1932

Abstract

Mixing is the operation by which a system evolves under stirring from one state of simplicity—the initial segregation of the constituents—to another state of simplicity—their complete uniformity. Between these extremes, patterns emerge, possibly interact, and die sooner or later. This review summarizes recent developments on the problem of mixing in its lamellar representation. This point of view visualizes a mixture as a set of stretched lamellae, or sheets, possibly interacting with each other. It relies on a near-exact formulation of the Fourier equation on a moving substrate and allows one to bridge the spatial structure and evolution of the concentration field with its statistical content in a direct way. Within this frame, one can precisely describe both the dynamics of the concentration levels in a mixture as a function of the intensity of the stirring motions at the scale of a single lamella and the interaction rule between adjacent lamellae, thus offering a detailed representation of the mixture content, its structure, and their evolution in time.

Loading

Article metrics loading...

/content/journals/10.1146/annurev-fluid-010518-040306
2019-01-05
2024-06-23
Loading full text...

Full text loading...

/deliver/fulltext/fluid/51/1/annurev-fluid-010518-040306.html?itemId=/content/journals/10.1146/annurev-fluid-010518-040306&mimeType=html&fmt=ahah

Literature Cited

  1. Allègre CJ, Turcotte DL 1986. Implications of a two-component marble-cake mantle. Nature 323:123–27
    [Google Scholar]
  2. Aref H, Blake JR, Budišić M, Cardoso SSS, Cartwright JHE et al. 2017. Frontiers of chaotic advection. Rev. Mod. Phys. 89:025007
    [Google Scholar]
  3. Arnold VI, Avez A 1967. Problèmes ergodiques de la mécanique classique Paris: Gauthier-Villars
    [Google Scholar]
  4. Ashurst WT, Kerstein AR, Kerr RM, Gibson CH 1987. Alignment of vorticity and scalar gradient with strain rate in simulated Navier–Stokes turbulence. Phys. Fluids 30:2343–53
    [Google Scholar]
  5. Audoly B, Berestycki H, Pomeau Y 2000. Réaction diffusion en écoulement stationnaire rapide. C. R. Acad. Sci. IIb 328:255–62
    [Google Scholar]
  6. Balmforth NJ, Young WR 2003. Diffusion-limited scalar cascades. J. Fluid Mech. 482:91–100
    [Google Scholar]
  7. Batchelor GK 1959. Small-scale variation of convected quantities like temperature in a turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5:113–33
    [Google Scholar]
  8. Beigie D, Leonard A, Wiggins S 1991. A global study of enhanced stretching and diffusion in chaotic tangles. Phys. Fluids A 3:1039–50
    [Google Scholar]
  9. Berg HC 2004. E. coli in Motion New York: Springer-Verlag
    [Google Scholar]
  10. Biferale L, Crisanti A, Vergassola M, Vulpiani A 1995. Eddy diffusivities in scalar transport. Phys. Fluids 7:2725–34
    [Google Scholar]
  11. Bohr T, Jensen MH, Paladin G, Vulpiani A 1998. Dynamical Systems Approach to Turbulence Cambridge, UK: Cambridge Univ. Press
    [Google Scholar]
  12. Bouchaud JP, Georges A 1990. Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195:127–293
    [Google Scholar]
  13. Brodkey RS 1967. The Phenomena of Fluid Motions Reading, MA: Addison-Wesley
    [Google Scholar]
  14. Carrier GF, Fendell FE, Marble FE 1975. The effect of strain rate on diffusion flames. SIAM J. Appl. Math. 28:463–500
    [Google Scholar]
  15. Carslaw HS, Jaeger JC 1959. Conduction of Heat in Solids Oxford: Clarendon. 2nd ed.
    [Google Scholar]
  16. Catrakis HJ, Dimotakis PE 1996. Mixing in turbulent jets: scalar measures and isosurface geometry. J. Fluid Mech. 317:369–406
    [Google Scholar]
  17. Celani A, Cencini M, Vergassola M, Villermaux E, Vincenzi D 2005. Shear effects on passive scalar spectra. J. Fluid Mech. 523:99–108
    [Google Scholar]
