A conservation-consistent boundary condition for nonlinear models of soluble-surfactant-laden falling films

TL;DR

This paper introduces a new boundary condition for nonlinear models of soluble surfactant falling films that guarantees exact surfactant mass conservation, correcting a mass drift issue in previous models.

Contribution

It proposes a conservation-consistent boundary condition that resolves mass drift in existing nonlinear models of surfactant-laden falling films.

Findings

The new boundary condition ensures exact surfactant mass conservation.

It corrects the mass drift observed in previous models.

Linear stability results are unaffected by the correction.

Abstract

A conservation-consistent boundary condition is proposed for nonlinear models of soluble-surfactant-laden falling films, ensuring exact conservation of total surfactant mass. The formulation resolves an inconsistency in widely used reduced models, Pascal et al. (PRF, 2019), D'Alessio et al. (JFM, 2020), which exhibit a gradual drift of mass during nonlinear evolution in a closed periodic domain. We show that this originates from an inconsistency in the surface transport reduction and derive a corrected boundary condition that removes this defect. As the discrepancy appears only at the nonlinear order, linear stability results remain unaffected, explaining why the issue has remained unnoticed.