Linear stability of ultrathin spherical coatings

We unravel the linear stability properties of an otherwise stagnant ultrathin non-wetting liquid film of thickness $h_o$ coating a spherical substrate of radius $R$. The configuration is known to be unstable due to the competition of the destabilizing van der Waals (vdW) forces and the stabilizing surface tension force. The governing equations of motion written in the Stokes limit of negligible liquid inertia and an accompanying lubrication model are linearised about the zero-velocity base state and decomposed into normal modes in order to obtain the temporal dispersion relation. Discrete unstable modes are identified and tracked as a function of a capillary number $Ca$ measuring the relative importance of surface tension to vdW forces and the film aspect ratio $\eta = R/h_o$. For small enough values of the capillary number only the first polar mode is unstable, and the corresponding maximum growth rate is shown to be a universal function of $\eta$. Lubrication theory is seen to provide a good quantitative prediction of the film stability properties for $\eta\gg 1$.