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Diffusion of swimming model micro-organisms in a semi-dilute suspension

Published online by Cambridge University Press:  24 September 2007

TAKUJI ISHIKAWA  and
T. J. PEDLEY
TAKUJI ISHIKAWA
Affiliation:
Department of Bioengineering and Robotics, Tohoku University, 6-6-01, Aoba, Aramaki, Aoba-ku, Sendai 980-8579, Japan
T. J. PEDLEY
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
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Abstract

The diffusive behaviour of swimming micro-organisms should be clarified in order to obtain a better continuum model for cell suspensions. In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, in which the centre of mass of the sphere may be displaced from the geometric centre (bottom-heaviness). Effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. The three-dimensional movement of 64 or 27 identical squirmers in a fluid otherwise at rest, contained in a cube with periodic boundary conditions, is dynamically computed, for random initial positions and orientations. The computation utilizes a database of pairwise interactions that has been constructed by the boundary element method. In the case of (non-bottom-heavy) squirmers, both the translational and the orientational spreading of squirmers is correctly described as a diffusive process over a sufficiently long time scale, even though all the movements of the squirmers were deterministically calculated. Scaling of the results on the assumption that the squirmer trajectories are unbiased random walks is shown to capture some but not all of the main features of the results. In the case of (bottom-heavy) squirmers, the diffusive behaviour in squirmers' orientations can be described by a biased random walk model, but only when the effect of hydrodynamic interaction dominates that of the bottom-heaviness. The spreading of bottom-heavy squirmers in the horizontal directions show diffusive behaviour, and that in the vertical direction also does when the average upward velocity is subtracted. The rotational diffusivity in this case, at a volume fraction c=0.1, is shown to be at least as large as that previously measured in very dilute populations of swimming algal cells (Chlamydomonas nivalis).