  18. Celani A, Villermaux E, Vergassola M 2014. Odor landscapes in turbulent environments. Phys. Rev. X 4:041015
    [Google Scholar]
  19. Chandrasekhar S 1943. Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15:1–89
    [Google Scholar]
  20. Childress S, Gilbert AD 1995. Stretch, Twist, Fold: The Fast Dynamo Berlin: Springer-Verlag
    [Google Scholar]
  21. Cocke W 1969. Turbulent hydrodynamic line stretching: consequences of isotropy. Phys. Fluids 12:2488–92
    [Google Scholar]
  22. Csanady GT 1973. Turbulent Diffusion in the Environment Dordrecht, Neth.: D. Reidel
    [Google Scholar]
  23. Curl RL 1963. Dispersed phase mixing: I. Theory and effect in simple reactors. AIChE J. 9:175–81
    [Google Scholar]
  24. Danckwerts PV 1952. The definition and measurement of some characteristics of mixtures. Appl. Sci. Res. A 3:279–96
    [Google Scholar]
  25. Danckwerts PV 1953. Theory of mixtures and mixing. Research 6:355–61
    [Google Scholar]
  26. de Rivas A, Villermaux E 2016. Dense spray evaporation as a mixing process. Phys. Rev. Fluids 1:014201
    [Google Scholar]
  27. Deseigne J, Cottin-Bizonne C, Stroock AD, Bocquet L, Ybert C 2014. How a “pinch of salt” can tune chaotic mixing of colloidal suspensions. Soft Matter 10:4795–99
    [Google Scholar]
  28. Donzis DA, Sreenivasan KR, Yeung PK 2005. Scalar dissipation rate and dissipative anomaly in isotropic turbulence. J. Fluid Mech. 532:199–216
    [Google Scholar]
  29. Duplat J, Innocenti C, Villermaux E 2010.a A non-sequential turbulent mixing process. Phys. Fluids 22:035104
    [Google Scholar]
  30. Duplat J, Jouary A, Villermaux E 2010.b Entanglement rules for random mixtures. Phys. Rev. Lett. 105:034504
    [Google Scholar]
  31. Duplat J, Villermaux E 2000. Persistency of material element deformation in isotropic flows and growth rate of lines and surfaces. Eur. Phys. J. B 18:353–61
    [Google Scholar]
  32. Duplat J, Villermaux E 2008. Mixing by random stirring in confined mixtures. J. Fluid Mech. 617:51–86
    [Google Scholar]
  33. Epstein IR 1990. Shaken, stirred—but not mixed. Nature 346:16–17
    [Google Scholar]
  34. Falkovich G, Gawedzki K, Vergassola M 2001. Particles and fields in fluid turbulence. Rev. Mod. Phys. 73:913–75
    [Google Scholar]
  35. Fannjiang A, Nonnenmacher S, Wolonski L 2004. Dissipation time and decay of correlations. Nonlinearity 17:1481–508
    [Google Scholar]
  36. Favre AE 1962. Mécanique de la turbulence Paris: Ed. Cent. Natl. Res. Sci.
    [Google Scholar]
  37. Feller W 1971. An Introduction to Probability Theory and Its Applications 2 New York: Wiley
    [Google Scholar]
  38. Figueroa A, Meunier P, Cuevas S, Villermaux E, Ramos E 2014. Chaotic advection at large Péclet number: electromagnetically driven experiments, numerical simulations, and theoretical predictions. Phys. Fluids 26:013601
    [Google Scholar]
  39. Fourier J 1822. Théorie analytique de la chaleur Paris: Firmin Didot
    [Google Scholar]
  40. Geri M, Keshavarz B, McKinley GH, Bush JWM 2017. Thermal delay of drop coalescence. J. Fluid Mech. 833:R3
    [Google Scholar]
  41. Gibbs JW 1902. Elementary Principles in Statistical Mechanics New Haven, CT: Yale Univ. Press
    [Google Scholar]
  42. Gibson CH, Libby PA 1972. On turbulent flows with fast chemical reactions. Part II. The distribution of reactants and products near a reacting surface. Combust. Sci. Technol. 6:29–35
    [Google Scholar]
  43. Gilbert AD 2006. Advected fields in maps—III. Passive scalar decay in baker's maps. Dyn. Syst. 21:25–71
    [Google Scholar]
  44. Giona M, Adrover A, Cerbelli S, Vitacolonna V 2004. Spectral properties and transport mechanisms of partially chaotic bounded flows in the presence of diffusion. Phys. Rev. Lett. 92:114101
    [Google Scholar]
