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

## TL;DR

This paper rigorously analyzes the asymptotic behavior of the thin film equation in the small slippage limit, establishing a connection to Tanner's law through a new approach involving the evolution of the droplet's apparent support.

## Contribution

Introduces a novel method to systematically relate the thin film equation to Tanner's law by deriving an evolution equation for the droplet's apparent support.

## Key findings

- Validation of Tanner's law in the small slippage regime
- Approximation of droplet evolution by a free boundary model
- New analytical framework for thin film and Tanner's law connection

## Abstract

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).