Channel Flow reference

Jonas Holdeman

Consider an incompressible conducting fluid in a 2D channel with electrically-conducting walls, with a magnetic field B perpendicular to the plane of the channel and an electrical current across the channel, produced by an applied voltage across the channel. There is a resulting force on the fluid along the channel proportional to the product j*B, where j is the electrical current density. I would think that the fluid flow would have the usual parabolic profile, with the flow proportional to j*B. The fluid might be a liquid metal or an ionic fluid like seawater.

This is a simple problem but I need to get the constants (or coefficients) involved straight. Can anyone give me a published reference to this elementary problem?