Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-19T06:25:44.605Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Cash, J. R. & Moore, D. R. 1980 A high order method for the numerical solution of two-point boundary value problems. BIT 20, 44–52.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
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 (referred to as CLS).Google Scholar
Childress, S. & Peyret, R. 1976 A numerical study of two-dimensional convection by motile particles. J. Méc. 15, 753–779.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Harashima, A., Watanabe, M. & Fujishiro, I. 1988 Evolution of bioconvection patterns in a culture of mobile flagellates. Phys. Fluids 31, 764–775.Google Scholar
Hurle, D. T. J., Jakeman, E. & Pike, E. R. 1967 On the solution of the Bénard problem with boundaries of finite conductivity. Proc. R. Soc. Lond. A 296, 469–475.Google Scholar
Ince, E. L. 1956 Ordinary Differential Equations. Dover.
Jeffrey, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.Google Scholar
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. Velande), pp. 241–248. Plenum.
Kessler, J. O. 1985 Cooperative and concentrative phenomena of swimming micro-organisms. Contempt. Phys. 26, 147–166.Google Scholar
Kessler, J. O. 1986 Individual and collective fluid dynamics of swimming cells. J. Fluid Mech. 173, 191–205.Google Scholar
Kevorkian, J. & Cole, J. D. 1981 Perturbation Methods in Applied Mathematics. Springer.
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, 223–238 (referred to as PHK).Google 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
Supplementary material: PDF

Hill et al. supplementary material

Supplementary Material

Download Hill et al. supplementary material(PDF)
PDF 4.5 MB