Instability between shear and surface tension for a thin-film flow surrounding an ellipse

We consider the thin-film flow of a free-surface fluid surrounding a two-dimensional elliptical substrate. The effects of shear, surface tension, mass injection and varying substrate curvature on the dynamics are modelled in a combined analysis of steady states, linear stability theory and general time-evolution simulations. We discover a symmetry-breaking instability in which the combined effect of surface tension and shear can transport fluid in a direction perpendicular to the shear flow when the elliptical curvature variation is sufficiently large. Symmetric steady solutions are shown to be linearly unstable to asymmetric temporal perturbations, and the resultant instability involves a breaking of symmetry of the fluid surface that arises from the competing effects of shear, surface tension and varying substrate curvature. Our work forms fundamental understanding of and a theoretical basis for further investigations into this symmetry-breaking instability.