On the nonlinear stability of parallel shear flow in the presence of a coplanar magnetic field

The nonlinear stability of laminar flow, in the xx-direction, between two parallel planes in the presence of a coplanar magnetic field has been studied using Lyapunov direct method with either rigid or stress-free boundary planes. By defining a Lyapunov function it is proved that the laminar solutions of the system are nonlinearly unconditionally and asymptotically stable for all Reynolds numbers and magnetic Reynolds numbers if the perturbations are two-dimensional and depend only on yy, zz and tt.