Sketch of rotating drum filled with viscous fluid and three beads. These beads are dragged upwards by the front wall of the drum until gravity pulls them away from the wall and they fall through the fluid. (Based on Ref. 30.)
Cascade speeds and plotted vs . The solid line represents , showing that is an approximate upper bound for the cascade speed.
The axes are defined such that represents the horizontal direction, is the vertical, and is the depth away from the front edge of the drum. Note that, with our current apparatus, there is no way to measure directly.
Typical examples of trajectories corresponding to the five regimes we observe. (a) Periodic trajectory with . (b) Stable doublet state with . (c) Stable triplet state with . (d) Biased chaotic state with . In such states, the center bead strongly interacts with only one of the outer beads. (e) Fully chaotic state with . For all figures, the varying thickness of the lines is a result of particle position uncertainty due to optical distortion from the curved drum walls. This distortion is exaggerated due to parallax near the end caps and minimized in the center of the drum.
Phase diagram showing the behaviors observed at various rotation rates. The symbols below the bar indicate individual observations.
In this experiment, at , the particle follows a seemingly disordered trajectory for , but then settles into a stable triplet state.
The approximate time taken for the initial transient behavior to die out is obtained by eye and plotted vs rotation rate. Symbols represent the long-term phase of each trajectory, after its transients have died out.
Kalliroscope images show the fluid behavior in various phases: (a) periodic , (b) doublet , and (c) chaotic .
Distributions of time between “collisions” where two or three beads come close together. These all correspond to fully disordered states with the drum rotation rates (a) rad/s, (b) rad/s, (c) rad/s.
(a) Disordered particle trajectories for an experiment with . Recall that we cannot distinguish the spheres from one another, so that by definition . (b) The same trajectories in reduced coordinates. The minima in these simplified trajectories indicate points where particle pairs approach one another.
A typical two dimensional histogram for a disordered trajectory at . The axes represent the distances between the pairs of particles, normalized by the particle diameter, and color represents the number of points that were counted in each bin. Darker shades indicate more prevalent configurations. This experiment has a calculated entropy .
Histograms for each phase of behavior clearly illustrate the amount of phase space they explore. (a) Schematic of typical areas occupied by the nondisordered trajectories. “Periodic” trajectories such as shown in Fig. 4(a) appear as a tight cluster of points near position 1. “Doublet” trajectories reside at one of the two locations marked as position 2. “Triplet” trajectories have all the particles at the same position, and thus the histogram for these states is a tight collection of points near the origin, marked as position 3. (b) A histogram for a biased disordered trajectory at . For this state, the entropy . [(c) and (d)] Histograms for disordered trajectories at , respectively. The entropies of these states are .
The entropy for each rotation rate, calculated from the 2D histograms. At some rotation rates ( and ), multiple states can be seen at the same or similar values of . This is due partially to motor stability and the presence of long transients (as discussed in Sec. III B) and partially because of the sensitivity to the exact value of . In some cases, different states are present for closely spaced values of .
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