Weakly nonlinear stability analysis of magnetohydrodynamic channel flow using an efficient numerical approach

J. Hagan, Janis Priede

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    Abstract

    We analyze weakly nonlinear stability of a flow of viscous conducting liquid driven by pressure gradient in the channel between two parallel walls subject to a transverse magnetic field. Using a non-standard numerical approach, we compute the linear growth rate correction and the first Landau coefficient, which in a sufficiently strong magnetic field vary with the Hartmann number as ?1~(0.814-i19.8)×10-3Ha and ?2~(2.73-i1.50)×10-5Ha-4. These coefficients describe a subcritical transverse velocity perturbation with the equilibrium amplitude |A|2 = R|?1|/R[?2](Rec - Re) ~ 29.8Ha5Rec - Re, which exists at Reynolds numbers below the linear stability threshold Rec~4.83×104Ha. We find that the flow remains subcritically unstable regardless of the magnetic field strength. Our method for computing Landau coefficients differs from the standard one by the application of the solvability condition to the discretized rather than continuous problem. This allows us to bypass both the solution of the adjoint problem and the subsequent evaluation of the integrals defining the inner products, which results in a significant simplification of the method. NOTE: some of the mathematical formulae do not display correctly on this platform, please see the original abstract at http://dx.doi.org/10.1063/1.4851275.
    Original languageEnglish
    Pages (from-to)124108
    JournalPhysics of Fluids
    Volume25
    Issue number12
    DOIs
    Publication statusPublished - 2013

    Bibliographical note

    Copyright (2013) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Hagan, J. and Priede, J. (2013) Weakly nonlinear stability analysis of magnetohydrodynamic channel flow using an efficient numerical approach. Physics of Fluids, volume 25 (12): 124108 and may be found at http://dx.doi.org/10.1063/1.4851275.

    Keywords

    • continuous problems
    • Landau coefficients
    • magnetic field strengths
    • numerical approaches
    • solvability conditions
    • strong magnetic fields
    • transverse magnetic field
    • weakly non-linear stabilities
    • magnetohydrodynamics
    • Reynolds number
    • magnetic fields

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