Analyses of dimensionless science

Diogo Bolster, Robert Hershberger, and Russell Donnelly have written a useful survey of the application of dimensional analysis (PHYSICS TODAY, September 2011, page 42 ). The Reynolds number, used to characterize the flow of viscous incompressible fluids, was referred to as the most fam-ous of the dimensionless numbers. In magnetohydrodynamics, the study of electrically conducting fluids, another Reynolds number arises, the magnetic Reynolds number. It is defined as R_m = UL/η, where U and L are the characteristic velocity and length, and η is the magnetic diffusivity.

It is often stated in the literature that the magnetic Reynolds number is a measure of the relative importance of magnetic convection to magnetic diffusion. ¹ That is not the case. ²

Consider a conducting disk rotating between the poles of a magnet, as found in some residential electricity meters. Currents induced in the disk by the magnet distort the magnetic field. The magnetic lines of force are dragged in the direction of rotation, but the disturbance is localized since the electrical conductivity is finite. The diffusion and convection processes must be equal in the steady state, irrespective of the value of the electrical conductivity of the disk and the value of the magnetic Reynolds number. This does not mean that the magnetic Reynolds number is meaningless; it is a measure of the distortion of the magnetic field due to the motion of the conductor.