On the behaviour of a suspension of conducting particles subjected to a time-periodic magnetic field

Abstract

The force F(x) and torque G(x) acting on a conducting body or particle suspended at position x in a magnetic field ${\rm Re} (\hat{\boldmath B}_0({\boldmath x}){\bf e}^{-{\rm i}\omega t})$ are determined to leading order in the ratio of the scale a of the particle to the scale L of the field. When the particle is spherical, F decomposes naturally into an irrotational ‘lift’ force F^L and a solenoidal ‘drag’ force F^D related to G by \[ F^{\rm D} + {\textstyle\frac{1}{2}}\nabla \wedge G = 0. \] This relationship is important when the action of the field on a suspension of such spheres is considered, because it implies that the net effective force per unit volume acting to generate bulk flow is zero in any region where the concentration c is uniform. However, non-uniformity is generated by the force ingredient F^L and bulk flow is then generated through interaction of G and ∇c. These effects are demonstrated for two examples involving rotating and travelling fields. Interactions of a finite number N of spheres are also considered, and in particular it is shown that when the field is a uniform rotating one, the governing dynamical system is at leading order Hamiltonian with four independent integral invariants. When N [ges ] 4, the system in general exhibits chaos.