Analysis of the flow of a thin liquid film on the surface of a rotating, curved, axisymmetric substrate

Spin coating is frequently used by the coating industry to achieve a very uniform final coating layer on a given substrate. Most of the research into this area has focused on flat substrates, but in this work we use scaling arguments and perturbation methods to derive the lubrication form of the equations governing the fluid motion of a thin liquid film on a curved, rotating, axisymmetric substrate. Though the substrate must be axisymmetric, the coating need not be. Though the slope of the substrate must be continuous, the curvature of the substrate need not be. One application for this work is the spin coating of food and beverage cans, most of which have a curved bottom due to structural reasons. Using an implicit finite difference scheme, we use our derivation to develop a numerical model to simulate the spin coating of the interior of a model soup can. We assume that the coating is initially uniform and model how centrifugal forces drive the coating outward past a series of axisymmetric undulations on the can bottom.