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We study the stability of two-dimensional inviscid flows in an annulusbetween two porous cylinders with respect to three-dimensional perturbations.The basic flow is irrotational, and both radial and azimuthal components of thevelocity are non-zero. The direction of the radial flow can be from the innercylinder to the outer one (the diverging flow) or from the outer cylinder tothe inner one (the converging flow). It had been shown earlier in Ref.\cite{IM2013a} that, independent of the direction of the radial flow, the basicflow can be unstable to small two-dimensional perturbations. In the presentpaper, we prove first that purely radial flow is stable and that flows withboth radial and azimuthal components are always stable to axisymmetricperturbations. Then we show that both the diverging and converging flows areunstable with respect to non-axisymmetric three-dimensional perturbationsprovided that the ratio of the azimuthal component of the velocity to theradial one is sufficiently large. Neutral curves in the space of parameters ofthe problem are computed and it is demonstrated that for any ratio of the radiiof the cylinders, the most unstable modes (corresponding to the smallest ratioof the azimuthal velocity to the radial one) are the two-dimensional ones. Wealso consider the corresponding viscous stability problem and construct anasymptotic expansion of its solutions for large radial Reynolds numbers. Wecompute the first-order viscous correction to inviscid eigenvalues and showthat the asymptotic results give a good approximation to the viscouseigenvalues even for moderate values of radial Reynolds number, which indicatesthat the instability may be observed in real flows.

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