog. Phys., vol. 68, 2005, p. 2495), we address here the limit of small contact angles at fixed drop volume and arbitrary $Bo$ . It readily leads to a pancake shape approximation, where the drop height and radius scale as $\alpha$ and $\alpha ^{-1/2}$ , respectively, with capillary forces being appreciable only near the edge. The pancake approximation breaks down for $Bo=\textrm {ord}(\alpha ^{2/3})$ . In that distinguished limit, capillary and gravitational forces are comparable throughout, and the drop height and radius scale as $\alpha ^{2/3}$ and $\alpha ^{-1/3}$ , respectively. For $Bo\ll \alpha ^{2/3}$ these scalings remain, with the drop shape turning into a spherical cap. The asymptotic results are compared with a numerical solution of the exact problem.