Three-dimensional lubrication model of a contact line corner singularity

Drops sliding along inclined planes in situations of partial wetting develop a "corner" at the back of the drop, similar to contact line shapes observed on plates withdrawn from a bath. We discuss, in the framework of lubrication theory, the 3D structure of this corner singularity: similarity solutions exist such that the interface adopts the shape of a cone, with well-defined relationships linking its two characteristic angles. Near both contact lines forming the corner, the solution reduces to the well-known lubrication solution for a single contact line, with an effective capillary number involving the normal component of the contact line velocity.