Influence of thin finite conducting walls on the linear stability of magnetohydrodynamic duct flow

A linear stability analysis has been performed for a liquid-metal flow in a square duct with thin electrically conducting walls in a uniform magnetic field. The analysis uses a vector-streamfunction-vorticity formulation, and the resulting eigenvalue problem is solved by a Chebyshev collocation method as described in previous studies for ideal cases [1, 2]. This method has been extended to apply for magnetohydrodynamic flows in ducts for engineering applications with thin walls of finite electrical conductivity. For strong magnetic fields, i.e. when the Hartmann number Ha is very high, such flows exhibit high velocity jets in narrow side layers of thickness δ_s ~ Ha^-1/2 along walls aligned with the magnetic field. Jets become unstable when the Reynolds number Re is increased beyond a critical value Re_c. For Ha ≫ 1 the critical Reynolds numbers apparently approach constant values depending on the conductivity of the duct walls, and an asymptotic dependence is found for the critical wave number as k_c ~ 0.5Ha^1/2.