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Streamwise dispersion of soluble matter in solvent flowing through a tube

Published online by Cambridge University Press:  02 February 2024

Mingyang Guan Open the ORCID record for Mingyang Guan [Opens in a new window]  and
Guoqian Chen Open the ORCID record for Guoqian Chen [Opens in a new window]
Mingyang Guan
Affiliation:
Laboratory of Systems Ecology and Sustainability Science, College of Engineering, Peking University, Beijing 100871, PR China
Guoqian Chen*
Affiliation:
Laboratory of Systems Ecology and Sustainability Science, College of Engineering, Peking University, Beijing 100871, PR China Macao Environmental Research Institute, Macau University of Science and Technology, Macao 999078, PR China
*
†Email address for correspondence: gqchen@pku.edu.cn
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Abstract

For the dispersion of soluble matter in solvent flowing through a tube as investigated originally by G.I. Taylor, a streamwise dispersion theory is developed from a Lagrangian perspective for the whole process with multi-scale effects. By means of a convected coordinate system to decouple convection from diffusion, a diffusion-type governing equation is presented to reflect superposable diffusion processes with a multi-scale time-dependent anisotropic diffusivity tensor. A short-time benchmark, complementing the existing Taylor–Aris solution, is obtained to reveal novel statistical and physical features of mean concentration for an initial phase with isotropic molecular diffusion. For long times, effective streamwise diffusion prevails asymptotically corresponding to the overall enhanced diffusion in Taylor's classical theory. By inverse integral expansions of local concentration moments, a general streamwise dispersion model is devised to match the short- and long-time asymptotic solutions. Analytical solutions are provided for most typical cases of point and area sources in a Poiseuille tube flow, predicting persistent long tails and skewed platforms. The theoretical findings are substantiated through Monte Carlo simulations, from the initial release to the Taylor dispersion regime. Asymmetries of concentration distribution in a circular tube are certified as originated from (a) initial non-uniformity, (b) unidirectional flow convection, and (c) non-penetration boundary effect. Peculiar peaks in the concentration cloud, enhanced streamwise dispersivity and asymmetric collective phenomena of concentration distributions are illustrated heuristically and characterised to depict the non-equilibrium dispersion. The streamwise perspective could advance our understanding of macro-transport processes of both passive solutes and active suspensions.

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Mass Transport: Dispersion

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JFM Papers
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Journal of Fluid Mechanics , Volume 980 , 10 February 2024 , A33
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© The Author(s), 2024. Published by Cambridge University Press

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References

Abbott, M.R., Denman, K.L., Powell, T.M., Richerson, P.J., Richards, R.C. & Goldman, C.R. 1984 Mixing and the dynamics of the deep chlorophyll maximum in Lake Tahoe. Limnol. Oceanogr. 29 (4), 862–878.CrossRefGoogle Scholar
Alessio, B.M., Shim, S., Gupta, A. & Stone, H.A. 2022 Diffusioosmosis-driven dispersion of colloids: a Taylor dispersion analysis with experimental validation. J. Fluid Mech. 942, A23.CrossRefGoogle Scholar
Aris, R. 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
Barry, M.T., Rusconi, R., Guasto, J.S. & Stocker, R. 2015 Shear-induced orientational dynamics and spatial heterogeneity in suspensions of motile phytoplankton. J. R. Soc. Interface 12 (112), 20150791.CrossRefGoogle ScholarPubMed
