Hostname: page-component-6b989bf9dc-jks4b Total loading time: 0 Render date: 2024-04-14T12:33:01.787Z Has data issue: false hasContentIssue false

Transient dispersion process of active particles

Published online by Cambridge University Press:  21 September 2021

Weiquan Jiang
Laboratory of Systems Ecology and Sustainability Science, College of Engineering, Peking University, Beijing 100871, PR China State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, PR China
Guoqian Chen*
Laboratory of Systems Ecology and Sustainability Science, College of Engineering, Peking University, Beijing 100871, PR China
Email address for correspondence:


Active particles often swim in confined environments. The transport mechanisms, especially the global one as reflected by the Taylor dispersion model, are of great practical interest to various applications. For the active dispersion process in confined flows, previous analytical studies focused on the long-time asymptotic values of dispersion characteristics. Only several numerical studies preliminarily investigated the temporal evolution. Extending recent studies of Jiang & Chen (J. Fluid Mech., vol. 877, 2019, pp. 1–34; vol. 899, 2020, A18), this work makes a semi-analytical attempt to investigate the transient process. The temporal evolution of the local distribution in the confined-section–orientation space, drift, dispersivity and skewness, is explored based on moments of distributions. We introduce the biorthogonal expansion method for solutions because the classic integral transform method for passive transport problems is not applicable due to the self-propulsion effect. Two types of boundary condition, the reflective condition and the Robin condition for wall accumulation, are imposed respectively. A detailed study on spherical and ellipsoidal swimmers dispersing in a plane Poiseuille flow demonstrates the influences of the swimming, shear flow, initial condition, wall accumulation and particle shape on the transient dispersion process. The swimming-induced diffusion makes the local distribution reach its equilibrium state faster than that of passive particles. Although the wall accumulation significantly affects the evolution of the local distribution and the drift, the time scale to reach the Taylor regime is not obviously changed. The shear-induced alignment of ellipsoidal particles can enlarge the dispersivity but impacts slightly on the drift and the skewness.

JFM Papers
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



Acién, F.G., Molina, E., Reis, A., Torzillo, G., Zittelli, G.C., Sepúlveda, C. & Masojídek, J. 2017 Photobioreactors for the production of microalgae. In Microalgae-Based Biofuels and Bioproducts (ed. C. Gonzalez-Fernandez & R. Muñoz), pp. 1–44. Woodhead Publishing.CrossRefGoogle Scholar
Alonso-Matilla, R., Chakrabarti, B. & Saintillan, D. 2019 Transport and dispersion of active particles in periodic porous media. Phys. Rev. Fluids 4 (4), 043101.CrossRefGoogle Scholar
Aminian, M., Bernardi, F., Camassa, R., Harris, D.M. & McLaughlin, R.M. 2016 How boundaries shape chemical delivery in microfluidics. Science 354 (6317), 12521256.CrossRefGoogle ScholarPubMed
