摘要：

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 Re = 10^5, where R_i and R_o are the radii of the inner and outer cylinders, respectively, Re ≡ overline 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 overline V_Z 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 hat η≈ 0.115. We also establish the connection of the linear stability of annular Poiseuille flow for 0 < η≤ hat 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 ≡Ω_i(R_o -R_i)^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 < hatη. 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 > η^2, and our earlier results for μ > η^2 at larger η.

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