The edge profile of liquid spills

Abstract

The Young–Laplace (Y–L) equation relates the curvature of the interface between two fluids to the pressure difference across that interface. The link between the surface curvature and pressure discontinuity underpins numerous phenomena in hydrostatics, including the profiles of sessile (immobile) drops, pendant drops, capillary bridges, and liquid pearls. In most cases, the application of the Y–L equation readily yields the governing differential equations of the fluid surface but often these equations have no analytic solutions; this is the case, for example, with the profile of a dew drop or a liquid pendant. There are, however, some cases for which the differential equations have closed form solutions; one is the edge profile of a liquid spill or puddle on a horizontal surface. Here, we review the Y–L equation and apply it to obtain the governing differential equations of the edge profile for a liquid spill. The solution of the differential equations results in a universal function that describes the edge profile of any fluid. One important finding is that the profile curve begins its descent very gradually from the flat top of the fluid far away from the edge and terminates at the point, where the angle between the profile curve and the horizontal matches the equilibrium contact angle of the liquid on the substrate.