Ao, X., Ghosh, P., Li, Y., Schmid, G., Hänggi, P. & Marchesoni, F. 2014 Active Brownian motion in a narrow channel. Eur. Phys. J. Spec. Top. 223 (14), 32273242.CrossRefGoogle Scholar
Apaza, L. & Sandoval, M. 2016 Ballistic behavior and trapping of self-driven particles in a Poiseuille flow. Phys. Rev. E 93 (6), 062602.CrossRefGoogle Scholar
Apaza, L. & Sandoval, M. 2020 Homotopy analysis and Padé approximants applied to active Brownian motion. Phys. Rev. E 101 (3), 032103.CrossRefGoogle ScholarPubMed
Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A 235 (1200), 6777.Google Scholar
Barton, N.G. 1983 On the method of moments for solute dispersion. J. Fluid Mech. 126, 205218.CrossRefGoogle Scholar
Bearon, R.N. 2013 Helical swimming can provide robust upwards transport for gravitactic single-cell algae; a mechanistic model. J. Math. Biol. 66 (7), 13411359.CrossRefGoogle ScholarPubMed
Bearon, R.N., Bees, M.A. & Croze, O.A. 2012 Biased swimming cells do not disperse in pipes as tracers: a population model based on microscale behaviour. Phys. Fluids 24 (12), 121902.CrossRefGoogle Scholar
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, 602635.CrossRefGoogle Scholar
Bechinger, C., Di Leonardo, R., Löwen, H., Reichhardt, C., Volpe, G. & Volpe, G. 2016 Active particles in complex and crowded environments. Rev. Mod. Phys. 88 (4), 045006.CrossRefGoogle Scholar
Bees, M.A. 2020 Advances in bioconvection. Annu. Rev. Fluid Mech. 52 (1), 449476.CrossRefGoogle Scholar
Bees, M.A. & Croze, O.A. 2010 Dispersion of biased swimming micro-organisms in a fluid flowing through a tube. Proc. R. Soc. Lond. A 466 (2119), 20572077.Google Scholar
Bees, M.A. & Croze, O.A. 2014 Mathematics for streamlined biofuel production from unicellular algae. Biofuels 5 (1), 5365.CrossRefGoogle Scholar
Berg, H.C. 1993 Random Walks in Biology, revised edn. Princeton University Press.Google Scholar
Berg, H.C. & Turner, L. 1990 Chemotaxis of bacteria in glass capillary arrays. Escherichia coli, motility, microchannel plate, and light scattering. Biophys. J. 58 (4), 919930.CrossRefGoogle ScholarPubMed
Berke, A.P., Turner, L., Berg, H.C. & Lauga, E. 2008 Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101 (3), 038102.CrossRefGoogle ScholarPubMed
Berlyand, L., Jabin, P.-E., Potomkin, M. & Ratajczyk, E. 2020 A kinetic approach to active rods dynamics in confined domains. Multiscale Model. Simul. 18 (1), 120.CrossRefGoogle Scholar
Bianchi, S., Saglimbeni, F. & Di Leonardo, R. 2017 Holographic imaging reveals the mechanism of wall entrapment in swimming bacteria. Phys. Rev. X 7 (1), 011010.Google Scholar
Brenner, H. 1982 A general theory of Taylor dispersion phenomena. II. An extension. Phys.-Chem. Hydrodyn. 3 (2), 139157.Google Scholar
Brenner, H. & Edwards, D.A. 1993 Macrotransport Processes. Butterworth-Heinemann.Google Scholar
Brezinski, C. 1991 Biorthogonality and its Applications to Numerical Analysis. Marcel Dekker.Google Scholar
Brosseau, Q., Usabiaga, F.B., Lushi, E., Wu, Y., Ristroph, L., Zhang, J., Ward, M. & Shelley, M.J. 2019 Relating rheotaxis and hydrodynamic actuation using asymmetric gold-platinum phoretic rods. Phys. Rev. Lett. 123 (17), 178004.CrossRefGoogle ScholarPubMed
Buchner, A.-J., Muller, K., Mehmood, J. & Tam, D. 2021 Hopping trajectories due to long-range interactions determine surface accumulation of microalgae. Proc. Natl Acad. Sci. 118 (20), e2102095118.CrossRefGoogle ScholarPubMed
