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

Abstract

For the radius ratio $\eta\,{\equiv}\,R_i/R_o\,{=}\,0.1$ and several rotation rate ratios $\mu\,{\equiv}\,\Omega_o/\Omega_i$, we consider the linear stability of spiral Poiseuille flow (SPF) up to ${\hbox {\it Re}}\,{=}\,10^5$, where $R_i$ and $R_o$ are the radii of the inner and outer cylinders, respectively, ${\hbox {\it Re}}\,{\equiv}\,\overline V_Z(R_o\,{-}R_i)/\nu$ is the Reynolds number, $\Omega_i$ and $\Omega_o$ are the (signed) angular speeds of the inner and outer cylinders, respectively, $\nu$ 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 $\mu\,{=}\,0$ 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 $\hat\eta\,{\approx}\,0.115$. We also establish the connection of the linear stability of annular Poiseuille flow for $0\,{<}\,\eta\,{\leq}\,\hat\eta$ 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 ${\hbox {\it Ta}}\,{\equiv}\,\Omega_i(R_o\,{-}R_i)^2/\nu$ 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 Tollmien–Schlichting-like instability of annular Poiseuille flow, since the latter instability does not exist for $\eta\,{<}\,\hat\eta$. 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 ${\hbox {\it Re}}\,{=}\,0$ 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 $\eta$.