We consider the magnetorotational instability (MRI) of a hydrodynamically stable Taylor-Couette flow with a helical external magnetic field in the inductionless approximation defined by a zero magnetic Prandtl number $(\mathrm{Pm}=0)$. This leads to a considerable simplification of the problem eventually containing only hydrodynamic variables. First, we point out that magnetic field adds more dissipation while it does not change the base flow which is the only source of energy for growing perturbations. Thus, it seems unclear from the energetic point of view how such a hydrodynamically stable flow can turn unstable in the presence of a helical magnetic field as it has been found recently by Hollerbach and Rüdiger [Phys. Rev. Lett. 95, 124501 (2005)]. We revisit this problem by using a Chebyshev collocation method to calculate the eigenvalue spectrum of the linearized problem. In this way, we confirm that a helical magnetic field can indeed destabilize the flow in the inductionless approximation. Second, we integrate the linearized equations in time to study the transient behavior of small amplitude perturbations, thus showing that the energy arguments are correct as well. However, there is no real contradiction between both facts. The linear stability theory predicts the asymptotic development of an arbitrary small-amplitude perturbation, while the energy stability theory yields the instant growth rate of any particular perturbation, but it does not account for the evolution of this perturbation. Thus, although switching on the magnetic field instantly increases the energy decay rate of the dominating hydrodynamic perturbation, in the same time this perturbation ceases to be an eigenmode in the presence of the magnetic field. Consequently, this perturbation is transformed with time and so becomes able to extract energy from the base flow necessary for the growth.

Four configurations of a rotating magnetic dipole-driven turbulent flow in an electrically conducting liquid cylinder are considered by spectral direct numerical simulation. These configurations differ by parallel or perpendicular orientation of the dipole rotation vector with respect to the nearest surface of the cylinder or its axis. The rotating dipole generates electromagnetic force in a thin outer liquid layer facing it. A concentrated vortex is driven when the dipole rotation vector is perpendicular to the nearest surface. This vortex closely resembles the rotating disk-driven flow. When the dipole rotation vector is parallel to the nearest surface, then a distributed vortex occurs akin of the translating wall-driven cavity flow. The characteristic velocity is comparably little influenced by dipole orientation despite the electromagnetic force magnitude varying by a factor of three. Perpendicular orientation of the magnetic dipole rotation vector with respect to the cylinder's axis causes secondary corner eddies increasing the overall turbulent fluctuation. The simulations are supplemented by an experiment featuring a deep and narrow funnel-shaped quasi-stationary free surface deformation above a concentrated vortex.

The stability of electrically conducting liquid flow in a cylindrical ring channel is studied numerically. The flow is driven by a rotating magnetic dipole placed at the ring’s center. Depending on ring’s width, two distinct flow regimes are observed. In a narrow ring, the flow itself and its instability resemble the related rotating magnetic field driven flow in a cylinder. This changes in a wide ring when an intense radial jet develops on the midplane. Within this jet, the driving magnetic force is overwhelmed by inertial and viscous forces similar to how it occurs in the boundary layer flow. The instability develops as an azimuthally periodic wave-like deformation of this jet. Non-uniform driving force and the viscous boundary layer at the inner side wall are supposed as the main ingredients of the jet formation.

Here, an original rotating permanent magnet (RPM) system placed coaxially with the liquid metal container is studied as an effective means of generating flow in shallow cylinders for potential application in aluminum metallurgy (e.g., for ladle stirring and metal dosing). The studied RPM system generates volume force with strong axial variation and force maximum near the radial midpoint. The numerical and experimental data show that, in the shallow cylinder case, the azimuthal velocity follows the force radial distribution. The resulting velocity maximum occurs near the radial midpoint, unlike in the traditional rotating magnetic field (RMF) stirrer systems, where the velocity maximum occurs near the outer radius. An analytical description is developed to explain the velocity radial distribution. The numerically calculated velocity distribution shows a peculiar result that the bottom-stirred liquid metal container results in peak angular momentum at the top of the container. The solution to asymptotic boundary layer equations shows that this is due to the force axial variation.