Bearon, R.N. & Hazel, A.L. 2015 The trapping in high-shear regions of slender bacteria undergoing chemotaxis in a channel. J. Fluid Mech. 771, R3.CrossRefGoogle Scholar
Bearon, R.N., Hazel, A.L. & Thorn, G.J. 2011 The spatial distribution of gyrotactic swimming micro-organisms in laminar flow fields. J. Fluid Mech. 680, 602–635.CrossRefGoogle Scholar
Bello, M.S., Rezzonico, R. & Righetti, P.G. 1994 Use of Taylor–Aris dispersion for measurement of a solute diffusion coefficient in thin capillaries. Science 266 (5186), 773–776.CrossRefGoogle ScholarPubMed
Biswas, R.R. & Sen, P.N. 2007 Taylor dispersion with absorbing boundaries: a stochastic approach. Phys. Rev. Lett. 98 (16), 164501.CrossRefGoogle ScholarPubMed
Brenner, H. & Edwards, D. 1993 Macrotransport Processes. Butterworth-Heinemann.Google Scholar
Chatwin, P.C. 1970 The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe. J. Fluid Mech. 43 (2), 321–352.CrossRefGoogle Scholar
Chatwin, P.C. 1972 The cumulants of the distribution of concentration of a solute dispersing in solvent flowing through a tube. J. Fluid Mech. 51 (1), 63–67.CrossRefGoogle Scholar
Chatwin, P.C. 1974 The dispersion of contaminant released from instantaneous sources in laminar flow near stagnation points. J. Fluid Mech. 66 (4), 753–766.CrossRefGoogle Scholar
Chatwin, P.C. 1976 The initial dispersion of contaminant in Poiseuille flow and the smoothing of the snout. J. Fluid Mech. 77 (3), 593–602.CrossRefGoogle Scholar
Chatwin, P.C. 1977 The initial development of longitudinal dispersion in straight tubes. J. Fluid Mech. 80 (1), 33–48.CrossRefGoogle Scholar
Chatwin, P.C. & Allen, C.M. 1985 Mathematical models of dispersion in rivers and estuaries. Annu. Rev. Fluid Mech. 17, 119–149.CrossRefGoogle Scholar
Chen, G. & Gao, Z. 1991 A transformation of the convective diffusion equation with corresponding finite difference method. Chin. J. Theor. Appl. Mech. 23 (4), 418–425.Google Scholar
Chen, G.Q., Gao, Z. & Yang, Z.F. 1993 A perturbational $h^4$ exponential finite difference scheme for the convective diffusion equation. J. Comput. Phys. 104 (1), 129–139.CrossRefGoogle Scholar
Chikwendu, S.C. 1986 Calculation of longitudinal shear dispersivity using an $N$-zone model as $N\rightarrow \infty$. J. Fluid Mech. 167, 19–30.CrossRefGoogle Scholar
Chikwendu, S.C. & Ojiakor, G.U. 1985 Slow-zone model for longitudinal dispersion in two-dimensional shear flows. J. Fluid Mech. 152, 15–38.CrossRefGoogle Scholar
Christov, I.C. & Stone, H.A. 2012 Resolving a paradox of anomalous scalings in the diffusion of granular materials. Proc. Natl Acad. Sci. USA 109 (40), 16012–16017.CrossRefGoogle ScholarPubMed
Chu, H.C.W., Garoff, S., Tilton, R.D. & Khair, A.S. 2021 Macrotransport theory for diffusiophoretic colloids and chemotactic microorganisms. J. Fluid Mech. 917, A52.CrossRefGoogle Scholar
Crank, J. 1975 The Mathematics of Diffusion, 2nd edn. Clarendon Press.Google Scholar
Debnath, S., Jiang, W., Guan, M. & Chen, G. 2022 Effect of ring-source release on dispersion process in Poiseuille flow with wall absorption. Phys. Fluids 34 (2), 027106.CrossRefGoogle Scholar
Deleanu, M., Hernandez, J.-F., Cipelletti, L., Biron, J.-P., Rossi, E., Taverna, M., Cottet, H. & Chamieh, J. 2021 Unraveling the speciation of $\beta$-amyloid peptides during the aggregation process by Taylor dispersion analysis. Anal. Chem. 93 (16), 6523–6533.CrossRefGoogle ScholarPubMed
Dewey, R.J. & Sullivan, P.J. 1982 Longitudinal-dispersion calculations in laminar flows by statistical analysis of molecular motions. J. Fluid Mech. 125, 203–217.CrossRefGoogle Scholar
Durham, W.M., Climent, E., Barry, M., De Lillo, F., Boffetta, G., Cencini, M. & Stocker, R. 2013 Turbulence drives microscale patches of motile phytoplankton. Nat. Commun. 4 (1), 2148.CrossRefGoogle ScholarPubMed