Camassa, R., Lin, Z. & McLaughlin, R.M. 2010 The exact evolution of the scalar variance in pipe and channel flow. Commun. Math. Sci. 8 (2), 601626.CrossRefGoogle 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), 321352.CrossRefGoogle Scholar
Chen, X., Zeng, L., Wu, Y., Gao, Y. & Zhao, Y. 2017 Swimming characteristics of gyrotactic microorganisms in low-Reynolds-number flow: Chlamydomonas reinhardtii. Energy Ecol. Environ. 2 (5), 289295.CrossRefGoogle Scholar
Chilukuri, S., Collins, C.H. & Underhill, P.T. 2014 Impact of external flow on the dynamics of swimming microorganisms near surfaces. J. Phys.: Condens. Matter 26 (11), 115101.Google ScholarPubMed
Chilukuri, S., Collins, C.H. & Underhill, P.T. 2015 Dispersion of flagellated swimming microorganisms in planar Poiseuille flow. Phys. Fluids 27 (3), 031902.CrossRefGoogle Scholar
Chisti, Y. 2007 Biodiesel from microalgae. Biotechnol. Adv. 25 (3), 294306.CrossRefGoogle ScholarPubMed
Contino, M., Lushi, E., Tuval, I., Kantsler, V. & Polin, M. 2015 Microalgae scatter off solid surfaces by hydrodynamic and contact forces. Phys. Rev. Lett. 115 (25), 258102.CrossRefGoogle ScholarPubMed
Costanzo, A., Di Leonardo, R., Ruocco, G. & Angelani, L. 2012 Transport of self-propelling bacteria in micro-channel flow. J. Phys.: Condens. Matter 24 (6), 065101.Google ScholarPubMed
Croze, O.A., Bearon, R.N. & Bees, M.A. 2017 Gyrotactic swimmer dispersion in pipe flow: testing the theory. J. Fluid Mech. 816, 481506.CrossRefGoogle Scholar
Croze, O.A., Sardina, G., Ahmed, M., Bees, M.A. & Brandt, L. 2013 Dispersion of swimming algae in laminar and turbulent channel flows: consequences for photobioreactors. J. R. Soc. Interface 10 (81), 20121041.CrossRefGoogle ScholarPubMed
Dehkharghani, A., Waisbord, N., Dunkel, J. & Guasto, J.S. 2019 Bacterial scattering in microfluidic crystal flows reveals giant active Taylor–Aris dispersion. Proc. Natl Acad. Sci. 116 (23), 1111911124.CrossRefGoogle ScholarPubMed
Doi, M. & Edwards, S.F. 1988 Brownian motion. In The Theory of Polymer Dynamics, pp. 46–90. Clarendon.Google Scholar
Drescher, K., Dunkel, J., Cisneros, L.H., Ganguly, S. & Goldstein, R.E. 2011 Fluid dynamics and noise in bacterial cell–cell and cell–surface scattering. Proc. Natl Acad. Sci. 108 (27), 1094010945.CrossRefGoogle ScholarPubMed
Durham, W.M. & Stocker, R. 2012 Thin phytoplankton layers: characteristics, mechanisms, and consequences. Annu. Rev. Mar. Sci. 4 (1), 177207.CrossRefGoogle ScholarPubMed
Duzgun, A. & Selinger, J.V. 2018 Active Brownian particles near straight or curved walls: pressure and boundary layers. Phys. Rev. E 97 (3), 032606.CrossRefGoogle ScholarPubMed
Elgeti, J. & Gompper, G. 2013 Wall accumulation of self-propelled spheres. Europhys. Lett. 101 (4), 48003.CrossRefGoogle Scholar
Elgeti, J. & Gompper, G. 2015 Run-and-tumble dynamics of self-propelled particles in confinement. Europhys. Lett. 109 (5), 58003.CrossRefGoogle Scholar
Enculescu, M. & Stark, H. 2011 Active colloidal suspensions exhibit polar order under gravity. Phys. Rev. Lett. 107 (5), 058301.CrossRefGoogle ScholarPubMed
Ezhilan, B. & Saintillan, D. 2015 Transport of a dilute active suspension in pressure-driven channel flow. J. Fluid Mech. 777, 482522.CrossRefGoogle Scholar
