'Phase diagram' of interfacial instabilities in a two-layer Couette flow and mechanism of the long-wave instability

A unified view is given of the instabilities that may develop in two-layer Couette flows, as a 'phase diagram' in the parameter space. This view is obtained from a preliminary study of the single-fluid Couette flow over a wavy bottom, which reveals three flow regimes for the disturbances created at the bottom, each regime being characterized by a typical penetration depth of the vorticity disturbances and an effective Reynolds number. It appears that the two-layer flow exhibits the same flow regimes for the disturbances induced by the perturbed interface, and that each type of instability can be associated with a flow regime. Typical curves giving the growth rate versus wavenumber are deduced from this analysis, and favourably compared with the existing literature. In the second part of this study, we propose a mechanism for the long wavelength instability, and provide simple estimates of the wave velocity and growth rate, for channel flows and for semi-bounded flows. In particular, an explanation is given for the 'thin-layer effect', which is typical of multi-layer flows such as pressure driven flows or gravity driven flows, and according to which the flow is stable if the thinner layer is the less viscous, and unstable otherwise.