Nonlinear Boundary-Value Problem for a Conducting Source Flow in an Inhomogeneous Magnetic Field

A closed form solution in terms of elliptic functions is given for a nonlinear boundary-value problem describing a conducting viscous fluid, which flows between plane divergent walls across an azimuthal magnetic field. The conducting fluid is injected through an inner circular section (source) and removed downstream through an outer circular section (sink). The magnetic field has its sources in an external current flowing parallel to the line at which the extended walls would intersect. It is shown that the flow exhibits regions of forward and backward fluid motion in the general case. The separation of the boundary layer occurs if the angle of inclination of the walls is larger than a critical value, which depends on the source or sink strengths of the flow and the magnetic field current. Separation is inhibited by the magnetic field at sufficiently large Hartmann numbers.