This reminds me of a emag flow meter. If this was a pipe you would get a voltage difference in the z out of plane direction, and that potential is proportional to the flow. That is it, never heard that it influences the flow otherwise emag flow meters would not work and trust me they do.

But MR seems different. This paper is OK, compare speed at centre of pipe for different B fields and you should see an effect if you model the material correctly (with emag properties)

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Magnetohydrodynamic Flow of Liquid-Metal in Circular Pipes for ...