Foister, R.T. & van de Ven, T.G.M. 1980 Diffusion of Brownian particles in shear flows. J. Fluid Mech. 96 (1), 105132.CrossRefGoogle Scholar
Frankel, I. & Brenner, H. 1989 On the foundations of generalized Taylor dispersion theory. J. Fluid Mech. 204, 97119.CrossRefGoogle Scholar
Fu, J., Perthame, B. & Tang, M. 2021 Fokker–Plank system for movement of micro-organism population in confined environment. J. Stat. Phys. 184 (1), 1.CrossRefGoogle Scholar
Fung, L., Bearon, R.N. & Hwang, Y. 2020 Bifurcation and stability of downflowing gyrotactic micro-organism suspensions in a vertical pipe. J. Fluid Mech. 902, A26.CrossRefGoogle Scholar
Ghosh, P.K., Misko, V.R., Marchesoni, F. & Nori, F. 2013 Self-propelled Janus particles in a ratchet: numerical simulations. Phys. Rev. Lett. 110 (26), 268301.CrossRefGoogle Scholar
Gill, W.N. & Sankarasubramanian, R. 1970 Exact analysis of unsteady convective diffusion. Proc. R. Soc. Lond. A 316 (1526), 341350.Google Scholar
Goldstein, R.E. 2015 Green algae as model organisms for biological fluid dynamics. Annu. Rev. Fluid Mech. 47 (1), 343375.CrossRefGoogle ScholarPubMed
Guan, M.Y., Zeng, L., Li, C.F., Guo, X.L., Wu, Y.H. & Wang, P. 2021 Transport model of active particles in a tidal wetland flow. J. Hydrol. 593, 125812.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
ten Hagen, B., van Teeffelen, S. & Löwen, H. 2011 a Brownian motion of a self-propelled particle. J. Phys.: Condens. Matter 23 (19), 194119.Google ScholarPubMed
ten Hagen, B., Wittkowski, R. & Löwen, H. 2011 b Brownian dynamics of a self-propelled particle in shear flow. Phys. Rev. E 84 (3), 031105.CrossRefGoogle ScholarPubMed
Heyes, D.M. & Melrose, J.R. 1993 Brownian dynamics simulations of model hard-sphere suspensions. J. Non-Newtonian Fluid Mech. 46 (1), 128.CrossRefGoogle Scholar
Hill, N.A. & Bees, M.A. 2002 Taylor dispersion of gyrotactic swimming micro-organisms in a linear flow. Phys. Fluids 14 (8), 25982605.CrossRefGoogle Scholar
Hill, N.A. & Pedley, T.J. 2005 Bioconvection. Fluid Dyn. Res. 37 (1–2), 120.CrossRefGoogle Scholar
Howse, J.R., Jones, R.A.L., Ryan, A.J., Gough, T., Vafabakhsh, R. & Golestanian, R. 2007 Self-motile colloidal particles: from directed propulsion to random walk. Phys. Rev. Lett. 99 (4), 048102.CrossRefGoogle ScholarPubMed
Hwang, Y. & Pedley, T.J. 2014 Stability of downflowing gyrotactic microorganism suspensions in a two-dimensional vertical channel. J. Fluid Mech. 749, 750777.CrossRefGoogle Scholar
Jeffery, G.B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102 (715), 161179.Google Scholar
Jiang, W. & Chen, G. 2019 a Dispersion of active particles in confined unidirectional flows. J. Fluid Mech. 877, 134.CrossRefGoogle Scholar
Jiang, W. & Chen, G. 2019 b Solute transport in two-zone packed tube flow: long-time asymptotic expansion. Phys. Fluids 31 (4), 043303.Google Scholar
Jiang, W. & Chen, G. 2020 Dispersion of gyrotactic micro-organisms in pipe flows. J. Fluid Mech. 889, A18.CrossRefGoogle Scholar
Kantsler, V., Dunkel, J., Polin, M. & Goldstein, R.E. 2013 Ciliary contact interactions dominate surface scattering of swimming eukaryotes. Proc. Natl Acad. Sci. 110 (4), 11871192.CrossRefGoogle ScholarPubMed
