Resistivity profile and instability of the plane sheet pinch

The stability of the quiescent ground state of an incompressible, viscous and electrically conducting fluid sheet, bounded by stress-free parallel planes and driven by an external electric field tangential to the boundaries, is studied numerically. The electrical conductivity varies as ${\cosh }^{\text{−2}}{\mathrm{(x}}_1\mathrm{/a)}$ , where $x_1$ is the cross-sheet coordinate and $a$ is the half width of a current layer centered about the midplane of the sheet. For $\text{a≲0.4L,}$ where $L$ is the distance between the boundary planes, the ground state is unstable to disturbances whose wavelengths parallel to the sheet lie between lower and upper bounds depending on the value of $a$ and on the Hartmann number. Asymmetry of the configuration with respect to the midplane of the sheet, modelled by the addition of an externally imposed constant magnetic field to a symmetric equilibrium field, acts as a stabilizing factor.