Thin three-dimensional droplets on an oscillating substrate with contact angle hysteresis

Recent experiments [P. Brunet, J. Eggers, and R. D. Deegan, Phys. Rev. Lett. 99, 144501 (2007)10.1103/PhysRevLett.99.144501] have shown that a liquid droplet on an inclined plane can be made to move uphill by sufficiently strong, vertical oscillations. In order to investigate this counterintuitive phenomenon we use a model in which liquid inertia and viscosity are assumed negligible so that the motion of the droplet is dominated by the applied acceleration due to the oscillation of the plate, gravity, and surface tension. We explain how the leading order motion of the droplet can be separated into a spreading mode and a swaying mode. For a linear contact line law, the maximum rise velocity occurs when these modes are in phase. We show that, both with and without contact angle hysteresis, the droplet can climb uphill and also that, for certain contact line laws, the motion of the droplet can produce footprints similar to experimental results. We show that if the two modes are out of phase when there is no contact angle hysteresis, the inclusion of hysteresis can force them into phase. This in turn increases the rise velocity of the droplet and can, in some cases, cause a sliding droplet to climb.