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Hydrodynamic dispersion in Hele-Shaw flows with inhomogeneous wall boundary conditions

Published online by Cambridge University Press:  23 August 2021

Sebastian Dehe Open the ORCID record for Sebastian Dehe [Opens in a new window] ,
Imke-Sophie Rehm Open the ORCID record for Imke-Sophie Rehm [Opens in a new window]  and
Steffen Hardt Open the ORCID record for Steffen Hardt [Opens in a new window]
Sebastian Dehe
Affiliation:
Fachbereich Maschinenbau, Technische Universität Darmstadt, 64287 Darmstadt, Germany
Imke-Sophie Rehm
Affiliation:
Fachbereich Maschinenbau, Technische Universität Darmstadt, 64287 Darmstadt, Germany
Steffen Hardt*
Affiliation:
Fachbereich Maschinenbau, Technische Universität Darmstadt, 64287 Darmstadt, Germany
*
†Email address for correspondence: hardt@nmf.tu-darmstadt.de
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Abstract

Single-phase flow inside a Hele-Shaw cell can exhibit inhomogeneous flow fields, for example when actuated by electroosmosis with varying wall mobilities, leading to internal pressure gradients. We derive a two-dimensional dispersion model for a dissolved species in such a non-uniform flow field, utilizing a multiple-scale perturbation approach. The resulting two-dimensional transport equation is an advection–diffusion equation containing an effective dispersion tensor field and additional advection-correction terms. It can be viewed as a generalization of the well-known Taylor–Aris dispersion model. The dispersion model allows for flow fields with both stationary and oscillatory components. For the special case of non-uniform flow induced by both pressure gradients and electroosmosis, we derive expressions for the flow field in the long-wavelength limit. These include arbitrary, time-dependent functions for both the driving field as well as the wall mobilities. We discuss the general characteristics of the model using a sinusoidally varying wall mobility, and derive analytical expressions for the dispersion tensor. Then, in order to validate the model, we compare three-dimensional Lagrangian particle tracing simulations with the dispersion model for several test cases, including stationary and oscillatory shear flow as well as a recirculating flow field. For each test case, a good agreement between the full three-dimensional simulations and the results of the two-dimensional dispersion model is obtained. The presented model has the potential to significantly simplify computations of mass transport in Hele-Shaw flows.

JFM classification

Low-Reynolds-number flows: Hele-Shaw flows Mass Transport: Dispersion
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 925 , 25 October 2021 , A11
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Adrover, A. 2013 Effect of secondary flows on dispersion in finite-length channels at high Péclet numbers. Phys. Fluids 25 (9), 093601.CrossRefGoogle Scholar
Ajdari, A. 1996 Generation of transverse fluid currents and forces by an electric field: electro-osmosis on charge-modulated and undulated surfaces. Phys. Rev. E 53 (5), 4996–5005.CrossRefGoogle ScholarPubMed
Ajdari, A. 2001 Transverse electrokinetic and microfluidic effects in micropatterned channels: lubrication analysis for slab geometries. Phys. Rev. E 65 (1), 016301.CrossRefGoogle ScholarPubMed
Ajdari, A., Bontoux, N. & Stone, H.A. 2006 Hydrodynamic dispersion in shallow microchannels: the effect of cross-sectional shape. Anal. Chem. 78 (2), 387–392.CrossRefGoogle ScholarPubMed
Ananthakrishnan, V., Gill, W.N. & Barduhn, A.J. 1965 Laminar dispersion in capillaries: part I. Mathematical analysis. AIChE J. 11 (6), 1063–1072.CrossRefGoogle Scholar
Arcos, J.C., Méndez, F., Bautista, E.G. & Bautista, O. 2018 Dispersion coefficient in an electro-osmotic flow of a viscoelastic fluid through a microchannel with a slowly varying wall zeta potential. J. Fluid Mech. 839, 348–386.CrossRefGoogle Scholar
Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.CrossRefGoogle Scholar
Aris, R. 1960 On the dispersion of a solute in pulsating flow through a tube. Proc. R. Soc. Lond. A 259 (1298), 370–376.Google Scholar
Aris, R. & Taylor, G.I. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A 235 (1200), 67–77.Google Scholar
Bacheva, V., Paratore, F., Rubin, S., Kaigala, G.V. & Bercovici, M. 2020 Tunable bidirectional electroosmotic flow for diffusion-based separations. Angew. Chem. 132 (31), 12994–12999.CrossRefGoogle Scholar
Bahga, S.S., Bercovici, M. & Santiago, J.G. 2012 Robust and high-resolution simulations of nonlinear electrokinetic processes in variable cross-section channels. Electrophoresis 33 (19–20), 3036–3051.CrossRefGoogle ScholarPubMed
Bandyopadhyay, S. & Mazumder, B.S. 1999 On contaminant dispersion in unsteady generalised Couette flow. Intl J. Engng Sci. 37 (11), 1407–1423.CrossRefGoogle Scholar
Barton, N.G. 1983 On the method of moments for solute dispersion. J. Fluid Mech. 126, 205–218.CrossRefGoogle Scholar
Boyko, E., Bacheva, V., Eigenbrod, M., Paratore, F., Gat, A.D., Hardt, S. & Bercovici, M. 2021 Microscale hydrodynamic cloaking and shielding via electro-osmosis. Phys. Rev. Lett. 126 (18), 184502.CrossRefGoogle ScholarPubMed
Boyko, E., Rubin, S., Gat, A.D. & Bercovici, M. 2015 Flow patterning in Hele-Shaw configurations using non-uniform electro-osmotic slip. Phys. Fluids 27 (10), 102001.CrossRefGoogle Scholar
Brenner, H. & Edwards, D.A. 1993 Macrotransport Processes, 1st edn. Butterworth-Heinemann.Google Scholar
Chatwin, P.C. 1975 On the longitudinal dispersion of passive contaminant in oscillatory flows in tubes. J. Fluid Mech. 71 (3), 513–527.CrossRefGoogle Scholar
Chu, H.C.W., Garoff, S., Przybycien, T.M., Tilton, R.D. & Khair, A.S. 2019 Dispersion in steady and time-oscillatory two-dimensional flows through a parallel-plate channel. Phys. Fluids 31 (2), 022007.CrossRefGoogle Scholar
Comsol Multiphysics 2019 COMSOL Multiphysics Reference Manual, Version 5.5.Google Scholar
Datta, R. & Kotamarthi, V.R. 1990 Electrokinetic dispersion in capillary electrophoresis. AIChE J. 36 (6), 916–926.CrossRefGoogle Scholar
Datta, S. & Choudhary, J.N. 2013 Effect of hydrodynamic slippage on electro-osmotic flow in zeta potential patterned nanochannels. Fluid Dyn. Res. 45 (5), 055502.CrossRefGoogle Scholar
Datta, S. & Ghosal, S. 2008 Dispersion due to wall interactions in microfluidic separation systems. Phys. Fluids 20 (1), 012103.CrossRefGoogle Scholar
Dehe, S., Rofman, B., Bercovici, M. & Hardt, S. 2020 Electro-osmotic flow enhancement over superhydrophobic surfaces. Phys. Rev. Fluids 5 (5), 053701.CrossRefGoogle Scholar
Fife, P.C. & Nicholes, K.R.K. 1975 Dispersion in flow through small tubes. Proc. R. Soc. Lond. A 344 (1636), 131–145.Google Scholar
Ghosal, S. 2002 Lubrication theory for electro-osmotic flow in a microfluidic channel of slowly varying cross-section and wall charge. J. Fluid Mech. 459, 103–128.CrossRefGoogle Scholar
Ghosal, S. 2003 The effect of wall interactions in capillary-zone electrophoresis. J. Fluid Mech. 491 (491), 285–300.CrossRefGoogle Scholar
Ghosh, U. & Chakraborty, S. 2015 Electroosmosis of viscoelastic fluids over charge modulated surfaces in narrow confinements. Phys. Fluids 27 (6), 062004.CrossRefGoogle Scholar
Gill, W.N., Ananthakrishnan, V. & Nunge, R.J. 1968 Dispersion in developing velocity fields. AIChE J. 14 (6), 939–946.CrossRefGoogle Scholar
Gill, W.N., Sankarasubramanian, R. & Taylor, G.I. 1971 Dispersion of a non-uniform slug in time-dependent flow. Proc. R. Soc. Lond. A 322 (1548), 101–117.Google Scholar
Giona, M., Adrover, A., Cerbelli, S. & Garofalo, F. 2009 Laminar dispersion at high Péclet numbers in finite-length channels: effects of the near-wall velocity profile and connection with the generalized Leveque problem. Phys. Fluids 21 (12), 123601.CrossRefGoogle Scholar
