Dynamics of a film bounded by a pinned contact line

Abstract

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 $h_t + (h^n h_{xxx})_x = 0$, 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 $\theta (t)$, which couples to the shape of the film. Starting from a regular initial condition $h_0(x)$, we investigate the dynamics of the drop both analytically and numerically, focusing on the contact angle. For short times $t\ll 1$, and if $n\ne 3$, the contact angle changes according to a power law $\displaystyle t^{\frac {n-2}{4-n}}$. In the critical case $n=3$, the dynamics become non-local, and $\dot {\theta }$ is now of order $\displaystyle {\rm{e}}^{-3/(2t^{1/3})}$. 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.