1985 Volume 54 Issue 5 Pages 1769-1781
It is shown that for the heterogeneous swirling flow with non-negative density-gradient and non-negative velocity-density-dependent factors, the complex angular phase-velocity of the unstable m, k mode must lie within the semicircle C(κ) whose diameter is equal to b−a+(β−α)|κ| and center is located at [(b+a+(β+α)κ)⁄2, 0]. Here m and k are the azimuthal and axial wavenumbers. κ=k⁄m. a, b and α, β are the lower and upper bounds of the rotating and the axial velocity. The domain of angular phase-velocity for any unstable mode is constructed as a sum area in the upper half-plane enclosed by the two semicircles C(κM) and C(κm) and the tangential lines t0κM and t0κm if exist. Here t0κ means the line tangential to C(0) and C(κ). The maximum κM and the minimum κm of κ are determined by the instability condition. If the effects of rotational stratification, Rayleigh-Synge’s discriminant and axial flow are incorporated the semicircle is transformed into a semi-ellipse.
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