A multiple-scale perturbation theory is developed to analyze the advection-diffusion transport of a passive solute through a parallel-plate channel. The fluid velocity comprises a steady and a time-oscillatory component, which may vary spatially in the transverse and streamwise directions, and temporally on the fast transverse diffusion timescale. A long-time asymptotic equation governing the evolution of the transverse averaged solute concentration is derived, complemented with Taylor dispersion coefficients and advection speed corrections that are functions of the streamwise coordinate. We demonstrate the theory with a two-dimensional flow in a channel comprising alternating shear-free and no-slip regions. For a steady flow, the dispersion coefficient changes from zero to a finite value when the flow transitions from plug-like in the shear-free section to parabolic in the no-slip region. For an oscillatory flow, the dispersion coefficient due to an oscillatory flow can be negative and two orders of magnitude larger than that due to a steady flow of the same amplitude. This motivates us to quantify the relative magnitude of the steady and oscillatory flow such that there is an overall positive dispersion coefficient necessary for an averaged (macrotransport) equation. We further substitute the transport coefficients into the averaged equation to compute the evolution of the concentration profile, which agrees well with that obtained by solving the full two-dimensional advection-diffusion equation. In a steady flow, we find that while the shear-free section suppresses band broadening, the following no-slip section may lead to a wider band compared with the dispersion driven by the same pressure gradient in an otherwise homogeneously no-slip channel. In an unsteady flow, we demonstrate that a naive implementation of the macrotransport theory with a (localized) negative dispersion coefficient will result in an aphysical finite time singularity (or “blow-up solution”), in contrast to the well-behaved solution of the full advection-diffusion equation.
REFERENCES
1.
G. I.
Taylor
, “Dispersion of soluble matter in solvent flowing slowly through a tube
,” Proc. R. Soc. London, Ser. A
219
, 186
–203
(1953
).2.
R.
Aris
, “On the dispersion of a solute in a fluid flowing through a tube
,” Proc. R. Soc. London, Ser. A
235
, 67
–77
(1956
).3.
M. T.
Blom
, E.
Chmela
, R. E.
Oosterbroek
, R.
Tijssen
, and A.
van den Berg
, “On-chip hydrodynamic chromatography separation and detection of nanoparticles and biomolecules
,” Anal. Chem.
75
, 6761
–6768
(2003
).4.
H.
Cottet
, J. P.
Biron
, and M.
Martin
, “Taylor dispersion analysis of mixtures
,” Anal. Chem.
79
, 9066
–9073
(2007
).5.
L.
Cipelletti
, J. P.
Biron
, M.
Martin
, and H.
Cottet
, “Measuring arbitrary diffusion coefficient distributions of nano-objects by Taylor dispersion analysis
,” Anal. Chem.
87
, 8489
–8496
(2015
).6.
R.
Smith
, “Contaminant dispersion in oscillatory flows
,” J. Fluid Mech.
114
, 379
–398
(1982
).7.
S. M.
Kashefipour
and R. A.
Falconer
, “Longitudinal dispersion coefficients in natural channels
,” Water Res.
36
, 1596
–1608
(2002
).8.
C. J. L.
Taylor
, “The effects of biological fouling control at coastal and estuarine power stations
,” Mar. Pollut. Bull.
53
, 30
–48
(2006
).9.
M. S.
Roberts
and M.
Rowland
, “A dispersion model of hepatic elimination: 2. Steady-state considerations-influence of hepatic blood flow, binding within blood, and hepatocellular enzyme activity
,” J. Pharmacokinet. Pharmacodyn.
14
, 261
–288
(1986
).10.
F.
Gentile
, M.
Ferrari
, and P.
Decuzzi
, “The transport of nanoparticles in blood vessels: The effect of vessel permeability and blood rheology
,” Ann. Biomed. Eng.
36
, 254
–261
(2007
).11.
J.
Tan
, A.
Thomas
, and Y.
