The equilibrium shape of an axisymmetric sessile drop subject to local stresses

The exact equation describing the shape of a fluid drop under the action of local surface stresses induced by colloidal interactions is derived without resorting to any of the approximations inherent in the profile equation currently employed in the literature. The exact equation implies, and numerical examples confirm, that repulsive external (i.e., positive) surface energies assist in stabilising the drop against deformation, while attractive (i.e., negative) energies destabilise the drop, promoting or enhancing deformation. An inherent singularity in the governing differential equation (absent from the approximate equations currently used) when the surface energy (surface tension) is identically matched by an external attractive energy represents an instability limit. Explicit bounds are established for a further instability criterion and for the hydrostatic pressure difference across the interface. An exact equation for the radial extent of the sessile drop and some numerical examples are also presented.