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Pattern formation in a suspension of swimming microorganisms: equations and stability theory

Published online by Cambridge University Press:  29 March 2006

S. Childress ,
M. Levandowsky  and
E. A. Spiegel
S. Childress
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York 10012
M. Levandowsky
Affiliation:
Haskins Laboratories, Pace University, New York 10038
E. A. Spiegel
Affiliation:
Department of Astronomy, Columbia University, New York 10027
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Abstract

A model for collective movement and pattern formation in layered suspensions of negatively geotactic micro-organisms is presented. The motility of the organism is described by an average upward swimming speed U and a diffusivity tensor D. It is shown that the equilibrium suspension is unstable to infinitesimal perturbations when either the layer depth or the mean concentration of the organisms exceeds a critical value. For deep layers the maximum growth rate determines a preferred pattern size explicitly in terms of U and D. The results are compared with observations of patterns formed by the ciliated protozoan Tetrahymena pyriformis.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 69 , Issue 3 , 10 June 1975 , pp. 591 - 613
Copyright
© 1975 Cambridge University Press

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References

Brinkman, K. 1968 Keine Geotaxis bei Euglena Z. Pflanzen Physiol 59, 12–16.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability, chap. 2. Oxford University Press.
Cole, J. D. 1968 Perturbation Methods in Applied Mathematics. Blaisdell.
Foster, T. D. 1965 Onset of convection in a layer of fluid cooled from above Phys. Fluids, 8, 1770–1774.Google Scholar
Foster, T. D. 1969 Onset of manifest convection in a layer of fluid with time-dependent surface temperature Phys. Fluids, 12, 2482–2487.Google Scholar
Hurle, D. T. G., Jakeman, E. & Pike, E. R. 1967 On the solution of the Bénard problem with boundaries of finite conductivity. Proc. Roy. Soc. A 296, 469–475.Google Scholar
Keller, E. F. & Segel, L. A 1970 Initiation of slime-mold aggregation viewed as an instability J. Theor. Biol. 26, 399–415.Google Scholar
Levandowsky, M., Childress, S., Spiegel, E. A. & Hutner, S. H. 1975 A mathematical model for pattern formation by swimming microorganisms J. Protozool. 22, 296–306.Google Scholar
Loeffer, J. B. & Mefferd, R. B. 1952 Concerning pattern formation by free-swimming microorganisms Am. Naturalist, 86, 325–329.Google Scholar
Nield, D. A. 1968 The Rayleigh-Jeffreys problem with boundary slab of finite conductivity J. Fluid Mech. 32, 393–398.Google Scholar
Platt, J. R. 1961 ‘Bioconvection patterns’ in cultures of free-swimming microorganisms. Science, 133, 1766–1767.Google Scholar
Plesset, M. S. & Whipple, C. G. 1974 Viscous effects in Rayleigh—Taylor instability Phys. Fluids, 17, 1–7.Google Scholar
Plesset, M. S. & Winet, H. 1974 Bioconvection patterns in swimming microorganism cultures as an example of Rayleigh—Taylor instability Nature, 248, 441–443.Google Scholar
Robbins, W. J. 1952 Patterns formed by motile Euglena gracilis var. Bacillaris. Bull. Torrey Bot. Club, 79, 107–109.Google Scholar
Roberts, A. M. 1970 Geotaxis in motile microorganisms J. Exp. Biol. 53, 687–699.Google Scholar
Wager, H. 1911 On the effect of gravity upon the movements and aggregation of Euglena viridis, Ehrb. and other micro-organisms. Phil. Trans. B 201, 333–390.Google Scholar
Wille, J. J. & Ehret, C. F. 1968 Circulation rhythm of pattern formation in populations of a free-swimming organism, Tetrahymena. J. Protozool. 15, 789–792.Google Scholar
Winet, H. 1969 The influence of gravity and origin of bioconvection in Tetrahymena pyriformis cultures. Ph.D. thesis, University of California, Los Angeles.
Winet, H. & Jahn, T. L. 1972 On the origin of bioconvective fluid instabilities in Tetrahymena culture systems. Biorheol. 9, 87–94.Google Scholar