Liu
, “Influence of red blood cells on nanoparticle targeted delivery in microcirculation
,” Soft Matter
8
, 1934
–1946
(2012
).12.
J. B.
Grotberg
, “Pulmonary flow and transport phenomena
,” Annu. Rev. Fluid Mech.
26
, 529
–571
(1994
).13.
G. A.
Ledezma
, A.
Folch
, S. N.
Bhatia
, U. J.
Balis
, M. L.
Yarmush
, and M.
Toner
, “Numerical model of fluid flow and oxygen transport in a radial-flow microchannel containing hepatocytes
,” J. Biomech. Eng.
121
, 58
–64
(1999
).14.
D. A.
Beard
and F.
Wu
, “Apparent diffusivity and Taylor dispersion of water and solutes in capillary beds
,” Bull. Math. Biol.
71
, 1366
–1377
(2009
).15.
K.
Gulabivala
, Y. L.
Ng
, M.
Gilbertson
, and I.
Eames
, “The fluid mechanics of root canal irrigation
,” Physiol. Meas.
31
, R49
–R84
(2010
).16.
W.
Ding
, W.
Li
, S.
Sun
, X.
Zhou
, P. A.
Hardy
, S.
Ahmad
, and D.
Gao
, “Three-dimensional simulation of mass transfer in artificial kidneys
,” Artif. Organs
39
, E79
–E89
(2015
).17.
R.
Aris
, “On the dispersion of a solute by diffusion, convection and exchange between phases
,” Proc. R. Soc. London, Ser. A
252
, 538
–550
(1959
).18.
C. C.
Mei
, J. L.
Auriault
, and C. O.
Ng
, “Some applications of the homogenization theory
,” Adv. Appl. Mech.
32
, 277
–348
(1996
).19.
P. C.
Chatwin
, “The approach to normality of the concentration distribution of solute in a solvent flowing along a straight pipe
,” J. Fluid Mech.
43
, 321
–352
(1970
).20.
N. G.
Barton
, “On the method of moments for solute dispersion
,” J. Fluid Mech.
126
, 205
–218
(1983
).21.
S.
Vedel
and H.
Bruus
, “Transient Taylor-Aris dispersion for time-dependent flows in straight channels
,” J. Fluid Mech.
691
, 95
–122
(2012
).22.
S.
Vedel
, E.
Howad
, and H.
Bruus
, “Time-dependent Taylor-Aris dispersion of an initial point concentration
,” J. Fluid Mech.
752
, 107
–122
(2014
).23.
S.
Debnath
, A. K.
Saha
, B. S.
Mazumder
, and A. K.
Roy
, “Laminar dispersion at low and high peclet numbers in finite-length patterned microtubes
,” Phys. Fluids
29
, 097107
(2017
).24.
M. J.
Lighthill
, “Initial development of diffusion in Poiseuille flow
,” J. Inst. Maths. Applics.
2
, 97
–108
(1966
).25.
W. N.
Gill
and R.
Sankarasubramanian
, “Exact analysis of unsteady convective diffusion
,” Proc. R. Soc. London, Ser. A
316
, 341
–350
(1970
).26.
R.
Smith
, “A delay-diffusion description for contaminant dispersion
,” J. Fluid Mech.
105
, 469
–486
(1981
).27.
A. N.
Stokes
and N. G.
Barton
, “The concentration distribution produced by shear dispersion of solute in Poiseuille flow
,” J. Fluid Mech.
210
, 201
–221
(1990
).28.
C. G.
Phillips
and S. R.
Kaye
, “A uniformly asymptotic approximation for the development of shear dispersion
,” J. Fluid Mech.
329
, 413
–443
(1996
).29.
W. N.
Gill
and R.
Sankarasubramanian
, “Dispersion of a non-uniform slug in time-dependent flow
,” Proc. R. Soc. London, Ser. A
322
, 101
–117
(1971
).30.
P. C.
Chatwin
, “On the longitudinal dispersion of passive contaminant in oscillatory flows in tubes
,” J. Fluid Mech.
