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The spatial distribution of gyrotactic swimming micro-organisms in laminar flow fields

Published online by Cambridge University Press:  13 June 2011

R. N. BEARON ,
A. L. HAZEL  and
G. J. THORN
R. N. BEARON*
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK
A. L. HAZEL
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK
G. J. THORN
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK
*
†Email address for correspondence: rbearon@liv.ac.uk
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Abstract

We compare the results of two-dimensional, biased random walk models of individual swimming micro-organisms with advection–diffusion models for the whole population. In particular, we consider the influence of the local flow environment (gyrotaxis) on the resulting motion. In unidirectional flows, the results of the individual and population models are generally in good agreement, even in flows in which the cells can experience a range of shear environments, and both models successfully predict the phenomena of gravitactic focusing. Numerical results are also compared with asymptotic expressions for weak and strong shear. Discrepancies between the models arise in two cases: (i) when reflective boundary conditions change the orientation distribution in the random walk model from that predicted by the long-term asymptotics used to derive the advection–diffusion model; (ii) when the spatial and temporal scales are not large enough for the advection–diffusion model to apply. We also use a simple two-dimensional flow containing a variety of flow regimes to explore what happens when there are localized regions in which the generalized Taylor dispersion theory used in the derivation of the population model does not apply. For spherical cells, we find good agreement between the models outside the ‘break-down’ regions, but comparison of the results within these regions is complicated by the presence of nearby boundaries and their influence on the random walk model. In contrast, for rod-shaped cells which are reorientated by both vorticity and strain, we see qualitatively different spatial patterns between individual and advection–diffusion models even in the absence of gyrotaxis, because cells are advected between regions of differing rates of strain.

