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Bioconvection in suspensions of oxytactic bacteria: linear theory

Published online by Cambridge University Press:  26 April 2006

A. J. Hillesdon  and
T. J. Pedley
A. J. Hillesdon
Affiliation:
Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, UK
T. J. Pedley
Affiliation:
Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT, UK Present address: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK.
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Abstract

When a suspension of the bacterium Bacillus subtilis is placed in a chamber with its upper surface open to the atmosphere, complex bioconvection patterns form. These arise because the cells (a) are denser than water, and (b) swim upwards on average so that the density of an initially uniform suspension becomes greater at the top than at the bottom. When the vertical density gradient becomes large enough an overturning instability occurs which evolves ultimately into the observed patterns. The cells swim upwards because they are oxytactic, i.e. they swim up gradients of oxygen, and they consume oxygen. These properties are incorporated in conservation equations for the cell and oxygen concentrations, which, for the pre-instability stage of the pattern formation process, have been solved in a previous paper (Hillesdon, Pedley & Kessler 1995). In this paper we carry out a linear instability analysis of the steady-state cell and oxygen concentration distributions. There are intrinsic differences between the shallow-and deep-chamber cell concentration distributions, with the consequence that the instability is non-oscillatory in shallow chambers, but must be oscillatory in deep chambers whenever the critical wavenumber is non-zero. We investigate how the critical Rayleigh number for the suspension varies with the three independent parameters of the problem and discuss the most appropriate definition of the Rayleigh number. Several qualitative aspects of the solution of the linear instability problem agree with experimental observation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 324 , 10 October 1996 , pp. 223 - 259
Copyright
© 1996 Cambridge University Press

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