The most general steady state solution is derived for the laminar flow of an incompressible, viscous and electrically conducting fluid in a one-dimensional channel under an applied transverse magnetic field. The channel can act as an electromagnetic flowmeter or pump. The effect of the conductivity of the walls is included. The solution has two unknown constants and, by choosing them properly, it can be made to fit the solution of all two-dimensional channels whose geometry approaches in the limit that of the one-dimensional channel. This is done in detail for the two-dimensional channels with rectangular and annular cross-section.

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