Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

Uniform resonant chaotic mixing in fluid flows

Nature volume 425, pages 376–380 (2003)Cite this article

Abstract

Laminar flows can produce particle trajectories that are chaotic1,2, with nearby tracers separating exponentially in time. For time-periodic, two-dimensional flows and steady three-dimensional (3D) flows, enhancements in mixing due to chaotic advection are typically limited by impenetrable transport barriers that form at the boundaries between ordered and chaotic mixing regions. However, for time-dependent 3D flows, it has been proposed theoretically3,4,5 that completely uniform mixing is possible through a resonant mechanism5 called singularity-induced diffusion; this is thought to be the case even if the time-dependent and 3D perturbations are infinitesimally small. It is important to establish the conditions for which uniform mixing is possible and whether or not those conditions are met in flows that typically occur in nature. Here we report experimental and numerical studies of mixing in a laminar vortex flow that is weakly 3D and weakly time-periodic. The system is an oscillating horizontal vortex chain (produced by a magnetohydrodynamic technique) with a weak vertical secondary flow that is forced spontaneously by Ekman pumping—a mechanism common in vortical flows with rigid boundaries, occurring in many geophysical, industrial and biophysical flows. We observe completely uniform mixing, as predicted3,4,5 by singularity-induced diffusion, but only for oscillation periods close to typical circulation times.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Sketch of a portion of the flow.
Figure 2: Numerical simulations of equations (1)–(3).
Figure 3: Experimental images of mixing for time-independent flow (b = 0).
Figure 4: Experimental images of mixing for time-periodic flow forced near resonance (b = 0.02, 0.10 Hz oscillation frequency, corresponding to a non-dimensional angular frequency ω = 2.5).
Figure 5: Experimental images of mixing for time-periodic flow forced away from a resonant band (b = 0.02, 0.160 Hz oscillation frequency, corresponding to a non-dimensional angular frequency ω = 4.2).

Similar content being viewed by others

References

  1. Aref, H. Stirring by chaotic advection. J. Fluid Mech. 143, 1–21 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  2. Aref, H. Fluid dynamics: Order in chaos. Nature 401, 756–758 (1999)

    Article  ADS  CAS  Google Scholar 

  3. Feingold, M., Kadanoff, L. P. & Piro, O. Passive scalars, 3-dimensional volume-preserving maps and chaos. J. Stat. Phys. 50, 529–565 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  4. Cartwright, J. H. E., Feingold, M. & Piro, O. Chaotic advection in three-dimensional unsteady incompressible laminar flow. J. Fluid Mech. 316, 259–284 (1996)

    Article  ADS  CAS  Google Scholar 

  5. Mezić, I. Break-up of invariant surfaces in action-angle-angle maps and flows. Physica D 154, 51–67 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  6. Rom-Kedar, V., Leonard, A. & Wiggins, S. An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394 (1990)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  7. Behringer, R. P., Meyers, S. D. & Swinney, H. L. Chaos and mixing in a geostrophic flow. Phys. Fluids A 3, 1243–1249 (1991)

    Article  ADS  Google Scholar 

  8. Pierrehumbert, R. T. Large scale horizontal mixing in planetary atmospheres. Phys. Fluids A 3, 1250–1260 (1991)

    Article  ADS  Google Scholar 

  9. Stroock, A. D., Dertinger, S. K. W., Mezić, I., Stone, H. A. & Whitesides, G. M. Chaotic mixer for microchannels. Science 295, 647–651 (2002)

    Article  ADS  CAS  Google Scholar 

  10. Karolyi, G., Pentek, A., Scheuring, I., Tel, T. & Toroczkai, Z. Chaotic flow: The physics of species coexistence. Proc. Natl Acad. Sci. USA 25, 13661–13665 (2000)

    Article  ADS  Google Scholar 

  11. Swanson, P. D., Muzzio, F. J., Annapragada, A. & Adjei, A. Numerical analysis of motion and deposition of particles in cascade impactors. Int. J. Pharm. 142, 33–51 (1996)

    Article  CAS  Google Scholar 

  12. Arnold, V. I. Sur la geometrie differentielle des groupes de lie de dimension infinie et ses applications I I'hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16, 316–361 (1966)

    Article  Google Scholar 

  13. Hénon, M. Sur la topologie des lignes de courant dans un cas particulier. C. R. Acad. Sci. Paris A 262, 312–314 (1966)

