Analysis and Control of Mixing with an Application to Micro and Macro Flow Processes pp 35-108 | Cite as

# Lectures on Mixing and Dynamical Systems

## Abstract

The subject of fluid mixing is, from the technological perspective, an old one. It is encountered almost daily when we pour milk into coffee or try to achieve a particular paint color. The yearly output of industrial products for which fluid mixing is a part of the production process is measured in billions of dollars. If two fluids are in contact, mixing can proceed purely by molecular diffusion, but it is most commonly achieved by a combination of stretching of fluid interfaces by the macroscopic velocity field and molecular diffusion. Within the subject of fluid mechanics this process has been studied for 100 years, beginning with a paper of Osborne Reynolds. A lot of work has been devoted to non-diffusive mixing in the applied dynamical systems community in the past 20 years, after a seminal paper by (1984), where he coined the term “chaotic advection” to mean *efficient non-di usive mixing in flows with simple time-dependence*. Since the 1990’s the field of chaotic advection has gained increased attention from (at least) three directions: 1) Scientific investigation of fluid processes at microscale, usually producing flows with simple time-dependence, accompanying the fast technological advances in miniaturization and applications to biology and medicine, 2) Recognition that mixing by large coherent structures can be described using the chaotic advection theory and 3) Increased interest in active control of fluid processes, including control of mixing.

## Keywords

Dimensional Manifold Invariant Torus Chaotic Advection Volume Preservation Dimensional Invariant Manifold## Preview

Unable to display preview. Download preview PDF.