Durham, W.M., Kessler, J.O. & Stocker, R. 2009 Disruption of vertical motility by shear triggers formation of thin phytoplankton layers. Science 323 (5917), 1067–1070.CrossRefGoogle ScholarPubMed
Elder, J.W. 1959 The dispersion of marked fluid in turbulent shear flow. J. Fluid Mech. 5 (4), 544–560.CrossRefGoogle Scholar
Ezhilan, B. & Saintillan, D. 2015 Transport of a dilute active suspension in pressure-driven channel flow. J. Fluid Mech. 777, 482–522.CrossRefGoogle Scholar
Fallon, M.S., Howell, B.A. & Chauhan, A. 2009 Importance of Taylor dispersion in pharmacokinetic and multiple indicator dilution modelling. Math. Med. Biol. 26 (4), 263–296.CrossRefGoogle ScholarPubMed
Fischer, H.B. 1972 Mass transport mechanisms in partially stratified estuaries. J. Fluid Mech. 53, 671–687.CrossRefGoogle Scholar
Fischer, H.B. 1973 Longitudinal dispersion and turbulent mixing in open-channel flow. Annu. Rev. Fluid Mech. 5 (1), 59–78.CrossRefGoogle Scholar
Fischer, H.B. 1976 Mixing and dispersion in estuaries. Annu. Rev. Fluid Mech. 8 (1), 107–133.CrossRefGoogle Scholar
Fischer, H.B. (Ed.) 1979 Mixing in Inland and Coastal Waters. Academic Press.Google Scholar
Foister, R.T. & van de Ven, T.G.M. 1980 Diffusion of Brownian particles in shear flows. J. Fluid Mech. 96 (1), 105–132.CrossRefGoogle Scholar
Frankel, I. & Brenner, H. 1989 On the foundations of generalized Taylor dispersion theory. J. Fluid Mech. 204, 97–119.CrossRefGoogle Scholar
Fung, L., Bearon, R.N. & Hwang, Y. 2022 A local approximation model for macroscale transport of biased active Brownian particles in a flowing suspension. J. Fluid Mech. 935, A24.CrossRefGoogle Scholar
Gekle, S. 2017 Dispersion of solute released from a sphere flowing in a microchannel. J. Fluid Mech. 819, 104–120.CrossRefGoogle Scholar
Gill, W. 1971 Dispersion of a non-uniform slug in time-dependent flow. Proc. R. Soc. Lond. A 322 (1548), 101–117.Google Scholar
Gill, W.N. 1967 A note on the solution of transient dispersion problems. Proc. R. Soc. Lond. A 298 (1454), 335–339.Google Scholar
Gill, W.N., Sankarasubramanian, R. & Taylor, G.I. 1970 Exact analysis of unsteady convective diffusion. Proc. R. Soc. Lond. A 316 (1526), 341–350.Google Scholar
Golestanian, R. 2012 Collective behavior of thermally active colloids. Phys. Rev. Lett. 108 (3), 038303.CrossRefGoogle ScholarPubMed
Guan, M., Jiang, W., Wang, B., Zeng, L., Li, Z. & Chen, G. 2023 Pre-asymptotic dispersion of active particles through a vertical pipe: the origin of hydrodynamic focusing. J. Fluid Mech. 962, A14.CrossRefGoogle Scholar
Guan, M., Zeng, L., Jiang, W., Guo, X., Wang, P., Wu, Z., Li, Z. & Chen, G. 2022 Effects of wind on transient dispersion of active particles in a free-surface wetland flow. Commun. Nonlinear Sci. Numer. Simul. 115, 106766.CrossRefGoogle Scholar
Guan, M., Zeng, L., Li, C., Guo, X., Wu, Y. & Wang, P. 2021 Transport model of active particles in a tidal wetland flow. J. Hydrol. 593, 125812.CrossRefGoogle Scholar
Guo, J. & Chen, G. 2022 Solute dispersion from a continuous release source in a vegetated flow: an analytical study. Water Resour. Res. 58, e2021WR030255.CrossRefGoogle Scholar
Guo, J., Jiang, W. & Chen, G. 2020 Transient solute dispersion in wetland flows with submerged vegetation: an analytical study in terms of time-dependent properties. Water Resour. Res. 56 (2), e2019WR025586.CrossRefGoogle Scholar
Haber, S. & Mauri, R. 1988 Lagrangian approach to time-dependent laminar dispersion in rectangular conduits. Part 1. Two-dimensional flows. J. Fluid Mech. 190, 201–215.CrossRefGoogle Scholar
Hahn, D.W. & Özişik, M.N. 2012 Heat Conduction. John Wiley & Sons.CrossRefGoogle Scholar
Hong, J., Wu, H., Zhang, R., He, M. & Xu, W. 2020 The coupling of Taylor dispersion analysis and mass spectrometry to differentiate protein conformations. Anal. Chem. 92 (7), 5200–5206.CrossRefGoogle ScholarPubMed
