MHD shear flows with non-constant transverse magnetic field

Viscous conducting flows parallel to a fixed plate are studied. In contrast with the Hartmann setting, the problem is not linearized near a fixed transverse magnetic field, although the field tends to be transversal far from the wall. While general solutions may be formally obtained for all cases, their behavior is far more clear when the magnetic Prandtl number equals one. We consider two different instances: a fixed magnetic field at the wall, or an insulating sheet. The evolution of the flow and the magnetic field both near the plate and far from it are detailed, analyzing the possibility of reverse flow and instability of the solutions.

► A conducting shear flow does not leave a transverse magnetic field invariant.
► Solutions are found for all cases, but these are more useful when kinetic and magnetic diffusivities coincide.
► Dirichlet and Neumann conditions on the magnetic field are studied.
► Reverse flow, and eventual instability, are possible.