Do liquid drops roll or slide on inclined surfaces?

TL;DR

This study analyzes the motion of liquid droplets on inclined surfaces, distinguishing between rolling and sliding behaviors using a diffuse interface model, and reveals how shape, viscosity contrast, and slip influence droplet dynamics.

Contribution

It introduces a kinematic decomposition method to differentiate rolling from sliding in droplet motion and uncovers universal behavior related to droplet rotation independent of several parameters.

Findings

Rolling dominates as droplet shape approaches a circle.

Droplet rotation follows a universal curve based on geometry.

Results inform design of surfaces for controlling droplet motion.

Abstract

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 analysed 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 characterised 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 towards a rational design of droplet-surface properties, both when rolling motion is desirable (as in self-cleaning hydrophobic droplets) or when it must be prevented (as in insecticide sprays on leaves).