71
, 513
–527
(1975
).31.
H.
Brenner
, “A general theory of Taylor dispersion phenomena. II. An extension
,” PhysicoChem. Hydrodyn.
3
, 139
–157
(1982
).32.
M.
Shapiro
and H.
Brenner
, “Taylor dispersion in the presence of time-periodic convection phenomena. Part II. Transport of transversely oscillating Brownian particles in a plane Poiseuille flow
,” Phys. Fluids
2
, 1744
–1753
(1990
).33.
R.
Smith
, “Longitudinal dispersion coefficients for varying channels
,” J. Fluid Mech.
130
, 299
–314
(1983
).34.
H. A.
Stone
and H.
Brenner
, “Dispersion in flows with streamwise variations of mean velocity: Radial flow
,” Ind. Eng. Chem. Res.
38
, 851
–854
(1999
).35.
W. N.
Gill
and U.
Guceri
, “Laminar dispersion in Jeffery-Hamel flows: Part 1. Diverging channels
,” AIChE J.
17
, 207
–214
(1971
).36.
M. D.
Bryden
and H.
Brenner
, “Multiple-timescale analysis of Taylor dispersion in converging and diverging flows
,” J. Fluid Mech.
311
, 343
–359
(1996
).37.
H.
Brenner
, “Dispersion resulting from flow through spatially periodic porous media
,” Proc. R. Soc. London, Ser. A
297
, 81
–133
(1980
).38.
A.
Adrover
and S.
Cerbelli
, “Laminar dispersion at low and high peclet numbers in finite-length patterned microtubes
,” Phys. Fluids
29
, 062005
(2017
).39.
A.
Adrover
, S.
Cerbelli
, and M.
Giona
, “Taming axial dispersion in hydrodynamic chromatography columns through wall patterning
,” Phys. Fluids
30
, 042002
(2018
).40.
D. L.
Koch
and J. F.
Brady
, “A non-local description of advection-diffusion with application to dispersion in porous media
,” J. Fluid Mech.
180
, 387
–403
(1989
).41.
S. P.
Neuman
, “Eulerian-Lagrangian theory of transport in space-time nonstationary velocity fields: Exact nonlocal formalism by conditional moments and weak approximation
,” Water Resour. Res.
29
, 633
–645
, https://doi.org/10.1029/92wr02306 (1993
).42.
S.
Ghosal
and S.
Datta
, “The effect of wall interactions in capillary-zone electrophoresis
,” J. Fluid Mech.
491
, 285
–300
(2003
).43.
S.
Datta
and S.
Ghosal
, “Dispersion due to wall interactions in microfluidic separation systems
,” J. Fluid Mech.
20
, 012103
(2008
).44.
J. P.
Gleeson
and H. A.
Stone
, “Taylor dispersion in electroosmotic flows with random zeta potentials
,” Technical Proceedings of the 2004 NSTI Nanotechnology Conference and Trade Show
(TechConnect
, 2004
) Vol. 2, pp. 375
–378
.45.
P. C.
Fife
and K. R. K.
Nicholes
, “Dispersion in flow through small tubes
,” Proc. R. Soc. London, Ser. A
344
, 131
–145
(1975
).46.
C. O.
Ng
, “Dispersion in steady and oscillatory flows through a tube with reversible and irreversible wall reactions
,” Proc. R. Soc. London, Ser. A
462
, 481
–515
(2006
).47.
C.
Vargas
, J.
Arcos
, O.
Bautista
, and F.
Mendez
, “Hydrodynamic dispersion in a combined magnetohydrodynamic-electroosmotic-driven flow through a microchannel with slowly varying wall zeta potentials
,” Phys. Fluids
29
, 092002
(2017
).48.
A.
Mukherjee
and B. S.
Mazumder
, “Dispersion of contaminant in oscillatory flows
,” Acta Mech.
74
, 107
–122
(1988
).49.
B. S.
Mazumder
and S. K.
Das
, “Effect of boundary reaction on solute dispersion in pulsatile flow through a tube
,” J. Fluid Mech.
