Steering droplets on substrates with plane-wave wettability patterns and deformations

Motivated by strategies for targeted microfluidic transport of droplets, we investigate how sessile droplets can be steered toward a preferred direction using travelling waves in substrate wettability or deformations of the substrate. To perform our numerical study, we implement the boundary-element method to solve the governing Stokes equations for the fluid flow field inside the moving droplet. In both cases we find two distinct modes of droplet motion. For small wave speed the droplet surfs with a constant velocity on the wave, while beyond a critical wave speed a periodic wobbling motion occurs, the period of which diverges at the transition. These observation can be rationalized by the nonuniform oscillator model and the transition described by a SNIPER bifurcation. For the travelling waves in wettability the mean droplet velocity in the wobbling state decays with the inverse wave speed. In contrast, for travelling-wave deformations of the substrate it is proportional to the wave speed at large speed values since the droplet always has to move up and down. To rationalize this behavior, the nonuniform oscillator model has to be extended. Since the critical wave speed of the bifurcation depends on the droplet radius, this dependence can be used to sort droplets by size.