Hostname: page-component-6b989bf9dc-jks4b Total loading time: 0 Render date: 2024-04-13T09:40:54.379Z Has data issue: false hasContentIssue false

Dispersion of active particles in confined unidirectional flows

Published online by Cambridge University Press:  16 August 2019

Weiquan Jiang
Laboratory of Systems Ecology and Sustainability Science, College of Engineering, Peking University, Beijing 100871, China
Guoqian Chen*
Laboratory of Systems Ecology and Sustainability Science, College of Engineering, Peking University, Beijing 100871, China
Email address for correspondence:


Transport of micro-organisms in confined flows can be characterized by a one-dimensional overall dispersion mechanism, of importance to various biotechnological applications. Based on Brenner’s generalized Taylor dispersion theory, an overall dispersion model is analytically studied in the present work for a dilute suspension of active particles in confined unidirectional flows. With the confined section of the channel and the swimming orientation space taken together as the local space and the longitudinal coordinate standing for the one-dimensional global space, this model is analytically accurate and possessed of wide adaptability in terms of the swimming Péclet number. The Robin boundary condition is introduced to account for wall accumulation of active particles, and compared with a typical reflection boundary condition. Complications associated with the boundary conditions for analytical derivation are removed respectively by a decomposition of the distribution function and an extension of the flow field. Interesting solutions are concretely found and intensively illustrated. Detailed case studies on the transport of spherical and rod-like particles to illustrate the dispersion mechanism are presented with respect to a Couette flow and a plane Poiseuille flow. Associated with the local distribution of particles, extensive descriptions are given for the dynamical system behaviours such as accumulation near both stable points/lines and boundaries, symmetric polarization structure, closed orbits, trapping effect, nematic alignments and bimodalization of swimming direction. For spherical particles, the accumulation is shown leading to a reduction of the overall dispersivity in both of the flows, while for rod-like active particles in the Couette flow, the accumulation can result in an enhancement of dispersion, due to the nematic alignments of particles towards streamlines.