JFM classification

Biological Fluid Dynamics: Micro-organism dynamics Complex Fluids: Suspensions

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Papers
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Journal of Fluid Mechanics , Volume 680 , 10 August 2011 , pp. 602 - 635
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Almog, Y. & Frankel, I. 1998 Rheology of dilute suspensions of Brownian dipolar axisymmetric particles. J. Fluid. Mech. 366, 289–310.Google Scholar
Batschelet, E. 1981 Circular Statistics in Biology. Academic.Google 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), 1552–1563.Google Scholar
Bearon, R. N. & Grünbaum, D. 2008 From individual behaviour to population models: A case study using swimming algae. J. Theor. Biol. 251 (4), 679–697.CrossRefGoogle ScholarPubMed
Bearon, R. N., Grünbaum, D. & Cattolico, R. A. 2006 Effects of salinity structure on swimming behavior and harmful algal bloom formation in Heterosigma akashiwo, a toxic raphidophyte. Mar. Ecol. Prog. Ser. 306, 153–163.Google 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. B 466 (2119), 2057–2077.Google Scholar
Bees, M. A., Hill, N. A. & Pedley, T. J. 1998 Analytical approximations for the orientation distribution of small dipolar particles in steady shear flows. J. Math. Biol. 36 (3), 269–298.CrossRefGoogle Scholar
Birch, D. A., Young, W. R. & Franks, P. J. S. 2008 Thin layers of plankton: Formation by shear and death by diffusion. Deep-Sea Res. Part 1 55 (3), 277–295.CrossRefGoogle Scholar
Brenner, H. 1979 Taylor dispersion in systems of sedimenting nonspherical Brownian particles 1. Homogeneous, centrosymmetric, axisymmetric particles. J. Colloid Interface Sci. 71 (2), 189–208.Google Scholar
Cisneros, L. H., Cortez, R., Dombrowski, C., Goldstein, R. E. & Kessler, J. O. 2007 Fluid dynamics of self-propelled microorganisms, from individuals to concentrated populations. Exp. Fluids 43 (5), 737–753.Google Scholar
Cisneros, L. H., Kessler, J. O., Ortiz, R., Cortez, R. & Bees, M. A. 2008 Unexpected bipolar flagellar arrangements and long-range flows driven by bacteria near solid boundaries. Phys. Rev. Lett. 101 (16), 168102.CrossRefGoogle ScholarPubMed
Demkowicz, L., Oden, J. T., Rachowicz, W. & Hardy, O. 1989 Toward a universal h-p adaptive finite element strategy, Part 1. Constrained approximation and data structure. Comput. Meth. Appl. Mech. Engng 77, 79–112.CrossRefGoogle Scholar
Demmel, J. W., Eisenstat, S. C., Gilbert, J. R., Li, X. S. & Liu, J. W. H. 1999 A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. Applics. 20, 720–755.CrossRefGoogle Scholar
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.Google Scholar
Ferreira de Sousa, P. J. S. A. & Pereira, J. C. F. 2009 Dynamics of passive scalars and tracers advected by a two-dimensional tripolar vortex. J. Fluid Mech. 634, 41–60.CrossRefGoogle Scholar
Frankel, I. & Brenner, H. 1989 On the foundations of generalized Taylor dispersion theory. J. Fluid Mech. 204, 97–119.CrossRefGoogle Scholar
Frankel, I. & Brenner, H. 1991 Generalized Taylor dispersion phenomena in unbounded homogeneous shear flows. J. Fluid Mech. 230, 147–181.CrossRefGoogle Scholar
Frankel, I. & Brenner, H. 1993 Taylor dispersion of orientable Brownian particles in unbounded homogeneous shear flows. J. Fluid Mech. 255, 129–156.CrossRefGoogle Scholar
Franks, P. J. S. 1995 Coupled physical–biological models in oceanography. Rev. Geophys. 33, 1177–1187.CrossRefGoogle Scholar
Ghorai, S. & Hill, N. A. 1999 Development and stability of gyrotactic plumes in bioconvection. J. Fluid Mech. 400, 1–31.CrossRefGoogle Scholar
Heil, M. & Hazel, A. L. 2006 oomph-lib – An object-oriented multi-physics finite-element library in fluid structure interaction. In Lecture Notes on Computational Science and Engineering (ed. Schafer, M. & Bungartz, H.-J.), pp. 19–49. Springer.Google Scholar
Hill, N. A. & Bees, M. A. 2002 Taylor dispersion of gyrotactic swimming micro-organisms in a linear flow. Phys. Fluids 14 (8), 2598–2605.CrossRefGoogle Scholar
Hill, N. A. & Häder, D. P. 1997 A biased random walk model for the trajectories of swimming micro-organisms. J. Theor. Biol. 186 (4), 503–526.Google Scholar
Hill, N. A. & Pedley, T. J. 2005 Bioconvection. Fluid Dyn. Res. 37 (1–2), 1–20.CrossRefGoogle Scholar
Horner, R. A., Garrison, D. L. & Plumley, F. G. 1997 Harmful algal blooms and red tide problems on the US west coast. Limnol. Oceanogr. 42 (5), 1076–1088.CrossRefGoogle Scholar
Ishikawa, T. 2009 Suspension biomechanics of swimming microbes. J. R. Soc. Interface 6 (39), 815–834.CrossRefGoogle ScholarPubMed
Kessler, J. O. 1985 Hydrodynamic focusing of motile algal cells. Nature 313 (5999), 218–220.CrossRefGoogle Scholar
Kim, S. & Karrila, S. J. 2005 Microhydrodynamics: Principles and Selected Applications. Dover.Google Scholar
Luchsinger, R., Bergesen, B. & Mitchell, J. G. 1999 Bacteria swimming strategies and turbulence. Biophys. J. 77, 2377–2386.CrossRefGoogle ScholarPubMed
Manela, A. & Frankel, I. 2003 Generalized Taylor dispersion in suspensions of gyrotactic swimming micro-organisms. J. Fluid Mech. 490, 99–127.Google Scholar
McManus, M. A., Alldredge, A. L., Barnard, A. H., Boss, E., Case, J. F., Cowles, T. J., Donaghay, P. L., Eisner, L. B., Gifford, D. J., Greenlaw, C. F., Herren, C. M., Holliday, D. V., Johnson, D., MacIntyre, S., McGehee, D. M., Osborn, T. R., Perry, M. J., Pieper, R. E., Rines, J. E. B., Smith, D. C., Sullivan, J. M., Talbot, M. K., Twardowski, M. S., Weidemann, A. & Zaneveld, J. R. 2003 Characteristics, distribution and persistence of thin layers over a 48 hour period. Mar. Ecol. Prog. Ser. 261, 1–19.Google Scholar
Morris, J. F. & Brady, J. F. 1996 Self-diffusion in sheared suspensions. J. Fluid Mech. 312, 232–252.CrossRefGoogle Scholar
Ottino, J. M. 1990 Mixing, chaotic advection, and turbulence. Annu. Rev. Fluid Mech. 22, 207–253.CrossRefGoogle Scholar
Pedley, T. J. & Kessler, J. O. 1990 A new continuum model for suspensions of gyrotactic microorganisms. J. Fluid Mech. 212, 155–182.Google Scholar
Pedley, T. J. & Kessler, J. O. 1992 Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24, 313–358.CrossRefGoogle Scholar
Raven, J. A. & Falkowski, P. G. 1999 Oceanic sinks for atmospheric CO2. Plant Cell. Environ. 22 (6), 741–755.Google Scholar
Sharples, J., Moore, C. M., Rippeth, T. P., Holligan, P. M., Hydes, D. J., Fisher, N. R. & Simpson, J. H. 2001 Phytoplankton distribution and survival in the thermocline. Limnol. Oceanogr. 46 (3), 486–496.CrossRefGoogle Scholar
Sobczyk, K. 1991 Stochastic Differential Equations: With Applications to Physics and Engineering. Kluwer.CrossRefGoogle Scholar
Steinbuck, J. V., Stacey, M. T., McManus, M. A., Cheriton, O. M. & Ryan, J. P. 2009 Observations of turbulent mixing in a phytoplankton thin layer: Implications for formation, maintenance, and breakdown. Limnol. Oceanogr. 54 (4), 1353–1368.Google Scholar
Thorn, G. J. & Bearon, R. N. 2010 Transport of gyrotactic organisms in general 3D flow fields. Phys. Fluids 22 (4), 041902.Google Scholar
Torney, C. & Neufeld, Z. 2007 Transport and aggregation of self-propelled particles in fluid flows. Phys. Rev. Lett. 99 (7), 078101.Google Scholar
Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240.Google Scholar
Zienkiewicz, O. C. & Zhu, J. 1992 a The superconvergent patch recovery and a posteriori error estimates. Part 1. The recovery technique. Intl J. Numer. Meth. Engng 33, 1331–1364.CrossRefGoogle Scholar
Zienkiewicz, O. C. & Zhu, J. 1992 b The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity. Intl J. Numer. Meth. Engng 33, 1365–1382.Google Scholar
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