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Unsteady convective–diffusive transport in semicircular microchannels with irreversible wall reaction

Published online by Cambridge University Press:  21 September 2022

Milad Azari Open the ORCID record for Milad Azari [Opens in a new window]  and
Arman Sadeghi Open the ORCID record for Arman Sadeghi [Opens in a new window]
Milad Azari
Affiliation:
Department of Mechanical Engineering, University of Kurdistan, Sanandaj 66177-15175, Iran
Arman Sadeghi*
Affiliation:
Department of Mechanical Engineering, University of Kurdistan, Sanandaj 66177-15175, Iran
*
†Email address for correspondence: armansadeghi@uok.ac.ir
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Abstract

The unsteady dispersion of a solute band by a steady pressure-driven flow in a semicircular microchannel is theoretically studied via the generalized dispersion model. Considering an irreversible first-order reaction at the curved wall while assuming a no-flux boundary condition at the flat wall, analytical solutions are obtained for the exchange, convection and dispersion coefficients as well as the dimensionless forms of solute concentration and mean solute concentration. The solutions are obtained assuming an initial solute band of arbitrary cross-sectional shape and axial distribution and the results are presented for both circular and semicircular shapes with the uniform distribution being a special case of the latter. Besides the general solutions, special solutions are also derived for uniform velocity and no-reaction cases. In the following, the influences of the initial concentration distribution and the Damköhler number, a measure of the reaction rate, on the transport coefficients and the concentration distribution are investigated in depth. It is demonstrated that the combination of the initial concentration distribution and the Damköhler number specifies the variations of the transport coefficients in the short term but the Damköhler number is the only parameter dominating the long-term values: the exchange and convection coefficients are increasing functions of the Damköhler number whereas the opposite is true for the dispersion coefficient. Moreover, we show that, provided the solute injection is appropriately positioned and shaped, liquid-phase transportation with little dispersion is possible in typical microchannels utilizing semicircular geometry.

JFM classification

Mass Transport: Dispersion Micro-/Nano-fluid dynamics: Microfluidics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 949 , 25 October 2022 , A1
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