## Bibliography

- H. Aref. Stirring by chaotic advection.
*Journal of Fluid Mechanics*, 143: 1–21, 1984.MATHCrossRefMathSciNetGoogle Scholar - R. Aris.
*Vectors, tensors, and the basic equations of fluid mechanics*. Prentice-Hall, Englewood Cliffs, N.J., 1962.MATHGoogle Scholar - V. I. Arnold. Small denominators and problems of stability of motion in classical and celestial mechanics.
*Russ. Math. Survey*, 18:85–192, 1963.CrossRefGoogle Scholar - V. I. Arnold. Sur la géométrie differentielle des groupes de lie de dimension infinie et ses applications á l’hydrodynamique des fluides parfaits.
*Ann. Inst Fourier*, 16:316–361, 1966.Google Scholar - V. I. Arnold.
*Mathematical Methods of Classical Mechanics*. Springer-Verlag, New York, 1978.MATHGoogle Scholar - V.I. Arnold.
*Ordinary Differential Equations*. Springer Verlag, Berlin, N.Y., 2006. Theorem and remark after Theorem 17, chapter 5, Sec. 3.3.Google Scholar - P. Ashwin and G. P. King. Azimuthally propagating ring vortices in a model for nonaxisymmetric Taylor vortex flow.
*Physical Review Letters*, 75:4610–4613, 1995a.CrossRefGoogle Scholar - P. Ashwin and G. P. King. Streamline topology in eccentric Taylor vortex flow.
*Journal of Fluid Mechanics*, 285:215–247, 1995b.MATHCrossRefMathSciNetGoogle Scholar - S. Balasuriya, C. K. R. T. Jones, and B. Sandstede. Viscous perturbations of vorticity-conserving flows and separatrix splitting.
*Nonlinearity*, 11: 47–77, 1997.CrossRefMathSciNetGoogle Scholar - G. K. Batchelor.
*An Introduction to Fluid Dynamics*. Cambridge University Press, Cambridge, 1967.MATHGoogle Scholar - F. Bottausci, I. Mezić, C. D. Meinhart, and C. Cardonne. Mixing in the shear superposition micromixer: three-dimensional analysis.
*Philosophical Transactions of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences*, 362:1001–1018, 2004.CrossRefMathSciNetGoogle Scholar - J. Branebjerg, B. Fabius, and P. Gravesen.
*Application of Miniature Analyzers from Microfluidic Components to μTAS*, pages 141–151. Proceedings of Micro Total Analysis System Conference, Twente, Netherlands. 1994.Google Scholar - J. Branebjerg, B. Fabius, and P. Gravesen.
*Fast mixing by lamination*, pages 441–446. Proceedings of the 9th Annual Workshop on Micro Electro Mechanical Systems, San Diego, CA. 1996.Google Scholar - J. H. E. Cartwright, M. Feingold, and O. Piro. Chaotic advection in three-dimensional unsteady incompressible laminar flow.
*Journal of Fluid Mechanics*, 316:259–284, 1996.MATHCrossRefGoogle Scholar - C.-Q. Cheng and Y.-S. Sun. Existence of invariant tori in three-dimensional measure-preserving mappings.
*Celestial Mechanics*, 47:275–292, 1990.MATHCrossRefMathSciNetGoogle Scholar - N. Chiem, C. Colyer, and Harrison.
*Microfluidic Systems for Clinical Diagnostics*, pages 183–186. International Conference on Solid State Sensors and Actuators, Chicago, IL, vol 1. 1997.CrossRefGoogle Scholar - A.J. Chorin and J.E. Marsden.
*A Mathematical Introduction to Fluid Mechanics*. Springer-Verlag, New York, 1998.Google Scholar - H.-P. Chou, M. A. Unger, and S. R. Quake. A microfabricated rotary pump.
*Biomedical Microdevices*, 3:323–330, 2001.CrossRefGoogle Scholar - D. D’Alessandro, M. Dahleh, and I. Mezić. Control of mixing in fluid flow: A maximum entropy approach.
*IEEE Transactions on Automatic Control*, 44:1852–1863, 1999.MATHCrossRefGoogle Scholar - J. Evans, D. Liepmann, and A. P. Pisano.
*Planar laminar mixer*, pages 96–101. 10th Annual Workshop of Micro Electro Mechanical System, Nagoya, Japan. 1997.Google Scholar - C.L. Fefferman. Existence & smoothness of the navierstokes equation. Description of Millenium Prize problems for Clay Mathematics Institute” 2000.Google Scholar
- M. Feingold, L. P. Kadanoff, and O. Piro. Passive scalars, 3-dimensional volume-preserving maps and chaos.
*Journal of Statistical Physics*, 50: 529–565, 1988.MATHCrossRefMathSciNetGoogle Scholar - G. O. Fountain, D. V. Khakhar, I. Mezić, and J. M. Ottino. Chaotic mixing in a bounded 3-D flow. In press, Journal of Fluid Mechanics, 2000.Google Scholar
- J. W. Gibbs.
*Elementary principles in statistical mechanics: developed with special reference to the rational foundation of thermodynamics*. Yale University Press, New Haven, 1902.Google Scholar - G. Haller and I. Mezić. Reduction of three-dimensional, volume-preserving flows by symmetry.
*Nonlinearity*, 11:319–339, 1998.MATHCrossRefMathSciNetGoogle Scholar - M. Hénon. Sur la topologie des lignes de courant dans un cas particulier.
*C.R. Acad. Sci. Paris A*, 262:312–314, 1966.Google Scholar - G.E. Karniadakis and A. Beskok.
*Micro Flows*. Springer-Verlag, New York, 2001.Google Scholar - T. C. Lackey and F. Sotiropoulos. Relationship between stirring rate and reynolds number in the chaotically advected steady flow in a container with exactly counter-rotating lids.
*Physics of Fluids*, 18:053601, 2006.CrossRefGoogle Scholar - H. Lamb.
*Hydrodynamics*. Dover, New York. 1932.MATHGoogle Scholar - Y. K. Lee, J. Deval, P. Tabeling, and C. M. Ho.