JFM Papers
© 2019 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. Gonzalez-Fernandez, C. & Muñoz, R.), pp. 144. Woodhead Publishing.Google Scholar
Alonso-Matilla, R., Chakrabarti, B. & Saintillan, D. 2019 Transport and dispersion of active particles in periodic porous media. Phys. Rev. Fluids 4 (4), 043101.10.1103/PhysRevFluids.4.043101Google Scholar
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, 205218.10.1017/S0022112083000117Google Scholar
Bearon, R. N. 2003 An extension of generalized Taylor dispersion in unbounded homogeneous shear flows to run-and-tumble chemotactic bacteria. Phys. Fluids 15 (6), 15521563.10.1063/1.1569482Google Scholar
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), 121902.10.1063/1.4772189Google 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, R3.10.1017/jfm.2015.198Google 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, 602635.10.1017/jfm.2011.198Google 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), 20572077.10.1098/rspa.2009.0606Google Scholar
Berke, A. P., Turner, L., Berg, H. C. & Lauga, E. 2008 Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101 (3), 038102.10.1103/PhysRevLett.101.038102Google Scholar
Brenner, H. 1982 A general theory of Taylor dispersion phenomena. II. An extension. Physico-Chem. Hydrodyn. 3 (2), 139157.Google Scholar
Brenner, H. & Edwards, D. A. 1993 Dispersion of nonreactive solutes in continuous media. In Macrotransport Processes, pp. 65155. Butterworth-Heinemann.10.1016/B978-0-08-051059-0.50007-7Google 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), 321352.10.1017/S0022112070002409Google 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), 6367.10.1017/S0022112072001077Google 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 Scholar
Chilukuri, S., Collins, C. H. & Underhill, P. T. 2015 Dispersion of flagellated swimming microorganisms in planar Poiseuille flow. Phys. Fluids 27 (3), 031902.10.1063/1.4914129Google Scholar
Chisti, Y. 2007 Biodiesel from microalgae. Biotechnol. Adv. 25 (3), 294306.10.1016/j.biotechadv.2007.02.001Google Scholar
Costanzo, A., Leonardo, R. D., Ruocco, G. & Angelani, L. 2012 Transport of self-propelling bacteria in micro-channel flow. J. Phys.: Condens. Matter 24 (6), 065101.Google Scholar
Croze, O. A., Bearon, R. N. & Bees, M. A. 2017 Gyrotactic swimmer dispersion in pipe flow: testing the theory. J. Fluid Mech. 816, 481506.10.1017/jfm.2017.90Google 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. Roy. Soc. Interface 10 (81), 20121041.Google Scholar
Doi, M. & Edwards, S. F. 1988 Brownian motion. In The Theory of Polymer Dynamics, pp. 4690. Oxford University Press.Google Scholar
Elgeti, J. & Gompper, G. 2013 Wall accumulation of self-propelled spheres. Europhys. Lett. 101 (4), 48003.10.1209/0295-5075/101/48003Google Scholar
Enculescu, M. & Stark, H. 2011 Active colloidal suspensions exhibit polar order under gravity. Phys. Rev. Lett. 107 (5), 058301.10.1103/PhysRevLett.107.058301Google Scholar
Ezhilan, B., Pahlavan, A. A. & Saintillan, D. 2012 Chaotic dynamics and oxygen transport in thin films of aerotactic bacteria. Phys. Fluids 24 (9), 091701.10.1063/1.4752764Google Scholar
Ezhilan, B. & Saintillan, D. 2015 Transport of a dilute active suspension in pressure-driven channel flow. J. Fluid Mech. 777, 482522.10.1017/jfm.2015.372Google Scholar
Frankel, I. & Brenner, H. 1989 On the foundations of generalized Taylor dispersion theory. J. Fluid Mech. 204, 97119.10.1017/S0022112089001679Google Scholar
Goldstein, R. E. 2015 Green algae as model organisms for biological fluid dynamics. Annu. Rev. Fluid Mech. 47 (1), 343375.10.1146/annurev-fluid-010313-141426Google Scholar
Guo, J., Wu, X., Jiang, W. & Chen, G. 2018 Contaminant transport from point source on water surface in open channel flow with bed absorption. J. Hydrol. 561, 295303.10.1016/j.jhydrol.2018.03.066Google Scholar
Hill, N. A. & Bees, M. A. 2002 Taylor dispersion of gyrotactic swimming micro-organisms in a linear flow. Phys. Fluids 14 (8), 25982605.10.1063/1.1458003Google Scholar
Hwang, Y. & Pedley, T. J. 2014 Bioconvection under uniform shear: linear stability analysis. J. Fluid Mech. 738, 522562.10.1017/jfm.2013.604Google Scholar
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.Google Scholar
Jiang, W. & Chen, G. 2019 Solute transport in two-zone packed tube flow: long-time asymptotic expansion. Phys. Fluids 31 (4), 043303.Google Scholar
Jiang, W. Q. & Chen, G. Q. 2018 Solution of Gill’s generalized dispersion model: solute transport in Poiseuille flow with wall absorption. Intl J. Heat Mass Transfer 127, 3443.10.1016/j.ijheatmasstransfer.2018.07.003Google 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), 745765.10.1017/S0022112072002125Google 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), 078101.10.1103/PhysRevLett.103.078101Google Scholar
