In homopolar motors and generators, large dc electric currents pass through the sliding electrical contacts between rotating copper disks (rotors) and static copper surfaces shrouding the rotor tips (stators). A liquid metal in the small radial gap between the rotor tip and concentric stator surface can provide a low‐resistance, low‐drag electrical contact. Since there is a strong magnetic field in the region of the electrical contacts, there are large electromagnetic body forces on the liquid metal. The primary, azimuthal motion consists of simple Couette flow, plus an electromagnetically driven flow with large extremes of the azimuthal velocity near the rotor corners. The secondary flow involves the radial and axial velocity components, is driven by the centrifugal force associated with the primary flow, and is opposed by the electromagnetic body force, so that the circulation varies inversely as the square of the magnetic‐field strength. Three flow regimes are identified as the angular velocity &OHgr; of the rotor is increased. For small &OHgr;, the primary flow is decoupled from the secondary flow. As &OHgr; increases, the secondary flow begins to convect the azimuthal‐velocity peaks radially outward, which in turn changes the centrifugal force driving the secondary flow. At some critical value of &OHgr;, the flow becomes periodic through the coupling of the primary and secondary flows. The azimuthal‐velocity peaks begin to move radially in and out with an accompanying oscillation in the secondary‐flow strength.