We analyse numerically the linear stability of the fully developed liquid metal flow in a square duct with insulating side walls and thin electrically conducting horizontal walls with the wall conductance ratio c=0.01⋯1 subject to a vertical magnetic field with the Hartmann numbers up to Ha=104. In a sufficiently strong magnetic field, the flow consists of two jets at the side walls walls and a near-stagnant core with the relative velocity ∼(cHa)−1. We find that for Ha≳300, the effect of wall conductivity on the stability of the flow is mainly determined by the effective Hartmann wall conductance ratio cHa. For c≪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 Ha≈30/c. In a stronger magnetic field with cHa≳30, the destabilizing effect vanishes and the asymptotic results of Priede et al. [J. Fluid Mech. 649, 115, 2010] for the ideal Hunt's flow with perfectly conducting Hartmann walls are recovered.
Arlt, T., Priede, J., & Bühler, L. (2017). The effect of finite-conductvity Hartmann walls on the linear stability of Hunt's flow. Journal of Fluid Mechanics, 822, 880-891. https://doi.org/10.1017/jfm.2017.322