Gleeson, J.P. & Stone, H.A. 2004 Taylor dispersion in electroosmotic flows with random zeta potentials. In Technical Proceedings of the 2004 NSTI Nanotechnology Conference and Trade Show, NSTI Nanotech, vol. 2, pp. 375–378. TechConnect.Google Scholar
Griffiths, S.K. & Nilson, R.t H. 2000 Electroosmotic fluid motion and late-time solute transport for large zeta potentials. Anal. Chem. 72 (20), 4767–4777.CrossRefGoogle ScholarPubMed
Hendy, S.C., Jasperse, M. & Burnell, J. 2005 Effect of patterned slip on micro- and nanofluidic flows. Phys. Rev. E 72 (1), 016303.CrossRefGoogle ScholarPubMed
Ismagilov, R.F., Stroock, A.D., Kenis, P.J.A., Whitesides, G. & Stone, H.A. 2000 Experimental and theoretical scaling laws for transverse diffusive broadening in two-phase laminar flows in microchannels. Appl. Phys. Lett. 76 (17), 2376–2378.CrossRefGoogle Scholar
Janssen, L.A.M. 1976 Axial dispersion in laminar flow through coiled tubes. Chem. Engng Sci. 31 (3), 215–218.CrossRefGoogle Scholar
Jiang, F., Drese, K.S., Hardt, S., Küpper, M. & Schönfeld, F. 2004 Helical flows and chaotic mixing in curved micro channels. AIChE J. 50 (9), 2297–2305.CrossRefGoogle Scholar
Joshi, C.H., Kamm, R.D., Drazen, J.M. & Slutsky, A.S. 1983 An experimental study of gas exchange in laminar oscillatory flow. J. Fluid Mech. 133, 245–254.CrossRefGoogle Scholar
Kamholz, A.E., Weigl, B.H., Finlayson, B.A. & Yager, P. 1999 Quantitative analysis of molecular interaction in a microfluidic channel: the T-sensor. Anal. Chem. 71 (23), 5340–5347.CrossRefGoogle Scholar
Kim, M. 2004 Effect of electrostatic, hydrodynamic, and Brownian forces on particle trajectories and sieving in normal flow filtration. J. Colloid Interface Sci. 269 (2), 425–431.CrossRefGoogle ScholarPubMed
Kumar, A., Datta, S. & Kalyanasundaram, D. 2016 Permeability and effective slip in confined flows transverse to wall slippage patterns. Phys. Fluids 28 (8), 082002.CrossRefGoogle Scholar
Lei, W., Rigozzi, M.K. & McKenzie, D.R. 2016 The physics of confined flow and its application to water leaks, water permeation and water nanoflows: a review. Rep. Prog. Phys. 79 (2), 025901.CrossRefGoogle ScholarPubMed
Lin, H., Storey, B.D. & Santiago, J.G. 2008 A depth-averaged electrokinetic flow model for shallow microchannels. J. Fluid Mech. 608, 43–70.CrossRefGoogle Scholar
Mei, C.C., Auriault, J.L. & Ng, C.O. 1996 Some applications of the homogenization theory. Adv. Appl. Mech. 32, 277–348.CrossRefGoogle Scholar
Mei, C.C. & Vernescu, B. 2010 Homogenization Methods for Multiscale Mechanics. World Scientific.CrossRefGoogle Scholar
Mukherjee, A. & Mazumder, B.S. 1988 Dispersion of contaminant in oscillatory flows. Acta Mech. 74 (1–4), 107–122.CrossRefGoogle Scholar
Muñoz, J., Arcos, J., Bautista, O. & Méndez, F. 2018 Slippage effect on the dispersion coefficient of a passive solute in a pulsatile electro-osmotic flow in a microcapillary. Phys. Rev. Fluids 3 (8), 084503.CrossRefGoogle Scholar
Ng, C.O. 2006 Dispersion in steady and oscillatory flows through a tube with reversible and irreversible wall reactions. Proc. R. Soc. A: Math. Phys. Engng Sci. 462 (2066), 481–515.CrossRefGoogle Scholar
Ng, C.O. & Chen, B. 2013 Dispersion in electro-osmotic flow through a slit channel with axial step changes of zeta potential. Trans. ASME J. Fluids Engng 135 (10), 101203.CrossRefGoogle Scholar
Ng, C.O. & Zhou, Q. 2012 a Dispersion due to electroosmotic flow in a circular microchannel with slowly varying wall potential and hydrodynamic slippage. Phys. Fluids 24 (11), 112002.CrossRefGoogle Scholar
Ng, C.O. & Zhou, Q. 2012 b Electro-osmotic flow through a thin channel with gradually varying wall potential and hydrodynamic slippage. Fluid Dyn. Res. 44 (5), 055507.CrossRefGoogle Scholar
Paratore, F., Bacheva, V., Kaigala, G.V. & Bercovici, M. 2019 a Dynamic microscale flow patterning using electrical modulation of zeta potential. Proc. Natl Acad. Sci. USA 116 (21), 10258–10263.CrossRefGoogle ScholarPubMed
