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On the Criterion for Long Wave Stability of Shear Flows

  • K. Reena Priya
  • V. Ganesh
Original Paper

Abstract

The present paper deals with the stability problem of incompressible, inviscid and density stratified fluid in a channel with arbitrary cross section. For this problem we derive a criterion for long wave stability namely, if the wave number is less than or equal to a critical wave number \(k_{c}\) then the disturbances is stable. The result is illustrated with plane Poiseuille flow and Couette flow basic flows. Furthermore, we have obtained an upper bound for the growth rate of an unstable mode, which is sharper than the known ones. Moreover, a parabolic instability region which does not depend on any condition and which intersect the known semielliptical instability region is derived under some conditions.

Keywords

Hydrodynamic stability Shear flows Variable topography Sea strait 

Mathematics Subject Classification

76E05 

Notes

Acknowledgements

We are thankful to the referees for their comments and suggestions which helped us to improve the presentation of the paper.

References

  1. 1.
    Deng, J., Pratt, L., Howard, L., Jones, C.: On stratified shear flow in sea straits of arbitrary cross section. Stud. Appl. Math. 111(4), 409–434 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Dou, H.S., Ganesh, V.: Short wave stability of homogeneous shear flows with variable topography. Appl. Math. Mech. 35(5), 541–548 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ganesh, V.: New upper bounds for the growth rate and a modified instability region for the extended Taylor–Goldstein problem of hydrodynamic stability. Indian J. Math. 52(2), 415–427 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ganesh, V., Subbiah, M.: On upper bounds for the growth rate in the extended Taylor–Goldstein problem of hydrodynamic stability. Proc. Indian Acad. Sci. 119(1), 119–135 (2009)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ganesh, V., Subbiah, M.: Series solutions and a perturbation formula for the extended Rayleigh problem of hydrodynamic stability. Proc. Math. Sci. 123(2), 293–302 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Pratt, L., Deese, H.E., Murray, S.P., Johns, W.: Continuous dynamical modes in straits having arbitrary cross sections with applications to the Bab al Mandab. J. Phys. Oceanogr 303, 2515–2534 (2000)CrossRefGoogle Scholar
  7. 7.
    Ramakrishna Reddy, V., Subbiah, M.: Long wave instability of shear flows in sea straits. In: Proceedings of 59 th Congress of ISTAM, pp. 1–10 (2014)Google Scholar
  8. 8.
    Reddy, V.R., Subbiah, M.: Stability of stratified shear flows in channels with variable cross section. Appl. Math. Mech. 36(1), 1459–1480 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Reenapriya, K., Ganesh, V.: Note on the long wave stability of shear flows with variable bottom topography. Int. J. Math. Anal. 10(16), 757–766 (2016)Google Scholar
  10. 10.
    Smeed, D.A.: Exchange through the Bab al Mandab. Deep Sea Res. Part II Top. Stud. Oceanogr. 51, 455–474 (2000)CrossRefGoogle Scholar
  11. 11.
    Sridevi, S., Ganesh, V.: On the long wave stability of Shear flows in a sea straits of arbitrary cross section. Int. J. Appl. Comput. Math. (2018).  https://doi.org/10.1007/s40819-017-0439-9
  12. 12.
    Sridevi, S., Dou, H.S., Ganesh, V.: A unified instability region for the extended Taylor–Goldstein problem of hydrodynamic stability. Adv. Appl. Math. Mech. 9(6), 1404–1419 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Subbiah, M., Ganesh, V.: Bounds on the phase speed and growth rate of extended Taylor–Goldstein problem. Fluid Dyn. Res. 40(5), 364–377 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Subbiah, M., Ganesh, V.: On the stability of homogeneous shear flows in sea straits of arbitrary cross section. Indian J. Pure Appl. Math. 38(1), 43–50 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2018

Authors and Affiliations

  1. 1.Research and Development CentreBharathiar UniversityCoimbatoreIndia
  2. 2.Department of MathematicsIFET College of EngineeringVillupuramIndia
  3. 3.Department of MathematicsRajiv Gandhi College of Engineering and TechnologyKirumampakkam, PondicherryIndia

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