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

For the radius ratio η\,≡\,Rᵢ/Rₒ\,=\,0.1 and several rotation rate ratios μ\,≡\,Ωₒ/Ωᵢ , we consider the linear stability of spiral Poiseuille flow (SPF) up to Re\,=\,10⁵ , where Rᵢ and Rₒ are the radii of the inner and outer cylinders, respectively, Re\,≡\, VZ(Rₒ\,-Rᵢ)/ν is the Reynolds number, Ωᵢ and Ωₒ are the (signed) angular speeds of the inner and outer cylinders, respectively, ν is the kinematic viscosity, and VZ is the mean axial velocity. The Re 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 η\,≈\,0.115 . We also establish the connection of the linear stability of annular Poiseuille flow for 0\,<\,η\,≤\,η at all Re to the linear stability of circular Poiseuille flow ( η\,=\,0 ) at all Re . 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 Ta\,≡\,Ωᵢ(Rₒ\,-Rᵢ)²/ν versus Re , show that the results are qualitatively different from those at larger η . For each μ , the centrifugal instability at small Re does not connect to a high- Re Tollmien–Schlichting-like instability of annular Poiseuille flow, since the latter instability does not exist for η\,<\,η . We find a range of Re 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 Re for which there exist three critical values of Ta , with the upper branch emanating from the Re\,=\,0 instability of Couette flow. For the co-rotating ( μ\,=\,0.2 ) case, there are two critical values of Ta for each Re in an apparently semi-infinite range of Re , with neither branch of the stability boundary intersecting the Re = 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 .