Eigenvalue Bounds in an Azimuthal Instability Problem of Inviscid Swirling Flows

We consider the eigenvalue problem of azimuthal instability of inviscid swirling ows between coaxial cylinders. It is shown that the complex eigenvalues corresponding to unstable azimuthal normal modes lie inside a semi-ellipse type region whose major axis coincides with the range of the angular velocity of the basic ow while its minor axis depends on the minimum Richardson number, the azimuthal wave number, and the width of the annular region between the coaxial cylinders.