Nonlinear Dynamics

, Volume 43, Issue 1–2, pp 117–126 | Cite as

Smart Baffle Placement for Chaotic Mixing

  • Michael J. Clifford
  • Stephen M. Cox


It is well known that fluid mixing can often be improved by the introduction of ‘baffles’ into the flow – the problem of baffle placement is examined here for chaotic fluid mixing of a highly viscous fluid. A simple model for a planetary mixer, with one stirring element, is modified by the introduction of one or more stationary baffles. Regular regions of poor mixing in the unbaffled flow are shown to be significantly reduced in size if the location of the baffles is chosen so that the flow necessarily generates ‘topological chaos’. By contrast, the positioning of baffles in superficially similar ways that do not generate such ‘topological chaos’ fails to provide a similar improvement.

Key Words

chaotic mixing topological chaos 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Lorenz, E. N., ‘Deterministic nonperiodic flow’, Journal of Atmospheric Sciences 20, 1963, 130–141.CrossRefADSGoogle Scholar
  2. 2.
    Ueda, Y., The Road to Chaos, Aerial Press, Santa Cruz, California, 1992.Google Scholar
  3. 3.
    Smith, L. A., ‘What might we learn from climate forecasts?’, Proceedings of the National Academy of Sciences 4, 2002, 2487–2492.Google Scholar
  4. 4.
    Thompson, J. M. T., Bishop S. R., and Leung L. M., ‘Fractal basins and chaotic bifurcations prior to escape from a potential well’, Physics Letters A 121, 1987, 116–120.CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Paulusa, M. P. and Braff, D. L., ‘Chaos and schizophrenia: Does the method fit the madness?’, Biological Psychiatry 53, 2003, 3–11.Google Scholar
  6. 6.
    Pogun, S., ‘Are attractors ‘strange’, or is life more complicated than the simple laws of physics?’, Biosystems 63, 2001, 101–114.CrossRefPubMedGoogle Scholar
  7. 7.
    Mackey, M. C. and Glass, L., ‘Oscillations and chaos in physiological control systems’, Science 197, 1977, 287–289.ADSPubMedGoogle Scholar
  8. 8.
    Holden, A. V., Winlow, W., and Haydon, P. G., ‘The induction of periodic and chaotic activity in a molluscan neurone’, Biological Cybernetics 43, 1982, 169–173.CrossRefPubMedMathSciNetGoogle Scholar
  9. 9.
    Calin, G. A., Vasilescub, C, Negrinia, M., and Barbanti–Brodanoc, G., ‘Genetic chaos and antichaos in human cancers’, Medical Hypotheses 60, 2003, 258–262.CrossRefPubMedGoogle Scholar
  10. 10.
    Wellsa, H., Straussb, E. G., Rutterc, M. A., and Wells, P. H., ‘Mate location, population growth and species extinction’, Biological Conservation 86, 1998, 317–324.Google Scholar
  11. 11.
    Keilis–Boroka, V., Ismail-Zadeh, A., Kossobokova, V., and Shebalina, P., ‘Non-linear dynamics of the lithosphere and intermediate-term earthquake prediction’, Tectonophysics 338, 2001, 247–260.Google Scholar
  12. 12.
    Tiwari, R. K., Srilakshmi S., and Rao, K. N. N., ‘Nature of earthquake dynamics in the central Himalayan region: A nonlinear forecasting analysis’, Journal of Geodynamics 35, 2003, 273–287.CrossRefGoogle Scholar
  13. 13.
    Tonona, F., Bernardinib, A., and Elishakoffc, I., ‘Concept of random sets as applied to the design of structures and analysis of expert opinions for aircraft crash’, Chaos, Solitons and Fractals 10, 1999, 1855–1868.Google Scholar
  14. 14.
    Rosser, J., From Catastrophe Theory to Chaos: A General Theory of Economic Discontinuities, Kluwer Academic Publishers, Norwell, MA, 1992.Google Scholar
  15. 15.
    Ionita, S., ‘A chaos theory perspective on system's failure’, Information Sciences 127, 2000, 193–215.CrossRefGoogle Scholar
  16. 16.
    Weisberg, H. F., ‘Nonlinear models of electoral change: The implications of political time and chaos theory for the study of mass political behaviour’, Electoral Studies 17, 1998, 369–382.CrossRefGoogle Scholar
  17. 17.
    Ott, E., Grebogi, C., and Yorke, J. A., ‘Controlling chaos’, Physical Review Letters 64, 1990, 1196–1199.ADSPubMedMathSciNetGoogle Scholar
  18. 18.
