Coating flows over curved substrates
Roy RV, Schwartz LW
Fluid Mechanics of Coating Processes , Proceedings of the Second European Coating Symposium, (Ed. P. Bourgin), 1997.

Abstract:
Certain defects in surface coatings arise from the flow of liquids. Many different physical factors are known to create flows with corresponding surface defects. Here we examine the influence of substrate curvature which can radically modify the long-term appearance of a coating. First we derive a mathematical model for the three-dimensional flow of a liquid layer on a curved substrate. By exploiting the thinness and slowness of the fluid layer, we arrive at a long-wave approximation of the flow dynamics under the influence of viscous, surface tension and gravitational forces. Our model systematically accounts for the curvature of the substrate and that of the surface of the film. Generic features of substrate curvature effects are demonstrated on two examples of curved surfaces.