Two-dimensional nonlinear travelling waves in magnetohydrodynamic channel flow

The present study is concerned with the stability of a flow of viscous conducting liquid driven by pressure gradient in the channel between two parallel walls subject to a transverse magnetic field. Although the magnetic field has a strong stabilizing effect, this flow, similarly to its hydrodynamic counterpart -- plane Poiseuille flow, is known to become turbulent significantly below the threshold predicted by linear stability theory. We investigate the effect of the magnetic field on 2D nonlinear travelling-wave states which are found at substantially subcritical Reynolds numbers starting from $Re_n=2939$ without the magnetic field and from $Re_n\sim6.50\times10^3Ha$ in a sufficiently strong magnetic field defined by the Hartmann number $Ha.$ Although the latter value is by a factor of seven lower than the linear stability threshold $Re_l\sim4.83\times10^4Ha$,it is still more by an order of magnitude higher than the experimentally observed value for the onset of turbulence in the MHD channel flow.