Latini, M. & Bernoff, A.J. 2001 Transient anomalous diffusion in Poiseuille flow. J. Fluid Mech. 441, 399411.CrossRefGoogle Scholar
Lauga, E. 2020 The Fluid Dynamics of Cell Motility. Cambridge University Press.CrossRefGoogle Scholar
Lauga, E., DiLuzio, W.R., Whitesides, G.M. & Stone, H.A. 2006 Swimming in circles: motion of bacteria near solid boundaries. Biophys. J. 90 (2), 400412.CrossRefGoogle ScholarPubMed
Lauga, E. & Powers, T.R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (9), 096601.CrossRefGoogle Scholar
Leal, L.G. & Hinch, E.J. 1972 The rheology of a suspension of nearly spherical particles subject to Brownian rotations. J. Fluid Mech. 55 (4), 745765.CrossRefGoogle Scholar
Li, G., Jiang, W., Wang, P., Guo, J., Li, Z. & Chen, G.Q. 2018 Concentration moments based analytical study on Taylor dispersion: open channel flow driven by gravity and wind. J. Hydrol. 562, 244253.CrossRefGoogle Scholar
Li, G., Tam, L.-K. & Tang, J.X. 2008 Amplified effect of Brownian motion in bacterial near-surface swimming. Proc. Natl Acad. Sci. 105 (47), 1835518359.CrossRefGoogle ScholarPubMed
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. IMA J. Appl. Maths 2 (1), 97108.CrossRefGoogle Scholar
Liu, L., Liu, D., Johnson, D.M., Yi, Z. & Huang, Y. 2012 Effects of vertical mixing on phytoplankton blooms in Xiangxi Bay of Three Gorges Reservoir: implications for management. Water Res. 46 (7), 21212130.CrossRefGoogle Scholar
Lushi, E., Goldstein, R.E. & Shelley, M.J. 2018 Nonlinear concentration patterns and bands in autochemotactic suspensions. Phys. Rev. E 98 (5), 052411.CrossRefGoogle Scholar
Lushi, E., Kantsler, V. & Goldstein, R.E. 2017 Scattering of biflagellate microswimmers from surfaces. Phys. Rev. E 96 (2), 023102.CrossRefGoogle ScholarPubMed
Lushi, E., Wioland, H. & Goldstein, R.E. 2014 Fluid flows created by swimming bacteria drive self-organization in confined suspensions. Proc. Natl Acad. Sci. 111 (27), 97339738.CrossRefGoogle ScholarPubMed
Makarchuk, S., Braz, V.C., Araújo, N.A.M., Ciric, L. & Volpe, G. 2019 Enhanced propagation of motile bacteria on surfaces due to forward scattering. Nat. Commun. 10 (1), 112.CrossRefGoogle ScholarPubMed
Makhnovskii, Y.A. 2019 Effect of particle size oscillations on drift and diffusion along a periodically corrugated channel. Phys. Rev. E 99 (3), 032102.CrossRefGoogle ScholarPubMed
Mathijssen, A.J.T.M., Figueroa-Morales, N., Junot, G., Clément, É., Lindner, A. & Zöttl, A. 2019 Oscillatory surface rheotaxis of swimming E. coli bacteria. Nat. Commun. 10 (1), 112.CrossRefGoogle ScholarPubMed
Morris, J.F. 2020 Shear thickening of concentrated suspensions: recent developments and relation to other phenomena. Annu. Rev. Fluid Mech. 52 (1), 121144.CrossRefGoogle Scholar
Nambiar, S., Phanikanth, S., Nott, P.R. & Subramanian, G. 2019 Stress relaxation in a dilute bacterial suspension: the active–passive transition. J. Fluid Mech. 870, 10721104.CrossRefGoogle Scholar
Nili, H., Kheyri, M., Abazari, J., Fahimniya, A. & Naji, A. 2017 Population splitting of rodlike swimmers in Couette flow. Soft Matt. 13 (25), 44944506.CrossRefGoogle ScholarPubMed
Ostapenko, T., Schwarzendahl, F.J., Böddeker, T.J., Kreis, C.T., Cammann, J., Mazza, M.G. & Bäumchen, O. 2018 Curvature-guided motility of microalgae in geometric confinement. Phys. Rev. Lett. 120 (6), 068002.CrossRefGoogle ScholarPubMed
