Viscosity stratification instability in two-layer, generalised Couette–Poiseuille flow

We investigate flow instability produced by viscosity and density discontinuities at the interface separating two Newtonian fluids in generalised Couette–Poiseuille (GCP) flow. The base flow, driven by counter-moving plates and an inclined pressure gradient at angle $0^\circ \leqslant \phi \leqslant 90^\circ$ , exhibits a twisted, two-component velocity profile across the layers, characterised by the Couette–Poiseuille magnitude parameter $0^\circ \leqslant \theta \leqslant 90^\circ$ . Plane Couette–Poiseuille (PCP) flow at $ \phi = 0^\circ$ is considered as a special case. Flow/geometry parameters are $(\phi ,\theta )$ , a Reynolds number $Re$ and the viscosity, depth and density ratios $(m,n,r)$ , respectively. A mapping from the GCP to PCP extended Orr–Sommerfeld equations is found that simplifies the numerical study of interfacial-mode instabilities, including determination of shear-mode critical parameters. For interfacial modes, unstable regions in $(m,n,r)$ space are delineated by three distinct surfaces found via long-wave analysis, with the exception of strict Couette flow where the $(m,n)$ surface asymptotically vanishes with $\theta \rightarrow 0^\circ$ . In interfacial stable regions but with unstable shear modes, one-layer PCP stability can be identified with a cut-off $\theta$ that conforms to canonical PCP stability. Competition between the interfacial-mode reversal phenomenon and the shear-mode cut-off behaviour is discussed. Extending to the full GCP configuration with the mapping algorithms applied, we systematically chart how pressure-gradient inclination and perturbation wavefront angle shift the balance between interfacial and shear instabilities in a specific case.