Instabilities of thin-film flow over a spinning disk

We study the dynamics of a thin, axisymmetric film of Newtonian fluid on a uniformly rotating disk with topography. The system is modelled via a thin-film approximation together with the Method of Weighted Residuals up to second order. The resulting model is a closed initial-value problem for the film thickness and the radial and azimuthal fluxes, including the effects of inertia, viscosity, centrifugation and capillarity. We determine simplified models in the far field to investigate the spatial and temporal stability and find that there exist three distinct regions that exhibit qualitatively different behaviours. We also study a family of substrate shapes with parameters controlling the asymmetry, smoothness, amplitude and frequency of the topography. The effect of the topography on the flow is quantified using an integral measure of the interfacial waviness amongst other measures. In particular, we find that the presence of topography can cause additional interfacial waves, increasing the surface area of the film.