  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.