Liao, Q., Li, L., Chen, R. & Zhu, X. 2014 A novel photobioreactor generating the light/dark cycle to improve microalgae cultivation. Bioresour. Technol. 161, 186191.10.1016/j.biortech.2014.02.119Google Scholar
Manela, A. & Frankel, I. 2003 Generalized Taylor dispersion in suspensions of gyrotactic swimming micro-organisms. J. Fluid Mech. 490, 99127.10.1017/S0022112003005147Google Scholar
Mata, T. M., Martins, A. A. & Caetano, N. S. 2010 Microalgae for biodiesel production and other applications: a review. Renewable Sustainable Energy Reviews 14 (1), 217232.10.1016/j.rser.2009.07.020Google Scholar
Muñoz, R. & Guieysse, B. 2006 Algal–bacterial processes for the treatment of hazardous contaminants: a review. Water Res. 40 (15), 27992815.10.1016/j.watres.2006.06.011Google 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.Google Scholar
Oncel, S. & Kose, A. 2014 Comparison of tubular and panel type photobioreactors for biohydrogen production utilizing Chlamydomonas reinhardtii considering mixing time and light intensity. Bioresour. Technol. 151, 265270.10.1016/j.biortech.2013.10.076Google Scholar
Pedley, T. J. & Kessler, J. O. 1990 A new continuum model for suspensions of gyrotactic micro-organisms. J. Fluid Mech. 212, 155182.10.1017/S0022112090001914Google Scholar
Pedley, T. J. & Kessler, J. O. 1992 Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24 (1), 313358.10.1146/annurev.fl.24.010192.001525Google Scholar
Posten, C. 2009 Design principles of photo-bioreactors for cultivation of microalgae. Engng Life Sci. 9 (3), 165177.10.1002/elsc.200900003Google Scholar
Rothschild 1963 Non-random distribution of bull spermatozoa in a drop of sperm suspension. Nature 198 (4886), 12211222.10.1038/1981221a0Google Scholar
Rusconi, R., Guasto, J. S. & Stocker, R. 2014 Bacterial transport suppressed by fluid shear. Nat. Phys. 10 (3), 212217.10.1038/nphys2883Google Scholar
Saintillan, D. & Shelley, M. J. 2008 Instabilities and pattern formation in active particle suspensions: kinetic theory and continuum simulations. Phys. Rev. Lett. 100 (17), 178103.10.1103/PhysRevLett.100.178103Google Scholar
Saintillan, D. & Shelley, M. J. 2013 Active suspensions and their nonlinear models. C. R. Physique 14 (6), 497517.10.1016/j.crhy.2013.04.001Google Scholar
Shapiro, M. & Brenner, H. 1987 Chemically reactive generalized Taylor dispersion phenomena. AIChE J. 33 (7), 11551167.10.1002/aic.690330710Google Scholar
Sipos, O., Nagy, K., Di Leonardo, R. & Galajda, P. 2015 Hydrodynamic trapping of swimming bacteria by convex walls. Phys. Rev. Lett. 114 (25), 258104.10.1103/PhysRevLett.114.258104Google Scholar
Spagnolie, S. E. & Lauga, E. 2012 Hydrodynamics of self-propulsion near a boundary: predictions and accuracy of far-field approximations. J. Fluid Mech. 700, 105147.10.1017/jfm.2012.101Google Scholar
Stark, H. 2016 Swimming in external fields. Eur. Phys. J. 225 (11), 23692387.Google Scholar
Stephenson, A. L., Kazamia, E., Dennis, J. S., Howe, C. J., Scott, S. A. & Smith, A. G. 2010 Life-cycle assessment of potential algal biodiesel production in the United Kingdom: a comparison of raceways and air-lift tubular bioreactors. Energy Fuels 24 (7), 40624077.10.1021/ef1003123Google Scholar
Suresh Kumar, K., Dahms, H.-U., Won, E.-J., Lee, J.-S. & Shin, K.-H. 2015 Microalgae – a promising tool for heavy metal remediation. Ecotoxicol. Environment. Safety 113, 329352.10.1016/j.ecoenv.2014.12.019Google Scholar
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
Volpe, G., Gigan, S. & Volpe, G. 2014 Simulation of the active Brownian motion of a microswimmer. Am. J. Phys. 82 (7), 659664.10.1119/1.4870398Google 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, 844852.10.1016/j.ijheatmasstransfer.2017.02.051Google 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, 196213.10.1017/jfm.2013.648Google Scholar
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, 358397.10.1017/jfm.2018.494Google Scholar
Zeng, L., Zhang, H. W., Wu, Y. H., Li, C. F. & Wang, P. 2019 Theoretical and numerical analysis of vertical distribution of active particles in a free-surface wetland flow. J. Hydrol. 573, 449455.10.1016/j.jhydrol.2019.03.085Google Scholar
Zöttl, A. & Stark, H. 2012 Nonlinear dynamics of a microswimmer in Poiseuille flow. Phys. Rev. Lett. 108 (21), 218104.10.1103/PhysRevLett.108.218104Google Scholar
Zöttl, A. & Stark, H. 2013 Periodic and quasiperiodic motion of an elongated microswimmer in Poiseuille flow. Eur. Phys. J. E 36 (1), 4.10.1140/epje/i2013-13004-5Google Scholar