On the amplification of small disturbances in a channel flow with a normal magnetic field

Abstract

Optimal perturbations are investigated in a magnetohydrodynamic flow bounded by perfectly insulating or conducting walls. The parallel channel flow submitted to uniform, normal magnetic field is taken as an example. The stability equations (linearized Navier–Stokes and Maxwell equations) are solved simultaneously, because of the natural existence of a coupling between them. Exponential instability is studied first to set ideas and to fix some reference magnetic Prandtl and magnetic Reynolds numbers. Then, optimal perturbations are searched for by employing the approach first proposed by Butler and Farrell [Phys. Fluids A 4, 1637 (1992)]. The shape of the optimally perturbed velocity is poorly affected by the magnetic field; however, the magnetic field is found to stabilize both exponential instability and algebraically growing perturbations. The critical Reynolds numbers in the presence of magnetic fields can be very large and it is thus possible to find very significant transient growth in subcritical condition. On the basis of the linear theory developed here, it is found that the gain experienced by the flow at the value of the Reynolds number at which transition is experimentally observed is constant, i.e., independent of the other dimensionless parameters.