  45. Gouillart E, Kuncio N, Dauchot O, Dubrulle B, Roux S, Thiffeault JL 2007. Walls inhibit chaotic mixing. Phys. Rev. Lett. 99:114501
    [Google Scholar]
  46. Hinch EJ 1999. Mixing: turbulence and chaos—an introduction. Mixing Chaos and Turbulence H Chaté, E Villermaux, J-M Chomaz37–56 New York: Springer Sci. Bus. Media
    [Google Scholar]
  47. Kalda J 2000. Simple model of intermittent passive scalar turbulence. Phys. Rev. Lett. 84:471–74
    [Google Scholar]
  48. Kalda J, Morozenko A 2008. Turbulent mixing: the roots of intermittency. New J. Phys. 10:093003
    [Google Scholar]
  49. Kitanidis PK 1994. The concept of the dilution index. Water Resour. Res. 30:2011–26
    [Google Scholar]
  50. Koehl MAR, Koseff JR, Crimaldi JP, McCay MG, Cooper T et al. 2001. Lobster sniffing: antennule design and hydrodynamic filtering of information in an odor plume. Science 294:1948–51
    [Google Scholar]
  51. Kraichnan RH 1994. Anomalous scaling of a randomly advected passive scalar. Phys. Rev. Lett. 72:1016
    [Google Scholar]
  52. Kree M, Villermaux E 2017. Scalar mixtures in porous media. Phys. Rev. Fluids 2:104502
    [Google Scholar]
  53. Kurtuldu H, Guasto JS, Johnson KA, Gollub JP 2011. Enhancement of biomixing by swimming algal cells in two-dimensional films. PNAS 108:10391–95
    [Google Scholar]
  54. Landau L, Lifshitz E 1987. Fluid Mechanics Oxford: Pergamon
    [Google Scholar]
  55. Le Borgne T, Dentz M, Villermaux E 2015. The lamellar description of mixing in porous media. J. Fluid Mech. 770:458–98
    [Google Scholar]
  56. Le Borgne T, Huck PD, Dentz M, Villermaux E 2017. Scalar gradients in stirred mixtures and the deconstruction of random fields. J. Fluid Mech. 812:578–610
    [Google Scholar]
  57. Levèque MA 1928. Les lois de la transmission de la chaleur par convection. Ann. Mines 13:201–39
    [Google Scholar]
  58. Mafra-Neto A, Cardé RT 1994. Fine-scale structure of pheromone plumes modulates upwind orientation of flying moths. Nature 369:142–44
    [Google Scholar]
  59. Marble FE 1964. Spacecraft propulsion Tech. Rep. ST-3 Calif. Inst. Technol. Pasadena, CA:
    [Google Scholar]
  60. Marble FE 1988. Mixing, diffusion and chemical reaction of liquids in a vortex field. Chemical Reactivity in Liquids: Fundamental Aspects M Moreau, P Turq581–606 New York: Plenum
    [Google Scholar]
  61. Marble FE, Broadwell JE 1977. The coherent flame model for turbulent chemical reactions Tech. Rep. TRW-9-PU, Project SQUID Purdue Univ. West Lafayette, IN:
    [Google Scholar]
  62. Martinez-Ruiz D, Meunier P, Favier B, Duchemin L, Villermaux E 2018. The diffusive sheet method for scalar mixing. J. Fluid Mech. 837:230–57
    [Google Scholar]
  63. Matheron G, de Marsilly G 1980. Is transport in porous media always diffusive? A counterexample. Water Resour. Res. 16:901–17
    [Google Scholar]
  64. Mathew G, Mezic I, Petzold L 2005. A multiscale measure for mixing. Physica D 211:23–46
    [Google Scholar]
  65. Maxwell JC 1867. On the dynamical theory of gases. Philos. Trans. R. Soc. Lond. 157:49–88
    [Google Scholar]
  66. McKenzie D 1979. Finite deformation during fluid flow. Geophys. J. R. Astron. Soc. 58:689–715
    [Google Scholar]
  67. Meunier P, Huck P, Nobili C, Villermaux E 2015. Transport and diffusion around a homoclinic point. Chaos, Complexity and Transport: Proceedings of the CCT '15 X Leoncini, C Eloy, G Boedec152–62 Singapore: World Sci.
    [Google Scholar]
  68. Meunier P, Villermaux E 2003. How vortices mix. J. Fluid Mech. 476:213–22
    [Google Scholar]
  69. Meunier P, Villermaux E 2007. Van Hove singularities in probability density functions of scalars. C. R. Méc. 335:162–67
    [Google Scholar]
  70. Meunier P, Villermaux E 2010. The diffusive strip method for scalar mixing in two dimensions. J. Fluid Mech. 662:134–72
