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Transient Taylor–Aris dispersion for time-dependent flows in straight channels

  • Søren Vedel (a1) and Henrik Bruus (a1)
Abstract

Taylor–Aris dispersion, the shear-induced enhancement of solute diffusion in the flow direction of the solvent, has been studied intensely in the past half century for the case of steady flow and single-frequency pulsating flows. Here, combining Aris’s method of moments with Dirac’s bra–ket formalism, we derive an expression for the effective solute diffusivity valid for transient Taylor–Aris dispersion in any given time-dependent, multi-frequency solvent flow through straight channels. Our theory shows that the solute dispersion may be greatly enhanced by the time-dependent parts of the flow, and it explicitly reveals how the dispersion coefficient depends on the external driving frequencies of the velocity field and the internal relaxation rates for mass and momentum diffusion. Although applicable to any type of fluid, we restrict the examples of our theory to Newtonian fluids, for which we both recover the known results for steady and single-frequency pulsating flows, and find new, richer structure of the dispersion as function of system parameters in multi-frequency systems. We show that the effective diffusivity is enhanced significantly by those parts of the time-dependent velocity field that have frequencies smaller than the fluid momentum diffusion rate and the solute diffusion rate.

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Email address for correspondence: soren.vedel@nanotech.dtu.dk
References
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1. Ajdari, A., Bontoux, N. & Stone, H. A. 2006 Hydrodynamic dispersion in shallow microchannels: the effect of cross-sectional shape. Analyt. Chem. 78, 387392.
2. Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A Mat. 235 (1200), 6777.
3. Aris, R. 1960 On the dispersion of solute in pulsating flow through a tube. Proc. R. Soc. Lond. A Mat. 259 (1298), 370376.
4. Bandyopadhyay, S. & Mazumder, B. S. 1999 Unsteady convective diffusion in a pulsatile flow through a channel. Acta Mechanica 134, 116.
5. Barton, N. G. 1983 On the method of moments for solute dispersion. J. Fluid Mech. 126, 205218.
6. Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
7. Bird, R. B., Stewart, W. E. & Lightfoot, E. N. 2006 Transport Phenomena, 2nd edn. John Wiley and Sons.
8. Bontoux, N., Pépin, A., Chen, Y., Ajdari, A. & Stone, H. A. 2006 Experimental characterization of hydrodynamic dispersion in shallow microchannels. Lab on a Chip 6, 930935.
9. Brenner, H. & Edwards, D. A. 1993 Macrotransport Processes. Butterworth–Heinemann.
10. Bruus, H. 2008 Theoretical Microfluidics. Oxford University Press.
11. Bruus, H. & Stone, A. D. 1994 Quantum chaos in a deformable billiard: applications to quantum dots. Phys. Rev. B 50 (24), 1827518287.
12. Camassa, R., Lin, Z. & McLaughlin, R. 2010 The exact evolution of scalar variance in pipe and channel flow. Commun. Math. Sci. 8 (2), 601626.
13. Chatwin, P. C. 1975 On the longitudinal dispersion of passive contaminant in oscillatory flows in tubes. J. Fluid Mech. 71, 513527.
14. Chatwin, P. C. 1977 Initial development of longitudinal dispersion in straight tubes. J. Fluid Mech. 80, 3348.
15. Chatwin, P. C. & Sullivan, P. J. 1982 The effects of aspect ratio on longitudinal diffusivity in rectangular channels. J. Fluid Mech. 120, 347358.
16. Dirac, P. A. M. 1981 The Principles of Quantum Mechanics, 4th edn. Oxford University Press.
17. Doshi, M. R., Daiya, P. M. & Gill, W. N. 1978 Three dimensional laminar dispersion in open and closed rectangular conduits. Chem. Engng Sci. 33, 795804.
