The stability of finite amplitude roll waves that may develop at a liquid free surface in inclined open channels of arbitrary cross-section is studied. In the framework of shallow water theory with turbulent friction the modulation equations for wave series are derived and a nonlinear stability criterion is obtained.

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The special class of periodic travelling waves which is known as roll waves is investigated for nonhomogeneous hyperbolic equations of gas dynamics type. In this Note these equations are applied to shallow water flows in inclined open channels, but the results obtained are more general and far-reaching. The necessary conditions for the existence of a roll wave are derived. It is shown that for a nonconvex pressure term, multi-shock configurations of roll waves of finite amplitude exist. A new type of periodic travelling wave, which corresponds to the slug flow regime in two-layer flows, is found.

In this paper I consider a system of equations describing two stratified fluids flowing in closed, slightly inclined ducts. In the framework of the shallow water approximation with turbulent friction acting on the wall and at the interface, I investigate a special class of periodic travelling waves containing stable moving jumps, namely, roll waves and periodic slug flows. The modulation equations of roll waves and slugs are derived and a nonlinear stability criterion is obtained. As for plug flow, only a gas-liquid system is considered. The stability criterion is expressed in terms of an integro-differential relation. For self-similar cross-section, this criterion is simplified to a relation for a function of one variable only.

Waves of finite amplitude on a thin layer of non-Newtonian fluid modelled as a power-law fluid are considered. In the long wave approximation, the system of equations taking into account the viscous and nonlinear effects has the hyper- bolic type. For the two-parameter family of periodic waves in the film flow on a vertical wall the modulation equations for nonlinear wave trains are derived and investigated. The stability criterium for roll waves based on the hyperbolicity of the modulation equations is suggested. It is shown that the evolution of stable roll waves can be described by self-similar solutions of the modulation equations

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,...