Inviscid instability of an incompressible flow between rotating porous cylinders to three-dimensional perturbations

Abstract

We consider the stability of the Couette–Taylor flow between porous cylinders with radial throughflow in the limit of high radial Reynolds number. It has already been shown earlier that this flow can be unstable to two-dimensional perturbations. In the present paper, we study its stability to general three-dimensional perturbations. In the limit of high radial Reynolds number, we show the following: (i) the purely radial flow is stable (for both possible directions of the flow); (ii) all rotating flows are stable with respect to axisymmetric perturbations; (iii) the instability occurs for both directions of the radial flow provided that the ratio of the azimuthal component of the velocity to the radial one at the cylinder, through which the fluid is pumped in, is sufficiently large; (iv) the most unstable modes are always two-dimensional, i.e. two-dimensional modes become unstable at the smallest ratio of the azimuthal velocity to the radial one; (v) the stability is almost independent of the rotation of the cylinder, through which the fluid is being pumped out. We extend these results to high but finite radial Reynolds numbers by means of an asymptotic expansion of the corresponding eigenvalue problem. Calculations of the first-order corrections show that small viscosity always enhances the flow stability. It is also shown that the asymptotic results give good approximations to the viscous eigenvalues even for moderate values of radial Reynolds number.