A derivation of viscous thin film flow equations on curved surfaces

Abstract

General equations are derived for slow viscous thin fluid film flows on curved surfaces through an extension of Leal's pedagogical approach, which leaves the characteristic velocity scale unspecified and employs a direct through-thickness integration of the continuity equation. The derivation neglects inertia, and includes gravitational, capillary, and Marangoni effects, the latter coupling the thickness 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 nongeneric cases. A few examples are briefly presented and literature comparisons made. The importance of gradients in curvature is emphasized, and it is suggested that nondimensionalization of geometric features might lead to further useful generalizations. This relatively simple formulation is intended as a starting point for exploring interactions between geometry, gravity, and surface tension.