Type
Papers
Information
Journal of Fluid Mechanics , Volume 588 , 10 October 2007 , pp. 437 - 462
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Acrivos, A. 1995 Bingham award lecture 1994: Shear-induced particle diffusion in concentrated suspensions of noncolloidal particles. J. Rheol. 39, 813–826.CrossRefGoogle Scholar
Batchelor, G. K. & Green, J. T. 1972 b The determination of the bulk stress in a suspension of spherical particles to order c2. J. Fluid Mech. 56, 401–427.CrossRefGoogle Scholar
Batchelor, G. K. 1976 Brownian diffusion of particles with hydrodynamic interaction. J. Fluid Mech. 75, 1–29.CrossRefGoogle Scholar
Bees, M. A. & Hill, N. A. 1998 Linear bioconvection in a suspension of randomly-swimming, gyrotactic micro-organisms. Phys. Fluids 10, 1864–1881.CrossRefGoogle Scholar
Blake, J. R. 1971 A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46, 199–208.CrossRefGoogle Scholar
Bossis, G. & Brady, J. F. 1984 Dynamic simulation of sheared suspensions. Part I. General method. J. Chem. Phys. 80, 5141–5154.CrossRefGoogle Scholar
Brady, J. F. & Bossis, G. 1985 The rheology of concentrated suspensions of spheres in simple shear flow by numerical simulation. J. Fluid Mech. 155, 105–129.CrossRefGoogle Scholar
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20, 111–157.CrossRefGoogle Scholar
Breedveld, V., Ende, D. V. D., Tripathi, A. & Acrivos, A. 1998 The measurement of the shear-induced particle and fluid tracer diffusivities in concentrated suspensions by a novel method. J. Fluid Mech. 375, 297–318.CrossRefGoogle Scholar
Brennen, C. & Winet, H. 1977 Fluid mechanics of propulsion by cilia and flagella. Annu. Rev. Fluid Mech. 9, 339–398.CrossRefGoogle Scholar
Brenner, H. & Weissman, M. H. 1972 Rheology of a dilute suspension of dipolar spherical particles in an external field. Part II. Effects of Rotary Brownian motion. J. Colloid Interface Sci. 41, 499–531.CrossRefGoogle Scholar
Childress, S., Levandowsky, M. & Spiegel, E. A. 1975 Pattern formation in a suspension of swimming micro-organisms: Equations and stability theory. J. Fluid Mech. 63, 591–613.CrossRefGoogle Scholar
Dratler, D. I. & Schowalter, W. R. 1996 Dynamic simulation of suspensions of non-Brownian hard spheres. J. Fluid Mech. 325, 53–77.CrossRefGoogle Scholar
Drazer, G., Koplik, J., Khusid, B. & Acrivos, A. 2002 Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions. J. Fluid Mech. 460, 307–335.CrossRefGoogle Scholar
Foss, D. & Brady, J. F. 1999 Self-diffusion in sheared suspensions by dynamic simulation. J. Fluid Mech. 401, 243–274.CrossRefGoogle Scholar
Foss, D. & Brady, J. F. 2000 Structure, diffusion and rheology of Brownian suspensions by Stokesian dynamics simulation. J. Fluid Mech. 407, 167–200.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, 503–526.CrossRefGoogle ScholarPubMed
Hillesdon, A. J., Pedley, T. J. & Kessler, J. O. 1995 The development of concentration gradients in a suspension of chemotactic bacteria. Bull. Math. Biol. 57, 299–344.CrossRefGoogle Scholar
Ishikawa, T. & Pedley, T. J. 2007 The rheology of a semi-dilute suspension of swimming model micro-organisms. J. Fluid Mech. 588, 399–435.CrossRefGoogle Scholar
Ishikawa, T., Simmonds, M. P. & Pedley, T. J. 2006 Hydrodynamic interaction of two swimming model micro-organisms. J. Fluid Mech. 568, 119–160.CrossRefGoogle Scholar
Kessler, J. O. 1985 a Hydrodynamic focusing of motile algal cells. Nature 313, 218–220.CrossRefGoogle Scholar
Kessler, J. O. 1985 b Co-operative and concentrative phenomena of swimming micro-organisms. Contemp. Phys. 26, 147–166.CrossRefGoogle Scholar
Kessler, J. O. 1986 a The external dynamics of swimming micro-organisms. In Progress in Phycological Research, vol. 4 (ed. Round, F. E. & Chapman, D. J.), pp. 257–307. Biopress.Google Scholar
Kessler, J. O. 1986 b Individual and collective dynamics of swimming cells. J. Fluid Mech. 173, 191–205.CrossRefGoogle Scholar
Kessler, J. O., Hill, N. A. & Häder, D.-P. 1992 Orientation of swimming flagellates by simultaneously acting external factors. J. Phycology 28, 816–822.CrossRefGoogle Scholar
Kim, S. & Karrila, S. J. 1992 Microhydrodynamics: Principles and Selected Applications. Butterworth–Heinemann.Google Scholar
Lighthill, M. J. 1952 On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun. Pure Appl. Maths. 5, 109–118.CrossRefGoogle Scholar
Magar, V., Goto, T. & Pedley, T. J. 2003 Nutrient uptake by a self-propelled steady squirmer. Q. J. Mech. Appl. Maths 56, 65–91.CrossRefGoogle Scholar
Marchiro, M. & Acrivos, A. 2001 Shear-induced particle diffusivities from numerical simulations. J. Fluid Mech. 443, 101–128.CrossRefGoogle Scholar
Metcalfe, A. M. & Pedley, T. J. 2001 Falling plumes in bacterial bioconvection. J. Fluid Mech. 445, 121–149.CrossRefGoogle Scholar
Metcalfe, A. M., Pedley, T. J. & Thingstad, T. F. 2004 Incorporating turbulence into a plankton foodweb model. J. Marine Systems 49, 105–122.CrossRefGoogle Scholar
O'Brien, R. W. 1979 A method for the calculation of the effective transport properties of suspensions of interacting particles. J. Fluid Mech. 91, 17–39.CrossRefGoogle Scholar
Pedley, T. J. & Kessler, J. O. 1987 The orientation of spheroidal microorganisms swimming in a flow field. Proc. R. Soc. Lond. B 231, 47–70.Google Scholar
Pedley, T. J. & Kessler, J. O. 1990 A new continuum model for suspensions of gyrotactic micro-organisms. J. Fluid Mech. 212, 155–182.CrossRefGoogle ScholarPubMed
Pedley, T. J. & Kessler, J. O. 1992 Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24, 313–358.CrossRefGoogle Scholar
Sierou, A. & Brady, J. F. 2004 Shear-induced self-diffusion in non-colloidal suspensions. J. Fluid Mech. 506, 285–314.CrossRefGoogle Scholar
Vladimirov, V. A., Denissenko, P. V., Pedley, T. J., Wu, M. & Moskalev, I. S. 2000 Algal motility measured by a laser-based tracking method. Mar. Freshwater Res. 51, 589–600.CrossRefGoogle Scholar
Vladimirov, V. A., Wu, M. S. C., Pedley, T. J., Denissenko, P. V. & Zakhidova, S. G. 2004 Measurement of cell velocity distributions in populations of motile algae. J. Expl Biol. 207, 1203–1216.CrossRefGoogle ScholarPubMed
Wang, Y., Murai, R. & Acrivos, A. 1998 Transverse shear-induced gradient diffusion in a dilute suspension of spheres. J. Fluid Mech. 357, 279–287.CrossRefGoogle Scholar