Quantifying vorticity in magnetic particle suspensions driven by symmetric and asymmetric multiaxial fields

We recently reported two methods of inducing vigorous fluid vorticity in magnetic particle suspensions. The first method employs symmetry-breaking rational fields. These fields are comprised of two orthogonal ac components whose frequencies form a rational number and an orthogonal dc field that breaks the symmetry of the biaxial ac field to create the parity required to induce deterministic vorticity. The second method is based on rational triads, which are fields comprised of three orthogonal ac components whose frequency ratios are rational (e.g., 1 : 2 : 3). For each method a symmetry theory has been developed that enables the prediction of the direction and sign of vorticity as functions of the field frequencies and phases. However, this theory has its limitations. It only applies to those particular phase angles that give rise to fields whose Lissajous plots, or principal 2-d projections thereof, have a high degree of symmetry. Nor can symmetry theory provide a measure of the magnitude of the torque density induced by the field. In this paper a functional of the multiaxial magnetic field is proposed that not only is consistent with all of the predictions of the symmetry theories, but also quantifies the torque density. This functional can be applied to fields whose Lissajous plots lack symmetry and can thus be used to predict a variety of effects and trends that cannot be predicted from the symmetry theories. These trends include the dependence of the magnitude of the torque density on the various frequency ratios, the unexpected reversal of flow with increasing dc field amplitude for certain symmetry-breaking fields, and the existence of off-axis vorticity for rational triads, such as 1 : 3 : 5, that do not have the symmetry required to analyze by symmetry theory. Experimental data are given that show the degree to which this functional is successful in predicting observed trends.