In this paper, we derive consistent shallow water equations for bi-layer flows of Newtonian fluids flowing down a ramp. We carry out a complete spectral analysis of steady flows in the low frequency regime and show the occurence of hydrodynamic insta
bilities, so called roll-waves, when steady flows are unstable.
This paper is concerned with the stability of periodic wave trains in a generalized Kuramoto-Sivashinski (gKS) equation. This equation is useful to describe the weak instability of low frequency perturbations for thin film flows down an inclined ramp
. We provide a set of equations, namely Whithams modulation equations, that determines the behaviour of low frequency perturbations of periodic wave trains. As a byproduct, we relate the spectral stability in the small wavenumber regime to properties of the modulation equations. This stability is always critical since 0 is a 0-Floquet number eigenvalue associated to translational invariance.
This paper is concerned with the detailed behaviour of roll-waves undergoing a low-frequency perturbation. We rst derive the so-called Whithams averaged modulation equations and relate the well-posedness of this set of equations to the spectral stabi
lity problem in the small Floquet-number limit. We then fully validate such a system and in particular, we are able to construct solutions to the shallow water equations in the neighbourhood of modulated roll-waves proles that exist for asymptotically large time.
