Stability Of Roll Waves Research Articles

Linear Stability of Viscous Roll Waves

The purpose of this paper is to study the linear stability of “viscous” roll waves. These are periodic continuous traveling waves solutions of viscous perturbations of inhomogeneous hyperbolic systems. We first study the scalar case for the Burgers equation and for an inhomogeneous hyperbolic equation. Then we analyze the stability of roll waves, solutions of the shallow water equations with a real viscosity. In both cases, we first analyze the Evans function and compute an asymptotic expansion in the low frequency regime. Under a strong spectral stability condition, we prove the linear stability of viscous roll waves, solutions of the Saint Venant equations, with pointwise estimates on the Green functions.

Stability of roll waves in open channel flows

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. To cite this article: A. Boudlal, V.Yu. Liapidevskii, C. R. Mecanique 330 (2002) 291–295.

Stability of Roll-Waves on Thin Laminar Flow down an Inclined Plane Wall

This paper considers the roll-wave phenomenon on thin laminar viscous flow down an inclined plane wall on the basis of hydraulic equations which are derived on the assumptions of long wave and parabolic velocity profile. Wave fronts are approximated by discontinuous jumps satisfying physical laws. The solution for the permanent roll-wave train obtained involves one free parameter, and is insufficient to explain observed results. Then, the stability of the said solution to infinitesimal disturbances is investigated. It is found that the roll-wave train is unstable if its wave length is shorter than a certain critical value. Furthermore, wave lengths of the most stable waves are favourably compared with those of waves observed in experiments.