    [Google Scholar]
  71. Miller PL, Dimotakis PE 1996. Measurements of scalar power spectra in high Schmidt number turbulent jets. J. Fluid Mech. 308:129–46
    [Google Scholar]
  72. Moffatt HK 1983. Transport effects associated with turbulence with particular attention to the influence of helicity. Rep. Prog. Phys. 46:621–64
    [Google Scholar]
  73. Mohr WD, Saxton RL, Jepson CH 1957. Mixing in laminar-flow systems. Ind. Eng. Technol. 49:1855–56
    [Google Scholar]
  74. Nagata S 1975. Mixing: Principles and Applications New York: Wiley
    [Google Scholar]
  75. Néel B, Villermaux E 2018. The spontaneous puncture of thick liquid films. J. Fluid Mech. 838:192–221
    [Google Scholar]
  76. Nicolleau FCGA, Elmaihy A 2004. Study of the development of three-dimensional sets of fluid particles and iso-concentration fields using kinematic simulation. J. Fluid Mech. 517:229–49
    [Google Scholar]
  77. Okubo A, Karweit M 1969. Diffusion from a continuous source in a uniform shear flow. Limnol. Oceanogr. 14:514–20
    [Google Scholar]
  78. Osborn TR 1980. Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10:83–89
    [Google Scholar]
  79. Ott E, Antonsen TM 1989. Fractal measures of passively convected vector fields and scalar gradients in chaotic fluid flows. Phys. Rev. A 39:3660–71
    [Google Scholar]
  80. Ottino JM 1982. Description of mixing with diffusion and reaction in terms of the concept of material surfaces. J. Fluid Mech. 114:83–103
    [Google Scholar]
  81. Pope SB 1985. PDF methods for turbulent reacting flows. Prog. Energy Combust. Sci. 11:119–92
    [Google Scholar]
  82. Poulain S, Villermaux E, Bourouiba L 2018. Ageing and burst of surface bubbles. J. Fluid Mech. 851:636–71
    [Google Scholar]
  83. Prieve DC, Anderson JL, Ebel JP, Lowell ME 1984. Motion of a particle generated by chemical gradients. Part 2. Electrolytes. J. Fluid Mech. 148:247–69
    [Google Scholar]
  84. Pumir A 1994. A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient. Phys. Fluids 6:2118–32
    [Google Scholar]
  85. Pumir A, Shraiman BI, Siggia ED 1991. Exponential tails and random advection. Phys. Rev. Lett. 66:2984–87
    [Google Scholar]
  86. Ranz WE 1979. Application of a stretch model to mixing, diffusion and reaction in laminar and turbulent flows. AIChE J. 25:41–47
    [Google Scholar]
  87. Raynal F, Bourgoin M, Cottin-Bizonne C, Ybert C, Volk R 2018. Advection and diffusion in a chemically induced compressible flow. J. Fluid Mech. 847:228–43
    [Google Scholar]
  88. Reif F 1965. Fundamentals of Statistical and Thermal Physics Long Grove, IL: Waveland
    [Google Scholar]
  89. Rhines PB, Young WR 1983. How rapidly is a passive scalar mixed within closed streamlines?. J. Fluid Mech. 133:133–45
    [Google Scholar]
  90. Richardson LF 1926. Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. Lond. A 110:709–37
    [Google Scholar]
  91. Rom-Kedar V, Leonard A, Wiggins S 1990. An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech 214:347–94
    [Google Scholar]
  92. Schnitzer MJ, Block SM, Berg HC, Purcell EM 1990. Strategies for chemotaxis. Biology of the Chemotactic Response JP Armitage, JM Lackie15–33 Cambridge, UK: Cambridge Univ. Press
    [Google Scholar]
  93. Schumacher J, Sreenivasan KR, Yeung PK 2005. Very fine structures in scalar mixing. J. Fluid Mech. 531:113–22
    [Google Scholar]
  94. Shin S, Shardt O, Warren PB, Stone HA 2017. Membraneless water filtration using CO2. Nat. Commun. 8:15181
    [Google Scholar]
  95. Shraiman BI 1987. Diffusive transport in a Rayleigh-Bénard convection cell. Phys. Rev. A 36:261–67
    [Google Scholar]
  96. Shraiman BI, Siggia ED 1994. Lagrangian path integrals and fluctuations in random flows. Phys. Rev. E 49:2912–27
    [Google Scholar]
