Liquid-metal flows in current collectors for homopolar machines: Fully developed solutions for the primary azimuthal velocity

Liquid metals in the small radial gaps between the rotors and stators of homopolar machines represent low-resistance electric current collectors. Design predictions for these liquid-metal current collectors require a thorough knowledge of liquid-metal flows in a narrow gap between a fixed and a moving surface, with a strong applied magnetic field and a free surface beyond each end of the gap. The radial and axial velocities in the secondary flow are reduced by a strong axial or radial magnetic field. For a sufficiently strong field, the azimuthal momentum transport by the secondary flow can be neglected. This assumption reduces the problem for the primary azimuthal velocity to a fully developed magnetohydrodynamic duct flow problem with a moving wall and two free surfaces. Asymptotic solutions for large Hartmann numbers are presented for skewed magnetic fields with both radial and axial components. Collectors without any electrical insulation or with insulation on the stator sides, or rotor sides, or both are considered. Solutions for a purely axial magnetic field and arbitrary Hartmann numbers are also presented