Evaporation of a sessile droplet on a slope

We theoretically examine the drying of a stationary liquid droplet on an inclined surface. Both analytical and numerical approaches are considered, while assuming that the evaporation results from the purely diffusive transport of liquid vapor and that the contact line is a pinned circle. For the purposes of the analytical calculations, we suppose that the effect of gravity relative to the surface tension is weak, i.e. the Bond number (Bo) is small. Then, we express the shape of the drop and the vapor concentration field as perturbation expansions in terms of Bo. When the Bond number is zero, the droplet is unperturbed by the effect of gravity and takes the form of a spherical cap, for which the vapor concentration field is already known. Here, the Young-Laplace equation is solved analytically to calculate the first-order correction to the shape of the drop. Knowing the first-order perturbation to the drop geometry and the zeroth-order distribution of vapor concentration, we obtain the leading-order contribution of gravity to the rate of droplet evaporation by utilizing Green's second identity. The analytical results are supplemented by numerical calculations, where the droplet shape is first determined by minimizing the Helmholtz free energy and then the evaporation rate is computed by solving Laplace's equation for the vapor concentration field via a finite-volume method. Perhaps counter-intuitively, we find that even when the droplet deforms noticeably under the influence of gravity, the rate of evaporation remains almost unchanged, as if no gravitational effect is present. Furthermore, comparison between analytical and numerical calculations reveals that considering only the leading-order corrections to the shape of the droplet and vapor concentration distribution provides estimates that are valid well beyond their intended limit of very small Bo.