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

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 ω ^ , the ratio of the frequency of the magnetic field and the strain rate, and , the ratio of the magnetic and hydrodynamic torques. As 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 1 , regions of odd rotation number nₒ are bound by the lines 8 (nₒ-1) ω ^ = 1 and 8 (nₒ+1) ω ^ = 1 , and there are discontinuous changes in the rotation number and mean and root-mean-square torque at these lines. For 1 , the domains of in-plane rotation of finite width in the ω ^ – plane extend into downward cusps at ω ^ = 1/2 nₒ . The orbits are quasi-periodic between these domains, where the rotation is out of plane.