×

Gravity-driven film flow down an inclined wall with three-dimensional corrugations. (English) Zbl 1124.76003

Summary: 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 parallel flow are calculated by carrying out an analysis in oblique coordinates, similar to that conducted for two-dimensional flow. We also consider the particular cases of flow down a wall with oblique two-dimensional, orthogonal three-dimensional, and hexagonal three-dimensional corrugations. The results illustrate the surface velocity field and the distribution of 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.

MSC:

76A20 Thin fluid films
76D45 Capillarity (surface tension) for incompressible viscous fluids
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[4] Blyth, M. G., Pozrikidis, C.: Film flow down an inclined plane over a three-dimensional obstacle. Phys. Fluids, in print (2006). · Zbl 1185.76437
[18] Scholle, M., Wierschem, A., Aksel, N.: Creeping film flow down an inclined wavy plane. Part I. ZAMM 81, S487-S488 (2001). · Zbl 1002.76005
[30] Wierschem, A., Scholle, M., Aksel, N.: Creeping film flow down an inclined wavy plane. Part II. ZAMM 81, S493–S494 (2001). · Zbl 1002.76006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.