A hyperbolic model for two-layer thin film flow with a perfectly soluble anti-surfactant

TL;DR

This paper develops a hyperbolic model for two-layer thin film flow with soluble anti-surfactant, deriving equations, analyzing hyperbolicity, and solving the Riemann problem with numerical methods.

Contribution

It introduces a new hyperbolic system for thin film flow with soluble anti-surfactant, including entropy analysis and explicit Riemann solutions.

Findings

The system is strictly hyperbolic for certain states.

An entropy function ensuring well-posedness is constructed.

Explicit solutions to the Riemann problem are obtained.

Abstract

We consider the motion of a two-layer thin film that consists of two immiscible viscous fluids and is endowed with an anti-surfactant solute. The presence of such solute particles induces variations of the surface tension and interfacial stress driving a Marangoni-type flow. We first analyze a lubrication limit and derive one-dimensional evolution equations for film heights and solute concentrations. Then, under the assumption that the capillarity and diffusion effects are negligible and the solute is perfectly soluble, we obtain a conservative first-order system in terms of film heights and concentration gradients. This reduced system is found to be strictly hyperbolic for a certain set of states and to admit an entire class of entropy/entropy-flux pairs. We also provide a strictly convex entropy for the hyperbolic system. Thus, the well-posedness for the Cauchy problem is given. Moreover, the system is almost a Temple-class system, which allows to compute explicit solutions of the Riemann problem. The paper concludes with numerical experiments using a Godunov-type finite volume method, which relies on the exact Riemann solver.