We analyse numerically the linear stability of the fully developed flow of a liquid metal in a rectangular duct subject to a transverse magnetic field. The walls of the duct perpendicular to the magnetic field are perfectly conducting whereas the parallel ones are insulating. In a sufficiently strong magnetic field, the flow consists of two jets at the insulating walls and a near-stagnant core. We use a vector stream function formulation and Chebyshev collocation method to solve the eigenvalue problem for small-amplitude perturbations. Due to the two-fold reflection symmetry of the base flow the disturbances with four different parity combinations over the duct cross-section decouple from each other. Magnetic field renders the flow in a square duct linearly unstable at the Hartmann number Ha 5.7 with respect to a disturbance whose vorticity component along the magnetic field is even across the field and odd along it. For this mode, the minimum of the critical Reynolds number Re_c 2018, based on the maximal velocity, is attained at Ha ~ 10. Further increase of the magnetic field stabilises this mode with Re_c growing approximately as Ha. For Ha>40, the spanwise parity of the most dangerous disturbance reverses across the magnetic field. At Ha ~ 46 a new pair of most dangerous disturbances appears with the parity along the magnetic field being opposite to that of the previous two modes. The critical Reynolds number, which is very close for both of these modes, attains a minimum Re_c ~ 1130 at Ha ~ 70 and increases as Re_c ~ Ha^1/2 for Ha >> 1. The asymptotics of the critical wavenumber is k_c ~ 0.525Ha^1/2 while the critical phase velocity approaches 0.475 of the maximum jet velocity.Comment: 20 pages, 13 figures, production version, to appear in J. Fluid Mec

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We analyse numerically the linear stability of a liquid metal flow in a rectangular duct with perfectly electrically conducting walls subject to a uniform transverse magnetic field. A non-standard three dimensional vector stream function/vorticity formulation is used with Chebyshev collocation method to solve the eigenvalue problem for smallamplitude perturbations. A relatively weak magnetic field is found to render the flow linearly unstable as two weak jets appear close to the centre of the duct at the Hartmann number Ha ≈ 9.6. In a sufficiently strong magnetic field, the instability following the jets becomes confined in the layers of characteristic thickness δ ∼ Ha −1/2 located at the walls parallel to the magnetic field. In this case the instability is determined by δ, which results in both the critical Reynolds and wavenumbers numbers scaling as ∼ δ −1 . Instability modes can have one of the four different symmetry combinations along and across the magnetic field. The most unstable is a pair of modes with an even distribution of vorticity along the magnetic field. These two modes represent strongly non-uniform vortices aligned with the magnetic field, which rotate either in the same or opposite senses across the magnetic field. The former enhance while the latter weaken one another provided that the magnetic field is not too strong or the walls parallel to the field are not too far apart. In a strong magnetic field, when the vortices at the opposite walls are well separated by the core flow, the critical Reynolds and wavenumbers for both of these instability modes are the same: Re c ≈ 642Ha 1/2 + 8.9 × 10 3 Ha −1/2 and k c ≈ 0.477Ha 1/2 . The other pair of modes, which differs from the previous one by an odd distribution of vorticity along the magnetic field, is more stable with approximately four times higher critical Reynolds number.

Three-dimensional buoyant convection in a rectangular cavity with a horizontal temperature gradient in a strong, uniform magnetic ÿeld is considered. The walls of the cavity are electrically insulating. An asymptotic solution to the problem in the inertialess approximation is obtained for high values of the Hartmann number, Ha. In the presence of either the vertical or the horizontal longitudinal ÿelds, the three-dimensional ow is characterised by high-velocity jets at the walls of the cavity parallel to the magnetic ÿeld. The velocity of the jets is O(Ha) times higher than in the bulk of the uid. On the other hand, in the presence of the horizontal transverse magnetic ÿeld, the velocity in the core is O(Ha) times higher than in the other two cases. However, no jets are present in the parallel layers. The analysis of the convective heat transfer for low values of the Peclet number shows that either the vertical, or the longitudinal ÿeld is the most e cient in damping of the convective heat transfer, depending on the dimensions of the cavity.

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