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Granular Matter

, 21:95 | Cite as

Push and pull: attractors and repellors of a dynamical system can localize inertial particles

  • Guy MetcalfeEmail author
Original Paper
  • 7 Downloads
Part of the following topical collections:
  1. In Memoriam of Robert P. Behringer, late Editor in Chief of Granular Matter

Abstract

Recent experiments in a laminar flow stirred tank found that particles, initially moving throughout the fluid domain, undergo an instability and cluster into subdomains of the fluid when the flow Reynolds number exceeds a critical value that depends on particle and fluid inertia. As the dynamical system for inertial particles in fluid flow has both attracting and repelling regions, I show how the interplay of these regions can localize particles and derive an expression for the instability boundary that is suitable for comparison to data. Moreover, I model a master curve for the particle clustering rate in terms of excess inertia past the instability.

Keywords

Localization Inertia Separation Chaotic advection 

Notes

Acknowledgements

GM had eight happy years in undergraduate and graduate research under Bob Behringer’s guidance and is tremendously grateful for the experience. This updated paper was originally to appear in the 2014 special issue for BobFest, but GM’s employer withdrew it, to file patents. A year later the employer killed off fluids research. GM had worked 20 years with the late CSIRO manufacturing fluids group.

Compliance with ethical standards

Conflict of interest

The author declare that they have no conflict of interest

References

  1. 1.
    Aref, H.: The development of chaotic advection. Phys. Fluids 14(4), 1315 (2002)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Sturman, R., Ottino, J.M., Wiggins, S.: The Mathematical Foundations of Mixing. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  3. 3.
    Metcalfe, G.: Applied fluid chaos: designing advection with periodically reoriented flows for micro to geophysical mixing and transport enhancement. In: Dewar, R.L., Detering, F. (eds.) Complex Physical, Biophysical and Econophysical Systems, pp. 187–239. World Scientific, Singapore (2010)CrossRefGoogle Scholar
  4. 4.
    Agricola, G.: De Re Metallica. Dover Publications, vol. 1556. Book 8, image 22 (1950)Google Scholar
  5. 5.
    Fountain, G.O., Khakhar, D.V., Mezić, I., Ottino, J.M.: Chaotic mixing in a bounded three-dimensional flow. J. Fluid Mech. 417, 265 (2000)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Takahashi, K., Motoda, M.: Chaotic mixing created by object inserted in a vessel agitated by an impeller. Chem. Eng. Res. Des. 87, 386 (2009)CrossRefGoogle Scholar
  7. 7.
    Wang, S., Wu, J., Bong, E.: Reduced IMRs in a mixing tank via agitation improvement. Chem. Eng. Res. Des. 91(6), 1009 (2013)CrossRefGoogle Scholar
  8. 8.
    Maxey, M.R., Riley, J.J.: Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883 (1983)ADSCrossRefGoogle Scholar
  9. 9.
    Metcalfe, G., Speetjens, M.F.M., Lester, D.R., Clercx, H.J.H.: Beyond passive: chaotic transport in stirred fluids. Adv. Appl. Mech. 45, 109 (2012).  https://doi.org/10.1016/B978-0-12-380876-9.00004-5 CrossRefGoogle Scholar
  10. 10.
    Wilkinson, M., Mehlig, B., Östlund, S., Duncan, K.P.: Unmixing in random flows. Phys. Fluids 19(11), 113303 (2007).  https://doi.org/10.1063/1.2766740 ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Bec, J., Chétrite, R.: Toward a phenomenological approach to the clustering of heavy particles in turbulent flows. New J. Phys. 9(3), 77 (2007).  https://doi.org/10.1088/1367-2630/9/3/077 ADSCrossRefGoogle Scholar
  12. 12.
