Gravity-driven film flow down an inclined wall with three-dimensional corrugations

The gravity-driven flow of a liquid film down an inclined wall with three-dimensional doubly periodic corrugations is investigated in the limit of vanishing Reynolds number. The film surface may exhibit constant or variable surface tension due to an insoluble surfactant. A perturbation analysis for small-amplitude corrugations is performed, wherein the wall geometry is expressed as a Fourier series consisting of a linear superposition of two-dimensional oblique waves defined by two base vectors. Each of the constituent perturbation flows over the individual oblique waves is further decomposed into a two-dimensional flow transverse to the oblique waves and a unidirectional flow parallel to the waves. Both the transverse and the parallel flow are calculated by carrying out an analysis in oblique coordinates, similar to that conducted for two-dimensional flow. The particular cases of flow down a wall with oblique two-dimensional, orthogonal three-dimensional, and hexagonal three-dimensional corrugations are considered. The results illustrate the surface velocity field and the distribution of the surfactant. The three-dimensional wall geometry is found to reduce the surface deformation with respect to its two-dimensional counterpart by increasing the effective wave numbers and decreasing the effective capillary number encapsulating the effect of surface tension.