Dynamics of a magnetic particle in an oscillating magnetic field subject to a shear flow

Abstract

The orientational dynamics of a spherical magnetic particle in linear shear flow subjected to an oscillating magnetic field in the flow plane is analysed in the viscous limit. The shear is in the $X$ – $Y$ plane, the magnetic field is in the $X$ direction and the vorticity is perpendicular to the flow in the $Z$ direction. The relevant dimensionless groups are $\omega ^\ast$ , the ratio of the frequency of the magnetic field and the strain rate, and $\varSigma$ , the ratio of the magnetic and hydrodynamic torques. As $\varSigma$ is decreased, there is a transition from in-plane rotation, where the rotation is in the flow ( $X$ – $Y$ ) plane, to out-of-plane rotation, where the orientation vector is not necessarily in the $X$ – $Y$ plane and the dynamics depends on the initial orientation. The particle rotation is phase-locked for in-plane rotation with discrete odd rotation number (number of rotations in one period of magnetic field oscillation), while the orbits are quasi-periodic with non-integer rotation number for out-of-plane rotation. For $\varSigma \gg 1$ , regions of odd rotation number $n_o$ are bound by the lines $8 (n_o-1) \varSigma \omega ^\ast = 1$ and $8 (n_o+1) \varSigma \omega ^\ast = 1$ , and there are discontinuous changes in the rotation number and mean and root-mean-square torque at these lines. For $\varSigma \ll 1$ , the domains of in-plane rotation of finite width in the $\omega ^\ast$ – $\varSigma$ plane extend into downward cusps at $\omega ^\ast = {1}/{2 n_o}$ . The orbits are quasi-periodic between these domains, where the rotation is out of plane.