Abstract
In order to give a theoretical description of two-dimensional waves in earlier experimental work [1, 2], a simplified system of equations has been proposed [3, 4] based on the long-wavelength approximation. It has been shown that this system gives a good description of the wave processes in films for moderate values of the Reynolds number [3, 5, 6]. In this paper, the previous approach [3] is developed for three-dimensional flow of a film. A system of equations is obtained which describes three-dimensional waves in a layer of viscous liquid flowing down a vertical wall under the action of gravity for moderate values of the Reynolds number. It is shown that the equation derived in previous work [7] by the long-band method is the limiting case for this system when the Reynolds number tends to zero. When Re → ∞, the system goes over to a hyperbolic system of the type describing long waves in shallow water. A small-parameter Galerkin expansion is constructed for numerical analysis of unsteady waves and the problem is solved over a wide range of flow parameters.
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Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 21–27, September–October, 1984.
The authors wish to thank the members of G. I. Petrov's seminar for useful discussions and personally to thank G. I. Petrov for his consideration of the paper as well as L. B. Postonogova and G. M. Shevtsova for carrying out the numerical experiments.
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Demekhin, E.A., Shkadov, V.Y. Three-dimensional waves in a liquid flowing down a wall. Fluid Dyn 19, 689–695 (1984). https://doi.org/10.1007/BF01093533
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DOI: https://doi.org/10.1007/BF01093533