Composite solutions to a liquid bilayer model

TL;DR

This paper derives explicit formulas for stationary solutions of a two-layer thin-film liquid model with intermolecular potential, classifies their stability, and explores their dynamic behaviors.

Contribution

It provides a comprehensive analytical and numerical classification of composite stationary solutions in a two-layer thin-film model with intermolecular interactions.

Findings

Explicit formulas for eleven types of stationary solutions.

Stable and weakly unstable solutions on finite intervals.

Numerical instability and complex coarsening dynamics for some solutions.

Abstract

This article continues the research initiated in [17]. We derive explicit formulae for the leading order profiles of eleven types of stationary solutions to a one-dimensional two-layer thin-film liquid model considered with an intermolecular potential depending on both layer heights. The found solutions are composed of the repeated elementary blocks (bulk, contact line and ultra-thin film ones) being consistently asymptotically matched together. We show that once considered on a finite interval with Neumann boundary conditions these stationary solutions are either dynamically stable or weakly translationally unstable. Other composite solutions are found to be numerically unstable and rather exhibit complex coarsening dynamics.