Linear stability analysis of generalized Couette–Poiseuille flow: the neutral surface and critical properties

We discuss the modal, linear stability analysis of generalized Couette–Poiseuille (GCP) flow between two parallel plates moving with relative speed in the presence of an applied pressure gradient vector inclined at an angle $0\leqslant \phi \leqslant 90^\circ$ to the plate relative velocity vector. All possible GCP flows can be described by a global Reynolds number $Re$ , $\phi$ and an angle $0\leqslant \theta \leqslant 90^\circ$ , where $\cos \theta$ is a measure of the relative weighting of Couette flow to the composite GCP flow. This provides a novel and uncommon group of generally three-dimensional base velocity fields with wall-normal twist, for which Squire's theorem does not generally apply, requiring study of oblique perturbations with wavenumbers $(\alpha,\beta )$ . With $(\theta,\phi )$ fixed, the neutral surface $f(\theta,\phi ;Re,\alpha,\beta )=0$ in $(Re, \alpha,\beta )$ space is discussed. A mapping from GCP to plane Couette–Poiseuille flow stability is found that suggests a scaling relation $Re^*\alpha /k = H(\theta ^*)$ that collapses all critical parameters, where ${Re}^*= Re\,({\alpha _1}/{\alpha })\,({\sin \theta }/{\sin {\theta }^*})$ and $\tan \theta ^*=({\alpha _{1}}/\alpha )\tan \theta$ , with $\alpha _1=\alpha \cos \phi +\beta \sin \phi$ . This analysis does not, however, directly reveal global critical properties for GCP flow. The global $Re_{cr}(\theta,\phi )$ shows continuous variation, while $\alpha _{cr}(\theta,\phi )$ and $\beta _{cr}(\theta,\phi )$ show complex behaviour, including discontinuities owing to jumping of critical states across neighbouring local valleys (in $Re$ ) or lobes of the neutral surface. The discontinuity behaviour exists for all low $\phi$ . For $\phi \gtrsim 21^\circ$ , variations of $\alpha _{cr}(\theta )$ and $\beta _{cr}(\theta )$ are generally smooth and monotonic.