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Analysis of optimal mixing in open-flow mixers with time-modulated vortex arrays

Bhargav Rallabandi1,*, Cheng Wang2, and Sascha Hilgenfeldt3,†

  • 1Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA
  • 2Department of Mechanical and Aerospace Engineering, Missouri University of Science and Technology, Rolla, Missouri 65409, USA
  • 3Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

  • *vbr@princeton.edu
  • †sascha@illinois.edu
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Phys. Rev. Fluids 2, 064501 – Published 26 June, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.064501

Abstract

In this work, a systematic approach to efficient open flow mixing is introduced, using general theoretical concepts to identify optimized parameters of a deliberately introduced unsteady flow component. The method is applied in detail to two-dimensional (2D) advective mixing in flows resulting from the superposition of a transport flow through a channel and secondary localized cross-flows, here the vortical streaming due to a microbubble array. A simple description of stirring in a steady 2D vortex identifies the characteristic time beyond which vortex stirring becomes ineffective, with slow algebraic decay of the mix-variance. Duty cycling of the vortices introduces flow unsteadiness, for which optimum duty cycling protocols are identified, following analytically from a few selected Eulerian properties of the combined transport and vortex stirring flow. In comparison with experiments and simulations, it is shown that this simple formalism allows for the accurate prediction of optimal advective mixing, exponential in time, in the microbubble streaming case and, by extension, for any open-flow mixer with modulated secondary flow. Taking into account the effect of diffusion, estimated residence times required for complete mixing in such optimized devices are obtained.

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References (65)

