Three-dimensional thin film flow over and around an obstacle on an inclined plane

Steady Stokes flow driven by gravity down an inclined plane over and around an attached obstacle is considered. The effects of the obstacle are examined for various flow configurations and results produced for flow over hemispherical obstacles. Comparison is made with previously published papers that assume that the obstacle is small and/or the free surface deflection and disturbance velocity are small. Values for the unit normal and curvature of the free surface are found using both finite difference approximations and Hermitian radial basis function interpolations, with the resulting solutions compared. Free surface profiles for thin film flows over hemispherical obstacles that approach the film surface are produced and the effects of near point singularities considered. All free surface profiles indicate an upstream peak, followed by a trough downstream of the obstacle with the peak decaying in a "horseshoe" shaped surface deformation. Flow profiles are governed by the plane inclination, the Bond number, and the obstacle geometry. An extension of this approach provides a new class of solutions where a thin film flows around a cylindrical obstacle. Notably, the static contact line angle between the free surface and the obstacle is introduced as an extra flow parameter and its effect investigated for a given set of flow parameters and fixed boundary conditions. Solutions are obtained where steady flow profiles can be found both over and around a cylindrical obstacle raising the awareness of possible multiple solutions.