On the limitations of the complex wave velocity for the heterogeneous swirling flows

Abstract The hydrodynamic instability of variable density swirling flows under gravity between two infinite coaxial cylinders is investigated for axisymmetric disturbances. It is shown that the complex wave velocity of any arbitrary unstable axisymmetric mode must lie within the semi-elliptical region whose minor axis depends on the density stratification parameter J ( r ). The stabilizing effect of density stratification is shown by reducing the semi-circular instability region of [G. K. Batchelor and A. E. Gill, “Analysis of the stability of axisymmetric jets,” J. Fluid Mech., vol. 14, p. 529, 1962]. Furthermore, we have obtained two parabolic instability regions which intersect and reduce the semi-elliptical instability region for density-stratified flows. These parabolic instability regions are uniformly valid for both variable density and density homogeneous flows also.