On the stability of the Hartmann layer

Abstract

In this paper we are concerned with the theoretical stability of the laminar Hartmann layer, which forms at the boundary of any electrically conducting fluid flow under a steady magnetic field at high Hartmann number. We perform both linear and energetic stability analyses to investigate the stability of the Hartmann layer to both infinitesimal and finite perturbations. We find that there is more than three orders of magnitude between the critical Reynolds numbers from these two analyses. Our interest is motivated by experimental results on the laminar–turbulent transition of ducted magnetohydrodynamics flows. Importantly, all existing experiments have considered the laminarization of a turbulent flow, rather than transition to turbulence. The fact that experiments have considered laminarization, rather than transition, implies that the threshold value of the Reynolds number for stability of the Hartmann layer to finite-amplitude, rather than infinitesimal, disturbances is in better agreement with the experimental threshold values. In fact, the critical Reynolds number for linear instability of the Hartmann layer is more than two orders of magnitude larger than experimentally observed threshold values. It seems that this large discrepancy has led to the belief that stability or instability of the Hartmann layer has no bearing on whether the flow is laminar or turbulent. In this paper, we give support to Lock’s hypothesis [Proc. R. Soc. London, Ser. A 233, 105 (1955)] that “transition” is due to the stability characteristics of the Hartmann layer with respect to large-amplitude disturbances.