Droplet spreading: Intermediate scaling law by PDE methods

We consider the spreading of a thin droplet of viscous liquid on a plane surface driven by capillarity. The standard lubrication approximation leads to an evolution equation for the film height h that is ill-posed when the spreading is limited by the no-slip boundary condition at the liquid-solid interface due to a singularity at the moving contact line. The most common relaxation of the no-slip boundary condition removes this singularity but introduces a new physical length scale: the slippage length b. It is believed that this microscopic-length scale only enters logarithmically in the effective (that is, macroscopic) spreading behavior.