Note on Instability Regions in the Circular Rayleigh Problem of Hydrodynamic Stability

• Published in: National Academy of Sciences, India. Proceedings. Section A. Physical Sciences India. Proceedings. Section A. Physical Sciences, 2021-03, Vol.91 (1), p.49-54
• Main Authors: Pavithra, P., Subbiah, M.
• Format: Article
• Language: English
• Subjects: Applied and Technical Physics; Atomic; Axial flow; Boundary conditions; Disturbances; Flow stability; Fluid flow; Incompressible flow; Molecular; Optical and Plasma Physics; Physics; Physics and Astronomy; Quantum Physics; Research Article; Wave velocity
• ISSN: 0369-8203; 2250-1762
• DOI: 10.1007/s40010-019-00654-z

Batchelor and Gill’s semicircular bound on the range of the complex wave velocity of an arbitrary unstable mode in the stability problem of inviscid incompressible homogeneous axial flows with respect to axisymmetric disturbances is further reduced. Significance In the circular Rayleigh problem of hydrodynamic stability, we have obtained a parabolic instability region. The significance of this result is that it depends on the discriminant Ψ ( r ) which plays a vital role in deciding whether a basic axial flow is stable or not to axisymmetric disturbances. However, our parabolic instability region is an unbounded one unlike the semicircular instability region of Batchelor and Gill, which has the weakness that it does not depend on Ψ ( r ) . So we have demonstrated that our parabola intersects the semicircle of Batchelor and Gill for two classes of basic flows, and consequently the semicircular instability region of Batchelor and Gill is further reduced. Moreover, we have stated the stability problem consisting of a second-order ODE and associated boundary conditions in three different but related variables. The significance of this is that further analytical results can be obtained for this problem, and some of these are presented in our subsequent paper.