Extended Semicircle and SemiEllipse Theorems for the Heterogeneous Swirling Flow of an Incompressible Fluid
Abstract
It is shown that for the heterogeneous swirling flow with nonnegative densitygradient and nonnegative velocitydensitydependent factors, the complex angular phasevelocity of the unstable m, k mode must lie within the semicircle C(κ) whose diameter is equal to ba+(βα)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 phasevelocity for any unstable mode is constructed as a sum area in the upper halfplane 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, RayleighSynge’s discriminant and axial flow are incorporated the semicircle is transformed into a semiellipse.
 Publication:

Journal of the Physical Society of Japan
 Pub Date:
 May 1985
 DOI:
 10.1143/JPSJ.54.1769
 Bibcode:
 1985JPSJ...54.1769S
 Keywords:

 Angular Velocity;
 Flow Geometry;
 Flow Stability;
 Incompressible Fluids;
 Phase Velocity;
 Vortices;
 Axial Flow;
 Heterogeneity;
 Stratification;
 Fluid Mechanics and Heat Transfer