Previous computations concerning the allowed magnetohydrodynamic steady states of a visco-resistive magnetofluid in a toroid are extended. The current is supported by an externally imposed toroidal electric field, and a scalar resistivity and viscosity are assumed. Emphasis is on the character of the necessary velocity fields (mass flows) that toroidal geometry demands. Non-ideal boundary conditions are imposed at the toroidal boundary. One of the more interesting results to emerge is the sensitive dependence of the flow pattern on the shape of the toroidal cross-section boundary: the dipolar poloidal flow that had appeared for cross sections that were symmetric about the midplane is seen to deform continuously into a monopolar pattern for a ‘D-shaped’ cross section as the viscous Lundquist number $M$ is increased. A net toroidal mass flow also develops. A boundary layer whose properties scale with fractional powers of $M$ is also studied. The interior of the magnetofluid is approximately force-free, with current densities and magnetic fields that are nearly parallel for JET-like parameters. Steep velocity derivatives and a steep pressure drop in this boundary layer become steeper with increasing $M$ (decreasing viscosity). The magnetic quantities do not reflect the rapid velocity and vorticity variations in the boundary layer. The maximum velocities, in the region where the viscosity is large enough for the numerics to work, are of the order of a few hundreds of centimetres per second. Measurements of velocity fields confined to the boundary layer would misrepresent the interior plasma conditions. Uncertainties in the magnetized plasma viscosity remain as an obstacle to unambiguous tests of the results in the case of real plasmas.