We analyse numerically the linear stability of a liquid metal flow in arectangular duct with perfectly electrically conducting walls subject to auniform transverse magnetic field. A non-standard three dimensional vectorstream function/vorticity formulation is used with Chebyshev collocation methodto solve the eigenvalue problem for small-amplitude perturbations. A relativelyweak magnetic field is found to render the flow linearly unstable as two weakjets appear close to the centre of the duct at the Hartmann number Ha \approx9.6. In a sufficiently strong magnetic field, the instability following thejets becomes confined in the layers of characteristic thickness \delta \simHa^{-1/2} located at the walls parallel to the magnetic field. In this case theinstability is determined by \delta, which results in both the criticalReynolds and wavenumbers numbers scaling as \sim \delta^{-1}. Instability modescan have one of the four different symmetry combinations along and across themagnetic field. The most unstable is a pair of modes with an even distributionof vorticity along the magnetic field. These two modes represent stronglynon-uniform vortices aligned with the magnetic field, which rotate either inthe same or opposite senses across the magnetic field. The former enhance whilethe latter weaken one another provided that the magnetic field is not toostrong or the walls parallel to the field are not too far apart. In a strongmagnetic field, when the vortices at the opposite walls are well separated bythe core flow, the critical Reynolds and wavenumbers for both of theseinstability modes are the same: Re_c \approx 642Ha^{1/2}+8.9x10^3Ha^{-1/2} andk_c \approx 0.477Ha^{1/2}. The other pair of modes, which differs from theprevious one by an odd distribution of vorticity along the magnetic field, ismore stable with approximately four times higher critical Reynolds number.
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