  1. E. L. Shock and M. D. Schulte, Organic synthesis during fluid mixing in hydrothermal systems, J. Geophys. Res. Planets 103, 28513 (1998).
  2. J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport (Cambridge University Press, Cambridge, 1989).
  3. M. Kakuta, F. G. Bessoth, and A. Manz, Microfabricated devices for fluid mixing and their application for chemical synthesis, Chem. Record 1, 395 (2001).
  4. J. M. Ottino, Mixing, chaotic advection, and turbulence, Annu. Rev. Fluid Mech. 22, 207 (1990).
  5. E. Villermaux and J. Duplat, Mixing as an Aggregation Process, Phys. Rev. Lett. 91, 184501 (2003).
  6. E. Villermaux, Unifying ideas on mixing and atomization, New J. Phys. 6, 125 (2004).
  7. N.-T. Nguyen and Z. Wu, Micromixers—A review, J. Micromech. Microeng. 15, R1 (2005).
  8. E. Villermaux and C. Innocenti, On the geometry of turbulent mixing, J. Fluid Mech. 393, 123 (1999).
  9. 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 (2002).
  10. J. M. Ottino and S. Wiggins, Introduction: Mixing in microfluidics, Philos. Trans. Roy. Soc. Lond. A 362, 923 (2004).
  11. R. F. Ismagilov, A. D. Stroock, P. J. A. Kenis, G. Whitesides, and H. A. Stone, Experimental and theoretical scaling laws for transverse diffusive broadening in two-phase laminar flows in microchannels, Appl. Phys. Lett. 76, 2376 (2000).
  12. T. M. Antonsen, Jr., Z. Fan, E. Ott, and E. Garcia-Lopez, The role of chaotic orbits in the determination of power spectra of passive scalars, Phys. Fluids 8, 3094 (1996).
  13. P. Tabeling, M. Chabert, A. Dodge, C. Jullien, and F. Okkels, Chaotic mixing in cross-channel micromixers, Philos. Trans. Roy. Soc. Lond. A 362, 987 (2004).
  14. Z. Lin, J.-L. Thiffeault, and C. R. Doering, Optimal stirring strategies for passive scalar mixing, J. Fluid Mech. 675, 465 (2011).
  15. H. Aref, Stirring by chaotic advection, J. Fluid Mech. 143, 1 (1984).
  16. R. Burton and R. W. Easton, in Ergodicity of Linked Twist Maps (Springer, Berlin, 1980), pp. 35–49.
  17. R. Sturman, J. M. Ottino, and S. Wiggins, The Mathematical Foundations of Mixing (Cambridge University Press, Cambridge, 2006).
  18. A. P. Sudarsan and V. M. Ugaz, Multivortex micromixing, Proc. Natl. Acad. Sci. USA 103, 7228 (2006).
  19. J. Melin, G. Giménez, N. Roxhed, W. van der Wijngaart, and G. Stemme, A fast passive and planar liquid sample micromixer, Lab Chip 4, 214 (2004).
  20. H.-Y. Lee and J. Voldman, Optimizing micromixer design for enhancing dielectrophoretic microconcentrator performance, Anal. Chem. 79, 1833 (2007).
  21. M. H. Oddy, J. G. Santiago, and J. C. Mikkelsen, Electrokinetic instability micromixing, Anal. Chem. 73, 5822 (2001).
  22. P.-H. Huang, Y. Xie, D. Ahmed, J. Rufo, N. Nama, Y. Chen, C. Y. Chan, and T. J. Huang, An acoustofluidic micromixer based on oscillating sidewall sharp-edges, Lab Chip 13, 3847 (2013).
  23. P.-H. Huang, N. Nama, Z. Mao, P. Li, J. Rufo, Y. Chen, Y. Xie, C.-H. Wei, L. Wang, and T. J. Huang, A reliable and programmable acoustofluidic pump powered by oscillating sharp-edge structures, Lab Chip 14, 4319 (2014).
  24. P. Agrawal, P. S. Gandhi, and A. Neild, Particle manipulation affected by streaming flows in vertically actuated open rectangular chambers, Phys. Fluids 28, 032001 (2016).
  25. D. J. Collins, Z. Ma, J. Han, and Y. Ai, Continuous micro-vortex-based nanoparticle manipulation via focused surface acoustic waves, Lab Chip 17, 91 (2017).
  26. D. J. Collins, B. L. Khoo, Z. Ma, A. Winkler, R. Weser, H. Schmidt, J. Han, and Y. Ai, Selective particle and cell capture in a continuous flow using micro-vortex acoustic streaming, Lab Chip 17, 1769 (2017).
  27. J. A. Levitan, S. Devasenathipathy, V. Studer, Y. Ben, T. Thorsen, T. M. Squires, and M. Z. Bazant, Experimental observation of induced-charge electro-osmosis around a metal wire in a microchannel, Colloids Surf. A 267, 122 (2005).
  28. C. Canpolat, M. Zhang, W. Rosen, S. Qian, and A. Beskok, Induced-charge electroosmosis around touching metal rods, J. Fluids Eng. 135, 021103 (2013).
  29. C. Park and S. T. Wereley, Rapid generation and manipulation of microfluidic vortex flows induced by ac electrokinetics with optical illumination, Lab Chip 13, 1289 (2013).
  30. P. Marmottant and S. Hilgenfeldt, Controlled vesicle deformation and lysis by single oscillating bubbles, Nature (London) 423, 153 (2003).
  31. C. Wang, S. V. Jalikop, and S. Hilgenfeldt, Size-sensitive sorting of microparticles through control of flow geometry, Appl. Phys. Lett. 99, 034101 (2011).
  32. C. Wang, S. V. Jalikop, and S. Hilgenfeldt, Efficient manipulation of microparticles in bubble streaming flows, Biomicrofluid. 6, 012801 (2012).
  33. R. Thameem, B. Rallabandi, and S. Hilgenfeldt, Particle migration and sorting in microbubble streaming flows, Biomicrofluid. 10, 014124 (2016).