239
, 523
–549
(1992
).50.
S.
Bandyopadhyay
and B. S.
Mazumder
, “On contaminant dispersion in unsteady generalised Couette flow
,” Int. J. Eng. Sci.
37
, 1407
–1423
(1999
).51.
S.
Bandyopadhyay
and B. S.
Mazumder
, “Unsteady convective diffusion in a pulsatile flow through a channel
,” Acta Mech.
134
, 1
–16
(1999
).52.
J. R.
Philip
, “Flows satisfying mixed no-slip and no-shear conditions
,” Z. Angew. Math. Phys.
23
, 353
–372
(1972
).53.
E.
Lauga
and H. A.
Stone
, “Effective slip in pressure-driven Stokes flow
,” J. Fluid Mech.
489
, 55
–77
(2003
).54.
T. M.
Squires
, “Electrokinetic flows over inhomogeneously slipping surfaces
,” Phys. Fluids
20
, 092105
(2008
).55.
C. O.
Ng
, H. C. W.
Chu
, and C. Y.
Wang
, “On the effects of liquid-gas interfacial shear on slip flow through a parallel-plate channel with superhydrophobic grooved walls
,” Phys. Fluids
22
, 102002
(2010
).56.
O. I.
Vinogradova
and A. V.
Belyaev
, “Wetting, roughness and flow boundary conditions
,” J. Phys.: Condens. Matter
23
, 184104
(2011
).57.
H. C. W.
Chu
and C. O.
Ng
, “Electroosmotic flow through a circular tube with slip-stick striped walls
,” J. Fluids Eng.
134
, 111201
(2012
).58.
H. C. W.
Chu
and C. O.
Ng
, “Oscillatory electro-osmotic flow through a slit channel with slipping stripes on walls
,” Fluid Dyn. Res.
45
, 022507
(2013
).59.
C. O.
Ng
and H. C. W.
Chu
, “Electrokinetic flows through a parallel-plate channel with slipping stripes on walls
,” Phys. Fluids
23
, 102002
(2011
).60.
R.
Aris
, “On the dispersion of a solute in pulsating flow through a tube
,” Proc. R. Soc. London, Ser. A
259
, 370
–376
(1960
).61.
R.
Smith
, “Dispersion of tracers in the deep ocean
,” J. Fluid Mech.
123
, 131
–142
(1982
).62.
R.
Smith
, “The contraction of contaminant distributions in reversing flows
,” J. Fluid Mech.
129
, 137
–151
(1983
).63.
E. J.
Watson
, “Diffusion in oscillatory pipe flow
,” J. Fluid Mech.
133
, 233
–244
(1983
).64.
H.
Yasuda
, “Longitudinal dispersion of matter due to the shear effect of steady and oscillatory currents
,” J. Fluid Mech.
148
, 383
–403
(1984
).65.
M. R.
Doshi
, P. M.
Daiya
, and W. N.
Gill
, “Three dimensional laminar dispersion in open and closed rectangular conduits
,” Chem. Eng. Sci.
33
, 795
–804
(1978
).66.
P. C.
Chatwin
and P. J.
Sullivan
, “The effect of aspect ratio on longitudinal diffusivity in rectangular channels
,” J. Fluid Mech.
120
, 347
–358
(1982
).67.
A.
Ajdari
, N.
Bontoux
, and H. A.
Stone
, “Hydrodynamic dispersion in shallow microchannels: The effect of cross-sectional shape
,” Anal. Chem.
78
, 387
–392
(2006
).68.
D.
Dutta
, A.
Ramachandran
, and D. T.
Leighton
, “Effect of channel geometry on solute dispersion in pressure-driven microfluidic systems
,” Microfluid. Nanofluid.
2
, 275
–290
(2006
).69.
S.
Datta
and S.
Ghosal
, “Characterizing dispersion in micro-fluidic channels
,” Lab Chip
9
, 2537
–2550
(2009
).© 2019 Author(s).
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