Similarity transformation and self-similar solutions are derived for the incompressible radial flow of conducting, viscous fluids across an external, homogeneous magnetic field (parallel to the apse line) in a diffuser with electrodes in the planes ϑ=±ϑ0 (cylindrical coordinate system). The conducting flow across the (axial) external and induced magnetic fields induces radial and azimuthal current densities. The azimuthal current density produces a net current flux I≠0 through the electrodes which are connected by an external circuit. The eigenvalue problems for the radial velocity and induced magnetic field amplitudes are solved analytically. The external and induced magnetic fields are shown to significantly change the pressure distribution but not the onset of flow separation. This is due to the irrotational nature of the Lorentz force, ∇× (∇×B×B) =0, in which the magnetic field B(r, ϑ) is axial (noncurved).