We develop a comprehensive three-dimensional theory to describe the hydrodynamics of magnetic nanoparticles exposed to an external time-dependent magnetic field. This theory extends the mobility matrix formalism developed by J. Happel, to (quasi-) magnetostatic interactions and incorporates Brownian motion as a background actuation source. Motions of both ferromagnetic and superparamagnetic nanorods are calculated for a large variety of magnetic fields: constant or sinusoidally varying, as well as fully or partially rotating. The complete three-dimensional dynamics of such nanoparticles, mostly in analytical forms, is obtained, revealing transient behaviors and details of the alignment dynamics of the nanorod along the direction of the field or into its plane of rotation. It can also be used to measure hydrodynamical parameters such as the dynamic viscosity of the fluid. Applying these results to biological tissues and attaining the intracellular spacescale resolution is of significant interest to elucidate biological processes occurring therein. Extension to incorporate viscoelastic effects is a next stage that seems fully compatible with this approach.

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• Biomolecular optics
• Magnetic nanoparticles
• Nanostructures