We consider Taylor dispersion for tracer particles in microfluidic planar channels with strong confinement. In this context, the channel walls modify the local diffusivity tensor and also interactions between the tracer particles and the walls become important. We provide a simple and general formula for the effective diffusion constant along the channel as well as the first non-trivial finite time correction for arbitrary flows along the channel, arbitrary interaction potentials with the walls, and arbitrary expressions for the diffusion tensor. The formulas are in particular amenable to a straightforward numerical implementation, rendering them extremely useful for comparison with experiments. We present a number of applications, notably for systems that have parabolically varying diffusivity profiles, to systems with attractive interactions with the walls as well as electro-osmotic flows between plates with differing surface charges within the Debye–Hückel approximation.
References
1.
G. I.
Taylor
, “Dispersion of soluble matter in solvent flowing slowly through a tube
,” Proc. R. Soc. Lond. A
219
, 186
(1953
).2.
R.
Aris
, “On the dispersion of a solute in a fluid flowing through a tube
,” Proc. R. Soc. Lond.
235
, 67
(1956
).3.
I.
Frankel
and H.
Brenner
, “On the foundations of generalized Taylor dispersion theory
,” J. Fluid Mech.
204
, 97
(1989
).4.
N. G.
Barton
, “On the method of moments for solute dispersion
,” J. Fluid. Mech.
126
, 205
(1983
).5.
S.
Vedel
, E.
Hovad
, and H.
Bruus
, “Time dependent Taylor Aris dispersion of an initial point concentration
,” J. Fluid Mech.
752
, 107
(2014
).6.
A.
Adrover
, C.
Venditti
, and M.
Giona
, “Laminar dispersion at low and high Peclet numbers in a sinusoidal microtube: Point-size versus finite-size particles
,” Phys. Fluids
31
(6
), 062003
(2019
).7.
A.
Adrover
, C.
Passaretti
, C.
Venditti
, and M.
Giona
, “Exact moment analysis of transient dispersion properties in periodic media
,” Phys. Fluids
31
(11
), 112002
(2019
).8.
H.
Brenner
and D. A.
Edwards
, Macrotransport Processes
(Butterworth-Heinemann
, Stoneham
, 1993
).9.
T.
Guérin
and D. S.
Dean
, “Force induced dispersion in heterogeneous media
,” Phys. Rev. Lett.
115
, 020601
(2015
).10.
T.
Guérin
and D. S.
Dean
, “Kubo formulas for dispersion in heterogeneous periodic nonequilibrium systems
,” Phys. Rev. E
92
, 062103
(2015
).11.
G. N.
Mercer
and A. J.
Roberts
, “A centre manifold description of contaminant dispersion in channels with varying flow properties
,” SIAM J. Appl. Math.
50
, 1547
(1990
).12.
G. N.
Mercer
and A. J.
Roberts
, “A complete model of shear dispersion in pipes
,” Jpn. J. Ind. Appl. Math.
11
, 499
(1994
).13.
V.
Balakotaiah
and H. C.
Chang
, “Dispersion of chemical solutes in chromatographs and reactors
,” Phil. Trans. R. Soc. Lond. A
351
, 39
(1995
).14.
M.
Levesque
, O.
Bénichou
, R.
Voituriez
, and B.
Rotenberg
, “Taylor dispersion with adsorption and desorption
,” Phys. Rev. E
86
, 036316
(2012
).15.
A. M.
Berezhkovskii
and A. T.
Skvortsov
, “Aris-Taylor dispersion with drift and diffusion of particles on the tube wall
,” J. Chem. Phys.
139
, 084101
(2013
).16.
J.
Happel
and H.
Brenner
, Low Reynolds Number Hydrodynamics
(Kluwer
, Dordrecht
, 1991
).17.
C. W.
Oseen
, Neuere Methoden und Ergebnisse in der Hydrodynamik
(Akademische Verlags-gesellschaft
, Leipzig
, 1927
).18.
P.
Ganatos
, S.
Weinbaum
, and R.
Pfeffer
, “A strong interaction theory for the creeping motion of a sphere between plane parallel boundaries. Part 1. Perpendicular motion
,” J. Fluid Mech.
99
, 739
(1980
).19.
E. R.
Dufresne
, D.
Altman
, and D. G.
Grier
, “Brownian dynamics of a sphere between parallel walls
,” Europhys. Lett.
53
, 264
(2001
).20.
H.
Brenner
and L. J.
Gaydos
, “The constrained Brownian movement of spherical particles in cylindrical pores of comparable radius: Models of the diffusive and convective transport of solute molecules in membranes and porous media
,” J. Colloid Interface Sci.
58
, 312
(1977
).21.
L. P.
Faucheux
and A. J.
Libchaber
, “Confined Brownian motion
,” Phys. Rev. E
49
, 5158
(1994
).22.
A.
Vilquin
, V.
Bertin
, P.
Soulard
, G.
Guyard
, E.
Raphaël
, F.
Restagno
, T.
Salez
, and J.
Mcgraw
, “Time dependence of advection-diffusion coupling for nanoparticle ensembles
,” arXiv:2007.08261 (2020
).23.
A. W. C.
Lau
and T. C.
Lubensky
, “State-dependent diffusion: Thermodynamic consistency and its path integral formulation
,” Phys. Rev. E
76
, 011123
(2007
).24.
Y.
Avni
, S.
Komura
, and D.
Andelman
, “Brownian motion of a charged colloid in restricted confinement
,” arXiv:2012.07070 (2020
).25.
S. K.
Griffiths
and R. H.
Nilson
, “Hydrodynamic dispersion of a neutral nonreacting solute in electroosmotic flow
,” Anal. Chem.
71
, 5522
(1999
).26.
G.
Barton
, Elements of Green's Functions and Propagation
(Clarendon Press
, Oxford
, 1989
).27.
A.
Singer
, Z.
Schuss
, A.
Osipov
, and D.
Holcman
, “Partially reflected diffusion
,” SIAM J. Appl. Math.
68
(68
), 844
–668
(2008
).28.
C.
Gardiner
, Stochastic Methods for Physics, and Handbook for the Natural and Social Sciences
(Springer Series in Synergetics
, 2009
).29.
D. S.
Dean
and G.
Oshanin
, “Approach to asymptotically diffusive behavior for Brownian particles in periodic potentials: Extracting information from transients
,” Phys. Rev. E
90
, 022112
(2014
).30.
D. S.
Dean
and T.
Guérin
, “Approach to asymptotically diffusive behavior for Brownian particles in media with periodic diffusivities
,” Phys. Rev. E
90
, 062114
(2014
).31.
J. H.
Masliyah
and S.
Bhattacharjee
, Electrokinetic and Colloid Transport Phenomena
(John Wiley and Sons
, Hoboken
, 2006
).32.
D.
Mampallil
and D.
van den Ende
, “Electroosmotic shear flow in microchannels
,” J. Colloid Interface Sci.
390
, 234
(2013
).33.
S. K.
Griffiths
and R. H.
Nilson
, “Electroosmotic fluid flow and late time solute transport for large zeta potentials
,” Anal. Chem.
72
, 4767
(2000
).© 2021 Author(s). Published under an exclusive license by AIP Publishing.
2021
Author(s)
You do not currently have access to this content.