Paratore, F., Boyko, E., Kaigala, G.V. & Bercovici, M. 2019 b Electroosmotic flow dipole: experimental observation and flow field patterning. Phys. Rev. Lett. 122 (22), 224502.CrossRefGoogle ScholarPubMed
Roht, Y.L., Auradou, H., Hulin, J. -P., Salin, D., Chertcoff, R. & Ippolito, I. 2015 Time dependence and local structure of tracer dispersion in oscillating liquid Hele-Shaw flows. Phys. Fluids 27 (10), 103602.CrossRefGoogle Scholar
Rubin, S., Tulchinsky, A., Gat, A.D. & Bercovici, M. 2017 Elastic deformations driven by non-uniform lubrication flows. J. Fluid Mech. 812, 841–865.CrossRefGoogle Scholar
Schönecker, C., Baier, T. & Hardt, S. 2014 Influence of the enclosed fluid on the flow over a microstructured surface in the Cassie state. J. Fluid Mech. 740, 168–195.CrossRefGoogle Scholar
Six, P. & Kamrin, K. 2013 Some exact properties of the effective slip over surfaces with hydrophobic patternings. Phys. Fluids 25 (2), 021703.CrossRefGoogle Scholar
Smith, R. 1982 Contaminant dispersion in oscillatory flows. J. Fluid Mech. 114 (1), 379–398.CrossRefGoogle Scholar
Squires, T.M. 2008 Electrokinetic flows over inhomogeneously slipping surfaces. Phys. Fluids 20 (9), 092105.CrossRefGoogle Scholar
Stone, H.A. & Brenner, H. 1999 Dispersion in flows with streamwise variations of mean velocity: radial flow. Ind. Engng Chem. Res. 38 (3), 851–854.CrossRefGoogle Scholar
Taylor, G.I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219 (1137), 186–203.Google Scholar
Taylor, G.I. 1954 Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion. Proc. R. Soc. Lond. A 225 (1163), 473–477.Google Scholar
Vargas, C., Arcos, J., Bautista, O. & Méndez, F. 2017 Hydrodynamic dispersion in a combined magnetohydrodynamic-electroosmotic-driven flow through a microchannel with slowly varying wall zeta potentials. Phys. Fluids 29 (9), 092002.CrossRefGoogle Scholar
Vedel, S. & Bruus, H. 2012 Transient Taylor–Aris dispersion for time-dependent flows in straight channels. J. Fluid Mech. 691, 95–122.CrossRefGoogle Scholar
Vedel, S., Hovad, E. & Bruus, H. 2014 Time-dependent Taylor–Aris dispersion of an initial point concentration. J. Fluid Mech. 752, 107–122.CrossRefGoogle Scholar
Watson, E.J. 1983 Diffusion in oscillatory pipe flow. J. Fluid Mech. 133, 233–244.CrossRefGoogle Scholar
Ybert, C., Barentin, C., Cottin-Bizonne, C., Joseph, P. & Bocquet, L. 2007 Achieving large slip with superhydrophobic surfaces: scaling laws for generic geometries. Phys. Fluids 19 (12), 123601.CrossRefGoogle Scholar
Zhao, H. & Bau, H.H. 2007 Effect of secondary flows on Taylor–Aris dispersion. Anal. Chem. 79 (20), 7792–7798.CrossRefGoogle ScholarPubMed
Zholkovskij, E.K. & Masliyah, J.H. 2004 Hydrodynamic dispersion due to combined pressure-driven and electroosmotic flow through microchannels with a thin double layer. Anal. Chem. 76 (10), 2708–2718.CrossRefGoogle ScholarPubMed
Zholkovskij, E.K., Masliyah, J.H. & Czarnecki, J. 2003 Electroosmotic dispersion in microchannels with a thin double layer. Anal. Chem. 75 (4), 901–909.CrossRefGoogle ScholarPubMed
Zholkovskij, E.K., Masliyah, J.H. & Yaroshchuk, A.E. 2013 Broadening of neutral analyte band in electroosmotic flow through slit channel with different zeta potentials of the walls. Microfluid Nanofluidics 15 (1), 35–47.CrossRefGoogle Scholar
Zholkovskij, E.K., Yaroshchuk, A.E., Masliyah, J.H. & de Pablo Ribas, J. 2010 Broadening of neutral solute band in electroosmotic flow through submicron channel with longitudinal non-uniformity of zeta potential. Colloids Surf. A 354 (1–3), 338–346.CrossRefGoogle Scholar
Zimmerman, W.B. & Homsy, G.M. 1991 Nonlinear viscous fingering in miscible displacement with anisotropic dispersion. Phys. Fluids A: Fluid Dyn. 3 (8), 1859–1872.CrossRefGoogle Scholar
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