Isogeometric analysis of insoluble surfactant spreading on a thin film

TL;DR

This paper applies isogeometric analysis to model surfactant spreading on thin films, directly solving fourth order PDEs and capturing complex phenomena like fingering instabilities.

Contribution

It introduces a direct IGA approach for fourth order PDEs in surfactant spreading, avoiding mixed methods and effectively modeling instabilities.

Findings

IGA reproduces results from mixed methods

Successfully captures fingering instabilities

Demonstrates high continuity benefits of IGA

Abstract

In this paper we tackle the problem of surfactant spreading on a thin liquid film in the framework of isogeometric analysis. We consider a mathematical model that describes this phenomenon as an initial boundary value problem (IBVP) that includes two coupled fourth order partial differential equations (PDEs), one for the film height and one for the surfactant concentration. In order to solve this problem numerically, it is customary to transform it into a mixed problem that includes at most second order PDEs. However, the higher-order continuity of the approximation functions in Isogeometric Analysis (IGA) allows us to deal with the weak form of the fourth order PDEs directly, without the need of resorting to mixed methods. We demonstrate numerically that the IGA solution is able to reproduce results obtained before with mixed approaches. Complex phenomena such as Marangoni-driven fingering instabilities triggered by perturbations are easily captured.