Making a fluid rotate: Circular flow of a weakly conducting fluid induced by a Lorentz body force

We describe the magnetohydrodynamic (MHD) flow of a viscous, electrically conducting, and incompressible fluid driven by the Lorentz body force inside the annulus between two stationary coaxial cylindrical electrodes under imposed electric and magnetic fields. The azimuthal velocity profile is found for axisymmetrical, fully developed laminar and steady flow at arbitrary annulus curvature and different values of the Hartmann number Ha. It is shown that for Ha⩽1, the MHD annular flow behaves as a conventional hydrodynamic fluid driven by an azimuthal Lorentz body force.