## Abstract

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}=10^{5}, 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 > η^{2}, and our earlier results for μ > η^{2} at larger η.

Original language | English (US) |
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Pages (from-to) | 1-20 |

Number of pages | 20 |

Journal | Journal of Fluid Mechanics |

Volume | 547 |

DOIs | |

State | Published - 2006 |

## ASJC Scopus subject areas

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering
- Applied Mathematics