Extended Semicircle and Semi-Ellipse Theorems for the Heterogeneous Swirling Flow of an Incompressible Fluid

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(κ_M) and C(κ_m) and the tangential lines t_0κM and t_0κm if exist. Here t_0κ means the line tangential to C(0) and C(κ). The maximum κ_M and the minimum κ_m 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.