Note on the stability of viscous roll waves

Abstract

In this note, we announce a complete classification of the stability of periodic roll-wave solutions of the viscous shallow water equations, from their onset at Froude number F≈2 up to the infinite Froude limit. For intermediate Froude numbers, we obtain numerically a particularly simple power-law relation between F and the boundaries of the region of stable periods, which appears potentially useful in hydraulic engineering applications. In the asymptotic regime F→2 (onset), we provide an analytic expression of the stability boundaries, whereas in the limit F→∞, we show that roll waves are always unstable.