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The effect of finite-conductivity Hartmann walls on the linear stability of Hunt’s flow

Published online by Cambridge University Press:  08 June 2017

Thomas Arlt Open the ORCID record for Thomas Arlt [Opens in a new window] ,
Jānis Priede Open the ORCID record for Jānis Priede [Opens in a new window]  and
Leo Bühler
Thomas Arlt*
Affiliation:
Institut für Kern- und Energietechnik, Karlsruhe Institute of Technology, Postfach 3640, 76021 Karlsruhe, Germany
Jānis Priede
Affiliation:
Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB, UK
Leo Bühler
Affiliation:
Institut für Kern- und Energietechnik, Karlsruhe Institute of Technology, Postfach 3640, 76021 Karlsruhe, Germany
*
†Email address for correspondence: Thomas.Arlt@kit.edu
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Abstract

We analyse numerically the linear stability of fully developed liquid metal flow in a square duct with insulating side walls and thin, electrically conducting horizontal walls. The wall conductance ratio $c$ is in the range of 0.01 to 1 and the duct is subject to a vertical magnetic field with Hartmann numbers up to $\mathit{Ha}=10^{4}$. In a sufficiently strong magnetic field, the flow consists of two jets at the side walls and a near-stagnant core with relative velocity ${\sim}(c\mathit{Ha})^{-1}$. We find that for $\mathit{Ha}\gtrsim 300,$ the effect of wall conductivity on the stability of the flow is mainly determined by the effective Hartmann wall conductance ratio $c\mathit{Ha}.$ For $c\ll 1$, the increase of the magnetic field or that of the wall conductivity has a destabilizing effect on the flow. Maximal destabilization of the flow occurs at $\mathit{Ha}\approx 30/c$. In a stronger magnetic field with $c\mathit{Ha}\gtrsim 30$, the destabilizing effect vanishes and the asymptotic results of Priede et al. (J. Fluid Mech., vol. 649, 2010, pp. 115–134) for ideal Hunt’s flow with perfectly conducting Hartmann walls are recovered.

JFM classification

MHD and Electrohydrodynamics: High-Hartmann-number flows Instability: Instability MHD and Electrohydrodynamics: MHD and Electrohydrodynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 822 , 10 July 2017 , pp. 880 - 891
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Copyright
© 2017 Cambridge University Press 

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