    Gualtieri, P., Picano, F., Casciola, C.M.: Anisotropic clustering of inertial particles in homogeneous shear flow. J. Fluid Mech. 629, 25 (2009).  https://doi.org/10.1017/S002211200900648X ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Wang, J.K.S.: Mixing of fluids and suspensions in a laminar stirred tank. Ph.D. thesis, Monash University (2014).  https://doi.org/10.4225/03/58a5351e77252. Advised by Jie Wu, David Boger, and Guy Metcalfe
  14. 14.
    Wang, S., Metcalfe, G., Stewart, R.L., Wu, J., Ohmura, N., Feng, X., Yang, C.: Solid–liquid separation by particle-flow-instability. Energy Environ. Sci. 7, 3982 (2014).  https://doi.org/10.1039/C4EE02841D CrossRefGoogle Scholar
  15. 15.
    Wang, S., Stewart, R., Metcalfe, G.: Visualization of the trapping of inertial particles in a laminar mixing tank. Chem. Eng. Sci. 143, 99 (2016).  https://doi.org/10.1016/j.ces.2015.12.023 CrossRefGoogle Scholar
  16. 16.
    Stewart, R.L., Šutalo, I.D., Wong, C.Y.: Three-dimensional tracking of sensor capsules mobilised by fluid flow. Meas. Sci. Technol. 26(3), 035302 (2015).  https://doi.org/10.1088/0957-0233/26/3/035302 ADSCrossRefGoogle Scholar
  17. 17.
    Stewart, R.L., Šutalo, I.D., Liovic, P.: Perceptive communicating capsules for fluid flow measurement and visualisation. Robotica 35(1), 143 (2017).  https://doi.org/10.1017/S0263574715000041 CrossRefGoogle Scholar
  18. 18.
    Rudman, M.: Mixing and particle dispersion in the wavy vortex regime of Taylor–Couette Flow. AIChE J. 44(5), 1015 (1998)CrossRefGoogle Scholar
  19. 19.
    Wereley, S.T., Lueptow, R.M.: Inertial particle motion in a Taylor Couette rotating filter. Phys. Fluids 11(2), 325 (1999)ADSCrossRefGoogle Scholar
  20. 20.
    Abatan, A.A., McCarthy, J.J., Vargas, W.L.: Particle migration in the rotating flow between co-axial disks. AIChE J. 52(6), 2039 (2006)CrossRefGoogle Scholar
  21. 21.
    Coussot, P., Ancey, C.: Rheophysical classification of concentrated suspensions and granular pastes. Phys. Rev. E 59(4), 4445 (1999)ADSCrossRefGoogle Scholar
  22. 22.
    Ahmed, S., John, S., Šutalo, I., Metcalfe, G., Liffman, K.: An experimental study of density segregation at end walls in a horizontal rotating cylinder filled with viscous fluid. Granul. Matter 14, 319 (2012).  https://doi.org/10.1007/s10035-012-0335-2 CrossRefGoogle Scholar
  23. 23.
    Babiano, A., Cartwright, J., Piro, O., Provenzale, A.: Dynamics of a small neutrally buoyant sphere in a fluid and targeting in Hamiltonian systems. Phys. Rev. Lett. 84(25), 5764 (2000)ADSCrossRefGoogle Scholar
  24. 24.
    Cartwright, J., Magnasco, M., Piro, O.: Bailout embeddings, targeting of invariant tori, and the control of Hamiltonian chaos. Phys. Rev. E 65, 045203(R) (2002)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Sapsis, T., Haller, G.: Instabilities in the dynamics of neutrally buoyant particles. Phys. Fluids 20, 017102 (2008)ADSCrossRefGoogle Scholar
  26. 26.
    Haller, G., Sapsis, T.: Where do inertial particles go in fluid flows? Physica D 237, 573 (2008)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Sapsis, T., Haller, G.: Clustering criterion for inertial particles in two-dimensional time-periodic and three-dimensional steady flows. Chaos 20, 017515 (2010)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Chong, M.S., Perry, A.E., Cantwell, B.J.: A general classification of three-dimensional flow fields. Phys. Fluids 2(5), 765 (1990)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Speetjens, M., Rudman, M., Metcalfe, G.: Flow regime analysis of non-Newtonian duct flows. Phys. Fluids 18, 013101 (2006)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Paul, E.L., Atiemo-Obeng, V., Kresta, S.M. (eds.): Handbook of Industrial Mixing. Wiley, Hoboken (2003)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Science, Engineering and TechnologySwinburne University of TechnologyHawthornAustralia

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