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Series solutions and a perturbation formula for the extended Rayleigh problem of hydrodynamic stability

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Abstract

We generalize Tollmien’s solutions of the Rayleigh problem of hydrodynamic stability to the case of arbitrary channel cross sections, known as the extended Rayleigh problem. We prove the existence of a neutrally stable eigensolution with wave number k s  > 0; it is also shown that instability is possible only for 0 < k < k s and not for k > k s . Then we generalize the Tollmien–Lin perturbation formula for the behavior of c i, the imaginary part of the phase velocity as the wave number k →k s − to the extended Rayleigh problem and subsequently, we use this formula to demonstrate the instability of a particular shear flow.

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Acknowledgements

The authors are thankful to the referee for his suggestions which helped in improving the presentation of their paper.

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Authors and Affiliations

  1. Department of Mathematics, Sri Manakula Vinayagar Engineering College, Madagadipet, Pondicherry, 605 107, India

    V GANESH

  2. Department of Mathematics, Pondicherry University, Kalapet, Pondicherry, 605 014, India

    M SUBBIAH

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  1. V GANESH
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  2. M SUBBIAH
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Correspondence to V GANESH.

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GANESH, V., SUBBIAH, M. Series solutions and a perturbation formula for the extended Rayleigh problem of hydrodynamic stability. Proc Math Sci 123, 293–302 (2013). https://doi.org/10.1007/s12044-013-0127-6

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  • DOI: https://doi.org/10.1007/s12044-013-0127-6

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