Separation of magnetic or magnetically labeled biological or synthetic particles in micro-fluidic setups has been recently achieved by subjecting them to an external magnetic field to produce controlled alterations in the flow-driven motion of target particles based on their size, shape and magnetic properties. We investigate Brownian noise effects on magnetic focusing of prolate and oblate spheroids carrying permanent magnetic dipoles in channel (Poiseuille) flow subject to a uniform magnetic field. The focusing is effected by the low-Reynolds-number wall-induced hydrodynamic lift which can be tuned via tilt angle of the field relative to the flow direction.
This mechanism is incorporated in a steady-state Smoluchowski equation that we solve numerically to analyze the noise effects through the joint position-orientation probability distribution function of spheroids within the channel. We complement this approach by analyzing stability of deterministic fixed points and a reduced onedimensional probabilistic theory which we introduce to semi-quantitatively explain noise-induced behavior of pinned spheroids under strong fields. The results reveal remarkable and even counterintuitive noise-induced phenomena that include fieldinduced defocusing upon strengthening the field. We map out focusing ‘phase’ diagrams based on the field strength and tilt angle to illustrate different regimes of behavior including centered focusing and defocusing in transverse/longitudinal fields and off-centered focusing in tilted fields. The latter encompasses two subregimes of optimal and shouldered focusing where spheroidal density profiles across the channel width display either an isolated off-centered peak or a skewed peak with a pronounced shoulder stretching toward the channel center. We also elucidate the implications of our results for efficient shape-based sorting of magnetic spheroids.