Abstract
This thesis explores the non-linear effects of gyrotaxis on the bioconvection patterns formed in a suspension of swimming micro-organisms. The cells are denser than the medium in which they swim and the patterns are formed spontaneously by aggregations of cells which drive bulk fluid motion. The micro-organisms under consideration are orientated by a balance between a gravitational torque, due to them being bottom heavy, and a viscous torque arising from local fluid velocity gradients. This mechanism is known as gyrotaxis. A wide range of investigative techniques are employed, from experiments in the laboratory to computer algebra and bifurcation analysis using amplitude equations.
Firstly, a series of experiments is described in which images of bioconvection patterns are captured and Fourier analysed. The most unstable pattern wavelength is extracted as a function of suspension concentration, depth and time. Ideas from surface geometry are exploited to produce a measure of pattern. Some other experiments are also discussed.
Secondly, a full linear analysis of a stochastic, gyrotactic continuum model in a
suspension of finite depth is conducted and an extension of the theory to include the random nature of the micro-organisms' swimming speeds is proposed.
Thirdly, an approximation to the steady Fokker-Planck equation describing the
stochastic nature of the micro-organism swimming direction using surface spherical harmonics is investigated. The limitations of this method are explored.
Finally, the non-linear mechanisms involved in a gyrotactic instability are elucidated by exploiting the long vertical scale for descending plumes in a deep suspension. Initially, a weakly non-linear analysis provides an amplitude equation that implies that the bifurcation to instability is supercritical. Secondly, non-linear solutions are seen to undergo a Hopf bifurcation when there is a weak background vorticity. The resulting limit cycle provides the basis for horizontally travelling, vertical plume solutions. Equations describing the slow vertical variations along plume solutions admit travelling waves, for which the wavespeed is found. The travelling waves are thought to describe the varicose instabilities seen on bioconvection plumes in experiments.
Firstly, a series of experiments is described in which images of bioconvection patterns are captured and Fourier analysed. The most unstable pattern wavelength is extracted as a function of suspension concentration, depth and time. Ideas from surface geometry are exploited to produce a measure of pattern. Some other experiments are also discussed.
Secondly, a full linear analysis of a stochastic, gyrotactic continuum model in a
suspension of finite depth is conducted and an extension of the theory to include the random nature of the micro-organisms' swimming speeds is proposed.
Thirdly, an approximation to the steady Fokker-Planck equation describing the
stochastic nature of the micro-organism swimming direction using surface spherical harmonics is investigated. The limitations of this method are explored.
Finally, the non-linear mechanisms involved in a gyrotactic instability are elucidated by exploiting the long vertical scale for descending plumes in a deep suspension. Initially, a weakly non-linear analysis provides an amplitude equation that implies that the bifurcation to instability is supercritical. Secondly, non-linear solutions are seen to undergo a Hopf bifurcation when there is a weak background vorticity. The resulting limit cycle provides the basis for horizontally travelling, vertical plume solutions. Equations describing the slow vertical variations along plume solutions admit travelling waves, for which the wavespeed is found. The travelling waves are thought to describe the varicose instabilities seen on bioconvection plumes in experiments.
| Original language | English |
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| Qualification | Doctor of Philosophy |
| Awarding Institution |
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| Supervisors/Advisors |
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| Award date | 1 Jun 1996 |
| Place of Publication | Leeds |
| Publication status | Published - 1996 |