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A new continuum model for suspensions of gyrotactic micro-organisms

  • T. J. Pedley (a1) and J. O. Kessler (a2)

A new continuum model is formulated for dilute suspensions of swimming microorganisms with asymmetric mass distributions. Account is taken of randomness in a cell's swimming direction, p, by postulating that the probability density function for p satisfies a Fokker–Planck equation analogous to that obtained for colloid suspensions in the presence of rotational Brownian motion. The deterministic torques on a cell, viscous and gravitational, are balanced by diffusion, represented by an isotropic rotary diffusivity Dr, which is unknown a priori, but presumably reflects stochastic influences on the cell's internal workings. When the Fokker-Planck equation is solved, macroscopic quantities such as the average cell velocity Vc, the particle diffusivity tensor D and the effective stress tensor Σ can be computed; Vc and D are required in the cell conservation equation, and Σ in the momentum equation. The Fokker-Planck equation contains two dimensionless parameters, λ and ε; λ is the ratio of the rotary diffusion time D-1r to the torque relaxation time B (balancing gravitational and viscous torques), while ε is a scale for the local vorticity or strain rate made dimensionless with B. In this paper we solve the Fokker–Planck equation exactly for ε = 0 (λ arbitrary) and also obtain the first-order solution for small ε. Using experimental data on Vc and D obtained with the swimming alga, Chamydomonas nivalis, in the absence of bulk flow, the ε = 0 results can be used to estimate the value of λ for that species (λ ≈ 2.2; Dr ≈ 0.13 s−1). The continuum model for small ε is then used to reanalyse the instability of a uniform suspension, previously investigated by Pedley, Hill & Kessler (1988). The only qualitatively different result is that there no longer seem to be circumstances in which disturbances with a non-zero vertical wavenumber are more unstable than purely horizontal disturbances. On the way, it is demonstrated that the only significant contribution to Σ, other than the basic Newtonian stress, is that derived from the stresslets associated with the cells’ intrinsic swimming motions.

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Arfken, G.: 1985 Mathematical Methods for Physicists, 3rd edn. Academic.
Baloch, K. M. & van de Ven, T. G. M.: 1989 Transient light scattering of suspensions of charged nonspherical particles subjected to an electric field. J. Colloid Interface Sci. 129, 91104.
Batchelor, G. K.: 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545570.
Batchelor, G. K.: 1977 The effect of Brownian motion on the bulk stress in a suspension of spherical particles. J. Fluid Mech. 83, 97117.
Batchelor, G. K. & Green, J. T., 1972 The determination of bulk stress in a suspension of spherical particles to O(c2). J. Fluid Mech. 56, 401427.
Berg, H. C. & Brown, D. A., 1972 Chemotaxis in Escherichia coli analysed by three-dimensional tracking. Nature 239, 500504.
Brenner, H.: 1972 Suspension rheology. Prog. in Heat Mass Transfer 5, 89129.
Brenner, H.: 1974 Rheology of a dilute suspension of axisymmetric Brownian particles. Intl J. Multiphase Flow 1, 195341.
Brenner, H. & Weissman, M. H., 1972 Rheology of a dilute suspension of dipolar spherical particles in an external field. II. Effect of rotary Brownian motion. J. Colloid Interface Sci. 41, 499531.
Childress, S., Levandowsky, M. & Spiegel, E. A., 1975 Pattern formation in a suspension of swimming micro-organisms. J. Fluid Mech. 69, 591613.
Gradshteyn, I. S. & Ryzhik, I. M., 1980 Table of Integrals, Series and Products. (English edn, ed. A. Jeffrey). Academic.
Guell, D. C., Brenner, H., Frankel, R. B. & Hartman, H., 1988 Hydrodynamic forces and band formation in swimming magnetotactic bacteria. J. Theor. Biol. 135, 525542.
Häder, D. & Hill, N. A. 1990 Tracking and averaging the swimming trajectories of Chlamydomonas nivalis (in preparation).
Happel, J. & Brenner, H., 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.
Hill, N. A., Pedley, T. J. & Kessler, J. O., 1989 The growth of bioconvection patterns in a suspension of gyrotactic micro-organisms in a layer of finite depth. J. Fluid Mech. 208, 509543.
Hinch, E. J. & Leal, L. G., 1972a The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles. J. Fluid Mech. 52, 683712.
Hinch, E. J. & Leal, L. G., 1972b Note on the rheology of a dilute suspension of dipolar spheres with weak Brownian couples. J. Fluid Mech. 56, 803813.
Jeffery, G. B.: 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161179.
Keller, E. F. & Segel, L. A., 1970 Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399415.
Keller, E. F. & Segel, L. A., 1971 Model for chemotaxis. J. Theor. Biol. 30, 224234.
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), pp. 241248. Plenum.
Kessler, J. O.: 1985a Hydrodynamic focusing of motile algal cells. Nature 313, 218220.
Kessler, J. O.: 1985b Co-operative and concentrative phenomena of swimming micro-organisms. Contemp. Phys. 26, 147166.
Kessler, J. O.: 1986a The external dynamics of swimming micro-organisms. In Progress in Phycological Research, vol. 4 (ed. F. E. Round), pp. 257307. Bristol Biopress.
Kessler, J. O.: 1986b Individual and collective dynamics of swimming cells. J. Fluid Mech. 173, 191205.
Kessler, J. O.: 1989 Path and pattern - the mutual dynamics of swimming cells and their environment. Comments Theor. Biol. 1, 85108.
Leal, L. G. & Hinch, E. J., 1971 The effect of weak Brownian rotations on particles in shear flow. J. Fluid Mech. 46, 685703.
Leal, L. G. & Hinch, E. J., 1972 The rheology of a suspension of nearly spherical particles subject to Brownian rotations. J. Fluid Mech. 55, 745765.
Pedley, T. J., Hill, N. A. & Kessler, J. O., 1988 The growth of bioconvection patterns in a uniform suspension of gyrotactic micro-organisms. J. Fluid Mech. 195, 223237 (referred to as PHK).
Pedley, T. J. & Kessler, J. O., 1987 The orientation of spheroidal micro-organisms swimming in a flow field. Proc. R. Soc. Lond. B 231, 4770.
Spormann, A. M.: 1987 Unusual swimming behaviour of a magnetotactic bacterium. FEMS Microbiol. Ecol. 45, 3745.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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