A boundary-integral model is derived to describe two-dimensional, weakly viscous, quasi-irrotational fluid flow at the interface between two fluids. Vortical flow is restricted to a thin boundary layer, allowing the interfacial fluid dynamics to be largely generated by a vortex sheet approximation. A theoretical analysis of the model reveals that small-amplitude interfacial waves have only conservative and dissipative modes of behaviour. In contrast, similar analyses of other models of weak vorticity reveal non-physical modes which increase in energy. Numerical simulations of our model demonstrate the appropriate conservation of key physical invariants.