  97. Shraiman BI, Siggia ED 2000. Scalar turbulence. Nature 405:639–46
    [Google Scholar]
  98. Solomon TH, Gollub JP 1988. Passive transport in steady Rayleigh–Bénard convection. Phys. Fluids 31:1372–79
    [Google Scholar]
  99. Souzy M, Lhuissier H, Villermaux E, Metzger B 2017. Stretching and mixing in sheared particulate suspensions. J. Fluid Mech. 812:611–35
    [Google Scholar]
  100. Souzy M, Zaier I, Lhuissier H, Le Borgne T, Metzger B 2018. Mixing lamellae in a shear flow. J. Fluid Mech. 838:R3
    [Google Scholar]
  101. Sreenivasan KR 1991. On local isotropy of passive scalars in turbulent shear flows. Proc. R. Soc. Lond. A 434:165–82
    [Google Scholar]
  102. Sturman R, Ottino JM, Wiggins S 2006. The Mathematical Foundations of Mixing Cambridge, UK: Cambridge Univ. Press
    [Google Scholar]
  103. Taylor GI 1953. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219:186–203
    [Google Scholar]
  104. Tél T, de Moura A, Grebogi C, Károlyi G 2005. Chemical and biological activity in open flows: a dynamical system approach. Phys. Rep. 413:91–196
    [Google Scholar]
  105. Thiffeault JL 2004. Stretching and curvature of material lines in chaotic flows. Physica D 198:169–81
    [Google Scholar]
  106. Thiffeault JL 2008. Scalar decay in chaotic mixing. Transport and Mixing in Geophysical Flows JB Weiss, A Provenzale3–36 Berlin: Springer-Verlag
    [Google Scholar]
  107. Thiffeault JL 2012. Using multiscale norms to quantify mixing and transport. Nonlinearity 25:R1
    [Google Scholar]
  108. Vidick B 1989. Critical mixing parameters for good control of cement slurry quality. J. Petrol. Technol. 42:7924–28
    [Google Scholar]
  109. Villermaux E 2012.a Mixing by porous media. C. R. Méc. 340:933–43
    [Google Scholar]
  110. Villermaux E 2012.b On dissipation in stirred mixtures. Adv. Appl. Mech. 45:91–107
    [Google Scholar]
  111. Villermaux E, Duplat J 2003. Mixing as an aggregation process. Phys. Rev. Lett. 91:184501
    [Google Scholar]
  112. Villermaux E, Duplat J 2006. Coarse grained scale of turbulent mixtures. Phys. Rev. Lett. 97:144506
    [Google Scholar]
  113. Villermaux E, Gagne Y 1994. Line dispersion in homogeneous turbulence: stretching, fractal dimensions, and micromixing. Phys. Rev. Lett. 73:252–55
    [Google Scholar]
  114. Villermaux E, Innocenti C 1999. On the geometry of turbulent mixing. J. Fluid Mech. 393:123–45
    [Google Scholar]
  115. Villermaux E, Moutte A, Amielh M, Meunier P 2017. Fine structure of the vapor field in evaporating dense sprays. Phys. Rev. Fluids 2:074501
    [Google Scholar]
  116. Villermaux E, Rehab H 2000. Mixing in coaxial jets. J. Fluid Mech. 425:161–85
    [Google Scholar]
  117. Villermaux E, Stroock AD, Stone HA 2008. Bridging kinematics and concentration content in a chaotic micromixer. Phys. Rev. E 77:015301
    [Google Scholar]
  118. von Smoluchowski M 1917. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Z. Phys. Chem. 92:129–68
    [Google Scholar]
  119. Warhaft Z 1984. The interference of thermal fields from line sources in grid turbulence. J. Fluid Mech. 144:363–87
    [Google Scholar]
  120. Warhaft Z 2000. Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32:203–40
    [Google Scholar]
  121. Weij JH, Bartolo D 2017. Mixing by unstirring: hyperuniform dispersion of interacting particles upon chaotic advection. Phys. Rev. Lett. 119:048002
    [Google Scholar]
  122. Zel'dovich YB 1937. The asymptotic law of heat transfer at small velocities in the finite domain problem. Zh. Eksp. Teor. Fiz. 7:1466–68
    [Google Scholar]
/content/journals/10.1146/annurev-fluid-010518-040306
Loading
/content/journals/10.1146/annurev-fluid-010518-040306
Loading

Data & Media loading...

  • Article Type: Review Article
This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error