Inertialess temporal and spatio-temporal stability analysis of the two-layer film flow with density stratification

Abstract

This paper presents a temporal, spatial, and spatio-temporal linear stability analysis of the two-layer film flow down a plate tilted at an angle θ. It is based on a zero Reynolds number approximation to the Orr-Sommerfeld equations and a zero surface tension approximation to both surface boundary conditions. The combined effects of density and viscosity stratifications are systematically investigated. The subtle influence of density stratification is first put into light by a temporal analysis for θ=0.2; when increasing/decreasing the density ratio (upper fluid/lower fluid), the two-layer film flow becomes much more unstable/stable with respect to the finite wavelength instability. Moreover, below a critical density ratio this finite wavelength instability even disappears, whatever the viscous ratio. Concerning the long wave instability, it becomes dominant when decreasing the density ratio below 1 and is even triggered in a region which was stable for equal density layers. The spatio-temporal analysis shows that the instability is convective for incline angles that are not too small as θ=0.2. The study of the local growth rates of the spatio-temporal instability as a function of the ray velocity V shows that there is a transition between long wave and short wave instabilities which has been determined by using the Briggs-Bers collision criterion. Accordingly, there exists a jump for the local oscillatory frequency, spatial amplification rate, and spatial wave number due to this transition. Due to the existence of the absolute Rayleigh-Taylor instability for θ=0, the transition from convective to absolute instability can be detected for values of θ smaller than 0.2, and absolute/convective instability boundary curves have been obtained for varying characteristic parameters.