Thin film equations on curved surfaces

We derive governing equations for viscous thin film flows on arbitrary curved surfaces, by extending Leal's pedagogical approach that does not initially prescribe the characteristic velocity scale, and employs a direct through-thickness integration of the continuity equation. We neglect inertia but include gravitational, capillary, and Marangoni effects, the latter coupling the height dynamics to free-surface transport of a dilute, non-diffusing surfactant. The resulting general expression incorporates the leading order terms of each type, as well as additional terms that become leading order for certain nongeneric, yet simple and common, geometries. This treatment collects and compares various results, and presents a simple equation useful for exploring balances between geometry, gravity, and surface tension effects.