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

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