Particle trajectories in a lattice of parallel magnetized fibers

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Abstract
Trajectories of paramagnetic particles in infinite rectangular and rhombic lattices of parallel fibers are calculated. The mathematical model for this analysis accounts for the magnetic and fluid perturbations of all the fibers in the lattice. The fluid field is approximated by potential flow, the viscous force on the particle is assumed to be the Stokes force, and particle diffusion and inertia are assumed to be negligible. For closely packed fibers, the particle collision cross section for a fiber in a lattice is found to differ significantly from the isolated fiber cross section. As one expects, they become equal as the filter porosity approaches unity. Channels are formed allowing particles to pass through the lattice. The channels can be made as narrow as desired by increasing the ratio of magnetic to viscous forces; however, they persist for arbitrarily large porosity. The differences between the filter efficiencies obtained from the isolated fiber model and the regular lattice model are compared. It is suggested that the isolated fiber theory, by neglecting shadows in the particle density cast by upstream fibers, predicts an efficiency that is higher than the efficiency of a real filter. On the other hand, the regularity within the lattice model maximizes the effects of shadows, hence the efficiency predicted by that model is too low.