The influence of viscosity stratification and counter-flow on instability in free shear layers

In this study, we undertake a comprehensive examination of the instability exhibited by free shear layers (specifically, mixing layers ) subject to viscosity stratification and counter-flow. We construct a base state where the two fluids share an identical density, employing boundary layer assumptions to model both the high and low-speed streams that form a mixing layer. Using self-similar base state profiles that satisfy the boundary layer equations, we perform a local linear stability analysis of the perturbed planar Navier-Stokes equations. While temporal stability of such flows has been examined previously, we demonstrate that for sufficiently high viscosity contrast between the high-speed and low-speed streams, absolutely unstable profiles can result even in the absence of counterflow, which is a new result. In addition, we have thoroughly examined absolute/convective instability transition boundary as a function of Reynolds number, viscosity ratio, and velocity ratio, which determines the nature of the mixing layer (co-flow and counter-flow). Our investigation delineates the intricate dynamics of free shear layers under viscosity stratification, and it is hoped that these numerical results stimulate an experimental realization that investigates instabilities in variable viscosity-free shear layers.