Shape of sessile drops at small contact angles

The shape of a sessile drop on a horizontal substrate depends upon the Bond number Bo and the contact angle α. 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 α and α-1/2, respectively, with capillary forces being appreciable only near the edge. The pancake approximation breaks down for Bo = ord(α2/3). In that distinguished limit, capillary and gravitational forces are comparable throughout, and the drop height and radius scale as α2/3 and α-1/3, respectively. For Bo ≪ α2/3 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.