*Chaotic mixing in electrokinetically and pressure-driven micro flows*, pages 483–486. Proceedings of the 14th IEEE Workshop on MEMS, Interlaken, Switzerland. 2001.Google Scholar - Z. Levnajić. Ucsb master’s thesis in mechanical engineering. 2006.Google Scholar
- R. H. Liu, K. V. Sharp, M. G. Olsen, M. A. Stremler, J. G. Santiago, R. J. Adrian, H. Aref, and D. J. Beebe. A passive micromixer: Three-dimensional serpentine microchannel.
*J. of MEMS*, 9(2), 2000.Google Scholar - R. Mane.
*Ergodic Theory and Differentiable Dynamics*. Springer-Verlag, 1987.Google Scholar - G. Mathew, I. Mezić, and L. Petzold. A multiscale measure for mixing.
*Physica D*, 211:23–46, 2005.MATHCrossRefMathSciNetGoogle Scholar - K. R. Meyer and G. R. Hall.
*Introduction to Hamiltonian systems and the N-body problem*. Springer-Verlag, New York, 1992.MATHGoogle Scholar - I. Mezić.
*On Geometrical and Statistical Properties of Dynamical Systems: Theory and Applications*. PhD thesis, California Institute of Technology, 1994.Google Scholar - I. Mezić. ABC flows as a paradigm for chaotic advection in 3-d. Preprint, 2000.Google Scholar
- I. Mezić and F. Sotiropoulos. Ergodic theory and experimental visualization of invariant sets in chaotically advected flows.
*Physics of Fluids*, 14:2235–2243, 2002.CrossRefMathSciNetGoogle Scholar - I. Mezić and S. Wiggins. On the integrability and perturbation of three dimensional fluid flows with symmetry.
*Journal of Nonlinear Science*, 4: 157–194, 1994.MATHCrossRefMathSciNetGoogle Scholar - I. Mezić and S. Wiggins. A method for visualization of invariant sets of dynamical systems based on the ergodic partition.
*Chaos*, 9:213–218, 1999.MATHCrossRefMathSciNetGoogle Scholar - R. Miyake, T. S. J. Lammerink, M. Elwenspoek, and J. H. J. Fluitman.
*Micro Mixer with fast diffusion*, pages 248–253. Proceedings of the IEEE Micro Electro Mechanical Workshop, Fort Lauderale, FL. 1993.Google Scholar - J. Moser. On the theory of quasiperiodic motion.
*SIAM Review*, 8:145–172, 1968.CrossRefGoogle Scholar - S. D. Mueller, I. Mezić, J. H. Walther, and P. Koumoutsakos. Transverse momentum micromixer optimization with evolution strategies. To appear in Computers and Fluids, 2003.Google Scholar
- X. Niu and Y. K. Lee. Efficient spatio-temporal chaotic mixing in microchannels.
*Journal of Micromechanics and Microengineering*, 13:454–462, 2003.CrossRefGoogle Scholar - J.M. Ottino.
*The Kinematics of Mixing: Stretching, Chaos and Transport*. Cambridge University Press, Cambridge, 1989.MATHGoogle Scholar - E. L. Paul, V. A. Atiemo-Obeng, and S. M. Kresta.
*Handbook of Industrial Mixing*. Wiley-Interscience, New York, 2004.Google Scholar - J. Pedlosky.
*Geophysical Fluid Dynamics*. Springer-Verlag, New York, 1987.MATHGoogle Scholar - O. Piro and M. Feingold. Diffusion in three-dimensional Liouvillian maps.
*Physical Review Letters*, 61:1799–1802, 1988.CrossRefMathSciNetGoogle Scholar - C. Siegel and J. Moser.
*Lectures on Celestial Mechanics*. Springer Verlag, Berlin, 1971.MATHGoogle Scholar - T. H. Solomon.
*Personal communication*, 1998.Google Scholar - T. H. Solomon and I. Mezić. Uniform resonant chaotic mixing in fluids.
*Nature*, 425:376–380, 2003.CrossRefGoogle Scholar - G. Sposito. On steady flows with Lamb surfaces.
*Int. J. Engng. Sci.*, 35: 197–209, 1997.MATHCrossRefMathSciNetGoogle Scholar - T. M. Squires and S. R. Quake. Microfluidics: Fluid physics at the nanoliter scale.
*Reviews of Modern Physics*, 77:977–1026, 2005.CrossRefGoogle Scholar - A. D. Stroock, S. K. W. Dertinger, A. Ajdari, I. Mezić, H. A. Stone, and G. M. Whitesides. Chaotic mixer for microchannels.
*Science*, 295:647–651, 2002.CrossRefGoogle Scholar - C. Truesdell.
*The Kinematics of Vorticity*. Indiana University Publications, Bloomington, Indiana, 1954.MATHGoogle Scholar - U. Vaidya and I. Mezić. Existence of invariant tori in action-angle-angle maps with degeneracy. 2006.Google Scholar
- R. A. Vijayendran, K. M. Motsegood, D. J. Beebe, and D. E. Leckband. Evaluation of a three-dimensional micromixer in a surface-based biosensor.
*Langmuir*, 19:1824–1828, 2003.CrossRefGoogle Scholar - M. Volpert, I. Mezić, C. D. Meinhart, and M. Dahleh. An actively controlled micromixer. pages 483–487, 1999. Proceedings of the ASME Mechanical engineering International Congress and Exposition, MEMS, Nashville, TN.Google Scholar
- S. Wiggins.
*Slowly Varying Oscillators*. PhD thesis, Cornell University, 1985.Google Scholar - S. Wiggins.
*Introduction to Applied Nonlinear Dynamical Systems and Chaos*. Springer-Verlag, New York, 1990.MATHGoogle Scholar - S. Wiggins.
*Chaotic Transport in Dynamical Systems*. Springer-Verlag, New York, 1992.MATHGoogle Scholar - S. Wiggins and P. Holmes. Periodic orbits in slowly varying oscillators.
*SIAM Journal of Mathematical Analysis*, 18:592–611, 1987.MATHCrossRefMathSciNetGoogle Scholar - T. Yannacopoulos, I. Mezić, G. King, and G. Rowlands. Eulerian diagnostics for Lagrangian chaos in three dimensional Navier-Stokes flows.
*Physical Review E*, 57:482–490, 1998.CrossRefMathSciNetGoogle Scholar - M. Yi and H. H. Bau.
*The kinematics of bend-induced stirring in microconduits*. Proceedings of MEMS-Vol. 2, Micro-Electro-Mechanical Systems, ASME, Orlando FL. 2000.Google Scholar