MHD vortex flow in liquid metal near a spherical particle with different conductivity

The flow of a liquid metal near a spherical particle, whose electrical conductivity differs from that of the liquid metal, is considered. A cylindrical vessel with metal is in an axial magnetic field and, accordingly, with an axial electric current flowing through it. If the conductivity of the particle is equal to the conductivity of the liquid, then the electric current flows along the magnetic field lines and there are no electromagnetic forces in the system. In the case of different conductivity, the electric field lines are either drawn to the particle or go around it, which causes the appearance of electromagnetic forces that generate a vortex flow of the metal. The flow consists of two toroidal vortices, in which the azimuthal motion develops in opposite directions. The poloidal flow in both vortices is arranged in such a way that the liquid on the axis of the cylinder always moves towards the particle. It is shown that the flow energy rapidly increases when the particle conductivity deviates from the liquid conductivity, and reaches asymptotes when the difference in conductivities turns out to be significant. So, with a relative difference in conductivity of only one percent, the energy of the azimuthal flow reaches 40% of the value corresponding to their dissimilarity by two orders of magnitude. At the same time, 80% of this value is achieved at a twofold difference. For a particle with reduced electrical conductivity, the effect is somewhat weaker than for a particle with increased electrical conductivity, but, on the whole, the structure of the emerging flow is similar. In the entire range of the considered values of the electromagnetic action parameter, the flow is unstable and generates fluctuations. As the impact grows, the emerging toroidal vortices become more compact, clinging to the particle, but the fluctuations intensify and capture an increasing volume around the particle.