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

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. Linearising 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 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 regularised by weak surface diffusion. Their existence highlights the need for careful treatment in computations of unsteady advection-dominated surfactant transport in confined domains.

Automated summary

This study investigates the advective Marangoni spreading of insoluble surfactant in a viscous fluid within a two-dimensional rectangular cavity, revealing singular flow structures near contact lines and logarithmically diverging pressure fields. The singularities at stationary contact lines are associated with dynamic compression of the surfactant monolayer, which can be regularised by weak surface diffusion. These findings underscore the importance of careful treatment in computations of unsteady advection-dominated surfactant transport in confined domains.