Linear stability of the hydromagnetic pipe flow subject to a transverse magnetic field

Despite decades of research, the problem of developping a fully consistent framework to explain the instability of Poiseuille pipe flow remains challenging. Surprisingly, little attention has been devoted to the linear stability analysis of the circular pipe in the hydromagnetic case and the only thorough numerical study available considers a parallel magnetic field acting on the flow.

We analyse here the linear stability of an electrically conducting circular pipe flow subject to a uniform transverse magnetic field. Both classical stability and transient growth theories are used to analyze the evolution of disturbances. We show that the elongation of the velocity profile in the direction of the magnetic field has a destabilizing effect on the flow. The situation is similar when considering the hydrodynamic flow through an elliptic pipe, which has been shown to be linearly unstable even when the aspect ratio of the pipe is just slightly larger than 1. In the hydromagnetic case, this destabilizing effect is however counterbalanced by the Joule dissipation resulting from the occurrence of induced currents. A thorough parametric study of the stability of the flow will be presented for Reynolds numbers up to Re=10^8 in the cases of perfectly conducting and insulating pipe wall.