Abstract
The velocity field of a liquid conductor will be planar, with all field quantities being invariant in the normal (z) direction, for two different categories of conditions. (1) Planar components of B and J are parallel. This poses a strong restriction on the electrical and magnetic boundary conditions. The magnetic field has a nontrivial influence on the flow field only when Jz is nonuniform. Bz has a precise analog in the laminar heat convection in a liquid with negligible viscous dissipation. (2) The total current field is normal to the planar velocity field. Consequently, Bz must be uniform. There is a space-charge density proportional to the product of Bz and vorticity. If the B field is both static and coplanar with the velocity field, then the electric field must be uniform parallel to the normal direction, and has only a higher-order influence on the velocity field, if the magnetic Reynolds number is small.
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