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 <= phi <= 90 degrees to the plate relative velocity vector. All possible GCP flows can be described by a global Reynolds number Re, phi and an angle 0 <= phi <= 90 degrees, 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., 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, f) 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 greater than or similar to 21 degrees, variations of alpha(cr)(theta) and beta(cr)(theta) are generally smooth and monotonic.