Chaotic mixing and mixing efficiency in a short time

Fluid Dynamics Research

Volumn 40, Issue 1, 2008, Pages 1-33

  Funakoshi, Mitsuaki ^a  

^a KYOTO UNIVERSITY   (Japan)

Author keywords

Chaotic mixing; Lagrangian chaos; Mixing efficiency; Stable and unstable manifolds; Static Mixer; Transport matrix

Indexed keywords

COMPUTER SIMULATION; MATRIX ALGEBRA; MIXERS (MACHINERY); STEADY FLOW;

CHAOTIC MIXING; MIXING EFFICIENCY;

MIXING;

COMPUTER SIMULATION; MATRIX ALGEBRA; MIXERS (MACHINERY); MIXING; STEADY FLOW;

EID: 36148970100     PISSN: 01695983     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.fluiddyn.2007.04.004     Document Type: Article

References (81)

1. Anderson P.D., Galaktionov O.S., Peters G.W.M., Meijer H.E.H., and Tucker III C.L. Material stretching in laminar mixing flows: extended mapping technique applied to the journal bearing flow. Int. J. Numer. Meth. Fluids 40 (2002) 189-196
2. Aref H. Stirring by chaotic advection. J. Fluid Mech. 143 (1984) 1-21
3. Aref H. Chaotic advection of fluid particles. Phil. Trans. R. Soc. Lond. A 333 (1990) 273-288
4. Aref H. Stochastic particle motion in laminar flows. Phys. Fluids A 3 (1991) 1009-1016
5. Aref H. The development of chaotic advection. Phys. Fluids 14 (2002) 1315-1325
6. Aref H., and Jones S.W. Enhanced separation of diffusing particles by chaotic advection. Phys. Fluids A 1 (1989) 470-474
7. Aref H., Jones S.W., Mofina S., and Zawadzki I. Vortices, kinematics and chaos. Physica D 37 (1989) 423-440
8. Atobe T., and Funakoshi M. Chaotic motion of fluid particles due to the alternate rotations of two eccentric cylinders. J. Phys. Soc. Jpn. 63 (1994) 1738-1753
9. Atobe T., Funakoshi M., and Inoue S. Orbital instability and chaos in the Stokes flow between two eccentric cylinders. Fluid Dyn. Res. 16 (1995) 115-129
10. Bajer K., and Moffatt H.K. On a class of steady confined Stokes flows with chaotic streamlines. J. Fluid Mech. 212 (1990) 337-363
11. Ballal B.Y., and Rivlin R.S. Flow of a Newtonian fluid between eccentric rotating cylinders: inertial effects. Arch. Rat. Mech. Anal. 62 (1977) 237-294
12. Beigie D., Leonard A., and Wiggins S. Invariant manifold templates for chaotic advection. Chaos, Solitons Fractals 4 (1994) 749-868
13. Boyland P.L., Aref H., and Stremler M.A. Topological fluid mechanics of stirring. J. Fluid Mech. 403 (2000) 277-304
14. Cartwright J.H.E., Feingold M., and Piro O. An introduction to chaotic advection. In: Chaté H., Villermaux E., and Chomaz J.-M. (Eds). Mixing: Chaos and Turbulence (1999), Kluwer Academic/Plenum Publishers, New York 307-342
15. Chaiken J., Chevray R., Tabor M., and Tan Q.M. Experimental study of Lagrangian turbulence in a Stokes flow. Proc. R. Soc. Lond. A 408 (1986) 165-174
16. Chew Y.T., Xia H.M., Shu C., and Wan S.Y.M. Techniques to enhance fluid micro-mixing and chaotic micromixers. Mod. Phys. Lett. B 19 (2005) 1567-1570
17. Childress S., and Gilbert A.D. Stretch, Twist, Fold: the Fast Dynamo (1995), Springer, Berlin
18. Danckwerts P.V. The definition and measurement of some characteristics of mixtures. Appl. Sci. Res. A 3 (1952) 279-296
19. Dean W.R. Note on the motion of fluid in a curved pipe. Phil. Mag. 4 (1927) 208-223
20. Dean W.R. The streamline motion of fluid in a curved pipe. Phil. Mag. 5 (1928) 673-693
21. Dittrich P.S., Tachikawa K., and Manz A. Micro total analysis systems. Latest advancements and trends. Anal. Chem. 78 (2006) 3887-3907
22. Doherty M.F., and Ottino J.M. Chaos in deterministic systems: strange attractors, turbulence, and applications in chemical engineering. Chem. Eng. Sci. 43 (1988) 139-183
23. Dombre T., Frisch U., Greene J.M., Hénon M., Mehr A., and Soward A.M. Chaotic streamlines in the ABC flows. J. Fluid Mech. 167 (1986) 353-391
24. Dutta P., and Chevray R. Inertial effects in chaotic mixing with diffusion. J. Fluid Mech. 285 (1995) 1-16
25. Finn M.D., and Cox S.M. Stokes flow in a mixer with changing geometry. J. Eng. Math. 41 (2001) 75-99
26. Finn M.D., Cox S.M., and Byrne H.M. Mixing measures for a two-dimensional chaotic Stokes flow. J. Eng. Math. 48 (2004) 129-155
