The interior of an insulating cylindrical container is supposed filled with an incompressible, electrically conducting, viscous fluid. An externally applied magnetic field is caused to rotate uniformly about an axis parallel to the cylinder generators (by applying two alternating components out of phase at right angles). Induced currents in the fluid give rise to a Lorentz force which drives a velocity field, which in general may have a steady and a fluctuating component. The particular case of a circular cylindrical container in a transverse magnetic field is studied in detail. Under certain reasonable assumptions, the resulting flow is shown to have only the steady component, and the distribution of this component is determined. Some conjectures are offered about the stability of this flow and about the corresponding flows in cavities of general shape.