Spatially-Periodic Cluster Pattern of Coupled Forced Oscillators

Abstract

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.

Figures (2)

• Figure 1: (a) Relationship between $k$ and $\lambda(k)$ for $D=0.25$ and 0.1. (b) Time evolution of $x_i$'s for $D=0.25$. (c) Time evolution of $x_i$'s for $D=0.1$. (d) Snapshot of $x_i$ for $i=1,2, \cdots,200$ at $t=5000$ for $D=0.1$.
• Figure 2: (a) Relationship between $k$ and $\lambda(k_x)$ for $k_y=6\pi/100\times l$ for $l=0,1,2, \cdots,10$. System parameters are $F=1$, $\omega=3$, $d=0.2$, $D=0.1$, and $\alpha=0.05$. (b) Snapshot of $x_{i,j}$ for $i=1,2,\cdots,40$ and $j=1,2, \cdots,20$ at $t=5000$.