Linear stability of magnetohydrodynamic flow in a square duct with thin conducting walls

This study is concerned with 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 which is valid not only for high but also moderate magnetic fields. This solution shows that for low wall conductance ratios $c\ll1,$ an extremely strong magnetic field with the Hartmann number $Ha\sim c^{-4}$ is required to attain the asymptotic flow regime considered in the previous studies. We use a vector stream function/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 $c=1,0.1,0.01$ and Hartmann numbers up to $10^{4}.$ As for similar flows, instability in a sufficiently strong magnetic field is found to occur in the side-wall jets with the characteristic thickness $\delta\sim Ha^{-1/2}.$ This results in the critical Reynolds number and wavenumber increasing asymptotically with the magnetic field as $Re_{c}\sim110Ha^{1/2}$ and $k_{c}\sim0.5Ha^{1/2}.$ The respective critical Reynolds number based on the total volume flux in a square duct with $c\ll1$ is $\bar{Re}_{c}\approx520.$ Although this value is somewhat larger than$\bar{Re}_{c}\approx313$ found by Ting et al. (1991) for the asymptotic side-wall 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 flows.