Modeling and Analysis of a Two-Phase Thin Film Model with Insoluble Surfactant

TL;DR

This paper develops and analyzes mathematical models for a two-phase thin film with insoluble surfactant, considering gravitational and capillary effects, and proves well-posedness and stability of these models.

Contribution

It introduces coupled second and fourth order parabolic models for thin films with insoluble surfactant, including gravitational and capillary effects, and establishes their well-posedness and stability.

Findings

Models are parabolic, degenerated, and strongly coupled.

Well-posedness of the models is proven.

Asymptotic stability of solutions is demonstrated.

Abstract

In this paper we consider a two-phase thin film consisting of two immiscible viscous fluids endowed with a layer of insoluble surfactant on the surface of the upper fluid. The governing equations for the two film heights and the surfactant concentration are derived using a lubrication approximation. Taking gravitational forces into account but neglecting capillary effects, the resulting system of evolution equations is parabolic, strongly coupled, of second order and degenerated in the equations for the two film heights. Incorporating on the contrary capillary forces and neglecting the effects of gravitation, the system of evolution equations is parabolic, degenerated and of fourth order for the film heights, strongly coupled to a second order transport equation for the surfactant concentration. Local well-posedness and asymptotic stability are shown for both systems.