Thin three-dimensional drops on a slowly oscillating substrate

We examine the evolution of a liquid drop on an inclined substrate oscillating vertically. The drop is assumed thin, and the oscillations are assumed weak and slow (the latter makes the liquid's inertia and viscosity negligible, so the drop's shape is determined by a balance of surface tension, gravity, and vibration-induced inertial force). On the basis of these approximations, asymptotic expressions are derived for the mean velocities of two- (2D) and three-dimensional (3D) drops. It is shown that, if the amplitude of the substrate's oscillations exceeds a certain threshold value, both 2D and 3D drops climb uphill. The two cases, however, exhibit different behaviors of the threshold amplitude of the oscillations on their frequency, in the low-frequency limit: to make 2D drops climb uphill, the oscillations must be much stronger than those in the 3D case. This difference is important, as the 2D behavior does not fit the existing experimental results.