  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).
