Dynamics of a film bounded by a pinned contact line

We consider the dynamics of a liquid film with a pinned contact line (for example, a drop), as described by the one-dimensional, surface-tension-driven thin-film equation ht+(hnhxxx)x=0, where h(x,t) is the thickness of the film. The case n=3 corresponds to a film on a solid substrate. We derive an evolution equation for the contact angle θ(t), which couples to the shape of the film. Starting from a regular initial condition h0(x), we investigate the dynamics of the drop both analytically and numerically, focusing on the contact angle. For short times t≪1 , and if n≠3, the contact angle changes according to a power law tn−24−n. In the critical case n=3, the dynamics become non-local, and θ˙is now of order e−3/(2t1/3) . This implies that, for n=3, the standard contact line problem with prescribed contact angle is ill posed. In the long time limit, the solution relaxes exponentially towards equilibrium.