The linear stability of the Kazhikhov–Smagulov model

Abstract

Using the Kazhikhov–Smagulov model, the linear stability of incompressible mixing layers and jets entailing large density variation is addressed analytically. The classical theorems of Squire, Rayleigh and Fjørtoft are extended to variable-density flows. It is shown that the bidimensional configuration is still the most unstable one, but the inflexion point is no longer a necessary condition for instability. Instead, a non trivial condition involving density and velocity gradient is identified. Dispersion relations are obtained for small wavenumbers as well as for piecewise linear base flow profiles. Additionally, an estimation of the threshold wavenumber that stabilises the flow is obtained. It is demonstrated that density variations modify the growth rate of the instability as well as the wavelength associated with the most unstable mode and the unstable wavenumber range. These results are in good agreement with numerical computations. Finally, it is observed that viscous effects are purely stabilising while molecular diffusion does not affect the stability.