Coating of bilayer thin liquid films on rotating cylinders

Motivated by the need to improve fundamental understanding of multilayer coating on discrete objects, a model problem involving the flow of bilayer thin liquid films on rotating cylinders is considered here. The layers are assumed to be immiscible, and the lubrication approximation is applied to derive two coupled nonlinear evolution equations describing the heights of the two layers as a function of time and the angular coordinate. In the limit of rapidly rotating cylinders, gravitational effects are negligible, and linear stability analysis and nonlinear simulations demonstrate that a more viscous, thicker inner film or higher liquid-liquid interfacial tension suppresses instabilities driven by centrifugal forces for fixed properties of the outer layer. When gravitational effects are significant, a parametric study reveals that the critical rotation rate required to cause motion of liquid lobes that form due to gravitational drainage is lowered for a more viscous and thicker inner film due to an increase in viscous forces. These properties of the inner layer also lead to a reduction in the amplitude of temporal oscillations in the film thickness. In contrast, the stabilizing effects of liquid-liquid interfacial tension are negligible near the critical rotation rate due to the largely uniform curvature of the inner film. Results from the lubrication-theory-based model are complemented by finite-element simulations of the full two-dimensional equations, which reveal that the lubrication model works well for thicker films provided that gravitational effects are sufficiently small. In addition to advancing fundamental understanding, the results reported here suggest strategies for improving the uniformity of coatings on discrete objects.