  34. R. H. Liu, J. Yang, M. Z. Pindera, M. Athavale, and P. Grodzinski, Bubble-induced acoustic micromixing, Lab Chip 2, 151 (2002).
  35. D. Ahmed, X. Mao, B. K. Juluri, and T. J. Huang, A fast microfluidic mixer based on acoustically driven sidewall-trapped microbubbles, Microfluid. Nanofluid. 7, 727 (2009).
  36. C. Wang, B. Rallabandi, and S. Hilgenfeldt, Frequency dependence and frequency control of microbubble streaming flows, Phys. Fluids 25, 022002 (2013).
  37. N. Bertin, T. A. Spelman, T. Combriat, H. Hue, O. Stéphan, E. Lauga, and P. Marmottant, Bubble-based acoustic micropropulsors: Active surfaces and mixers, Lab Chip 17, 1515 (2017).
  38. B. Rallabandi, C. Wang, and S. Hilgenfeldt, Two-dimensional streaming flows driven by sessile semicylindrical microbubbles, J. Fluid Mech. 739, 57 (2014).
  39. C. Wang, Microbubble streaming flows for non-invasive particle manipulation and liquid mixing, Ph.D. thesis, University of Illinois at Urbana-Champaign (2013).
  40. A. Marin, M. Rossi, B. Rallabandi, C. Wang, S. Hilgenfeldt, and C. J. Kähler, Three-Dimensional Phenomena in Microbubble Acoustic Streaming, Phys. Rev. Appl. 3, 041001 (2015).
  41. S. M. Davidson, M. Wessling, and A. Mani, On the dynamical regimes of pattern-accelerated electroconvection, Sci. Rep. 6, 22505 (2016).
  42. J. C. McDonald, D. C. Duffy, J. R. Anderson, D. T. Chiu, H. Wu, O. J. A. Schueller, and G. M. Whitesides, Fabrication of microfluidic systems in poly(dimethylsiloxane), Electrophoresis 21, 27 (2000).
  43. F. Bottausci, C. Cardonne, C. Meinhart, and I. Mezić, An ultrashort mixing length micromixer: The shear superposition micromixer, Lab Chip 7, 396 (2007).
  44. J. Aubin, M. Ferrando, and V. Jiricny, Current methods for characterising mixing and flow in microchannels, Chem. Eng. Sci. 65, 2065 (2010).
  45. D. Gobby, P. Angeli, and A. Gavriilidis, Mixing chacteristics of T-type microfluidic mixers, J. Micromech. Microeng. 11, 126 (2001).
  46. S. S. Wang, Z. J. Jiao, X. Y. Huang, C. Yang, and N. T. Nguyen, Acoustically induced bubbles in a microfluidic channel for mixing enhancement, Microfluid. Nanofluid. 6, 847 (2009).
  47. N. Riley, Steady streaming, Annu. Rev. Fluid Mech. 33, 43 (2001).
  48. N. A. Mortensen, F. Okkels, and H. Bruus, Reexamination of Hagen-Poiseuille flow: Shape dependence of the hydraulic resistance in microchannels, Phys. Rev. E 71, 057301 (2005).
  49. R. Thameem, Quantifying particle sorting in microbubble streaming flows, Master's thesis, University of Illinois at Urbana-Champaign (2015).
  50. L. Cortelezzi, A. Adrover, and M. Giona, Feasibility, efficiency and transportability of short-horizon optimal mixing protocols, J. Fluid Mech. 597, 199 (2008).
  51. O. Gubanov and L. Cortelezzi, Towards the design of an optimal mixer, J. Fluid Mech. 651, 27 (2010).
  52. D. Saintillan and M. J. Shelley, Emergence of coherent structures and large-scale flows in motile suspensions, J. Roy. Soc. Interface 9, 571 (2012).
  53. G. Mathew, I. Mezić, and L. Petzold, A multiscale measure for mixing, Physica D 211, 23 (2005).
  54. G. Mathew, I. Mezić, S. Grivopoulos, U. Vaidya, and L. Petzold, Optimal control of mixing in stokes fluid flows, J. Fluid Mech. 580, 261 (2007).
  55. G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. Roy. Soc. A 219, 186 (1953).
  56. R. Aris, On the dispersion of a solute in a fluid flowing through a tube, Proc. Roy. Soc. A 235, 67 (1956).
  57. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer Science & Business Media, New York, 2003), Vol. 2.
  58. P. Meunier and E. Villermaux, How vortices mix, J. Fluid Mech. 476, 213 (2003).
  59. J.-L. Thiffeault, Using multiscale norms to quantify mixing and transport, Nonlinearity 25, R1 (2012).
  60. S. Wiggins and J. M. Ottino, Foundations of chaotic mixing, Philos. Trans. Roy. Soc. Lond. A 362, 937 (2004).
  61. D. V. Khakhar, H. Rising, and J. M. Ottino, Analysis of chaotic mixing in two model systems, J. Fluid Mech. 172, 419 (1986).
  62. V. V. Meleshko and H. Aref, A blinking rotlet model for chaotic advection, Phys. Fluids 8, 3215 (1996).
  63. B. Rallabandi, A. Marin, M. Rossi, C. J. Kähler, and S. Hilgenfeldt, Three-dimensional streaming flow in confined geometries, J. Fluid Mech. 777, 408 (2015).
  64. P. B. Rhines and W. R. Young, How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133 (1983).
  65. E. J. Watson, Diffusion in oscillatory pipe flow, J. Fluid Mech. 133, 233 (1983).

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Phys. Rev. Fluids 2, 064501– Published 26 June, 2017
  • Received 22 December 2016
Article part of CHORUS

©2017 American Physical Society

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