Linear stability of spiral and annular Poiseuille flow for small radius ratio

For the radius ratio η ≡ R_i/R_o=0.1 and several rotation rate ratios μ ≡ Ω_o/Ω_i we consider the linear stability of spiral Poiseuille flow (SPF) up to R_e=10⁵, where R_i and R_o are the radii of the inner and outer cylinders, respectively, R_e ≡ V_z (R_o-R_i/ν is the Reynolds number, Ω_i and Ω_o are the (signed) angular speeds of the inner and outer cylinders, respectively, ν is the kinematic viscosity, and V_z is the mean axial velocity. The R_e range extends more than three orders of magnitude beyond that considered in the previous μ=0 work of Recktenwald et al. (Phys. Rev. E, vol. 48, 1993, p. 444). We show that in the non-rotating limit of annular Poiseuille flow, linear instability does not occur below a critical radius ratio ̂ηap;0.115. We also establish the connection of the linear stability of annular Poiseuille flow for 0<η≤̂η at all R_e to the linear stability of circular Poiseuille flow (η=0) at all R_e. For the rotating case, with μ=-1, -0.5, -0.25, 0 and 0.2 the stability boundaries, presented in terms of critical Taylor number T_a ≡Ω_i(R_o-R_i)²/ν versus R_e, show that the results are qualitatively different from those at larger η. For each η the centrifugal instability at small R_e does not connect to a highR_e Tollmien-Schlichting-like instability of annular Poiseuille flow, since the latter instability does not exist for η<̂η. We find a range of R_e for which disconnected neutral curves exist in the k-Ta plane, which for each non-zero μ considered, lead to a multi-valued stability boundary, corresponding to two disjoint ranges of stable Ta. For each counter-rotating (μ<0) case, there is a finite range of R_e for which there exist three critical values of Ta, with the upper branch emanating from the R_e instability of Couette flow. For the co-rotating (μ=0.2) case, there are two critical values of Ta for each R_e in an apparently semi-infinite range of R_e, with neither branch of the stability boundary intersecting the R_e = 0 axis, consistent with the classical result of Synge that Couette flow is stable with respect to all small disturbances if μ > η², and our earlier results for μ > η² at larger η.