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Experimental study and nonlinear dynamic analysis of time-periodic micro chaotic mixers

Published online by Cambridge University Press:  07 March 2007

YI-KUEN LEE ,
CHIANG SHIH ,
PATRICK TABELING  and
CHIH-MING HO
YI-KUEN LEE
Affiliation:
Department of Mechanical Engineering, The Hong Kong University of Science and Technology, Kowloon, Hong Kong
CHIANG SHIH
Affiliation:
Department of Mechanical Engineering, Florida State University, Tallahassee, FL 32310, USA
PATRICK TABELING
Affiliation:
MMN, ESPCI, 10, rue Vauquelin, 75005 Paris, France
CHIH-MING HO
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095, USA
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Abstract

The efficiency of MEMS-based time-periodic micro chaotic mixers is experimentally and theoretically investigated in this study. A time-periodic flow perturbation was realized using digitally controlled solenoid valves to activate a source and sink alternately, acting together as a pair, with different driving frequencies. Working fluids with and without fluorescent dye were used in the micromixing experiments. The spatio-temporal variation of the mixing concentration during the mixing process was characterized at different Strouhal numbers, ranging from 0.03 to 0.74, using fluorescence microscopy. A simple kinematical model for the micromixer was used to demonstrate the presence of chaotic mixing. Specific stretching rate, Lyapunov exponent, and local bifurcation and Poincaré section analyses were used to identify the emergence of chaos. Two different numerical methods were employed to verify that the maximum Lyapunov exponent was positive in the proposed micromixer model. A simplified analytical analysis of the effect of Strouhal number is presented. Kolmogorov–Arnold–Mose (KAM) curves, which are mixing barriers, were also found in Poincaré sections. From a comparative study of the experimental results and theoretical analysis, a finite-time Lyapunov exponent (FTLE) was shown to be a more practical mixing index than the classical Lyapunov exponent because the time spent in mixing is the main concern in practical applications, such as bio-medical diagnosis. In addition, the FTLE takes into account both fluid stretching in terms of the stretching rate and fluid folding in terms of curvature.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 575 , March 2007 , pp. 425 - 448
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.CrossRefGoogle Scholar
Aref, H. 2002 The development of chaotic advection. Phys. Fluids 14, 1315–1325.CrossRefGoogle Scholar
Beigie, D., Leonard, A. & Wiggins, S. 1994 Invariant manifold templates for chaotic advection. Chaos Solitons Fractals 4, 749.CrossRefGoogle Scholar
Benettin, G., Galgani, L., Giorgilli, A. & Strelcyn, J. M. 1980 Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Parts I and II. Meccanica 15, 9–30.CrossRefGoogle Scholar
Bökenkamp, D., Desai, A., Yang, X., Tai, Y.-C., Marzluff, E. M. & Mayo, S. L. 1998 Microfabricated silicon mixers for submillisecond quench-flow analysis. Anal. Chem. 70, 232–236.CrossRefGoogle Scholar
vonBremen, H. F. Bremen, H. F., Udwadia, F. E. & Proskurowski, W. 1997 An efficient QR based method for the computation of Lyapunov exponents. Physica D 101, 1–16.Google Scholar
Burns, M. A., Johnson, B. N., Brahmasandra, S. N., Handique, K., Webster, J. R., Krishnan, M., Sammarco, T. S., Man, P. M., Jones, D., Heldsinger, D., Mastrangelo, C. H. & Burke, D. T. 1998 An Integrated Nanoliter DNA Analysis Device. Science 282, 484–487.CrossRefGoogle ScholarPubMed
D'alessandro, D., Dahleh, M. & Mezic, I. 1999 Control of mixing in fluid flow: a maximum entropy approach. IEEE T. Automat. Control 44, 1852–1863.CrossRefGoogle Scholar
Dieci, L. & vanVleck, E. Vleck, E. 1995 Computation of a few Lyapunov exponents for continuous and discrete dynamical systems. Appl. Numer. Math. 17, 275–291.CrossRefGoogle Scholar
Eringen, A. C. 1967 Mechanics of Continua. Wiley.Google Scholar
Evensen, H. T., Meldrum, D. R. & Cunningham, D. L. 1998 Automated fluid mixing in glass capillaries. Rev. Sci. Instrum. 69, 519–526.CrossRefGoogle Scholar
Figeys, D. & Pinto, D. 2000 Lab-on-a-chip: A revolution in biological and medical sciences. Anal. Chem. 72 (11), 330A–335A.CrossRefGoogle ScholarPubMed
Franjione, J. G. & Ottino, J. M. 1987 Feasibility of numerical tracking of material lines and surface in chaotic flows. Phys. Fluids 30, 3641–3643.CrossRefGoogle Scholar