Houseworth, J.E. 1984 Shear dispersion and residence time for laminar flow in capillary tubes. J. Fluid Mech. 142, 289–308.CrossRefGoogle Scholar
Jiang, W. & Chen, G. 2018 Solution of Gill's generalized dispersion model: solute transport in Poiseuille flow with wall absorption. Intl J. Heat Mass Transfer 127, 34–43.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. 2021 Transient dispersion process of active particles. J. Fluid Mech. 927, A11.CrossRefGoogle Scholar
Jiang, W., Zeng, L., Fu, X. & Wu, Z. 2022 Analytical solutions for reactive shear dispersion with boundary adsorption and desorption. J. Fluid Mech. 947, A37.CrossRefGoogle Scholar
Kendall, M.G. & Stuart, A. 1958 The Advanced Theory of Statistics, vol. 1. Griffin.Google Scholar
Kessler, J.O. 1985 Hydrodynamic focusing of motile algal cells. Nature 313 (5999), 218–220.CrossRefGoogle Scholar
Latini, M. & Bernoff, A.J. 2001 Transient anomalous diffusion in Poiseuille flow. J. Fluid Mech. 441, 399–411.CrossRefGoogle Scholar
Li, G. 2018 An extended Taylor–Aris method in dispersion theory and its applications. PhD thesis, Peking University.Google Scholar
Li, G. & Tang, J.X. 2009 Accumulation of microswimmers near a surface mediated by collision and rotational Brownian motion. Phys. Rev. Lett. 103 (7), 078101.CrossRefGoogle Scholar
Lighthill, M.J. 1966 Initial development of diffusion in Poiseuille flow. J. Inst. Maths Applics. 2, 97–108.CrossRefGoogle Scholar
Masoud, H. & Stone, H.A. 2019 The reciprocal theorem in fluid dynamics and transport phenomena. J. Fluid Mech. 879, P1.CrossRefGoogle Scholar
Mercer, G.N. & Roberts, A.J. 1990 A centre manifold description of contaminant dispersion in channels with varying flow properties. SIAM J. Appl. Maths 50 (6), 1547–1565.CrossRefGoogle Scholar
Mercer, G.N. & Roberts, A.J. 1994 A complete model of shear dispersion in pipes. Japan J. Ind. Appl. Maths 11 (3), 499.CrossRefGoogle Scholar
Morris, J.F. 2020 Shear thickening of concentrated suspensions: recent developments and relation to other phenomena. Annu. Rev. Fluid Mech. 52 (1), 121–144.CrossRefGoogle Scholar
Moser, M.R. & Baker, C.A. 2021 Taylor dispersion analysis in fused silica capillaries: a tutorial review. Anal. Methods 13 (21), 2357–2373.CrossRefGoogle ScholarPubMed
Nakad, M., Witelski, T., Domec, J.C., Sevanto, S. & Katul, G. 2021 Taylor dispersion in osmotically driven laminar flows in phloem. J. Fluid Mech. 913, A44.CrossRefGoogle Scholar
Pasmanter, R.A. 1985 Exact and approximate solutions of the convection–diffusion equation. Q. J. Mech. Appl. Maths 38 (1), 1–26.CrossRefGoogle Scholar
Pedley, T.J. & Kessler, J.O. 1992 Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24 (1), 313–358.CrossRefGoogle Scholar
Phillips, C.G. & Kaye, S.R. 1997 The initial transient of concentration during the development of Taylor dispersion. Proc. R. Soc. Lond. A 453 (1967), 2669–2688.CrossRefGoogle Scholar
Rusconi, R., Guasto, J.S. & Stocker, R. 2014 Bacterial transport suppressed by fluid shear. Nat. Phys. 10 (3), 212–217.CrossRefGoogle Scholar
Saffman, P.G. 1960 On the effect of the molecular diffusivity in turbulent diffusion. J. Fluid Mech. 8 (2), 273–283.CrossRefGoogle Scholar
Smith, R. 1981 A delay-diffusion description for contaminant dispersion. J. Fluid Mech. 105, 469–486.CrossRefGoogle Scholar
Smith, R. 1982 a Contaminant dispersion in oscillatory flows. J. Fluid Mech. 114 (-1), 379.CrossRefGoogle Scholar
Smith, R. 1982 b Gaussian approximation for contaminant dispersion. Q. J. Mech. Appl. Maths 35 (3), 345–366.CrossRefGoogle Scholar
Sokolov, A. & Aranson, I.S. 2016 Rapid expulsion of microswimmers by a vortical flow. Nat. Commun. 7, 11114.CrossRefGoogle ScholarPubMed
Stokes, A.N. & Barton, N.G. 1990 The concentration distribution produced by shear dispersion of solute in Poiseuille flow. J. Fluid Mech. 210, 201–221.CrossRefGoogle Scholar
