Stability of Parallel Flow in an Oscillating Magnetic Field

Equations are derived for the stability of unsteady plane parallel flow of a viscous electrically-conducting incompressible fluid in an unsteady parallel magnetic field. A Squire transformation is valid, but, with general considerations of parametric instability, it implies in general that some three-dimensional disturbance is the most unstable when the basic flow is periodic in time. Some sufficient conditions of stability for general flows are found from the energy equation. The parametric instability of a steady vortex sheet of inviscid fluid in an oscillating magnetic field is studied in detail. This special case is reduced to a Floquet system, and stability diagrams are calculated. They show that, with given maximum intensity of field, it is less efficient to stabilize the vortex sheet with an oscillatory field than a steady one.