On the final configuration of a plane magnetic field dragged by a highly conducting fluid and anchored at the boundary

• Magnetic field lines are dragged by a highly conducting flow.
• They tend to become constant along the streamlines.
• If these lines are anchored at the boundary, a boundary layer correction is needed.
• The value of the magnetic potential at each streamline is found.
• Its behavior is different depending on the presence of stagnation or corner points at the boundary.

The final configuration of the magnetic field dragged by a plane conducting flow such that the feet of the field lines are fixed at the boundary is studied by asymptotic analysis on the small magnetic diffusivity. The first order approximation yields that the streamlines become also magnetic field lines and the magnetic potential satisfies an ordinary differential equation on the transversal variable whose boundary values are found by the addition of a boundary layer. It turns out that these values correspond to certain averages along the boundaries, except when there exist stagnation points, which dominate the magnetic potential diffusion. Corners of the boundary curves behave differently, because stagnation points there disappear after straightening the curve by a change of variables that also kills the zero of the velocity.