Absence of magnetohydrodynamic activity in the voltage-driven sheet pinch

We have numerically studied the bifurcation properties of a sheet pinch with impenetrable stress-free boundaries. An incompressible, electrically conducting fluid with spatially and temporally uniform kinematic viscosity and magnetic diffusivity is confined between planes at ${\mathit{x}}_1$=0 and 1. Periodic boundary conditions are assumed in the ${\mathit{x}}_2$ and ${\mathit{x}}_3$ directions and the magnetofluid is driven by an electric field in the ${\mathit{x}}_3$ direction, prescribed on the boundary planes. There is a stationary basic state with the fluid at rest and a uniform current J=(0,0,${\mathit{J}}_3$). Surprisingly, this basic state proves to be stable and apparently to be the only time-asymptotic state, no matter how strong the applied electric field and irrespective of the other control parameters of the system, namely, the magnetic Prandtl number, the spatial periods ${\mathit{L}}_2$ and ${\mathit{L}}_3$ in the ${\mathit{x}}_2$ and ${\mathit{x}}_3$ directions, and the mean values ${\mathit{B}\mathit{¯}}_2$ and ${\mathit{B}\mathit{¯}}_3$ of the magnetic-field components in these directions. © 1996 The American Physical Society.