Ajdari, A., Bontoux, N. & Stone, H.A. 2006 Hydrodynamic dispersion in shallow microchannels: the effect of cross-sectional shape. Analyt. Chem. 78 (2), 387–392.CrossRefGoogle ScholarPubMed
Alassar, R.S. 2014 Fully developed forced convection through semicircular ducts. J. Thermophys. Heat Transfer 28 (3), 560–565.CrossRefGoogle Scholar
Alassar, R.S. & Abushoshah, M. 2012 Hagen-Poiseuille flow in semi-elliptic microchannels. J. Fluids Engng 134 (12), 124502.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
Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 235 (1200), 67–77.Google Scholar
Asadi-Saghandi, H., Karimi-Sabet, J., Ghorbanian, S. & Moosavian, S.M.A. 2022 Dimensionless analysis on liquid–liquid two-phase flow patterns in a numbered-up microfluidic device. Chem. Engng J. 429, 132428.CrossRefGoogle Scholar
Azari, M., Sadeghi, A. & Chakraborty, S. 2020 Electroosmotic flow and heat transfer in a heterogeneous circular microchannel. Appl. Math. Model. 87, 640–654.CrossRefGoogle Scholar
Cheng, D., Feng, X., Cheng, J., Yang, C. & Mao, Z.-S. 2015 Experimental study on the dispersed phase macro-mixing in an immiscible liquid–liquid stirred reactor. Chem. Engng Sci. 126, 196–203.CrossRefGoogle Scholar
Constanda, C. 2016 Solution Techniques for Elementary Partial Differential Equations, 3rd ed. CRC Press.CrossRefGoogle Scholar
Dehe, S., Rehm, I.-S. & Hardt, S. 2021 Hydrodynamic dispersion in Hele-Shaw flows with inhomogeneous wall boundary conditions. J. Fluid Mech. 925, A11.CrossRefGoogle Scholar
Dutta, D. 2007 Electroosmotic transport through rectangular channels with small zeta potentials. J. Colloid Interface Sci. 315 (2), 740–746.CrossRefGoogle ScholarPubMed
Fridjonsson, E.O., Seymour, J.D. & Codd, S.L. 2014 Anomalous preasymptotic colloid transport by hydrodynamic dispersion in microfluidic capillary flow. Phys. Rev. E 90 (1), 010301.CrossRefGoogle ScholarPubMed
Garcia, A.L., Ista, L.K., Petsev, D.N., O'Brien, M.J., Bisong, P., Mammoli, A.A. & López, G.P. 2005 Electrokinetic molecular separation in nanoscale fluidic channels. Lab on a Chip 5 (11), 1271–1276.CrossRefGoogle ScholarPubMed
Gill, W. & Sankarasubramanian, R. 1970 Exact analysis of unsteady convective diffusion. Proc. R. Soc. Lond. A. Math. Phys. Sci. 316 (1526), 341–350.Google Scholar
Gill, W.-N. & Sankarasubramanian, R. 1972 Dispersion of non-uniformly distributed time-variable continuous sources in time-dependent flow. Proc. R. Soc. Lond. A. Math. Phys. Sci. 327 (1569), 191–208.Google 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. Math. Phys. Sci. 322 (1548), 101–117.Google Scholar
Hisamoto, H., Shimizu, Y., Uchiyama, K., Tokeshi, M., Kikutani, Y., Hibara, A. & Kitamori, T. 2003 Chemicofunctional membrane for integrated chemical processes on a microchip. Analyt. Chem. 75 (2), 350–354.CrossRefGoogle ScholarPubMed
Hoshyargar, V., Ashrafizadeh, S.N. & Sadeghi, A. 2017 Mass transport characteristics of diffusioosmosis: potential applications for liquid phase transportation and separation. Phys. Fluids 29 (1), 012001.CrossRefGoogle Scholar
Hoshyargar, V., Khorami, A., Ashrafizadeh, S.N. & Sadeghi, A. 2018 Solute dispersion by electroosmotic flow through soft microchannels. Sensors Actuators B: Chem. 255, 3585–3600.CrossRefGoogle Scholar
Jiang, W. & Chen, G. 2019 Dispersion of active particles in confined unidirectional flows. J. Fluid Mech. 877, 1–34.CrossRefGoogle Scholar
Jiang, W. & Chen, G. 2020 Dispersion of gyrotactic micro-organisms in pipe flows. J. Fluid Mech. 889, A18.CrossRefGoogle Scholar
Karniadakis, G., Beskok, A. & Aluru, N. 2005 Microflows and Nanoflows, Fundamentals and Simulation. Springer.Google Scholar
Lee, G., Luner, A., Marzuola, J. & Harris, D.M. 2021 Dispersion control in pressure-driven flow through bowed rectangular microchannels. Microfluid Nanofluid 25 (4), 34.CrossRefGoogle Scholar
Matsunaga, T., Lee, H.-J. & Nishino, K. 2013 An approach for accurate simulation of liquid mixing in a T-shaped micromixer. Lab on a Chip 13 (8), 1515–1521.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
Paine, M.A., Carbonell, R.G. & Whitaker, S. 1983 Dispersion in pulsed systems—I: heterogenous reaction and reversible adsorption in capillary tubes. Chem. Engng Sci. 38 (11), 1781–1793.CrossRefGoogle Scholar
Patel, H., Panchal, D.R., Patel, U., Brahmbhatt, T. & Suthar, M. 2011 Matrix type drug delivery system: a review. J. Pharm. Sci. Biosci. Res. 1 (3), 143–151.Google Scholar
Paul, S. & Ng, C.-O. 2012 On the time development of dispersion in electroosmotic flow through a rectangular channel. Acta Mechanica Sin. 28 (3), 631–643.CrossRefGoogle Scholar
Phillips, C.G. & Kaye, S.R. 1998 Approximate solutions for developing shear dispersion with exchange between phases. J. Fluid Mech. 374, 195–219.CrossRefGoogle Scholar
Sadeghi, A. 2016 Analytical solutions for species transport in a T-sensor at low peclet numbers. AIChE J. 62 (11), 4119–4130.CrossRefGoogle Scholar
Sadeghi, M., Sadeghi, A. & Saidi, M.H. 2016 Electroosmotic flow in hydrophobic microchannels of general cross section. J. Fluids Engng 138 (3), 031104.CrossRefGoogle Scholar
Sadeghi, M., Saidi, M.H., Moosavi, A. & Sadeghi, A. 2020 Unsteady solute dispersion by electrokinetic flow in a polyelectrolyte layer-grafted rectangular microchannel with wall absorption. J. Fluid Mech. 887, A13.CrossRefGoogle Scholar
Sadeghi, A., Yavari, H., Saidi, M.H. & Chakraborty, S. 2011 Mixed electroosmotically and pressure-driven flow with temperature-dependent properties. J. Thermophys. Heat Transfer 25 (3), 432–442.CrossRefGoogle Scholar
Sankarasubramanian, R. & Gill, W.-N. 1972 Dispersion from a prescribed concentration distribution in time variable flow. Proc. R. Soc. Lond. A. Math. Phys. Sci. 329 (1579), 479–492.Google Scholar
Sankarasubramanian, R. & Gill, W.N. 1973 Unsteady convective diffusion with interphase mass transfer. Proc. R. Soc. Lond. A. Math. Phys. Sci. 333 (1592), 115–132.Google Scholar
Siepmann, J. & Siepmann, F. 2013 Mathematical modeling of drug dissolution. Intl J. Pharm. 453 (1), 12–24.CrossRefGoogle ScholarPubMed
Taghizadeh, E., Valdés-Parada, F.J. & Wood, B.D. 2020 Preasymptotic Taylor dispersion: evolution from the initial condition. J. Fluid Mech. 889, A5.CrossRefGoogle Scholar
Talebi, R., Ashrafizadeh, S.N. & Sadeghi, A. 2021 Hydrodynamic dispersion by electroosmotic flow in soft microchannels: consideration of different properties for electrolyte and polyelectrolyte layer. Chem. Engng Sci. 229, 116058.CrossRefGoogle Scholar
Taylor, G.I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 219 (1137), 186–203.Google Scholar
Thomas, D., Nair, V.V., Latha, M. & Thomas, K.K. 2019 Theoretical and experimental studies on theophylline release from hydrophilic alginate nanoparticles. Future J. Pharm. Sci. 5 (1), 2.CrossRefGoogle Scholar
Tripathi, A., Bozkurt, O. & Chauhan, A. 2005 Dispersion in microchannels with temporal temperature variations. Phys. Fluids 17 (10), 103607.CrossRefGoogle Scholar
Ziaie, B., Baldi, A. & Atashbar, M.Z. 2010 Introduction to micro-/nanofabrication. In Springer Handbook of Nanotechnology (ed. B. Bhushan), pp. 231–269. Springer.CrossRefGoogle Scholar
Ziaie, B., Baldi, A., Lei, M., Gu, Y. & Siegel, R.A. 2004 Hard and soft micromachining for BioMEMS: review of techniques and examples of applications in microfluidics and drug delivery. Adv. Drug Deliv. Rev. 56 (2), 145–172.CrossRefGoogle ScholarPubMed