Do Liquid Drops Roll or Slide on Inclined Surfaces?

We study the motion of a two-dimensional droplet on an inclined surface, under the action of gravity, using a diffuse interface model which allows for arbitrary equilibrium contact angles. The kinematics of motion is analyzed by decomposing the gradient of the velocity inside the droplet into a shear and a residual flow. This decomposition helps in distinguishing sliding versus rolling motion of the drop. Our detailed study confirms intuition, in that rolling motion dominates as the droplet shape approaches a circle, and the viscosity contrast between the droplet and the ambient fluid becomes large. As a consequence of kinematics, the amount of rotation in a general droplet shape follows a universal curve characterized by geometry, and independent of Bond number, surface inclination and equilibrium contact angle, but determined by the slip length and viscosity contrast. Our results open the way toward a rational design of droplet-surface properties, both when rolling motion is desirable (as in self-cleaning hydrophobic droplets) and when it must be prevented (as in insecticide sprays on leaves).