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Instability of a simple steady flow between rotating porous cylinders

The stability of a two-dimensional viscous flow in an annulus with permeable walls with respect to small two-dimensional perturbations will be considered. The basic steady flow is the most general rotationally-invariant solution of the Navier-Stokes equations in which the velocity has both radial and azimuthal components, and the azimuthal velocity profile depends on the radial Reynolds number. It will be shown that for a wide range of parameters of the problem, the basic flow is unstable to small two-dimensional perturbations and that the stability properties of this flow are determined by the azimuthal velocity at the inner cylinder when the direction of the radial flow is from the inner cylinder to the outer one and by the azimuthal velocity at the outer cylinder when the direction of the radial flow is reversed. The mechanism of the instability will also be discussed.