Abstract
The hydromagnetic stability of inviscid compressible flows with axial and azimuthal velocity components in an annular channel has been studied to axisymmetric disturbances. An estimate for the growth rate of arbitrary unstable disturbances is obtained. It is shown that the complex phase-velocity of any unstable magneto-subsonic axisymmetric mode must lie in a semiellipse-type region, which depends on the parameters of compressibility, stratification and the magnetic field. For a monotonic axial velocity profile, this semiellipse-type instability region reduces to the line \(c_{i}=0\) as the minimum Richardson number \({\widetilde{J}}_{m}\rightarrow \frac{1}{4}-\) in accord with the Richardson number criterion. Further, an semi-elliptical instability region depending on the minimum local Richardson number is also obtained for a class of magneto-supersonic axisymmetric disturbances.
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References
Dandapat BS, Gupta AS (1975) On the stability flow in magnetogasdynamics. Q Appl Math 33(2):182–186
Zhang Y, Ding N (2006) Effects of compressibility on the magneto-Rayleigh Taylor instability in Z-pinch implosions with sheared axial flows. Phys Plasmas 13:022701
Howard LN (1973) On the stability of compressible swirling flows. Stud Appl Math 52(1):39–43
Fung YT (1986) A Richardson criterion for compressible swirling flows in a tordial magnetic field. Phys Fluids 29:1326–1328
Miles JW (1961) On the stability of heterogeneous shear flows. J Fluid Mech 10:496–508
Howard LN (1961) Note on a paper of John W. Miles. J Fluid Mech 10:509–512
Waleffe F (2019) Semicircle theorem for streak instability. Fluid Dyn Res 51(1):011403. https://doi.org/10.1088/1873-7005/aadb58
Deguchi K (2021) Eigenvalue bounds for compressible stratified magneto-shear flows varying in two transverse directions. J Fluid Mech 920:A55
Howard LN, Gupta AS (1962) On the hydrodynamic and hydromagnetic stability of swirling flows. J Fluid Mech 14(3):463–476
Singh SN, Hankey WL (1981) On vortex breakdown and instability. AFWAL-TR 81–3021, pp 1–25
Lalas DP (1975) The Richardson criterion for compressible swirling flows. J Fluid Mech 69(1):65–72
Thulasi V, Subbiah M (1989) On the stability of compressible swirling flows. Indian J Pure Appl Math 20(11):1159–1171
Albayrak A, Juniper M, Polifke W (2019) Propagation speed of inertial waves in cylindrical flows. J Fluid Mech 879:85–120
Chandrasekhar S (1961) Hydrodynamic and hydromagnetic instability. Clarendon Press, Oxford
Fung YT (1984) On the stability of vortex motions in the presence of magnetic fields. Phys Fluids 27:838–847
Sasakura Y (1984) Semi-ellipse theorem for the heterogeneous swirling flow in an azimuthal magnetic field with respect to axisymmetric disturbances. J Phys Soc Japan 53(6):2012–2017
Iype MSA, Subbiah M (2008) Eigen value bounds in the stability problem of hydromagnetic swirling flows. Indian J Pure Appl Math 39:211–225
Iype MSA, Subbiah M (2010) On the hydrodynamic and hydromagnetic stability of inviscid flows between coaxial cylinders. Inter J Fluid Mech Research 37:1–15
Prakash S, Subbiah M (2021) Bounds on complex eigenvalues in a hydromagnetic stability problem. J Anal 29:1137–1150
Fung YT (1983) Non-axisymmetric instaility of a rotating layer of fluid. J Fluid Mech 127:83–90
Dipierro B, Abid M (2012) Rayleigh-Taylor instability in variable density swirling flows. Eur Phys J B 85:69–76
Dixit HN, Govindarajan R (2011) Stability of a vortex in radial density stratification: role of wave interactions. J Fluid Mech 679:582–615
Dattu H, Subbiah M (2015) A note on the stability of swirling flows with radius dependent density with respect to infinitesimal azimuthal disturbances. ANZIAM J 56:209–232
Prakash S, Subbiah M (2022) Eigenvalue bounds in an azimuthal instability problem of inviscid swirling flows. J Indian Math Soc 89(3–4):387–405
Jacques C, Dipierro B, Alizard F, Buffat M (2023) Stability of variable density rotating flows: inviscid case and viscous effects in the limit of large Reynolds numbers. Phys Rev Fluids 8:033901. https://doi.org/10.1103/PhysRevFluids.8.033901
Leibovich S, Stewartson K (1983) A sufficient condition for the instability of columnar vortices. J Fluid Mech 126:335–356
Srinivasan V, Kandaswamy P, Debnath L (1984) Hydromagnetic stability of rotating stratified compressible fluid flows. ZAMP 35:728–738
Bhattacharyya SN, Gupta AS, Ganguly K (1985) Hydromagnetic stability of some non-dissipative flows subject to non-axisymmetric disturbances. Astrophys Space Sci 115:51–60
Barston EM (1980) Circle theorem for inviscid steady flows. Int J Eng Sci 18:477–489
Iype MSA (2010) Some analytical and numerical results on the hydrodynamic and hydromagnetic stability of inviscid swirling flows. Pondicherry University, Pondicherry
Shumlak U, Roderick NF (1998) Mitigation of the Rayleigh-Taylor instability by sheared axial flows. Phys Plasmas 5(6):2384–2389
Kouznetsov A, Freidberg JP, Kesner J (2007) The effect of sheared axial flow on the interchange mode in a hard-core Z pinch. Phys Plasmas 14(1):012503. https://doi.org/10.1063/1.2423014
Pavithra P, Subbiah M (2021) On compressible circular Rayleigh problem of hydrodynamic stability. Sadhana-Indian Acad Sci 46(178):1–12. https://doi.org/10.1007/s12046-021-01696-z
Kochar GT, Jain RK (1979) Note on Howard’s semicircle theorem. J Fluid Mech 91:489–491
Drazin PG, Reid WH (1981) Hydrodynamic stability. Cambridge University Press, UK
Jain RK, Kochar GT (1983) Stability of stratified shear flows. J Math Anal Appl 96:269–282
Acheson DJ, Hide R (1973) Hydromagnetics of rotating fluids. Rep Prog Phys 36:159–221
Rudiger G, Gellert M, Hollerbach R, Schultz M, Stefani F (2018) Stability and instability of hydromagnetic Taylor - Couette flows. Phys Reports 741:1–89
Raphaldini B, Dikpati M, Raupp CFM (2023) Quasi-geostropic MHD equation: hamiltonian formulation and non linear stability. Comp Appl Math 42(1):57. https://doi.org/10.1007/s40314-023-02192-2
Acknowledgments
S. Prakash acknowledges with thanks to Council of Scientific and Industrial Research (CSIR), India for supporting this research work under Grant No. 09/559(0134)/2019-EMR-I.
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Prakash, S., Subbiah, M. On the Hydromagnetic Stability Analysis of Inviscid Compressible Annular Flows. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 93, 601–611 (2023). https://doi.org/10.1007/s40010-023-00848-6
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DOI: https://doi.org/10.1007/s40010-023-00848-6
Keywords
- Hydromagnetic stability
- Compressible flows
- Axial velocity
- Azimuthal magnetic field
- Axisymmetric disturbances