Title:

Mathematical modelling of droplets climbing an oscillating plane

Recent experiments [P. Brunet, J. Eggers, and R. Deegan, Phys. Rev. Lett. 99, 114501 (2007)] 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 will derive three different models that qualitatively reproduce the main features of the experiment. For the first model the liquid's 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 and that the droplet is thin. 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 the frequencies of oscillation of the two 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. For the second model we use a twodimensional flow where the Reynolds number is assumed large enough for viscosity to be neglected. We show that the leading order motion of the droplet can be separated into the same two modes and the net motion of the droplet is an oscillatory function of the frequency. For increasingly nonwetting droplets we discover that the rise velocity begins to oscillate very rapidly as a function of the static contact angle. What we also discover is that the change in the free surface of the droplet is actually a wave travelling travelling across the droplet, and the amount of modes present coincide with the rapid change in the rise velocity. Using a cubic contact line law and contact angle hysteresis we observe a droplet that can climb uphill for parameter values similar to that of the experiment. With the addition of a time dependent term within the contact line law we show that it is possible to obtain a multivalued relationship between the velocity of the contact line and the respective contact angles, reproducing experimental observations seen for unsteady, moving contact lines. For the third model we again assume that the liquid's viscosity is negligible, similar to model 2, only now for a threedimensional, thin droplet. For very small amplitudes the motion of the droplet is a combination of a swaying mode and a spreading mode that interact causing a net motion of the droplet. This motion is found to be an oscillatory function of the driving frequency and the magnitude of the peak rise velocity is proportional to one over the frequency squared. By examining the velocity of the centre of the droplet and the displacement of the contact line we see that the absolute maximums of both of these, over one period of oscillation, contain natural frequencies, which are evenly spaced with respect to the square root of the frequency of the oscillation.
