Field-aligned flow of a conducting fluid past a source of magnetism

Summary

A prescribed source of magnetism moves at constant speed through a viscous conducting incompressible fluid with an aligned uniform magnetic field. The velocity and magnetic fields induced at a distance from the source are calculated. The induced fields are also calculated for the case in which the applied field is absent. Although no special symmetry or alignment is assumed, the source is “ideal” in the sense that enclosures (wires or magnets) are infinitesimal in at least two dimensions. Dynamical interactions will occur in a viscous fluid and their effect in the far field is estimated.

As a consequence of finite conductivity and viscosity, the usual wakes are present which trail or lead the source depending upon the sign of (1−A ²), where A is the ratio of the source speed to the Alfvén speed in the undisturbed fluid. Outside the wake the total perturbation magnetic field due to the source is the static field plus a monopole field, divided by (1−A ²).

An estimate is also made of the rate at which energy is dissipated as a consequence of viscous interactions and ohmic heating throughout the fluid, outside the immediate vicinity of the source.