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 effects of inertia, viscosity, centrifugation and capillarity. We find the spatial stability depends on the position relative to the inlet: close to the inlet the flow is convectively unstable while far from it the flow is absolutely unstable. We investigate the temporal stability in the far field and find there exist three distinct regions that exhibit different behaviors: no growth or decay, conditionally stable and unconditionally stable. We study a family of topographies with parameters controlling the asymmetry, smoothness, amplitude and frequency of the topography. The effect of topography on the flow is determined using an integral measure of the interfacial waviness. In particular, we find that the presence of topography can cause additional interfacial waves that increase the surface area of the film.