On the Nonlinear Stability of Plane Parallel Shear Flow in a Coplanar Magnetic Field

Abstract

Lyapunov direct method has been used to study the nonlinear stability of laminar flow between two parallel planes in the presence of a coplanar magnetic field for streamwise perturbations with stress-free boundary planes. Two Lyapunov functions are defined. By means of the first, it is proved that the transverse components of the perturbations decay unconditionally and asymptotically to zero for all Reynolds numbers and magnetic Reynolds numbers. By means of the second, it is showed that the other components of the perturbations decay conditionally and exponentially to zero for all Reynolds numbers and the magnetic Reynolds numbers below \(\frac{\pi ^2}{2M}\), where M is the maximum of the absolute value of the velocity field of the laminar flow.