On the linear stability of spiral flow between rotating cylinders

A numerical study is made of the effects of both axisymmetric and non-axisymmetric disturbances on the stability of spiral flow between rotating cylinders. If we let $\Omega_1$ and $\Omega_2$ be the angular speeds of the inner and outer cylinders, and R$_1$ and R$_2$ be their respective radii, then for fixed values of $\eta$ = R$_1$/R$_2$ and $\mu$ = $\Omega_2$/$\Omega_1$, the onset of instability depends on both the Taylor number T and the axial Reynolds number R. Here R is based on the gap width between the cylinders and the average axial velocity of the basic flow, while T is based on the average angular speeds of the cylinders. Using the compound matrix method, we have computed the complete stability boundary in the R, T-plane for axisymmetric disturbances with $\eta$ = 0.95 and $\mu$ = 0. We find that, for sufficiently high Reynolds numbers, there are two distinct axisymmetric modes corresponding to the usual shear and rotational instabilities. We have also obtained the stability boundaries for non-axisymmetric disturbances for R $\leqslant$ 6000 for $\eta$ = 0.95 and 0.77 with $\mu$ = 0. These last results are found to be in substantial agreement with the experimental observations of Snyder (1962, 1965), Nagih (1972) and Mavec (1973) in the low and moderate axial Reynolds number regimes.