Onset of instability and travelling wave states in MHD pipe flow subject to a transverse magnetic field

Linear stability theory suggests that all three-dimensional perturbations of small amplitude superposed on a flow of viscous incompressible fluid in a circular pipe decay exponentially in the asymptotic limit of large times. However, the experiments indicate that the flow undergoes laminar-turbulent transition at a Reynolds number Re=2000. There were numerous attempts to resolve this inconsistency. In particular, more recent studies focus on finding non-linear equilibrium solutions of the equations of motion. They correspond to the coherent states observed in a turbulent pipe flow and are labeled travelling waves. Despite significant advancement in understanding the instability mechanism in the Hagen-Poiseuille flow, little is known about the onset of turbulence in a liquid metal flow in a circular pipe subject to a transverse magnetic field. Since the classic experiments of Hartmann and Lazarus in 1937, it is well known that an applied magnetic field can stabilize otherwise unstable flow. However, it may as well introduce anisotropy and create velocity jets with inflection points. The present study consists in two parts and addresses the evolution of small disturbances and the formation of the travelling waves states in the MHD pipe flow subject to a transverse magnetic field.