On the self-similar exact MHD jet solution

We consider an axisymmetric steady flow of a viscous incompressible conducting fluid. The flow is induced by the point source of momentum and point electrode discharging the electric current, both of which are located at the end of a thin semi-infinite insulated wire. We seek the solution in the conical self-similar class where the velocity and magnetic field decrease as the inverse distance from the origin. The solution is obtained for various parameters of the problem, namely the Reynolds number, dimensionless electric current and Batchelor number (magnetic Prandtl number). A reverse flow along the wire occurs, leading to the confinement of the current density in the direction of the jet. If the Batchelor number is zero, the solution obtains a singularity at finite values of the current leading to its breakdown; otherwise, the solution exists at all parameter values. We derive the boundary-layer equations near the wire for large current values and obtain the solution. The pitchfork bifurcation with non-zero poloidal magnetic field occurs and causes the rotation of the fluid, which eliminates the current confinement effect. We describe the conditions when the solution for the swirling jet exists. The connection of this problem to the ones considered previously is discussed.