Droplet motion with contact-line friction: long-time asymptotics in complete wetting

TL;DR

This paper analyzes the long-time behavior of thin-film equations with contact-line friction, characterizing how solutions spread over time depending on friction strength, supported by numerical simulations.

Contribution

It provides a formal asymptotic analysis of solutions in the perfect wetting regime, including profile characterization and spreading rates influenced by contact-line friction.

Findings

Characterized solution profiles and spreading rates based on friction strength

Identified corrections due to dynamical free boundary conditions

Validated asymptotic results with numerical simulations

Abstract

We consider the thin-film equation for a class of free boundary conditions modelling friction at the contact line, as introduced by E and Ren. Our analysis focuses on formal long-time asymptotics of solutions in the perfect wetting regime. In particular, through the analysis of quasi-self-similar solutions, we characterize the profile and the spreading rate of solutions depending on the strength of friction at the contact line, as well as their (global or local) corrections, which are due to the dynamical nature of the free boundary conditions. These results are complemented with full transient numerical solutions of the free boundary problem.