In this paper we consider theoretical and experimental aspects of axisymmetric, swirling flow which is generated in a column of liquid metal by a rotating magnetic field. Two cases are discussed, one in which there is no axial variation in the stirring force, and one where the body force is restricted to a relatively short length of the column. The latter case is of considerable practical interest in continuous casting.
One-dimensional stirring, where the swirl is independent of z and θ, is well understood. The magnetic body force is balanced by shear, all inertial forces being zero (except for the centripetal acceleration). However, in two-dimensional axisymmetric stirring, the axial variation in swirl drives a strong secondary poloidal flow. The principal local force balance is between the magnetic torque and inertia. The body force spins up the fluid as it passes through the forced region and the secondary flow sweeps this angular momentum into the unforced region. Consequently, the size and distribution of the swirl is controlled by the secondary flow.
The role of wall friction is considered and shown to control the length of the recirculating eddy. An approximate solution of the inviscid equations of motion, based on the angular momentum integral, is derived for the flow in the forced region. This is compared with the results of numerical experiments.
The analysis predicts that the swirl velocity scales on {B(σ/ρω)½}ωR, has a maximum at the bottom of the driven region, and penetrates an axial distance of the order ℝR away from the forced region. (For turbulent flow the Reynolds number ℝ must be based on an effective eddy viscosity.) All these features were reproduced experimentally.