    Raoa, R. K. A. and Yeragani, V. K., ‘Decreased chaos and increased nonlinearity of heart rate time series in patients with panic disorder’, Autonomic Neuroscience 88, 2001, 99–108.Google Scholar
  19. 19.
    Yeragani, V. K., Rao, K. A. R. K., Smitha, M. R., Pohl, R. B., Balon, R., and Srinivasan, K., ‘Diminished chaos of heart rate time series in patients with major depression’, Biological Psychiatry 51, 2002, 733–744.CrossRefPubMedGoogle Scholar
  20. 20.
    Palacios, A. and Juarez, H., ‘Cryptography with cycling chaos’, Physics Letters A 303, 2002, 345–351.CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    van den Bleek, C. M., Coppensa, M.-O., and Schoutenb, J. C., ‘Application of chaos analysis to multiphase reactors’, Chemical Engineering Science 57, 2002, 4763–4778.Google Scholar
  22. 22.
    Clifford, M. J., Cox, S. M., and Roberts, E. P. L., ‘Lamellar modelling of reaction, diffusion and mixing in a two-dimensional flow’, Chemical Engineering Journal 71, 1998, 49–56.CrossRefGoogle Scholar
  23. 23.
    Ottino, J. M., The Kinematics of Mixing: Stretching, Chaos and Transport, Cambridge University Press, Cambridge, 1989.Google Scholar
  24. 24.
    Boyland, P. L., Aref, H., and Stremler, M. A., ‘Topological fluid mechanics of stirring’, Journal of Fluid Mechanics 403, 2000, 277–304.CrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Acheson, D. J., Elementary Fluid Dynamics, Clarendon Press, Oxford, 1996.Google Scholar
  26. 26.
    Milne-Thomson, L. M., Theoretical Hydrodynamics, Macmillan, New York, 1968.Google Scholar
  27. 27.
    Brown, M. G. and Samelson, R. M., ‘Particle motion in vorticity-conserving, two-dimensional incompressible flows’, Physics of Fluids 6, 1994, 2875–2876.CrossRefADSGoogle Scholar
  28. 28.
    Finn, M. D., Cox, S. M., and Byrne, H. M., ‘Mixing measures for a two-dimensional chaotic Stokes flow’, Journal of Engineering Mathematics 48, 2004, 129–155.CrossRefMathSciNetGoogle Scholar
  29. 29.
    Aref, H., ‘Stirring by chaotic advection’, Journal of Fluid Mechanics 143, 1984, 1–21.ADSMATHMathSciNetGoogle Scholar
  30. 30.
    Aref, H., ‘The development of chaotic advection’, Physics of Fluids 14, 2002, 1315–1325.ADSMathSciNetGoogle Scholar
  31. 31.
    Aref, H. and Balachandar, S., ‘Chaotic advection in a Stokes flow’, Physics of Fluids 29, 1986, 3515–3521.CrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Finn, M. D. and Cox, S. M., ‘Stokes flow in a mixer with changing geometry’, Journal of Engineering Mathematics 41, 2001, 75–99.CrossRefMathSciNetGoogle Scholar
  33. 33.
    Finn, M. D., Cox, S. M., and Byrne, H. M., ‘Topological chaos in inviscid and viscous mixers’, Journal of Fluid Mechanics 493, 2003, 345–361.CrossRefADSMathSciNetGoogle Scholar
  34. 34.
    Tufillaro, N. B., Abbot, T., and Reilly, J., An Experimental Approach to Nonlinear Dynamics and Chaos, Addison Wesley, Redwood City, CA, 1992.Google Scholar
  35. 35.
    Hobbs, D. M., Swanson, P. D., and Muzzio, F. J., ‘Numerical characterization of low Reynolds number flow in the Kenics static mixer’, Chemical Engineering Science 53, 1998, 1565–1584.Google Scholar
  36. 36.
    Rauline, D., Tanguy, P. A., Le Blévec, J.–M., and Bousquet, J., ‘Numerical investigation of the performance of several static mixers’, Canadian Journal of Chemical Engineers 76, 1998, 527–535.Google Scholar
  37. 37.
    Zalc, J. M., Szalai, E. S., Muzzio, F. J., and Jaffer, S., ‘Characterization of flow and mixing in an SMX static mixer’, Journal of American Institute of Chemical Engineers 48, 2002, 427–436.Google Scholar
  38. 38.
    Vikhansky, A., ‘On the applicability of topological chaos to mixer design’, Chemical Engineering Science 2003 (submitted).Google Scholar
  39. 39.
    Finn, M. D., Cox, S. M., and Byrne, H. M., ‘Chaotic advection in a braided pipe mixer’, Physics of Fluids 15, 2003, 77–80.CrossRefADSGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.School of Mechanical, Materials and Manufacturing EngineeringThe University of NottinghamNottinghamU.K.
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia

Personalised recommendations