Droplets spreading with contact-line friction: lubrication approximation and traveling wave solutions

We consider the spreading, driven by surface tension, of a thin liquid droplet on a plane solid surface. In lubrication approximation, this phenomenon may be modeled by a class of free boundary problems for fourth order nonlinear degenerate parabolic equations, the free boundary being defined as the contact line where liquid, solid and vapor meet. Our interest is on an effective free boundary condition which has been recently proposed by Ren and E: it includes into the model the effect of frictional forces at the contact line, which arises from the deviation of the contact angle from its equilibrium value. In this note we outline the lubrication approximation of this condition, we describe the dissipative structure and the traveling wave profiles of the resulting free boundary problem, and we prove existence and uniqueness of a class of traveling wave solutions which naturally emerges from the formal asymptotic analysis.