Surfactant spreading in a two-dimensional cavity and emergent contact-line singularities

TL;DR

This paper models the spreading of insoluble surfactant in a 2D cavity, revealing contact-line singularities and their regularization, which are crucial for accurate simulations of surfactant dynamics.

Contribution

It introduces a linearized modal decomposition approach to analyze surfactant spreading and identifies singular flow structures near contact lines, highlighting the importance of regularization.

Findings

Singular flow structures near contact lines cause diverging pressure fields.

Eigenfunctions reveal oscillatory shear stress patterns.

Weak surface diffusion regularizes contact-line singularities.

Abstract

We model the advective Marangoni spreading of insoluble surfactant at the free surface of a viscous fluid that is confined within a two-dimensional rectangular cavity. Interfacial deflections are assumed small, with contact lines pinned to the walls of the cavity, and inertia is neglected. Linearizing the surfactant transport equation about the equilibrium state allows a modal decomposition of the dynamics, with eigenvalues corresponding to decay rates of perturbations. Computation of the family of mutually orthogonal two-dimensional eigenfunctions reveals singular flow structures near each contact line, resulting in spatially oscillatory patterns of wall shear stress and a pressure field that diverges logarithmically. These singularities at a stationary contact line are associated with dynamic compression of the surfactant monolayer; we show how they can be regularized by weak surface diffusion. Their existence highlights the need for careful treatment in computations of unsteady advection-dominated surfactant transport in confined domains.