27. Fountain G.O., Khakhar D.V., and Ottino J.M. Visualization of three-dimensional chaos. Science 281 (1998) 683-686
28. Fountain G.O., Khakhar D.V., Mezić I., and Ottino J.M. Chaotic mixing in a bounded three-dimensional flow. J. Fluid Mech. 417 (2000) 265-301
29. Franjione J.G., Leong C.-W., and Ottino J.M. Symmetries within chaos: a route to effective mixing. Phys. Fluids A 1 (1989) 1772-1783
30. Funakoshi M. Lagrangian chaos and mixing of fluids. Japan J. Indust. Appl. Math. 18 (2001) 613-626
31. Galaktionov O.S., Anderson P.D., Kruijt P.G.M., Peters G.W.M., and Meijer H.E.H. A mapping approach for three-dimensional distributive mixing analysis. Comput. Fluids 30 (2001) 271-289
32. Galaktionov O.S., Anderson P.D., Peters G.W.M., and Meijer H.E.H. Analysis and optimization of Kenics static mixers. Int. Polym. Process. (2003) 138-150
33. Giona M., Adrover A., Muzzio F.J., Cerbelli S., and Alvarez M.M. The geometry of mixing in time-periodic chaotic flows. I. Asymptotic directionality in physically realizable flows and global invariant properties. Physica D 132 (1999) 298-324
34. Giona M., Adrover A., Muzzio F.J., and Cerbelli S. The geometry of mixing in 2-d time-periodic chaotic flows. Chem. Eng. Sci. 55 (2000) 381-389
35. Hardt S., Pennemann H., and Schönfeld F. Theoretical and experimental characterization of a low-Reynolds number split-and-recombine mixer. Microfluid Nanofluid 2 (2006) 237-248
36. Hessel V., Löwe H., and Schönfeld F. Micromixers-a review on passive and active mixing principles. Chem. Eng. Sci. 60 (2005) 2479-2501
37. Hobbs D.M., and Muzzio F.J. The curvature of material lines in a three-dimensional chaotic flow. Phys. Fluids 10 (1998) 1942-1952
38. Ichida, Y., Funakoshi, M., 1998. Chaotic mixing under the flow between two eccentric cylinders, Res. Inst. Math. Sci., Kyoto Univ., Kokyuroku No.1029, 31-52 (in Japanese).
39. Ishii K., and Iwatsu R. Numerical simulation of the Lagrangian flow structure in a driven-cavity. In: Moffatt H.K., and Tsinobar A. (Eds). Topological Fluid Mechanics (1989), Cambridge University Press, Cambridge 54-63
40. Iwatsu, R., Ishii, K., Kawamura, T., Kuwahara, K., Hyun, J.M., 1989. Simulation of transition to turbulence in a cubic cavity. AIAA Pap. No. 89-0040.
41. Jeffery G.B. The rotation of two circular cylinders in a viscous fluid. Proc. R. Soc. Lond. A 101 (1922) 169-174
42. Jones S.W., and Aref H. Chaotic advection in pulsed source-sink systems. Phys. Fluids 31 (1988) 469-485
43. Jones S.W., Thomas O.M., and Aref H. Chaotic advection by laminar flow in a twisted pipe. J. Fluid Mech. 209 (1989) 335-357
44. Kaper T.J., and Wiggins S. An analytical study of transport in Stokes flows exhibiting large-scale chaos in the eccentric journal bearing. J. Fluid Mech. 253 (1993) 211-243
45. Kawazoe, H., 2004. Mixing and deformation of fluid elements by the flow between two eccentric cylinders. Master thesis, Graduate School of Informatics, Kyoto University.
46. Kawazoe, H., Funakoshi, M., Kaneko, Y., 2004. Indices of the mixing efficiency in a short time by a flow between two eccentric cylinders. Res. Inst. Math. Sci., Kyoto University, Kokyuroku No. 1406, 62-79 (in Japanese).
47. Khakhar D.V., Franjione J.G., and Ottino J.M. A case study of chaotic mixing in deterministic flows: the partitioned-pipe mixer. Chem. Eng. Sci. 42 (1987) 2909-2926
48. King G.P., Rowlands G., Rudman M., and Yannacopoulos A.N. Predicting chaotic dispersion with Eulerian symmetry measures: Wavy Taylor-vortex flow. Phys. Fluids 13 (2001) 2522-2528
49. Kruijt P.G.M., Galaktionov O.S., Anderson P.D., Peters G.W.M., and Meijer H.E.H. Analyzing mixing in periodic flows by distribution matrices: mapping method. AIChE J. 47 (2001) 1005-1015
50. Kusch H.A., and Ottino J.M. Experiments on mixing in continuous chaotic flows. J. Fluid Mech. 236 (1992) 319-348
51. Leonard A. Curvature and torsion of material lines in chaotic flows. Fluid Dyn. Res. 36 (2005) 261-275
52. Leong C.W., and Ottino J.M. Experiments on mixing due to chaotic advection in a cavity. J. Fluid Mech. 209 (1989) 463-499
