  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 small-amplitude 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 \approx
9.6. In a sufficiently strong magnetic field, the instability following the
jets becomes confined in the layers of characteristic thickness \delta \sim
Ha^{-1/2} located at the walls parallel to the magnetic field. In this case the
instability is determined by \delta, which results in both the critical
Reynolds and wavenumbers numbers scaling as \sim \delta^{-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 \approx 642Ha^{1/2}+8.9x10^3Ha^{-1/2} and
k_c \approx 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.

Fluid Dynamics