Modulation equations for wave flows of a fluid film along a vertical wall

Abstract

We considered the flow of a thin layer of a viscous Newtonian fluid along a vertical wall. In the long wave approximation, the set of equations describing the influence of the combined effects of nonlinearity and viscosity on the development of the wave mode of flow is of the hyperbolic type. For the two parametrical family of periodic wave flows (roll waves), we investi gated modulation equations and formulated the crite rion of nonlinear stability of wave packets. The self similar solutions of the modulation equations are used for explaining the features of the evolution of nonlin ear wave packets at the forced mode of development of perturbations in the fluid film.1. We consider the set of equations describing the wave motion of a thin fluid film during its flow down a vertical wall:(1)Eqs. (1) are written in dimensionless variables, h is the film thickness, u is the average velocity of the flowing fluid, t is the time, and x is the coordinate directed ver tically downwards. The coefficient k is related to the self similar profile of velocity in a fluid film. Further, it is assumed that k = 1, 2, which corresponds to the parabolic velocity profile satisfying the sticking condi tions for a fluid on a solid wall and the condition of no stress at the free boundary. Model (1) is derived by integrating the boundary layer equations [1] and is aspecial case of the Shkadov model disregarding the capillary effects [2, 3] and also the models of flow of power fluids [4]. The interest in model (1) is caused by the fact that it describes the influence of nonlinear effects on the development of long wave perturbations in thin fluid layers. Equations (1) represent the hyperbolic set of equa tions with real characteristics(2)An arbitrary stationary solution of Eqs. (1) h ≡ h 0 , u ≡ u 0 = is unstable with respect to small perturba tions. Similar to flows in open channels for the high Reynolds numbers, almost periodic wave flow modes (roll waves) can be formed in thin films. In [3,4], it is shown that the roll waves form a two parametrical family of periodic solutions for Eqs. (1) containing discontinuities as in the Dressler theory [5] for turbu lent flows in open channels. At the fixed average flow rate of a fluid, the free parameter determining the peri odic solutions is the wavelength, which can vary from zero to infinity. The experiments and numerical calcu lations show [1,4,6] that the periodic small ampli tude perturbations during the evolution are trans formed into roll waves of finite amplitude, and the fur ther increase of waves stops when achieving a certain "saturation" condition. One of the unsolved problems of the theory of roll waves consists in determining the boundaries of nonlinear stability of wave packets. In [4], the condition of stability of discontinuous solution (1) based on the nondecrease in the energy of flow behind the discontinuity was introduced, and, on its basis, the amplitude of the minimum allowable roll wave, for which the total energy does not decrease when passing through the discontinuity,...