53. Lichtenberg A.J., and Lieberman M.A. Regular and Chaotic Dynamics. second ed (1992), Springer, New York
54. Ling F.H. Chaotic mixing in a spatially periodic continuous mixer. Phys. Fluids A 5 (1993) 2147-2160
55. Liu M., and Muzzio F.J. The curvature of material lines in chaotic cavity flows. Phys. Fluids 8 (1996) 75-83
56. Malyuga V.S., Meleshko V.V., Speetjens M.F.M., Clercx H.J.H., and van Heijst G.J.F. Mixing in the Stokes flow in a cylindrical container. Proc. R. Soc. Lond. A 458 (2002) 1867-1885
57. Mandelbrot B.B. The Fractal Geometry of Nature (1982), W. H. Freeman, New York
58. Mathew G., Mezić I., and Petzold L. A multiscale measure for mixing. Physica D 211 (2005) 23-46
59. Meleshko V.V., Galaktionov O.S., Peters G.W.M., and Meijer H.E.H. Three-dimensional mixing in Stokes flow: the partitioned pipe mixer problem revisited. Eur. J. Mech. B/Fluids 18 (1999) 783-792
60. Mizuno Y., and Funakoshi M. Chaotic mixing due to a spatially periodic three-dimensional flow. Fluid Dyn. Res. 31 (2002) 129-149
61. Mizuno Y., and Funakoshi M. Chaotic mixing caused by an axially periodic steady flow in a partitioned-pipe mixer. Fluid Dyn. Res. 35 (2004) 205-227
62. Mizuno Y., and Funakoshi M. Reynolds number dependences of velocity field and fluid mixing in partitioned-pipe mixer. J. Phys. Soc. Jpn. 74 (2005) 1479-1489
63. Muzzio F.J., Swanson P.D., and Ottino J.M. Mixing distributions produced by multiplicative stretching in chaotic flows. Int. J. Bifurcations and Chaos 2 (1992) 37-50
64. Ottino J.M. The Kinematics of Mixing: Stretching, Chaos, and Transport (1989), Cambridge University Press, Cambridge
65. Ottino J.M. Mixing, chaotic advection, and turbulence. Annu. Rev. Fluid Mech. 22 (1990) 207-253
66. Ottino J.M., and Shinbrot T. Comparing extremes: mixing of fluids, mixing of solids. In: Chaté H., Villermaux E., and Chomaz J.-M. (Eds). Mixing: Chaos and Turbulence (1999), Kluwer Academic/Plenum Publishers, New York 163-186
67. Ottino J.M., and Wiggins S. Introduction: mixing in microfluidics. Phil. Trans. R. Soc. Lond. A 362 (2004) 923-935
68. Rom-Kedar V., Leonard A., and Wiggins S. An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214 (1990) 347-394
69. Speetjens M.F.M., Clercx H.J.H., and van Heijst G.J.F. Inertia-induced coherent structures in a time-periodic viscous mixing flow. Phys. Fluids 18 (2006) 083603
70. Spencer R.S., and Wiley R.M. The mixing of very viscous liquids. J. Colloid Sci. 6 (1951) 133-145
71. Stremler M.A., Haselton F.R., and Aref H. Designing for chaos: applications of chaotic advection at the microscale. Phil. Trans. R. Soc. Lond. A 362 (2004) 1019-1036
72. Sturman R., Ottino J.M., and Wiggins S. The Mathematical Foundations of Mixing-The Linked Twist Map as a Paradigm in Applications: Micro to Macro, Fluids to Solids- (2006), Cambridge University Press, Cambridge
73. Swanson P.D., and Ottino J.M. A comparative computational and experimental study of chaotic mixing of viscous fluids. J. Fluid Mech. 213 (1990) 227-249
74. Tang X.Z., and Boozer A.H. Finite time Lyapunov exponent and advection-diffusion equation. Physica D 95 (1996) 283-305
75. Thiffeault J.-L. Stretching and curvature of material lines in chaotic flows. Physica D 198 (2004) 169-181
76. Thiffeault J.-L., and Boozer A.H. Geometrical constraints on finite-time Lyapunov exponents in two and three dimensions. Chaos 11 (2001) 16-28
77. Thiffeault J.-L., and Finn M.D. Topology, braids and mixing in fluids. Phil. Trans. R. Soc. A 364 (2006) 3251-3266
78. Wiggins S. Chaotic Transport in Dynamical Systems (1992), Springer, New York
79. Wiggins S., and Ottino J.M. Foundations of chaotic mixing. Phil. Trans. R. Soc. Lond. A 362 (2004) 937-970
80. Wilson M.C.T., Summers J.L., Kapur N., and Gaskell P.H. Stirring and transport enhancement in a continuously modulated free-surface flow. J. Fluid Mech. 565 (2006) 319-351
81. Wonhas A., and Vassilicos J.C. Mixing in frozen and time-periodic two-dimensional vortical flows. J. Fluid Mech. 442 (2001) 359-385