Flow instability originating from particle configurations using the two-dimensional optimal velocity model

The two-dimensional optimal velocity model has potential applications to pedestrian dynamics and the collective motion of animals. In this paper, we extend the linear stability analysis presented in a previous paper [A Nakayama et al., Phys. Rev. E. 77, 016105 (2008)] and investigate the effects of particle configuration on the stability of several wave modes of collective oscillations of moving particles. We find that, when a particle moves without interacting with particles that are positioned in a diagonally forward or backward direction, the stable region of the particle flow is completely removed by the elliptically polarized mode.

Figure 1

Particle configurations and magnified views of selected segments expressed as polygons. The triangles and triangle tips indicate the particles and their directions of travel, respectively. $r$ is the average distance between the particles, and $s$ and $u$ indicate the relative positions of the surrounding particles. The particle indexes are used in the linear analysis. (a–d) Hexagonal 1 (H1), hexagonal 2 (H2), square 1 (S1), and square (S2) configurations, respectively.

Figure 2

Phase diagrams of stability conditions for H1, H2, S1, and S2 configurations at parameter $c=0$. The solid and dashed curves are the critical curves for the longitudinal and transverse modes along the $x$ axis, respectively. The dotted lines are the critical lines for the modes along the $y$ axis, while the dash-dotted lines represent the critical lines for the elliptically polarized modes. The gray regions labeled “S” represent the parameter regions in which a homogeneous flow is stable. Panels (a), (b), (c), and (d) are phase diagrams of the H1, H2, S1, and S2 configurations, respectively. In the S2 configuration, there is no parameter region in which a stable homogeneous flow is obtained, as the elliptically polarized mode is always unstable.