A pinned contact line

We consider the dynamics of a drop with a pinned contact line $h(0,t) = 0$, as described by the thin film equation $h_t + (h^n h_{xxx})_x = 0$. Starting from an arbitrary initial condition at $t=0$, the profile relaxes toward an equilibrium parabolic profile. Our focus is on the early-time dynamics, which for $n\ne 3$ is described by a local similarity solution. As a result, the contact angle changes like a power law as a function of the time $t$. In the case $n=3$, which corresponds to a no slip boundary condition, the problem becomes non-local, and the change in contact angle is of the form $e^{-A/t^\alpha}$ (with $A$ complex). We consider the implications of this result for Huh and Scriven's contact line paradox for the case $n=3$. (joint work with Marco Fontelos)