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Chaos and threshold for irreversibility in sheared suspensions


Systems governed by time reversible equations of motion often give rise to irreversible behaviour1,2,3. The transition from reversible to irreversible behaviour is fundamental to statistical physics, but has not been observed experimentally in many-body systems. The flow of a newtonian fluid at low Reynolds number can be reversible: for example, if the fluid between concentric cylinders is sheared by boundary motion that is subsequently reversed, then all fluid elements return to their starting positions4. Similarly, slowly sheared suspensions of solid particles, which occur widely in nature and science5, are governed by time reversible equations of motion. Here we report an experiment showing precisely how time reversibility6 fails for slowly sheared suspensions. We find that there is a concentration dependent threshold for the deformation or strain beyond which particles do not return to their starting configurations after one or more cycles. Instead, their displacements follow the statistics of an anisotropic random walk7. By comparing the experimental results with numerical simulations, we demonstrate that the threshold strain is associated with a pronounced growth in the Lyapunov exponent (a measure of the strength of chaotic particle interactions). The comparison illuminates the connections between chaos, reversibility and predictability.

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Figure 1: Particle displacements and trajectories.
Figure 2: Experimental diffusivities.
Figure 3: Threshold strain amplitudes for the onset of irreversibility as a function of volume fraction.
Figure 4: Diffusivities and Lyapunov exponents.


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We appreciate discussions with L. G. Leal and G. Homsy. K. Knipmeyer and E. Knowlton provided assistance with data acquisition and reduction. 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. Author Contributions D.J.P and J.P.G. were responsible for the experiments; J.F.B. and A.M.L. did the numerical simulations.

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Correspondence to D. J. Pine or J. F. Brady.

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Supplementary information

Supplementary Video 1

This is a continuously sampled movie clip showing the motion of tracer particles in an oscillatory shear flow. (MOV 3413 kb)

Supplementary Video 2

This movie is a periodically sampled clip showing the reversible motion of tracer particles in an oscillatory shear flow. The shear strain amplitude is 1.0, which is just below the irreversibility threshold. (MOV 5151 kb)

Supplementary Video 3

This movie is a periodically sampled clip showing the irreversible motion of tracer particles in an oscillatory shear flow with a shear strain amplitude of 2.5, which is above the irreversibility threshold. (MOV 4751 kb)

Supplementary Video Legends

This provides information about the experimental conditions under which Supplementary Videos 1, 2, and 3 were obtained. (PDF 26 kb)

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Pine, D., Gollub, J., Brady, J. et al. Chaos and threshold for irreversibility in sheared suspensions. Nature 438, 997–1000 (2005).

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