Numerical simulations of magnetic suspensions with hydrodynamic and dipole-dipole magnetic interactions

Abstract

This work describes a numerical model to compute the translational and rotational motion of N spherical magnetic particles settling in a quiescent viscous fluid under creeping flow condition. The motion of the particles may be produced by the action of gravitational forces, Brownian thermal fluctuations, magnetic dipole-dipole interactions, external magnetic field, and hydrodynamic interactions. In order to avoid particle overlap, we consider a repulsive force based on a variation of a screened-Coulomb potential mixed with Hertz contact forces. The inertia of the particles is neglected so that a mobility approach to describe the hydrodynamic interactions is used. The magnetic dipoles are fixed with respect to the particles themselves. Thus they can only interact magnetically between them and with an external applied magnetic field. Therefore the effect of magnetic field moment rotation relative to the particle as a consequence of a finite amount of particle anisotropy is neglected in this work. On the other hand, the inclusion of particle viscous hydrodynamic interactions and dipolar interactions is considered in our model. Both long-range hydrodynamic and magnetic interactions are accounted by a sophisticated technique of lattice sums. This work considers several possibilities of periodic and non-periodic particle interaction schemes. This paper intends to show the benefits and disadvantages of the different approaches, including a hybrid possibility of computing periodic and non-periodic particle interactions. The well-known mean sedimentation velocity and the equilibrium magnetization of the suspension are computed to validate the numerical scheme. The comparison is performed with the existent theoretical models valid for dilute suspensions and several empirical correlations available in the current literature. In the presence of dipole-dipole particle interactions, the simulations show a non-monotonic behavior of the mean sedimentation velocity as the particle volume fraction increased. This work is the first involving a magnetic suspension under the influence of both magnetic and hydrodynamic particle interactions. The mean sedimentation velocity and the suspension magnetization are examined under the steady-state condition over several realizations. Simulation results for the fluid magnetization are compared with a modified mean field theory, and a very good agreement for semi-dilute suspensions is observed. Additionally, the motion and shape transition of an initially spherical blob composed of magnetic spherical particles are investigated by computer simulations. We show the existence of velocity fluctuations due to the interplay of magnetically induced aggregates and their hydrodynamic dispersion. We find that the collective hydrodynamic interactions play a dispersive role opposite to the aggregative contribution of the magnetic dipole-dipole interactions.