Sullivan, P.J. 1971 Longitudinal dispersion within a two-dimensional turbulent shear flow. J. Fluid Mech. 49 (3), 551–576.CrossRefGoogle Scholar
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
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 a 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
Taylor, G.I. 1954 b Diffusion and mass transport in tubes. Proc. Phys. Soc. B 67 (12), 857.CrossRefGoogle Scholar
Taylor, G.I. 1954 c The dispersion of matter in turbulent flow through a pipe. Proc. R. Soc. Lond. A 223 (1155), 446–468.Google Scholar
Taylor, G.I. 1959 The present position in the theory of turbulent diffusion. Adv. Geophys. 6, 101–112.CrossRefGoogle Scholar
Townsend, A.A. & Taylor, G.I. 1951 The diffusion of heat spots in isotropic turbulence. Proc. R. Soc. Lond. A 209 (1098), 418–430.Google 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
Vilquin, A., Bertin, V., Raphaël, E., Dean, D.S., Salez, T. & McGraw, J.D. 2023 Nanoparticle Taylor dispersion near charged surfaces with an open boundary. Phys. Rev. Lett. 130 (3), 038201.CrossRefGoogle ScholarPubMed
Wang, B., Jiang, W. & Chen, G. 2022 a Gyrotactic trapping of micro-swimmers in simple shear flows: a study directly from the fundamental Smoluchowski equation. J. Fluid Mech. 939, A37.CrossRefGoogle Scholar
Wang, B., Jiang, W. & Chen, G. 2023 Dispersion of a gyrotactic micro-organism suspension in a vertical pipe: the buoyancy–flow coupling effect. J. Fluid Mech. 962, A39.CrossRefGoogle Scholar
Wang, B., Jiang, W., Chen, G. & Tao, L. 2022 b Transient dispersion in a channel with crossflow and wall adsorption. Phys. Rev. Fluids 7 (7), 074501.CrossRefGoogle Scholar
Wang, P. & Chen, G. 2016 a Transverse concentration distribution in Taylor dispersion: Gill's method of series expansion supported by concentration moments. Intl J. Heat Mass Transfer 95, 131–141.CrossRefGoogle Scholar
Wang, P. & Chen, G.Q. 2016 b Solute dispersion in open channel flow with bed absorption. J. Hydrol. 543, 208–217.CrossRefGoogle Scholar
Wang, P. & Chen, G. 2017 Basic characteristics of Taylor dispersion in a laminar tube flow with wall absorption: exchange rate, advection velocity, dispersivity, skewness and kurtosis in their full time dependance. Intl J. Heat Mass Transfer 109, 844–852.CrossRefGoogle Scholar
Wang, P. & Cirpka, O.A. 2021 Surface transient storage under low-flow conditions in streams with rough bathymetry. Water Resour. Res. 57 (12), e2021WR029899.CrossRefGoogle Scholar
Wu, Z. & Chen, G. 2014 Approach to transverse uniformity of concentration distribution of a solute in a solvent flowing along a straight pipe. J. Fluid Mech. 740, 196–213.CrossRefGoogle Scholar
Wu, Z., Furbish, D. & Foufoula-Georgiou, E. 2020 Generalization of hop distance-time scaling and particle velocity distributions via a two-regime formalism of bedload particle motions. Water Resour. Res. 56 (1), e2019WR025116.CrossRefGoogle Scholar
Wu, Z., Jiang, W., Zeng, L. & Fu, X. 2023 Theoretical analysis for bedload particle deposition and hop statistics. J. Fluid Mech. 954, A11.CrossRefGoogle Scholar
Yasa, O., Erkoc, P., Alapan, Y. & Sitti, M. 2018 Microalga-powered microswimmers toward active cargo delivery. Adv. Mater. 30 (45), 1804130.CrossRefGoogle ScholarPubMed
Yasuda, H. 1984 Longitudinal dispersion of matter due to the shear effect of steady and oscillatory currents. J. Fluid Mech. 148, 383–403.CrossRefGoogle Scholar
Young, W.R. & Jones, S. 1991 Shear dispersion. Phys. Fluids 3 (5), 1087–1101.CrossRefGoogle Scholar
Zeng, L., Jiang, W. & Pedley, T.J. 2022 Sharp turns and gyrotaxis modulate surface accumulation of microorganisms. Proc. Natl Acad. Sci. USA 119 (42), e2206738119.CrossRefGoogle ScholarPubMed
Zhang, L., Hesse, M.A. & Wang, M. 2017 Transient solute transport with sorption in Poiseuille flow. J. Fluid Mech. 828, 733–752.CrossRefGoogle Scholar
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