Spontaneous rotation in the exact solution of magnetic hydrodynamic equations for flow between two stationary impermeable disks

Magnetohydrodynamic (MHD) flow of a viscous electroconductive incompressible fluid between two stationary impermeable disks is considered. A homogeneous electric current density vector along the normal to the surface is specified on the upper disk, and the lower disk is non-conductive. The exact von Karman solution of the complete system of MHD equations is studied in which the axial velocity and the magnetic field depend only on the axial coordinate. The problem contains two dimensionless parameter: the electric current density on the upper plate Y and the Batchelor number (the magnetic Prandtl number). It is assumed that the external source producing the axial magnetic field is absent. The problem is solved for the Batchelor number in the range of $0$ – $2$. Fluid flow is produced by electric current. It is shown that for small values of $Y$, the fluid velocity vector of the has only axial and radial components. The rate of motion increases with increasing $Y$, and at a critical value of $Y$, there is a bifurcation of a new stable flow regime with fluid rotation, while the flow without rotation becomes unstable. A feature of the obtained new exact solution is the absence of an axial magnetic field necessary for the occurrence of an azimuthal component of the ponderomotive force, as is the case in the MHD dynamo. A new mechanism of the bifurcation of rotation in the MHD flow is found.