The flow of a liquid film along a periodic wall

The creeping flow of a liquid film along an inclined periodic wall of arbitrary geometry is considered. The problem is formulated using the boundary-integral method for Stokes flow. This method is extended to two-dimensional flows involving free surfaces, and is implemented in an iterative numerical procedure. Detailed calculations for flow along a sinusoidal wall are perfomed. The free-surface profile is studied as a function of flow rate, inclination angle, wave amplitude, and surface tension, and is compared with previous asymptotic solutions. The results include streamline patterns, velocity profiles and wall-shear-stress distributions, and establish criteria for flow reversal. For specified wall geometry, the asymptotic behaviour for very small flow rates is shown to be a strong function of surface tension. It is demonstrated that these results are valid in a qualitative sense for general wall geometries. The analogy between gravity-driven flow and the flow of a liquid layer on a rotating disk (spin coating) is also discussed.