    Google Scholar 

  14. Bajer, K. & Moffatt, H. K. On a class of steady confined Stokes flows with chaotic streamlines. J. Fluid Mech. 212, 337–363 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  15. Stone, H. A., Nadim, A. & Strogatz, S. H. Chaotic streaklines inside drops immersed in steady linear flows. J. Fluid Mech. 232, 629–646 (1991)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  16. Holm, D. D. & Kimura, Y. Zero-helicity Lagrangian kinematics of three dimensional advection. Phys. Fluids A 3, 1033–1038 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  17. Mackay, R. S. Transport in 3-D volume-preserving flows. J. Nonlinear Sci. 4, 329–354 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  18. Mezić, I. & Wiggins, S. On the integrability and perturbation of three dimensional fluid flows with symmetry. J. Nonlinear Sci. 4, 157–194 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  19. Ashwin, P. & King, G. P. Streamline topology in eccentric Taylor vortex flow. J. Fluid Mech. 285, 215–247 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  20. Fountain, G. O., Khakhar, D. V., Mezić, I. & Ottino, J. M. Chaotic mixing in a bounded 3D flow. J. Fluid Mech. 417, 265–301 (2000)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  21. Sotiropoulos, F., Ventikos, Y. & Lackey, T. C. Chaotic advection in 3-D stationary vortex-breakdown bubbles: Šil'nikov's chaos and the devil's staircase. J. Fluid Mech. 444, 257–297 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  22. Fountain, G. O., Khakhar, D. V. & Ottino, J. M. Visualization of three-dimensional chaos. Science 281, 683–686 (1998)

    Article  ADS  CAS  Google Scholar 

  23. Shinbrot, T., Alvarez, M. M., Zalc, J. M. & Muzzio, F. J. Attraction of minute particles to invariant regions of volume preserving flows by transients. Phys. Rev. Lett. 86, 1207–1210 (2001)

    Article  ADS  CAS  Google Scholar 

  24. Vainshtein, D. L., Vasiliev, A. A. & Neishtadt, A. J. Changes in the adiabatic invariant and streamline chaos in confined incompressible Strokes flow. Chaos 6, 67–77 (1996)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  25. Clever, R. M. & Busse, F. H. Transition to time-dependent convection. J. Fluid Mech. 65, 625–645 (1974)

    Article  ADS  Google Scholar 

  26. Solomon, T. H. & Gollub, J. P. Chaotic particle transport in time-dependent Rayleigh-Bénard convection. Phys. Rev. A. 38, 6280–6286 (1988)

    Article  ADS  CAS  Google Scholar 

  27. Chandresekhar, S. Hydrodynamic and Hydromagnetic Stability 39 (Clarendon, Oxford, 1961)

    Google Scholar 

  28. Solomon, T. H., Tomas, S. & Warner, J. L. The role of lobes in chaotic mixing of miscible and immiscible impurities. Phys. Rev. Lett. 77, 2682–2685 (1996)

    Article  ADS  CAS  Google Scholar 

  29. Solomon, T. H., Tomas, S. & Warner, J. L. Chaotic mixing of immiscible and immiscible impurities in a two-dimensional flow. Phys. Fluids 10, 342–350 (1998)

    Article  ADS  CAS  Google Scholar 

  30. Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems (Cambridge Univ. Press, Cambridge, 1984)

    Google Scholar 

Download references

Acknowledgements

This work was supported by the US National Science Foundation, AFOSR, and by a Sloan Fellowship grant.

Author information

Authors and Affiliations

  1. Department of Physics, Bucknell University, Lewisburg, Pennsylvania, 17837, USA

    T. H. Solomon

  2. Department of Mechanical and Environmental Engineering, University of California at Santa Barbara, Santa Barbara, California, 93106, USA

    Igor Mezić

Authors
  1. T. H. Solomon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Igor Mezić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to T. H. Solomon or Igor Mezić.

Ethics declarations

Competing interests

The authors declare that they have no competing financial interests.

Supplementary information

Rights and permissions

Reprints and permissions

About this article

Cite this article

Solomon, T., Mezić, I. Uniform resonant chaotic mixing in fluid flows. Nature 425, 376–380 (2003). https://doi.org/10.1038/nature01993

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature01993

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing