Hydrodynamic reversibility
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It is well known that low Reynolds number flows are reversible. This is dramatically demonstrated in a film clip (Fig. 1) made by G.I. Taylor in which a colored droplet is introduced into a viscous liquid contained between two transparent concentric cylinders (an arrangement known as a Couette cell). When the inner cylinder is rotated through several revolutions, the colored droplet is sheared with the rest of the liquid and is stretched into a barely visible pink ribbon winding around the Couette cell (see Fig. 1 film clip). When the direction of the inner cylinder is reversed, the thin pink ribbon of fluid reforms the original spherical droplet, dramatically illustrating the reversibility of the flow.
In a series of experiments, we investigate reversibility in non-Brownian particle suspensions. We employ a Couette cell filled with a viscous liquid similar to that used for the film clip in Fig. 1. However, we make one important change: we load the liquid with small (0.22 millimeter) spheres. The spheres are density matched to the liquid, to avoid settling, and index matched, so that they are invisible. We adjust the concentration of the particles so that they occupy about 30% of the total (liquid + particle) volume.
In order to follow the motion of the suspended particles, a very small fraction of the particles are dyed black (shown as black dots in Fig. 2), rendering them visible while the vast majority of particles remain invisible.
To test whether the flow is reversible, we rotate the inner cylinder of the Couette cell a small distance, and then reverse direction and rotate it back to its original position. We then continue rotating the inner cylinder back and forth in an oscillatory fashion, as illustrated by the animation in Fig. 2. Note how the particles nearer the oscillating inner cylinder move farther than those near the stationary outer cylinder.
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Figure 3 shows a film clip of our actual Couette cell viewed from the side looking through the outer (transparent) cylinder. The black circles are the dyed particles, which represent only a small fraction of the total number of particles. The particles (both visible and invisible) occupy 30% of the volume of this sample. The faster moving particles in the clip are those nearer the inner cylinder of the Couette cell (and farthest from the camera).
In order to see if the flow is really reversible, we strobe the camera so that it takes only one picture per cycle. We take a picture at the same point in each cycle and then string the pictures together in a movie. If the particle motion is reversible, each particle should return to the same position every cycle – the movie, made from periodically sampled picture frames, should show particles that do not move. This is exactly what we observe in Fig. 4. In this case, the travel of the inner cylinder back and forth is equal to the gap between the inner and outer cylinders (corresponding to a strain amplitude of 1.0). Thus, we see that the system is reversible for a strain amplitude on 1.0.
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The movie clip in Fig. 5 shows what happens when the travel of the inner cylinder back and forth is 2.5 times the gap between the inner and outer cylinders (a strain amplitude of 2.5). As can be seen from the film clip, the particles do not return to the same positions they occupied one cycle earlier. In this case, the particle movement (and fluid motion) is irreversible, contrary to the equations governing low Reynolds number flow.
The random motion of the particles in the movie clip in Figure 5 is reminiscent of Brownian motion, even though it arises by shearing the suspension rather than from random thermal motion. In fact, by tracing the motion of the dyed particles, we find that the mean square particle displacement increases linearly in time, just as it does for a diffusive process. In this case, the diffusivities are different in different directions (i.e. the diffusion is anisotropic). Figure 6 shows the means square displacements as a function of the total accumulated strain, which is the non-dimensional time.
What is particularly striking about these experiments is the rapid onset of hydrodynamic irreversibility. For strain amplitudes less than 1.0, the system is reversible, while for large strain amplitudes, the system is highly irreversible, with random irreversible particle motions of more than a particle radius per cycle for strain amplitudes of 2.0 or more. The rapid onset of irreversible behavior is evident in the plot of the diffusion coefficients vs. strain amplitude shown in Figure 7, which shows that the diffusivities are finite above a strain amplitude of 1 but negligible below 1. Numerical work shows that the origin of the rapid onset of irreversibility in the suspensions is a large increase in the strength of chaotic many-particle hydrodynamic interactions. That is, the trajectories of the particles are sensitive to even the smallest perturbations when many particles interact. Thus, when the concentration of particles is reduced below 30% volume fraction, there is rapid increase in the strain amplitude required for irreversible motion, consistent with the requirement of many-particle interactions for irreversible behavior.
These experiments were done in a collaboration with Jerry Gollub, Professor of Physics at Haverford College. Numerical work to help understand the onset of irreversibility was done by John Brady, Professor of Chemical Engineering at Caltech in Pasadena, California, and by Alex Leshansky, Professor of Chemical Engineering at the Technion Israel Institute of Technology in Haifa, Israel.
You can obtain more information by contacting Professor David Pine at the address below or by contacting any of his collaborators listed above.
You can also read a more complete account of our work published in the journal Nature 38, 997-1000 (2005).
The work described on this page was featured in the Science Times section of the 20 December 2005 issue of the New York Times. [LINK]
This work was supported by the Keck Foundation (D.J.P.), the National Science Foundation (J.P.G.) and the US-Israel Binational Science Foundation (A.M.L.). The work was initiated during a granular physics workshop hosted by the Kavli Institute for Theoretical Physics at UCSB.
