The connection between centrifugal instability and Tollmien-Schlichting-like instability for spiral Poiseuille flow

For spiral Poiseuille flow with radius ratio η ≡ R_i/R_o = 0.5, we have computed complete linear stability boundaries for several values of the rotation rate ratio μ ≡ Ω_o/Ω_i, where R_i and R_o are the inner and outer cylinder radii, respectively, and Ω_i and Ω_o are the corresponding (signed) angular speeds. The analysis extends the previous range of Reynolds number Re studied computationally by more than eightyfold, and accounts for arbitrary disturbances of infinitesimal amplitude over the entire range of Re for which spiral Poiseuille flow is stable for some range of the Taylor number Ta. We show how the centrifugally driven instability (beginning with steady or azimuthally travelling-wave bifurcation of circular Couette flow at Re = 0 when μ < η²) connects, as conjectured by Reid (1961) in the narrow-gap limit, to a non-axisymmetric Tollmien-Schlichting-like instability of non-rotating annular Poiseuille flow at Ta = 0. For μ > η², we show that there is no instability for 0 ≤ Re ≤ Re_min. For μ = 0.5, Re_min corresponds to a turning point, beyond which exists a range of Re for which there are two critical values of Ta, with spiral Poiseuille flow being stable below the lower one and above the upper one, and unstable in between. For the special case μ = 1, with the two cylinders having the same angular velocity, Re_min corresponds to a vertical asymptote smaller than found by Meseguer & Marques (2002), whose results for μ > η² fail to account for disturbances with a sufficiently wide range of azimuthal wavenumbers.