Thin layer flow along arbitrary curved surfaces

To model the inertia and gravity driven stationary flow of a thin layer of water along a curved surface (for example, in a washing bowl, a toilet, or other technical applications) the equations for free-surface potential flow (Laplace, Bernoulli, b.c.) are rewritten in surface-bound, curvilinear orthogonal coordinates. Assuming a small parameter, measuring the thin fluid layer, a systematic perturbation analysis is made, producing, to leading order, equations limilar to the eikonal and energy equation in ray theory. These hyperbolic equations can be integrated along streamlines, with explicit results for some geometries. In these equations the smoothing backreaction of the pressure is decoupled, leading to the possibility of singular lines, being the envelope of crossing streamlines (caustics).