Transition of a moving contact line from smooth to angular

Abstract

We consider the motion of a small droplet sliding under gravity down an inclined plane. Experimentally [see T. Podgorski, Thèse, Université Paris 6 (October 2000); T. Podgorski et al., Phys. Rev. Lett. 87, 036102 (2001)] it has been observed that such a droplet will develop an angular point in the contact line at its rear if its velocity is sufficiently high (i.e., if the plane is inclined sufficiently steeply to the horizontal). The angular point first appears at some critical, or threshold speed, Uc, and the angular discontinuity observed increases monotonically from zero at U=Uc. We seek to describe this phenomenon using a simple mathematical model based on lubrication theory. In the subcritical regime we find an exact solution to our model: a drop of circular perimeter sliding steadily down the inclined plane. This allows us to predict the formation of a nonanalyticity in the contact line at a well-defined critical droplet speed. For speeds just beyond this, we construct a local solution valid near the singular point, for which the angle in the contact line predicted by our theory agrees with the experimental results. We also construct a local solution in the fully developed critical regime, for sliding speeds well above critical, which is suggestive of a further bifurcation, also observed in the experiments [T. Podgorski, Thèse, Université Paris 6 (October 2000)].