Motion of paramagnetic particles in a viscous fluid under a uniform magnetic field: benchmark solutions

Numerical simulations of the two-dimensional motion of multiple paramagnetic particles suspended in a viscous fluid subjected to a uniform magnetic field are presented. Both the magnetic field and flow field can be described efficiently with simple series in local coordinates attached to each particle. The coefficients of the series can be obtained with fast convergence when only a few leading coefficients are treated implicitly. Numerical results for the flow field are validated by comparing the data with those given by an asymptotic solution for a pair of particles separated by a small distance. The numerical results of the magnetic field are validated by comparison with the solutions in bipolar coordinates. Simulations of the motion of multiple particles reveal interesting phenomena and shed light on the fundamental mechanism of particles clustering into a straight chain. The data presented in this paper can be used as a benchmark solution for verifying codes for simulating the motion of paramagnetic particles in a magnetic field.