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Growth of bioconvection patterns in a suspension of gyrotactic micro-organisms in a layer of finite depth

Published online by Cambridge University Press:  26 April 2006

N. A. Hill ,
T. J. Pedley  and
J. O. Kessler
N. A. Hill
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
T. J. Pedley
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
J. O. Kessler
Affiliation:
Department of Physics, University of Arizona, Tucson, AZ 85721, USA
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Abstract

The effect of gyrotaxis on the linear stability of a suspension of swimming, negatively buoyant micro-organisms is examined for a layer of finite depth. In the steady basic state there is no bulk fluid motion, and the upwards swimming of the cells is balanced by diffusion resulting from randomness in their shape, orientation and swimming behaviour. This leads to a bulk density stratification with denser fluid on top. The theory is based on the continuum model of Pedley, Hill & Kessler (1988), and employs both asymptotic and numerical analysis. The suspension is characterized by five dimensionless parameters: a Rayleigh number, a Schmidt number, a layer-depth parameter, a gyrotaxis number G, and a geometrical parameter measuring the ellipticity of the micro-organisms. For small values of G, the most unstable mode has a vanishing wavenumber, but for sufficiently large values of G, the predicted initial wavelength is finite, in agreement with experiments. The suspension becomes less stable as the layer depth is increased. Indeed, if the layer is sufficiently deep an initially homogeneous suspension is unstable, and the equilibrium state does not form. The theory of Pedley, Hill & Kessler (1988) for infinite depth is shown to be appropriate in that case. An unusual feature of the model is the existence of overstable or oscillatory modes which are driven by the gyrotactic response of the micro-organisms to the shear at the rigid boundaries of the layer. These modes occur at parameter values which could be realized in experiments.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 208 , November 1989 , pp. 509 - 543
Copyright
© 1989 Cambridge University Press

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