Flow of a liquid layer over heated topography

The flow of a viscous liquid layer over an inclined uneven wall heated from below is considered. The flow is assumed to occur at zero Reynolds number and the thermal Peclet number is taken to be sufficiently small that the temperature field inside the layer is governed by Laplace's equation. With a prescribed wall temperature distribution and Newton's Law of cooling imposed at the layer surface, the emphasis is placed on describing the surface profile of the liquid layer and, in particular, on studying how this is affected by wall heating. A linearized theory, valid when the amplitude of the wall topography is small, is derived and this is complemented by some nonlinear results computed using the boundary element method. It is shown that for flow over a sinusoidally shaped wall the liquid layer can be completely flattened by differential wall heating. For flow over a flat wall with a downwards step, it is demonstrated how the capillary ridge that has been identified by previous workers may be eliminated by suitable localized wall cooling in the vicinity of the step.