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

2For the radius ratio eta = R(i)/R(o) = 0.1 and several rotation rate ratios mu =Omega(o)/Omega(i), we consider the linear stability of spiral Poiseuille flow (SPF) Lip to Re = 10(5), where Ri and R(o) are the radii of the inner and outer cylinders, respectively, Re equivalent to (V) over bar (Z)(R(o) -R(i))/v is the Reynolds number, Omega(i) and Omega(o), are the (signed) angular speeds of the inner and outer cylinders, respectively, v is the kinematic viscosity, and (V) over bar (Z) is the mean axial velocity. The Re range extends more than three orders of magnitude beyond that considered in the previous mu = 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 (eta) over cap approximate to 0. 115. We also establish the connection of the linear stability of annular Poiseuille flow for 0 < eta <= (eta) over cap 7 at all Re to the linear stability of circular Poiseuille flow (eta = 0) at all Re. For the rotating case, with mu = -1, -0.5, -0.25, 0 and 0.2, the stability boundaries, presented in terms of critical Taylor number Ta equivalent to Omega(i)(R(o)-R(i))(2)/v versus Re, show that the results are qualitatively different from those at larger eta. For each mu, the centrifugal instability at small Re does not connect to a high-Re Tollimen-Schlichting-like instability of annular Poiseuille flow, since the latter instability does not exist for eta < (eta) over cap. We find a range of Re for which disconnected neutral curves exist in the k-Ta plane, which for each non-zero mu considered, lead to a Multi-valued stability boundary, corresponding to two disjoint ranges of stable Ta. For each counter-rotating (mu < 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 (mu = 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 mu > eta(2), and our earlier results for mu > eta(2) at larger eta.