Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T13:39:50.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Formation of coherent structures by fluid inertia in three-dimensional laminar flows

Published online by Cambridge University Press:  17 June 2010

Z. POURANSARI ,
M. F. M. SPEETJENS  and
H. J. H. CLERCX
Z. POURANSARI
Affiliation:
Fluid Dynamics Laboratory, Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
M. F. M. SPEETJENS
Affiliation:
Energy Technology Laboratory, Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
H. J. H. CLERCX*
Affiliation:
Fluid Dynamics Laboratory, Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
*
‡Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Mixing under laminar flow conditions is key to a wide variety of industrial fluid systems of size extending from micrometres to metres. Profound insight into three-dimensional laminar mixing mechanisms is essential for better understanding of the behaviour of such systems and is in fact imperative for further advancement of (in particular, microscopic) mixing technology. This insight remains limited to date, however. The present study concentrates on a fundamental transport phenomenon relevant to laminar mixing: the formation and interaction of coherent structures in the web of three-dimensional paths of passive tracers due to fluid inertia. Such coherent structures geometrically determine the transport properties of the flow and thus their formation and topological structure are essential to three-dimensional mixing phenomena. The formation of coherent structures, its universal character and its impact upon three-dimensional transport properties is demonstrated by way of experimentally realizable time-periodic model flows. Key result is that fluid inertia induces partial disintegration of coherent structures of the non-inertial limit into chaotic regions and merger of surviving parts into intricate three-dimensional structures. This response to inertial perturbations, though exhibiting great diversity, follows a universal scenario and is therefore believed to reflect an essentially three-dimensional route to chaos. Furthermore, a first outlook towards experimental validation and investigation of the observed dynamics is made.

JFM classification

Mixing: Chaotic advection Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
Papers
Information
Journal of Fluid Mechanics , Volume 654 , 10 July 2010 , pp. 5 - 34
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Present address: Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden

References

REFERENCES

Aref, H. 2002 The development of chaotic advection. Phys. Fluids 14, 1315–1325.CrossRefGoogle Scholar
Arnold, V. I. 1978 Mathematical Methods of Classical Mechanics. Springer.CrossRefGoogle Scholar
Bajer, K. 1994 Hamiltonian formulation of the equations of streamlines in three-dimensional steady flows. Chaos Solitons Fractals 4, 895–911.CrossRefGoogle Scholar
Beebe, D. J., Mensing, G. A. & Walker, G. M. 2002 Physics and applications of microfluidics in biology. Annu. Rev. Biomed. Engng 4, 261.CrossRefGoogle Scholar
Bertsch, A., Heimgartner, S., Cousseau, P. & Renaud, P. 2001 Static micromixers based on large-scale industrial mixer geometry. Lab Chip 1, 56.CrossRefGoogle ScholarPubMed
Branicki, M. & Wiggins, S. 2009 An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems. Physica D 238, 1625.CrossRefGoogle Scholar
Broer, H. W., Huitema, G. B. & Sevryuk, M. B. 1996 Quasi-periodic Motions in Families of Dynamical Systems: Order Amidst Chaos. Springer.Google Scholar
Cartwright, J. H. E., Feingold, M. & Piro, O. 1996 Chaotic advection in three-dimensional unsteady incompressible laminar flow. J. Fluid Mech. 316, 259–284.CrossRefGoogle Scholar
Cheng, C. Q. & Sun, Y. S. 1990 Existence of invariant tori in three-dimensional measure preserving mappings. Celest. Mech. 47, 275–292.CrossRefGoogle Scholar
Fountain, G. O., Khakhar, D. V., Mezić, I. & Ottino, J. M. 2000 Chaotic mixing in a bounded three-dimensional flow. J. Fluid Mech. 417, 265–301.CrossRefGoogle Scholar
Fountain, G. O., Khakhar, D. V. & Ottino, J. M. 1998 Chaotic mixing in a bounded three-dimensional flow. Science 281, 683–686.CrossRefGoogle Scholar
Franjione, J. G., Leong, C.-W. & Ottino, J. M. 1989 Symmetries within chaos: a route to effective mixing. Phys. Fluids A 11, 1772–1783.CrossRefGoogle Scholar
Gómez, A. & Meiss, J. D. 2002 Volume-preserving maps with an invariant. Chaos 12, 289.CrossRefGoogle ScholarPubMed
Jaluria, Y. 2003 Thermal processing of materials: from basic research to engineering. ASME J. Heat Transfer 125, 957.CrossRefGoogle Scholar
Litvak-Hinenzon, A. & Rom-Kedar, V. 2002 Parabolic resonances in 3 degree of freedom near-integrable Hamiltonian systems. Physica D 164, 213.CrossRefGoogle Scholar
Luethi, B., Tsinober, A. & Kinzelbach, W. 2005 Lagrangian measurement of vorticity dynamics in turbulent flow. J. Fluid Mech. 528, 87.CrossRefGoogle Scholar
MacKay, R. S. 1994 Transport in 3d volume-preserving flows. J. Nonlinear Sci. 4, 329–354.CrossRefGoogle Scholar
Malyuga, V. S., Meleshko, V. V., Speetjens, M. F. M., Clercx, H. J. H. & van Heijst, G. J. F. 2002 Mixing in the stokes flow in a cylindrical container. Proc. R. Soc. Lond. A 458, 1867–1885.CrossRefGoogle Scholar
Meagher, R. J., Hatch, A. V., F.Renzi, R. & Singh, A. K. 2008 An integrated microfluidic platform for sensitive and rapid detection of biological toxins. Lab Chip 8, 2046.CrossRefGoogle ScholarPubMed
Meier, S. W., Lueptow, R. M. & Ottino, J. M. 2007 A dynamical systems approach to mixing and segregation of granular materials in tumblers. Adv. Phys. 56, 757–827.CrossRefGoogle Scholar
Meleshko, V. V. & Peters, G. W. M. 1996 Periodic points for two-dimensional stokes flow in a rectangular cavity. Phys. Lett. A 216, 87–96.CrossRefGoogle Scholar
Mezić, I. 2001 Break-up of invariant surfaces in action-angle-angle maps and flows. Physica D 154, 51.CrossRefGoogle Scholar
Mezić, I. & Wiggins, S. 1994 On the integrability and perturbation of three-dimensional fluid flows with symmetry. J. Nonlinear Sci. 4, 157–194.CrossRefGoogle Scholar
Mullowney, P., Julien, K. & Meiss, J. D. 2008 Chaotic advection and the emergence of tori in the Küppers–Lortz state. Chaos 18, 033104.CrossRefGoogle ScholarPubMed
Nguyen, N.-T. & Wu, Z. 2005 Micro-mixers – a review. J. Micromech. Microengng 15, R1.CrossRefGoogle Scholar
Ott, E. 1993 Chaos in Dynamical Systems. Cambridge University Press.Google Scholar
Ottino, J. M. 1989 The Kinematic of Mixing: Stretching, Chaos and Transport. Cambridge University Press.Google Scholar
Ottino, J. M., Jana, S. C. & Chakravarthy, V. S. 1994 From Reynolds's stretching and folding to mixing studies using horseshoe maps. Phys. Fluids 6, 685–699.CrossRefGoogle Scholar
Ottino, J. M. & Wiggins, S. 2004 Introduction: mixing in microfluidics. Phil. Trans. R. Soc. Lond. A 362, 923.CrossRefGoogle ScholarPubMed
Shankar, P. N. 1997 Three-dimensional eddy structure in a cylindrical container. J. Fluid Mech. 342, 97–118.CrossRefGoogle Scholar
Speetjens, M. F. M. 2001 Three-Dimensional Chaotic Advection in a Cylindrical Domain. PhD thesis, Eindhoven University of Technology.Google Scholar
Speetjens, M. F. M. & Clercx, H. J. H. 2005 A spectral solver for the Navier–Stokes equations in the velocity-vorticity formulation. Intl J. Comput. Fluid Dyn. 19, 191–209.CrossRefGoogle Scholar
Speetjens, M. F. M., Clercx, H. J. H. & van Heijst, G. J. F. 2004 A numerical and experimental study on advection in three-dimensional stokes flows. J. Fluid Mech. 514, 77–105.CrossRefGoogle Scholar
Speetjens, M. F. M., Clercx, H. J. H. & van Heijst, G. J. F. 2006 a Inertia-induced coherent structures in a time-periodic viscous mixing flow. Phys. Fluids 18, 083603.CrossRefGoogle Scholar
Speetjens, M. F. M., Clercx, H. J. H. & vanHeijst, G. J. F. Heijst, G. J. F. 2006 b Merger of coherent structures in time-periodic viscous flows. Chaos 16, 0431104.CrossRefGoogle ScholarPubMed
Speetjens, M., Metcalfe, G. & Rudman, M. 2006 c Topological mixing study of non-Newtonian duct flows. Phys. Fluids 18, 103103.CrossRefGoogle Scholar
Squires, T. M. & Quake, S. R. 2005 Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77, 977.CrossRefGoogle Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381.CrossRefGoogle Scholar
Sturman, R., Meier, S. W., Ottino, J. M. & Wiggins, S. 2008 Linked twist map formalism in two and three dimensions applied to mixing in tumbled granular flows. J. Fluid Mech. 602, 129–174.CrossRefGoogle Scholar
Sturman, R., Ottino, J. M. & Wiggins, S. 2006 The Mathematical Foundations of Mixing. In Cambridge Monographs on Appl. and comp. Math. 22, Cambridge University Press.Google Scholar
Sunden, B. & Shah, R. K. 2007 Advances in Compact Heat Exchangers. Springer.Google Scholar
Tourovskaia, A., Figueroa-Masot, X. & Folch, A. 2005 Differentiation-on-a-chip: a microfluidic platform for long-term cell culture studies. Lab Chip 5, 14.CrossRefGoogle ScholarPubMed
Vainchtein, D. L., Neishtadt, A. I. & Mezić, I. 2006 On passage through resonances in volume-preserving systems. Chaos 16, 043123.CrossRefGoogle ScholarPubMed
Voth, G. A., Haller, G. & Gollub, J. P. 2002 Experimental measurements of stretching fields in fluid mixing. Phys. Rev. Lett. 88, 254501.CrossRefGoogle ScholarPubMed
Weigl, B., Labarre, G., Domingo, P. & Gerlach, J. 2008 Towards non- and minimally instrumented, microfluidics-based diagnostic devices. Lab Chip 8, 1999.CrossRefGoogle ScholarPubMed
Wiggins, S. & Ottino, J. M. 2004 Foundations of chaotic mixing. Phil. Trans. R. Soc. Lond. A 362, 937.CrossRefGoogle ScholarPubMed