18. Dutta, D., Ramachandran, A & Leighton, D. T. 2006 Effect of channel geometry on solute dispersion in pressure-driven microfluidic systems. Microfluid Nanofluid 2, 275290.
19. Erdogan, M. E. & Chatwin, P. C. 1967 Effects of curvature and buoyancy on laminar dispersion of solute in a horizontal tube. J. Fluid Mech. 29 (3), 465484.
20. Fan, L. T. & Wang, C. B. 1966 Dispersion of matter in non-Newtonian laminar flow through a circular tube. Proc. R. Soc. Lond. A Mat. 292 (1429), 203208.
21. Gleeson, J. P. 2002 Electroosmotic flows with random zeta potential. J. Colloid Interface Sci. 249 (1), 217226.
22. Goddard, J. D. 1993 The Green’s function for passive scalar diffusion in a homogeneously sheared continuum. Phys. Fluids A 5, 22952297.
23. Harris, H. G. & Goren, S. L. 1967 Axial diffusion in a cylinder with pulsed flow. Chem. Engng Sci. 22, 15711576.
24. Jansons, K. M. 2006 On Taylor dispersion in oscillatory channel flows. Proc. R. Soc. Lond. A Mat. 462, 35013509.
25. Latini, M & Bernoff, AJ 2001 Transient anomalous diffusion in Poiseuille flow. J. Fluid Mech. 441, 399411.
26. Leighton, D. T. 1989 Diffusion from an intial point distribution in an unbounded oscillating simple shear flow. Physico-Chem. Hydrodyn. 11, 377386.
27. Lide (editor-in-chief), D. R. 1995 CRC Handbook of Chemistry and Physics, 75th edn. CRC Press.
28. Mehta, M. L. 2004 Random Matrices, 3rd edn. Pure and Applied Mathematics , vol. 142. Elsevier/Academic Press.
29. Molloy, R. F. & Leighton, D. T. 1998 Binary oscillatory cross-flow electrophoresis: theory and experiments. J. Pharma. Sci. 87, 12701281.
30. Mortensen, N. A. & Bruus, H. 2006 Universal dynamics in the onset of a Hagen–Poiseuille flow. Phys. Rev. E 74 (1), 017301.
31. Mortensen, N. A., Olesen, L. H. & Bruus, H. 2006 Transport coefficients for electrolytes in arbitrarily shaped nano and micro-fluidic channels. New J. Phys. 8, 3751.
32. Mukherjee, A. & Mazumder, B. S. 1988 Dispersion of contaminant in oscillatory flows. Acta Mechanica 74, 107.
33. Paul, S. & Mazumder, B. S. 2008 Dispersion in unsteady Couette–Poiseuille flows. Intl J. Engng Sci. 46, 12031217.
34. Probstein, R. F. 1994 Physicochemical Hydrodynamics. An Introduction, 2nd edn. John Wiley and Sons.
35. Sankarasubramanian, R. & Gill, W. N. 1973 Unsteady convective diffusion with interphase mass-transfer. Proc. R. Soc. Lond. A Mat. 333 (1592), 115132.
36. Skafte-Pedersen, P., Sabourin, D., Dufva, M. & Snakenborg, D. 2009 Multi-channel peristaltic pump for microfluidic applications featuring monolithic PDMS inlay. Lab on a Chip 9, 30033006.
37. Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A Mat. 219 (1137), 186.
38. Taylor, H. M. & Leonard, E. F. 1965 Axial dispersion during pulsating pipe flow. AIChE J. 11 (4), 686689.
39. Thomas, A. M. & Narayanan, R. 2001 Physics of oscillatory flow and its effect on the mass transfer and separation of species. Phys. Fluids 13 (4), 859866.
40. van den Broeck, C. 1982 A stochastic description of longitudinal dispersion in uniaxial flows. Physica A 112, 343352.
41. Vedel, S., Olesen, L. H. & Bruus, H. 2010 Pulsatile microfluidics as an analytical tool for determining the dynamic characteristics of microfluidic systems. J. Micromech. Microengng 20, 035026.
42. Vikhansky, A. & Wang, W. 2011 Taylor dispersion in finite-length capillaries. Chem. Engng Sci. 66 (4), 642649.
43. Watson, E. J. 1983 Diffusion in oscillatory pipe flow. J. Fluid Mech. 133, 233244.
44. Womersley, J. R. 1955 Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127, 553563.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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