Glazing of Doughnuts: Non-Normal Instability and Asymmetric Pattern Formation

The flow of a thin Newtonian film coating a solid substrate with complex curvature is a challenging problem of interest to many fields, especially biophysics. By a combination of multiple-scale analysis and center manifold techniques, Roy, Roberts and Simpson (2002) were able to develop a general nonlinear interface equation, whose solutions reveal the mechanism responsible for film thinning near regions of high substrate curvature. Here we examine the perturbative response of this equation by investigating an interfacial instability involving capillary driven flow of a thin film on a toroidal substrate. We coin this system the doughnut glazing problem. Results of a generalized linear stability analysis based on non-normal growth of a non-uniform liquid layer reveal the migration patterns of growing modes to the interior surface of the torus. Contrary to expectation, as the torii radii approach infinity, the wavelength of the maximally unstable mode does not asymptote to that of the well-known sinusoidal instablity for a thin film on a solid cylinder (Goren 1962). We demonstrate how the asymmetry inherent in the torus geometry biases the flow to help steer the formation, evolution and growth of unstable non-normal modes.