Grayson, A. C. R., Shawgo, R. S., Johnson, A. M., Flynn, N. T., Yawen, L. I., Cima, M. J. & Langer, R. 2004 A BioMEMS review: MEMS technology for physiologically integrated devices. P. IEEE 92, 6–21.CrossRefGoogle Scholar
Jones, S. & Aref, H. 1988 Chaotic advection in pulsed source-sink systems. Phys. Fluids 31, 469–485.CrossRefGoogle Scholar
Khakhar, D. V., Rising, H. & Ottino, J. M. 1986 Analysis of chaotic mixing in two model systems. J. Fluid Mech. 172, 419–451.CrossRefGoogle Scholar
Lee, Y.-K., Tabeling, P., Shih, C. & Ho, C. M. 2000 Characterization of a MEMS-Fabricated Mixing Device. In Proc. ASME IMECE 2000, Orlando, FL, USA, Nov 5–10, pp. 505–511.Google Scholar
Levenspiel, O. 1972 Chemical Reaction Engineering. Wiley.Google Scholar
Lichtenberg, A. J. & Lieberman, M. A. 1982 Regular and Stochatic Motion. Springer.Google Scholar
Liu, R. H., Stremler, M. A., Sharp, K. V., Olsen, M. G., Santiago, J. G., Adrian, R. J., Aref, H. & Beebe, D. J. 2000 Passive mixing in a three-dimensional serpentine microchannel. J. MEMS 9, 190–197.CrossRefGoogle Scholar
Mathew, G., Mezić, I. & Petzold, L. 2005 A multiscale measure for mixing. Physica D 211, 23–46.CrossRefGoogle Scholar
Miyake, R., Lammerink, T. S. J., Elwenspoek, M. & Fluitman, J. H. J. 1993 Micromixer with fast diffusion. In Proc. IEEE MEMS'93, Fort Lauderdale, FL, pp. 248–253.Google Scholar
Muzzio, F. J., Swanson, P. D. & Ottino, J. M. 1991 The statistics of stretching and stirring in chaotic flows. Phys. Fluids 3, 822–834.CrossRefGoogle Scholar
Nguyen, N.-T. & Wu, Z. 2005 Micromixers – a Review. J. Micromech. Microengng 15, R1–R16.CrossRefGoogle Scholar
Niu, X. & Lee, Y.-K. 2003 Efficient spatial-temporal chaotic mixing in microchannels. J. Micromech. Microengng 13, 454–462.CrossRefGoogle Scholar
Niu, X. & Lee, Y.-K. 2003 Finite time Lyapunov exponent for micro chaotic mixer design. In Proc. MEMS, ASME IMECE'03, Washington DC, Nov. 16–21 (Paper No: IMECE2003-41389).Google Scholar
Okkels, F. & Tabeling, P. 2004 Spatiotemporal resonances in mixing of open viscous fluids, Phys. Rev. Lett. 92, 038301.CrossRefGoogle ScholarPubMed
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.Google Scholar
Ottino, J. M., Muzzio, F. J., Tjahjadi, M., Franjione, J. G., Jana, S. C. & Kusch, H. A. 1992 Chaos, symmetry, and self-similarity: Exploiting order and disorder in mixing processes. Science 257, 754–760.CrossRefGoogle ScholarPubMed
Ottino, J. M. & Wiggins, S. 2004 Introduction: mixing in microfluidics. Phil. Trans. R. Soc. Lond. A 362, 923–935.CrossRefGoogle ScholarPubMed
Parker, T. S. & Chua, L. O. 1989 Practical Numerical Algorithms for Chaotic Systems. Springer.CrossRefGoogle Scholar
Pierrehumbert, R. T. 1991 Large-scale horizontal mixing in planetary atmospheres. Phys. Fluids A 3, 1250–1260.CrossRefGoogle Scholar
Raynal, F. & Gence, J.-N. 1997 Energy saving in chaotic laminar mixing. Intl J. Heat Mass Transfer 40, 3267–3273.CrossRefGoogle Scholar
Regenfuss, P., Clegg, R. M., Fulwyler, M. J., Barrantes, F. J. & Jovin, T. M. 1985 Mixing liquids in microseconds. Rev. Sci. Instrum. 56, 283–290.CrossRefGoogle Scholar
Sprott, J. C. 2003 Chaos and Time-Series Analysis pp. 116–117. Oxford University Press.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, 381–411.CrossRefGoogle Scholar
Stroock, A. D., Dertinger, S. K. W., Ajdari, A., Mezic, I., Stone, H. A., Whitesides, G. M. 2002 Chaotic mixer for microchannels. Science. 295, 647–651.CrossRefGoogle ScholarPubMed
Tabeling, P. Chabert, M., Dodge, A., Jullien, M.-C. & Okkels, F. 2004 Chaotic mixing in cross-channel micromixers. Phil. Trans. R. Soc. Lond. A 362, 987–1000.CrossRefGoogle ScholarPubMed
Tabor, M. 1989 Chaos and Integrability in Nonlinear Dynamics: An Introduction. Wiley.Google Scholar
Tang, X. Z. & Boozer, A. H. 1996 Finite time Lyapunov exponent and advection–diffusion equation. Physica D 95, 283–305.CrossRefGoogle Scholar
Thiffeault, J. L. 2004 Stretching and curvature of material lines in chaotic flows. Physica D 198, 169–181.CrossRefGoogle Scholar
Tufillaro, N. B., Abbott, T. & Reilly, J. 1992 An Experimental Approach to Nonlinear Dynamics and Chaos. Addison–Wesley.Google Scholar
Volpert, M., Meinhart, C. D., Mezic, I. & Dahelh, M. 1999 An actively controlled micromixer. In Proc. MEMS, ASME IMECE, Nashville, TN, USA, Nov., pp. 483–487.Google Scholar
Voth, G. A., Haller, G. & Gollub, J. P. 2002 Experimental measurements of stretching fields in fluid mixing. Phys. Rev. Lett. 88, 254501.CrossRefGoogle ScholarPubMed
Wiggins, S. 1992 Chaotic Transport in Dynamical Systems. Springer.CrossRefGoogle Scholar
Yang, H. 1994 On the relative importance between chaotic mixing and diffusion. Phys. Lett. A 185, 191–195.CrossRefGoogle Scholar