Stationary solutions of liquid two-layer thin film models

TL;DR

This paper analyzes stationary solutions of a two-layer liquid thin-film model, establishing existence, asymptotic behavior, and contact-angle conditions through energetic and asymptotic methods.

Contribution

It provides a rigorous analysis of contact-angle conditions and stationary solutions, including asymptotic and variational approaches, for two-layer thin-film models.

Findings

Existence of stationary solutions proven.

Asymptotic analysis yields a sharp-interface model.

Uniqueness of energetic minimizers established.

Abstract

We investigate stationary solutions of a thin-film model for liquid two-layer flows in an energetic formulation that is motivated by its gradient flow structure. The goal is to achieve a rigorous understanding of the contact-angle conditions for such two-layer systems. We pursue this by investigating a corresponding energy that favors the upper liquid to dewet from the lower liquid substrate, leaving behind a layer of thickness $h_*$. After proving existence of stationary solutions for the resulting system of thin-film equations we focus on the limit $h_*\to 0$ via matched asymptotic analysis. This yields a corresponding sharp-interface model and a matched asymptotic solution that includes logarithmic switch-back terms. We compare this with results obtained using $\Gamma$-convergence, where we establish existence and uniqueness of energetic minimizers in that limit.