Spatially-Periodic Cluster Pattern of Coupled Forced Oscillators

We propose a simple model for periodic clustering of particles under forced oscillation. Effective viscosity is assumed to increase owing to neighboring particles by analogy with the Einstein viscosity law. The linear stability analysis and numerical simulations show that the uniform distribution is unstable, and spatially-periodic and stripe patterns appear respectively in one and two dimensions.

💡 Research Summary

The paper introduces a minimalist phenomenological model to explain the spontaneous formation of spatially periodic clusters of particles subjected to a sinusoidal external forcing. The key hypothesis is that the effective viscous damping experienced by each particle increases with the presence of neighboring particles, an idea borrowed from Einstein’s classic viscosity law for suspensions (η = η₀(1 + 2.5 φ)). Translating this into a dynamical system, the authors write the equation of motion for particle i as

 d²x_i/dt² = F sin ωt − α ẋ_i − ∑_j exp