Pedley, T.J. & Kessler, J.O. 1990 A new continuum model for suspensions of gyrotactic micro-organisms. J. Fluid Mech. 212, 155182.CrossRefGoogle ScholarPubMed
Pedley, T.J. & Kessler, J.O. 1992 Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24 (1), 313358.CrossRefGoogle Scholar
Peng, Z. & Brady, J.F. 2020 Upstream swimming and Taylor dispersion of active Brownian particles. Phys. Rev. Fluids 5 (7), 073102.CrossRefGoogle Scholar
Posten, C. 2009 Design principles of photo-bioreactors for cultivation of microalgae. Engng Life Sci. 9 (3), 165177.CrossRefGoogle Scholar
Romanczuk, P., Bär, M., Ebeling, W., Lindner, B. & Schimansky-Geier, L. 2012 Active Brownian particles. Eur. Phys. J. Spec. Top. 202 (1), 1162.CrossRefGoogle Scholar
Rothschild, 1963 Non-random distribution of bull spermatozoa in a drop of sperm suspension. Nature 198 (4886), 12211222.CrossRefGoogle Scholar
Rusconi, R., Guasto, J.S. & Stocker, R. 2014 Bacterial transport suppressed by fluid shear. Nat. Phys. 10 (3), 212217.CrossRefGoogle Scholar
Saintillan, D. 2018 Rheology of active fluids. Annu. Rev. Fluid Mech. 50 (1), 563592.CrossRefGoogle Scholar
Saintillan, D. & Shelley, M.J. 2013 Active suspensions and their nonlinear models. C. R. Phys. 14 (6), 497517.CrossRefGoogle Scholar
Sandoval, M. & Dagdug, L. 2014 Effective diffusion of confined active Brownian swimmers. Phys. Rev. E 90 (6), 062711.CrossRefGoogle ScholarPubMed
Sandoval, M., Marath, N.K., Subramanian, G. & Lauga, E. 2014 Stochastic dynamics of active swimmers in linear flows. J. Fluid Mech. 742, 5070.CrossRefGoogle Scholar
Schweitzer, F. 2003 Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences. Springer.Google Scholar
Shah, R.K. & London, A.L. 1978 Laminar Flow Forced Convection in Ducts: A Source Book for Compact Heat Exchanger Analytical Data. Academic Press.Google Scholar
Sipos, O., Nagy, K., Di Leonardo, R. & Galajda, P. 2015 Hydrodynamic trapping of swimming bacteria by convex walls. Phys. Rev. Lett. 114 (25), 258104.CrossRefGoogle ScholarPubMed
Spagnolie, S.E. & Lauga, E. 2012 Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations. J. Fluid Mech. 700, 105147.CrossRefGoogle Scholar
Strand, S.R., Kim, S. & Karrila, S.J. 1987 Computation of rheological properties of suspensions of rigid rods: stress growth after inception of steady shear flow. J. Non-Newtonian Fluid Mech. 24 (3), 311329.CrossRefGoogle Scholar
Su, T.-W., Xue, L. & Ozcan, A. 2012 High-throughput lensfree 3D tracking of human sperms reveals rare statistics of helical trajectories. Proc. Natl Acad. Sci. 109 (40), 1601816022.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
Takatori, S.C. & Brady, J.F. 2017 Superfluid behavior of active suspensions from diffusive stretching. Phys. Rev. Lett. 118 (1), 018003.CrossRefGoogle ScholarPubMed
Taylor, G. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219 (1137), 186203.Google Scholar
Taylor, G. 1954 The dispersion of matter in turbulent flow through a pipe. Proc. R. Soc. Lond. A 223 (1155), 446468.Google Scholar
Uspal, W.E., Popescu, M.N., Dietrich, S. & Tasinkevych, M. 2015 Rheotaxis of spherical active particles near a planar wall. Soft Matt. 11 (33), 66136632.CrossRefGoogle Scholar
