The Rayleigh-Taylor instability for a thin film on the inside of a horizontal cylinder

Thin films on curved surfaces are widely observed in coating and painting processes and wetting problems. We consider a thin film on a curved substrate under the effect of gravitational, viscous, and surface tension forces. When the film is on the underside of the substrate, gravity works as a destabilizing force, and a Rayleigh-Taylor type instability is expected. We consider the stability of a uniform thin film coating the inside of a horizontal circular cylinder. Using asymptotic methods, we find that instabilities are of a transient nature, thus showing that curvature helps stabilize the film. We also find that these ``instabilities'' occur primarily in the angular direction with the axial perturbations only appearing as higher-order corrections. These results seem to agree well with experiments (H. Kim et al., this conference).