We investigate the collective motion of magnetic rotors suspended in a viscous fluid under a uniform rotating magnetic field. We report there are three collective rotational patterns: synchronized, oscillating and chessboard patterns.

We investigate the collective motion of magnetic rotors suspended in a viscous fluid under an uniform rotating magnetic field. The rotors are positioned on a square lattice, and low Reynolds hydrodynamics is assumed. For a 3 × 3 array of magnets, we observe three characteristic dynamical patterns as the external field strength is varied: a synchronized pattern, an oscillating pattern, and a chessboard pattern. The relative stability of these depends on the competition between the energy due to the external magnetic field and the energy of the magnetic dipole-dipole interactions among the rotors. We argue that the chessboard pattern can be understood as an alternation in the stability of two degenerate states, characterized by striped and spin-ice configurations, as the applied magnetic field rotates. For larger arrays, we observe propagation of slip waves that are similar to metachronal waves. The rotor arrays have potential as microfluidic devices that can mix fluids and create vortices of different sizes.

We investigate the collective motion of rotlets that are placed in a single plane. Due to the hydrodynamic interactions, the particles move through the two-dimensional (2D) plane and we analyze these diffusive motions. By analyzing the scaling of the values, we predict that the diffusion coefficient scales with ϕ 0.5, the average velocity with ϕ, and relaxation time of the velocity autocorrelation function with ϕ –1.5, where ϕ is the area fraction of the particles. In this paper, we find that the predicted scaling could be seen only when the initial particle position is homogeneous. The particle collective motions are different by starting the simulation from random initial positions, and the diffusion coefficient is the largest at a minimum volume fraction of our parameter range, ϕ = 0.05. The deviations based on two initial positions can be explained by the frequency of the collision events. The particles collide during their movements and the inter-particle distances gradually increase. When the area fraction is large, the particles will result in relatively homogeneous configurations regardless of the initial positions because of many collision events. When the area fraction is small (ϕ < 0.25), on the other hand, two initial positions would fall into different local solutions because the rare collision events would not modify the inter-particle distances drastically. By starting from the homogeneous initial positions, the particles show the maximum diffusion coefficient at ϕ ≈ 0.20. The diffusion coefficient starts to decrease from this area fraction because the particles start to collide and hinder each other from a critical fraction ∼ 23 %. We believe our current work contributes to a basic understanding of the collective motion of rotating units.

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