Vedel, S., Hovad, E. & Bruus, H. 2014 Time-dependent Taylor–Aris dispersion of an initial point concentration. J. Fluid Mech. 752, 107122.CrossRefGoogle Scholar
Vennamneni, L., Nambiar, S. & Subramanian, G. 2020 Shear-induced migration of microswimmers in pressure-driven channel flow. J. Fluid Mech. 890, A15.CrossRefGoogle Scholar
Vicsek, T. & Zafeiris, A. 2012 Collective motion. Phys. Rep. 517 (3), 71140.CrossRefGoogle Scholar
Volpe, G., Buttinoni, I., Vogt, D., Kümmerer, H.-J. & Bechinger, C. 2011 Microswimmers in patterned environments. Soft Matt. 7 (19), 88108815.CrossRefGoogle Scholar
Volpe, G., Gigan, S. & Volpe, G. 2014 Simulation of the active Brownian motion of a microswimmer. Am. J. Phys. 82 (7), 659664.CrossRefGoogle Scholar
Wang, B., Jiang, W., Chen, G., Tao, L. & Li, Z. 2021 Vertical distribution and longitudinal dispersion of gyrotactic microorganisms in a horizontal plane Poiseuille flow. Phys. Rev. Fluids 6 (5), 054502.CrossRefGoogle Scholar
Wang, P. & Chen, G.Q. 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, 844852.CrossRefGoogle Scholar
Wu, Z. & Chen, G.Q. 2014 Approach to transverse uniformity of concentration distribution of a solute in a solvent flowing along a straight pipe. J. Fluid Mech. 740, 196213.CrossRefGoogle Scholar
Wu, Z., Foufoula-Georgiou, E., Parker, G., Singh, A., Fu, X. & Wang, G. 2019 Analytical solution for anomalous diffusion of bedload tracers gradually undergoing burial. J. Geophys. Res. 124 (1), 2137.CrossRefGoogle Scholar
Wu, Z., Singh, A., Foufoula-Georgiou, E., Guala, M., Fu, X. & Wang, G. 2021 A velocity-variation-based formulation for bedload particle hops in rivers. J. Fluid Mech. 912, A33.CrossRefGoogle Scholar
Xiao, Z., Wei, M. & Wang, W. 2019 A review of micromotors in confinements: pores, channels, grooves, steps, interfaces, chains, and swimming in the bulk. ACS Appl. Mater. Interfaces 11 (7), 66676684.CrossRefGoogle ScholarPubMed
Yang, Y., Tan, S.W., Zeng, L., Wu, Y.H., Wang, P. & Jiang, W.Q. 2020 Migration of active particles in a surface flow constructed wetland. J. Hydrol. 582, 124523.CrossRefGoogle Scholar
Yariv, E. & Schnitzer, O. 2014 Ratcheting of Brownian swimmers in periodically corrugated channels: a reduced Fokker-Planck approach. Phys. Rev. E 90 (3), 032115.CrossRefGoogle ScholarPubMed
Yasa, O., Erkoc, P., Alapan, Y. & Sitti, M. 2018 Microalga-powered microswimmers toward active cargo delivery. Adv. Mater. 30 (45), 1804130.CrossRefGoogle ScholarPubMed
Zeng, L. & Pedley, T.J. 2018 Distribution of gyrotactic micro-organisms in complex three-dimensional flows. Part 1. Horizontal shear flow past a vertical circular cylinder. J. Fluid Mech. 852, 358397.CrossRefGoogle Scholar
Zheng, X., ten Hagen, B., Kaiser, A., Wu, M., Cui, H., Silber-Li, Z. & Löwen, H. 2013 Non-Gaussian statistics for the motion of self-propelled Janus particles: experiment versus theory. Phys. Rev. E 88 (3), 032304.CrossRefGoogle ScholarPubMed
Zöttl, A. & Stark, H. 2012 Nonlinear dynamics of a microswimmer in Poiseuille flow. Phys. Rev. Lett. 108 (21), 218104.CrossRefGoogle ScholarPubMed
Zöttl, A. & Stark, H. 2013 Periodic and quasiperiodic motion of an elongated microswimmer in Poiseuille flow. Eur. Phys. J. E 36 (1), 4.CrossRefGoogle ScholarPubMed