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.

This study is concerned with the numerical linear stability analysis of liquid-metal flow in a square duct with thin electrically conducting walls subject to a uniform transverse magnetic field. We derive an asymptotic solution for the base flow that is valid for not only high but also moderate magnetic fields. This solution shows that, for low wall conductance ratios $c\ll 1$ , an extremely strong magnetic field with Hartmann number $\mathit{Ha}\sim c^{-4}$ is required to attain the asymptotic flow regime considered in previous studies. We use a vector streamfunction–vorticity formulation and a Chebyshev collocation method to solve the eigenvalue problem for three-dimensional small-amplitude perturbations in ducts with realistic wall conductance ratios $c=1$ , 0.1 and 0.01 and Hartmann numbers up to $10^{4}$ . As for similar flows, instability in a sufficiently strong magnetic field is found to occur in the sidewall jets with characteristic thickness ${\it\delta}\sim \mathit{Ha}^{-1/2}$ . This results in the critical Reynolds number and wavenumber increasing asymptotically with the magnetic field as $\mathit{Re}_{c}\sim 110\mathit{Ha}^{1/2}$ and $k_{c}\sim 0.5\mathit{Ha}^{1/2}$ . The respective critical Reynolds number based on the total volume flux in a square duct with $c\ll 1$ is $\overline{\mathit{Re}}_{c}\approx 520$ . Although this value is somewhat larger than $\overline{\mathit{Re}}_{c}\approx 313$ found by Ting et al. (Intl J. Engng Sci., vol. 29 (8), 1991, pp. 939–948) for the asymptotic sidewall jet profile, it still appears significantly lower than the Reynolds numbers at which turbulence is observed in experiments as well as in direct numerical simulations of this type of flow.

In the EVOLVE concept for a nuclear fusion blanket a pool boiling scenario has been proposed where a number of permanent vertical vapour channels are formed in a horizontal layer of liquid lithium. Similar situations occur during laser beam welding where a relatively long vapour capillary is observed. The present analysis focuses on the flow of the electrically conducting liquid phase in the presence of a strong uniform horizontal magnetic field. The cross-section of vapour channels is circular if surface tension dominates magnetic forces. In the opposite case a stretching of the liquid–vapour interface along magnetic field lines is observed and contours become possible where a major part of the interface is straight and aligned with the field. For strong magnetic fields the liquid flow exhibits several distinct subregions. Most of the liquid domain is occupied by inviscid cores. These are separated from each other by parallel layers that spread along the field lines which are tangential to the vapour channel. In one core, which is located between two parallel layers, the flow direction is preferentially oriented along magnetic field lines, while in the other cores the flow is perpendicular to the field.