Class of periodic and quasiperiodic trajectories of particles settling under gravity in a viscous fluid

We investigate regular configurations of a small number of non-Brownian particles settling under gravity in a viscous fluid. The particles do not touch each other and can move relative to each other. The dynamics is analyzed in the point-particle approximation. A family of regular configurations is found with periodic oscillations of all the settling particles. The oscillations are shown to be robust under some out-of-phase rearrangements of the particles. In the presence of an additional particle above such a regular configuration, the particle periodic trajectories are horizontally repelled from the symmetry axis, and flattened vertically. The results are used to propose a mechanism of how a spherical cloud, made of a large number of particles distributed at random, evolves and destabilizes.

Figure 1

Initial positions of 16 particles (dots) for $c=0.8$: side, top and 3D views. Solid lines at the middle panel: trajectories.

Figure 2

Trajectories of 16 particles initially located as shown in Fig. 1, with $c=0.8$ (side view).

Figure 3

Trajectories in the center-of-mass frame for $c=0.1$.

Figure 4

Trajectories in the center-of-mass frame, for $c=2.2,1.4,0.9,0.6,0.2$. The smaller $c$, the shorter the trajectory and the smaller its width, $(d_{\text{max}}-d_{\text{min}})/2$.

Figure 5

Trajectories in the center-of-mass frame, for small values of $c=0.1,0.05,0.047$. The smaller $c$, the longer and wider the trajectory. Note a different scale on each axis.

Figure 6

The maximal and minimal width of the group trajectories, $d_{\text{max}}$ and $d_{\text{min}}$, versus $c$, for $c\ge 0.04481$. Inset: for $c\ge 0.8$, the ratio $d_{\text{min}}/d_{\text{max}}$ (symbols) scales as $0.095/p+0.22$ (solid line).

Figure 7

The aspect ratio $p$ of the group trajectories as a function of $c$ (symbols). Straight lines: dashed red, $p=0.44c+0.09$, and solid blue, $p=c-c_0$.

Figure 8

The time-dependent interparticle distance between the twin particles, $r_{19}$, and between the closest particles from the polygon, $r_{12}$. Top: $c=0.9$. Bottom: $c=0.1$.

Figure 9

The period $T$ and the cluster width $d_{\text{max}}$ versus $c-c_0$ (symbols). Straight lines: $T=0.0013/{(c-c_0)}^{1.5}$ (solid) and $d_{\text{max}}=c_0/(c-c_0)$ (dashed).

Figure 10

Two types of the particle trajectories: periodic oscillations ($c=0.05,0.049,0.045$) and separation into pairs without oscillations ($c=0.04,0.03,0.02$).

Figure 11

In the center-of-mass frame, the particle trajectories with the perturbed initial positions, Eq. (10), (solid lines), and with the unperturbed ones, Eq. (5), (dashed lines), are superimposed for $c=0.9$ and differ from each other significantly for $c=0.1$.

Figure 12

The time-dependent distance between the slower twin particles, $r_{19}$, (solid line) and between the faster twin particles, $r_{2,10}$, (dashed line), for the initial positions given by Eq. (11) with $2N=16$ and $c=0.1$.

Figure 13

The quasiperiodic particle trajectory in the center-of-mass frame, obtained for the initial positions given by Eq. (11) with $2N=16$ and $c=0.9$, during time $0\le t\le 10$.

Figure 14

Evolution of 17 particles, initially at the positions (indicated by dots) described by Eq. (5) with $c=0.9$ plus ${\mathbf{r}}_{17}=(0,0,1.7)$, in the center-of-mass frame of the regular configuration made of 16 particles. Trajectories of the other 12 particles look the same in their planes of the motion.

Figure 15

Schematic explanation of the horizontal expansion and vertical flattening of the particle trajectories inside the group. Arrows (sketch): this part of the group-particle velocity, which comes from its interaction with the tail-particle.

Figure 16

Trajectories of centers of two twin spheres from the regular group. Dashed line: without a tail. Solid line: in the presence of a tail-sphere, initially centered at the distance 1.25 above the group center of mass. The sphere diameter $d=0.19$. The group center-of-mass frame is taken.