Mathematical model of the mean electromagnetic force induced by a rotating magnetic fteld in a finite-length cylindrical liquid column

A mathematical model is developed for the mean electromagnetic force produced by a rotating magnetic field driving a flow of a conducting liquid in a cylindrical cavity of finite length. The model takes into account the presence of insulating end-walls, causing the induced electric currents to close within the liquid volume. For cavities with a realistic aspect ratio this effect can result in considerable deviations both in the distribution and of the net value of the torque from a convectional case of an infinite cylinder. We deal with an approximation of an ideal inductor generating a uniform rotating magnetic field along the height of the container. The magnetic field rotational frequency is supposed to be sufficiently low for the skin-effect to be negligible. Moreover, it is assumed that the contribution to the liquid flow of the oscillating part of the electromagnetic force is negligible compared to that of the time-averaged part. The validity range of the underlying approximations is estimated using order of magnitude considerations. Eventually, the mean electromagnetic force is defined by means of a single axisymmetric amplitude of the electrostatic potential. This amplitude actually is shown to be independent of the liquid flow, and therefore, to be determined solely by the boundary conditions on the electric current.