Influence of gravity on the spreading of two-dimensional droplets over topographical substrates

Abstract

The influence of gravity on the motion of a two-dimensional droplet of a partially wetting fluid over a topographical substrate is considered. The spreading dynamics is modeled under the assumption of small contact angles in which case the long-wave expansion in the Stokes-flow regime can be employed to derive a single equation for the evolution of the droplet thickness. The relative importance of gravity to capillarity in the equation is measured by the Bond number which is taken to be low to moderate. In this regime, the flow in the vicinity of the contact line is matched asymptotically through a singular perturbation approach to the flow in the bulk of the droplet to yield a set of coupled integrodifferential equations for the location of the two droplet fronts. The matching procedure is verified through direct comparisons with numerical solutions to the full problem. The equations obtained by asymptotic matching are analyzed in the phase plane and the effects of Bond number on the droplet dynamics and its equilibria are scrutinized.