Linear temporal and spatiotemporal stability analysis of two-layer falling films with density stratification

Abstract

A detailed temporal and spatiotemporal stability analysis of two-layer falling films with density and viscosity stratification is performed by using the Chebyshev collocation method to solve the full system of linear stability equations. From the neutral curves Re(k) for the surface mode and the interface mode of instability, obtained for different density ratios gamma of the upper layer to the lower layer, it is found that smaller density ratios make the surface mode and the short-wave interface mode much more stable, and can even make the short-wave interfacial instability disappear. Moreover, through the study of the local growth rates of the spatiotemporal instability as a function of the ray velocity V , it is found that for not too small incline angles like theta=0.2, the two-layer flow is always convectively unstable, and there is a transition between long- and short-wave instabilities which is determined by the Briggs-Bers collision criterion. Due to the existence of the absolute Rayleigh-Taylor instability for gamma>0 and theta=0, a transition from convective to absolute instability can be detected at small incline angles, and the corresponding boundary curves are plotted for different Reynolds numbers, viscosity ratios, and incline angles. It is found that there exists a limit Reynolds number above which the two-layer film flow can only be convectively unstable for a fixed small incline angle. The spatial amplification properties of the convective waves are finally presented for both surface and interface modes.