On the relationship between the thin film equation and Tanner's law

This paper is devoted to the asymptotic analysis of a thin film equation which describes the evolution of a thin liquid droplet on a solid support driven by capillary forces. We propose an analytic framework to rigorously investigate the connection between this model and Tanner's law [22] which claims: the edge velocity of a spreading thin film on a pre-wetted solid is approximately proportional to the cube of the slope at the inflection. More precisely, we investigate the asymptotic limit of the thin film equation when the slippage coefficient is small and at an appropriate time scale (see Equation (8)). We show that the evolution of the droplet can be approximated by a moving free boundary model (the so-called quasi-static approximation) and we present some results pointing to the validity of Tanner's law in that regime. Several papers [5, 6, 10] have previously investigated a similar connection between the thin film equation and Tanner's law either formally or for particular solutions. Our main contribution is the introduction of a new approach to systematically study this problem by finding an equation for the evolution of the apparent support of the droplet (described mathematically by a nonlinear function of the solution).