Rayleigh-Taylor instability under a spherical substrate

We investigate the Rayleigh-Taylor instability of a thin viscous film coating the inside of a spherical substrate. The aim of this work is to find and characterize the instability pattern in this spherical geometry. In contrast to the Rayleigh-Taylor instability under a planar substrate, where the interface is asymptotically unstable with respect to infinitesimal perturbations, the drainage induced by the component of gravity tangent to a curved substrate stabilizes the liquid interface, making the system linearly asymptotically stable. By performing a linear optimal transient growth analysis we show that the double curvature of a spherical substrate yields a critical Bond number, prescribing the ratio between gravitational and capillary forces, before an initial growth of perturbations is possible two-times larger than for a circular cylindrical substrate. This linear transient growth analysis however does not yield any selection principle for an optimal azimuthal wave number and we have to resort to a fully nonlinear analysis. By numerically solving the nonlinear lubrication equation we find that the most amplified azimuthal wave number increases with the Bond number. Nonlinear interactions are responsible for the transfer of energy to higher-order harmonics. The larger the Bond number and the farther away from the apex of the sphere, the richer the wave-number spectrum.