Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-23T09:43:05.378Z Has data issue: false hasContentIssue false

Pre-asymptotic dispersion of active particles through a vertical pipe: the origin of hydrodynamic focusing

Published online by Cambridge University Press:  27 April 2023

Mingyang Guan
Laboratory of Systems Ecology and Sustainability Science, College of Engineering, Peking University, Beijing 100871, PR China
Weiquan Jiang
Macao Environmental Research Institute, Macau University of Science and Technology, Macao 999078, PR China
Bohan Wang
Laboratory of Systems Ecology and Sustainability Science, College of Engineering, Peking University, Beijing 100871, PR China
Li Zeng
State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100038, PR China
Zhi Li
Laboratory of Systems Ecology and Sustainability Science, College of Engineering, Peking University, Beijing 100871, PR China
Guoqian Chen*
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:


When motile algal cells are exposed to gyrotactic torques, their swimming directions are guided to form radial accumulation, well known as hydrodynamic focusing. The origin of hydrodynamic focusing from the effects of active swimming, ambient flow and particle anisotropy is elucidated in the present study on the pre-asymptotic dispersion of active particles through a vertical pipe. With an extension of the Galerkin method to pipe flows, time-dependent solutions directly from the Smoluchowski equation in the position and orientation space are derived by series expansions of spherical harmonics and Bessel functions. Ballistic and diffusive scaling laws are examined with the predominance of self-propelled swimming, and computation is validated against an explicit benchmark solution and Lagrangian particle simulation. In the limit of extreme shear, the competitive roles of shear dispersion and Brownian rotation are reflected concretely in the pre-asymptotic phase of hydrodynamic focusing. For flows with various shear strengths, a concentration peak in near-wall regions with a smooth transition to hydrodynamic focusing is illustrated with richer phenomena in upwelling and downwelling flows. A newly observed regime through a vertical pipe, named transient effective trapping, is revealed as a transitional mode towards hydrodynamic focusing. The pre-asymptotic approach to hydrodynamic focusing is elaborated intensively through extensive solutions of concentration moments and macroscopic transport coefficients characterised by swimming and flow Péclet numbers. The unique findings for the origin of hydrodynamic focusing could provide insight into related micro-algae reactor technology and contribute to flow control and biomass transfer in confined environments.

JFM Papers
© The Author(s), 2023. 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.)


