Periodic and quasiperiodic motions of many particles falling in a viscous fluid

The dynamics of regular clusters of many nontouching particles falling under gravity in a viscous fluid at low Reynolds number are analyzed within the point-particle model. The evolution of two families of particle configurations is determined: two or four regular horizontal polygons (called “rings”) centered above or below each other. Two rings fall together and periodically oscillate. Four rings usually separate from each other with chaotic scattering. For hundreds of thousands of initial configurations, a map of the cluster lifetime is evaluated in which the long-lasting clusters are centered around periodic solutions for the relative motions, and they are surrounded by regions of chaotic scattering in a similar way to what was observed by Janosi et al. [Phys. Rev. E. 56, 2858 (1997)] for three particles only. These findings suggest that we should consider the existence of periodic orbits as a possible physical mechanism of the existence of unstable clusters of particles falling under gravity in a viscous fluid.

Figure 1

The system of $M$ particles that form 2 rings. (a) $M=256$. The initial configuration as in Eq. (15). (b) $M=20000$. In the center-of-mass frame, the trajectory of each particle has the same shape (not drawn to scale). Different shapes correspond to the indicated values of $C$.

Figure 2

Initially, $M$ particles form 4 rings, as specified in Eq. (19). Here, $M=256$.

Figure 3

Evolution of $M=256$ particles that form 4 rings with $C=1.5$ and $R_2,R_4$ as indicated. Trajectories of four particles (each from a different ring) in the center-of-mass reference frame until time $t=500$. The initial positions are marked by dots. (a) “3+1” type of decay. One of the rings separates from the cluster. (b) “2+2” type of decay. The group breaks up into two pairs of oscillating rings—one of them falls faster than the other. (c) “2+1+1” type of decay. The cluster breaks up into one pair of oscillating rings and two single rings. (d) “4”: lack of decay. All rings oscillate.

Figure 4

The cluster lifetime for around $420000$ initial configurations with $C=1.5$ and different values of $R_2$ and $R_4$ (drawn to scale). The darkest red corresponds to the cluster lifetime larger than the simulation time $t=5000.$ The resolution of the map is $0.002\times 0.002$ and $M=256$.

Figure 5

Cross sections (now in linear scale) through the map of the cluster lifetime from Fig. 4: (a) $R_2=0.3$, (b) $R_4=0.87$. Sensitivity to initial conditions is clearly visible. Inset: Scaling down the resolution of $R_2$ and $R_4$ from $0.002$ to ${10}^{-4}$, a self-similar structure is found.

Figure 6

Sensitivity of patterns of the cluster decay to the initial parameters $R_2$ and $R_4$, for $C=1.5$ and $M=256$. White means no decay. (a) The types of decay: into “3+1,” “2+2,” “2+1+1,” or “1+1+1+1” rings (see Sec. 4c for the exact definitions). (b) “3+1” type of decay: the color indicates the label of the ring that separates from the others at time $\tau$. (c) “2+1+1” type of decay: the colors indicate the interacting pair of the rings at time $\tau$. (d) ‘2+2” type of decay: the colors indicate which rings form interacting pairs at time $\tau$.

Figure 7

The average deviation $\mathrm{\Delta }$ of particle trajectories for clusters with lifetimes $\tau \ge 5000,C=1.5$, and $M=256$. Resolution: (a) $0.001\times 0.001$; (b) and (c) $0.002\times 0.002$. The minima in (a), (b), and (c) correspond to the solutions presented in Figs. 8, 8, and 8, respectively.

Figure 8

Periodic trajectories of four particles (each from a different ring) in the center-of-mass frame of a cluster with $M=256$ particles. Dots: the initial particle positions, with $C=1.5$. Solutions in (a)–(c) correspond to the minima of $\mathrm{\Delta }$ in Figs. 7–7(c), respectively.

Figure 9

Periodic solutions of type 1 for different values of $C$ and $M=256$.

Figure 10

Periodic trajectories of four particles (each from a different ring) in the center-of-mass frame of a cluster with $M=256$ particles. Dots: the initial particle positions. Here, $C=2.0$ (compare with Fig. 8, where $C=1.5$).