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Chaos and threshold for irreversibility in sheared suspensions

Abstract

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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References

  1. Tolman, R. C. The Principles of Statistical Mechanics (Oxford Univ. Press, London, 1938)

    MATH  Google Scholar 

  2. Boffeta, G., Cencini, M., Falcioni, M. & Vulpiani, A. Predictability: a way to characterize complexity. Phys. Rep. 356, 367–474 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  3. Wang, G. M., Sevick, E. M., Mittag, E., Searles, D. J. & Evans, D. J. Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales. Phys. Rev. Lett. 89, 050601 (2002)

    Article  ADS  CAS  Google Scholar 

  4. Leal, L. G. Laminar Flow and Convective Transport Processes: Scaling Principles and Asymptotic Analysis (Butterworth-Heinemann, Boston, 1992)

    Google Scholar 

  5. Morrison, I. D. & Ross, S. Colloidal Dispersions: Suspensions, Emulsions, and Foams (Wiley-Interscience, Hoboken, 2003)

    Google Scholar 

  6. Lamb, J. S. W. & Roberts, J. A. G. Time-reversal symmetry in dynamical systems: a survey. Physica D 112, 1–39 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  7. Berg, H. C. Random Walks in Biology (Princeton Univ. Press, Princeton, New Jersey, 1993)

    Google Scholar 

  8. Taylor, G. I. & Friedman, J. Low Reynolds Number Flows (National Committee on Fluid Mechanics Films, Encyclopedia Britannica Educational Corp., United States, 1966); available as ‘G. I. Taylor and Kinematic Reversibility’ in Multi-Media Fluid Mechanics CD-ROM (eds Homsy, G. M.) (Cambridge Univ. Press, Cambridge, 2000).

  9. Aref, H. Chaotic advection of fluid particles. Phil. Trans. R. Soc. Lond. A 333, 273–288 (1990)

    Article  ADS  Google Scholar 

  10. Drazer, D., Koplik, J., Khusid, B. & Acrivos, A. Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions. J. Fluid Mech. 460, 307–335 (2002)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  11. Jánosi, I. M., Tél, T., Wolf, D. E. & Gallas, J. A. C. Chaotic particle dynamics in viscous flows: The three-particle Stokeslet problem. Phys. Rev. E 56, 2858–2866 (1997)

    Article  ADS  Google Scholar 

  12. Lighthill, J. The recently recognized failure of predictability in Newtonian dynamics. Proc. R. Soc. Lond. A 407, 35–48 (1986)

    Article  ADS  Google Scholar 

  13. Eckstein, E. C., Bailey, D. G. & Shapiro, A. H. Self-diffusion of particles in shear-flow of a suspension. J. Fluid Mech. 79, 191–208 (1977)

    Article  ADS  Google Scholar 

  14. Leighton, D. & Acrivos, A. Measurement of shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 177, 109–131 (1987)

    Article  ADS  CAS  Google Scholar 

  15. Breedveld, V., van den Ende, D., Tripathi, A. & Acrivos, A. The measurement of the shear-induced particle and fluid tracer diffusivities by a novel method. J. Fluid Mech. 375, 297–318 (1998)

    Article  ADS  CAS  Google Scholar 

  16. Breedveld, V., van den Ende, D. & Jongschaap, R. Shear-induced diffusion and rheology of noncolloidal suspensions: Time scales and particle displacements. J. Chem. Phys. 114, 5923–5936 (2001)

    Article  ADS  CAS  Google Scholar 

  17. Zarraga, I. E. & Leighton, D. T. Measurement of an unexpectedly large shear-induced self-diffusivity in a dilute suspension of spheres. Phys. Fluids 14, 2194–2201 (2002)

    Article  ADS  CAS  Google Scholar 

  18. Marchioro, M. & Acrivos, A. Shear-induced particle diffusivities from numerical simulations. J. Fluid Mech. 443, 101–128 (2001)

    Article  ADS  Google Scholar 

  19. Sierou, A. & Brady, J. F. Shear-induced self-diffusion in non-colloidal suspensions. J. Fluid Mech. 506, 285–314 (2004)

    Article  ADS  Google Scholar 

  20. Segrè, P. N., Herbolzheimer, E. & Chaikin, P. M. Long-range correlations in sedimentation. Phys. Rev. Lett. 79, 2574–2577 (1997)

    Article  ADS  Google Scholar 

  21. Segrè, P. N., Liu, F., Umbanhowar, P. & Weitz, D. A. An effective gravitational temperature for sedimentation. Nature 409, 594–597 (2001)

    Article  ADS  Google Scholar 

  22. Voth, G. A. et al. Ordered clusters and dynamical states of particles in a vibrated fluid. Phys. Rev. Lett. 88, 234301 (2002)

    Article  ADS  Google Scholar 

  23. Krishnan, G. P., Beimfohr, S. & Leighton, D. T. Shear-induced radial segregation in bidisperse suspensions. J. Fluid Mech. 321, 371–393 (1996)

    Article  ADS  Google Scholar 

  24. Crocker, J. C. & Grier, D. G. Methods of digital video microscopy for colloidal studies. J. Colloid Interf. Sci. 179, 298–310 (1996)

    Article  ADS  CAS  Google Scholar 

  25. Taylor, G. I. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186–203 (1953)

    Article  ADS  CAS  Google Scholar 

  26. Sussman, G. J. & Wisdom, J. Chaotic evolution of the solar system. Science 257, 56–62 (1992)

    Article  ADS  MathSciNet  CAS  Google Scholar 

Download references

Acknowledgements

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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Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare no competing financial interests.

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). https://doi.org/10.1038/nature04380

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