Nonaxisymmetric flow in a finite-length cylinder with a rotating magnetic field

This paper treats the flow of an electrically conducting liquid in an insulating cylinder with a spatially uniform, transverse, rotating magnetic field. The frequency of the externally applied magnetic field is sufficiently low that this field penetrates throughout the liquid. Previous researchers have treated the steady, axisymmetric flow driven by the azimuthal average of the electromagnetic body force. This paper presents an analytical solution for the nonaxisymmetric flow perturbation driven by the deviation of the body force from its azimuthal average. For the magnetic field strengths and frequencies currently used in crystal-growth processes, the values of the interaction parameter N and the Reynolds number Re, both based on the frequency of the magnetic field, are small and large, respectively. The results of the appropriate asymptotic analysis show that the ratio of the nonaxisymmetric flow to the axisymmetric one is extremely small, thus validating the common neglect of the nonaxisymmetric flow. Therefore a rotating magnetic field may provide beneficial stirring without disturbing the desirable axisymmetric distribution of additives in the liquid.