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
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
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
Barton, N.G. 1983 On the method of moments for solute dispersion. J. Fluid Mech. 126, 205218.CrossRefGoogle Scholar
Baskaran, A. & Marchetti, M.C. 2009 Statistical mechanics and hydrodynamics of bacterial suspensions. Proc. Natl Acad. Sci. USA 106 (37), 1556715572.CrossRefGoogle ScholarPubMed
Bearon, R.N. 2022 When do shape changers swim upstream? J. Fluid Mech. 950, F1.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. A 466 (2119), 20572077.CrossRefGoogle Scholar
Berg, H.C. 1993 Random Walks in Biology. Princeton University Press.Google Scholar
Brenner, H. 1964 The Stokes resistance of an arbitrary particle – IV. Arbitrary fields of flow. Chem. Engng Sci. 19 (10), 703727.CrossRefGoogle Scholar
Brenner, H. & Edwards, D. 1993 Macrotransport Processes. Butterworth-Heinemann.Google Scholar
Brezinski, C. 1991 Biorthogonality and its Applications to Numerical Analysis. Marcel Dekker.Google Scholar
Brillinger, D.R. 1997 A particle migrating randomly on a sphere. J. Theor. Probab. 10 (2), 429443.CrossRefGoogle Scholar
Cates, M.E. & Tailleur, J. 2015 Motility-induced phase separation. Annu. Rev. Condens. Matter Phys. 6 (1), 219244.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
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
Chirikjian, G.S. 2009 Stochastic Models, Information Theory, and Lie Groups, vol. 1. Birkhäuser.CrossRefGoogle Scholar
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. USA 116 (23), 1111911124.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), 10671070.CrossRefGoogle ScholarPubMed
Durham, W.M. & Stocker, R. 2012 Thin phytoplankton layers: characteristics, mechanisms, and consequences. Annu. Rev. Mar. Sci 4 (1), 177207.CrossRefGoogle ScholarPubMed
Elgeti, J. & Gompper, G. 2015 Run-and-tumble dynamics of self-propelled particles in confinement. Europhys. Lett. 109 (5), 58003.CrossRefGoogle Scholar
Ezhilan, B. & Saintillan, D. 2015 Transport of a dilute active suspension in pressure-driven channel flow. J. Fluid Mech. 777, 482522.CrossRefGoogle Scholar
Fenchel, T. & Finlay, B.J. 1984 Geotaxis in the ciliated protozoon Loxodes. J. Expl Biol. 110 (1), 1733.CrossRefGoogle Scholar
Fenchel, T. & Finlay, B.J. 1986 Photobehavior of the ciliated protozoon Loxodes: taxic, transient, and kinetic responses in the presence and absence of oxygen. J. Protozool. 33 (2), 139145.CrossRefGoogle Scholar
Foister, R.T. & Ven, T.G.M.V.D. 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
Frankel, I. & Brenner, H. 1993 Taylor dispersion of orientable Brownian particles in unbounded homogeneous shear flows. J. Fluid Mech. 255, 129156.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
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
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
Guan, M.Y., Zeng, L., Jiang, W.Q., Guo, X.L., Wang, P., Wu, Z., Li, Z. & Chen, G.Q. 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.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. & Chen, G. 2022 Solute dispersion from a continuous release source in a vegetated flow: an analytical study. Water Resour. Res. 58, e2021WR030255.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
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
Jakuszeit, T., Croze, O.A. & Bell, S. 2019 Diffusion of active particles in a complex environment: role of surface scattering. Phys. Rev. E 99 (1), 012610.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 Dispersion of active particles in confined unidirectional flows. J. Fluid Mech. 877, 134.CrossRefGoogle Scholar
Jiang, W. & Chen, G. 2020 Dispersion of gyrotactic micro-organisms in pipe flows. J. Fluid Mech. 889, A18.CrossRefGoogle Scholar
Jiang, W. & Chen, G. 2021 Transient dispersion process of active particles. J. Fluid Mech. 927, A11.CrossRefGoogle Scholar
Kessler, J.O. 1984 Gyrotactic buoyant convection and spontaneous pattern formation in algal cell cultures. In Nonequilibrium Cooperative Phenomena in Physics and Related Fields (ed. M.G. Velarde), vol. 116, pp. 241–248. Springer.CrossRefGoogle Scholar
Kessler, J.O. 1985 a Co-operative and concentrative phenomena of swimming micro-organisms. Contemp. Phys. 26 (2), 147166.CrossRefGoogle Scholar
Kessler, J.O. 1985 b Hydrodynamic focusing of motile algal cells. Nature 313 (5999), 218220.CrossRefGoogle Scholar
Kessler, J.O. 1986 Individual and collective fluid dynamics of swimming cells. J. Fluid Mech. 173, 191205.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., Gong, Z., Jiang, W., Zhan, J., Wang, B., Fu, X., Xu, M. & Wu, Z. 2023 Environmental transport of gyrotactic microorganisms in an open-channel flow. Water Resour. Res. 59, e2022WR033229.CrossRefGoogle Scholar
Liao, S. 2004 Beyond Perturbation: Introduction to the Homotopy Analysis Method. CRC Series–Modern Mechanics and Mathematics, vol. 2. Chapman & Hall/CRC.Google Scholar
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
Olver, F.W., Lozier, D.W., Boisvert, R. & Clark, C.W. 2010 The NIST Handbook of Mathematical Functions. Cambridge University Press.Google Scholar
Pedley, T.J. & Kessler, J.O. 1987 The orientation of spheroidal microorganisms swimming in a flow field. Proc. R. Soc. B 231 (1262), 4770.Google Scholar
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
Peruani, F. & Morelli, L.G. 2007 Self-propelled particles with fluctuating speed and direction of motion in two dimensions. Phys. Rev. Lett. 99 (1), 010602.CrossRefGoogle ScholarPubMed
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., 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
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
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.I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219 (1137), 186203.Google Scholar
Theillard, M. & Saintillan, D. 2019 Computational mean-field modeling of confined active fluids. J. Comput. Phys. 397, 108841.CrossRefGoogle Scholar
Thorn, G.J. & Bearon, R.N. 2010 Transport of spherical gyrotactic organisms in general three-dimensional flow fields. Phys. Fluids 22 (4), 041902.CrossRefGoogle Scholar
Uhlenbeck, G.E. & Ornstein, L.S. 1930 On the theory of the Brownian motion. Phys. Rev. 36 (5), 823841.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
Volpe, G., Gigan, S. & Volpe, G. 2014 Simulation of the active Brownian motion of a microswimmer. Am. J. Phys. 82 (7), 659664.CrossRefGoogle Scholar
Walker, B.J., Ishimoto, K., Moreau, C., Gaffney, E.A. & Dalwadi, M.P. 2022 Emergent rheotaxis of shape-changing swimmers in Poiseuille flow. J. Fluid Mech. 944, R2.CrossRefGoogle Scholar
Wang, B., Jiang, W. & Chen, G. 2022 Gyrotactic trapping of micro-swimmers in simple shear flows: a study directly from the fundamental Smoluchowski equation. J. Fluid Mech. 939, A37.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
Yan, W. & Brady, J.F. 2015 The force on a boundary in active matter. J. Fluid Mech. 785, R1.CrossRefGoogle Scholar
Zade, S., Costa, P., Fornari, W., Lundell, F. & Brandt, L. 2018 Experimental investigation of turbulent suspensions of spherical particles in a square duct. J. Fluid Mech. 857, 748783.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
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