Three-dimensional waves in a liquid flowing down a wall

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.