ALBERT

Description: A new continuum model is formulated for dilute suspensions of swimming microorganisms with asymmetric mass distributions. Account is taken of randomness in a cell's swimming direction, p, by postulating that the probability density function for p satisfies a Fokker-Planck equation analogous to that obtained for colloid suspensions in the presence of rotational Brownian motion. The deterministic torques on a cell, viscous and gravitational, are balanced by diffusion, represented by an isotropic rotary diffusivity Dr, which is unknown α priori, but presumably reflects stochastic influences on the cell's internal workings. When the Fokker-Planck equation is solved, macroscopic quantities such as the average cell velocity Vc, the particle diffusivity tensor D and the effective stress tensor Ʃ can be computed Vcand D are required in the cell conservation equation, and Ʃ in the momentum equation. The Fokker-Planck equation contains two dimensionless parameters, λ and ͼ A is the ratio of the rotary diffusion time Dr-1to the torque relaxation time B (balancing gravitational and viscous torques), while ͼ is a scale for the local vorticity or strain rate made dimensionless with B. In this paper we solve the Fokker-Planck equation exactly for ͼ = 0 (λ arbitrary) and also obtain the first-order solution for small c. Using experimental data on Vcand D obtained with the swimming alga, Chlamydornonas nivalis, in the absence of bulk flow, the ͼ — 0 results can be used to estimate the value of A for that species (λ = 2.2 Dr = 0.13s-1). The continuum model for small e is then used to reanalyse the instability of a uniform suspension, previously investigated by Pedley, Hill & Kessler (1988). The only qualitatively different result is that there no longer seem to be circumstances in which disturbances with a non-zero vertical wavenumber are more unstable than purely horizontal disturbances. On the way, it is demonstrated that the only significant contribution to Ʃ, other than the basic Newtonian stress, is that derived from the stresslets associated with the cells’ intrinsic swimming motions. © 1990, Cambridge University Press. All rights reserved.

Description: ‘Bioconvection’ is the name given to pattern-forming convective motions set up in suspensions of swimming micro-organisms. ‘Gyrotaxis describes the way the swimming is guided through a balance between the physical torques generated by viscous drag and by gravity operating on an asymmetric distribution of mass within the organism. When the organisms are heavier towards the rear, gyrotaxis turns them so that they swim towards regions of most rapid downflow. The presence of gyrotaxis means that bioconvective instability can develop from an initially uniform suspension, without an unstable density stratification. In this paper a continuum model for suspensions of gyrotactic micro-organisms is proposed and discussed; in particular, account is taken of the fact that the organisms of interest are nonspherical, so that their orientation is influenced by the strain rate in the ambient flow as well as the vorticity. This model is used to analyse the linear instability of a uniform suspension. It is shown that the suspension is unstable if the disturbance wavenumber is less than a critical value which, together with the wavenumber of the most rapidly growing disturbance, is calculated explicitly. The subsequent convection pattern is predicted to be three-dimensional (i.e. with variation in the vertical as well as the horizontal direction) if the cells are sufficiently elongated. Numerical results are given for suspensions of a particular algal species (Chlamy-domonas nivalis); the predicted wavelength of the most rapidly growing disturbance is 5—6 times larger than the wavelength of steady-state patterns observed in experiments. The main reasons for the difference are probably that the analysis describes the onset of convection, not the final, nonlinear steady state, and that the experimental fluid layer has finite depth. © 1988, Cambridge University Press. All rights reserved.

Description: 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. © 1989, Cambridge University Press. All rights reserved.

Description: The velocity distribution of swimming micro-organisms depends on directional cues supplied by the environment. Directional swimming within a bounded space results in the accumulation of organisms near one or more surfaces. Gravity, gradients of chemical concentration and illumination affect the motile behaviour of individual swimmers. Concentrated populations of organisms scatter and absorb light or consume molecules, such as oxygen. When supply is one-sided, consumption creates gradients; the presence of the population alters the intensity and the symmetry of the environmental cues. Patterns of cues interact dynamically with patterns of the consumer population. In suspensions, spatial variations in the concentration of organisms are equivalent to variations of mean mass density of the fluid. When organisms accumulate in one region whilst moving away from another region, the force of gravity causes convection that translocates both organisms and dissolved substances. The geometry of the resulting concentration-convection patterns has features that are remarkably reproducible. Of interest for biology are (1) the long-range organisation achieved by organisms that do not communicate, and (2) that the entire system, consisting of fluid, cells, directional supply of consumables, boundaries and gravity, generates a dynamic that improves the organisms' habitat by enhancing transport and mixing. Velocity distributions of the bacterium Bacillus subtilis have been measured within the milieu of the spatially and temporally varying oxygen concentration which they themselves create. These distributions of swimming speed and direction are the fundamental ingredients required for a quantitative mathematical treatment of the patterns. The quantitative measurement of swimming behaviour also contributes to our understanding of aerotaxis of individual cells.

Description: A new continuum model is formulated for dilute suspensions of swimming micro-organisms with asymmetric mass distributions. Account is taken of randomness in a cell's swimming direction, p, by postulating that the probability density function for p satisfies a Fokker-Planck equation analogous to that obtained for colloid suspensions in the presence of rotational Brownian motion. The deterministic torques on a cell, viscous and gravitational, are balanced by diffusion, represented by an isotropic rotary diffusivity Dr, which is unknown a priori, but presumably reflects stochastic influences on the cell's internal workings. When the Fokker-Planck equation is solved, macroscopic quantities such as the average cell velocity Vc, the particle diffusivity tensor D and the effective stress tensor sigma can be computed; Vc and D are required in the cell conservation equation, and sigma in the momentum equation. The Fokker-Planck equation contains two dimensionless parameters, lambda and epsilon; lambda is the ratio of the rotary diffusion time Dr-1 to the torque relaxation time B (balancing gravitational and viscous torques), while epsilon is a scale for the local vorticity or strain rate made dimensionless with B. In this paper we solve the Fokker-Planck equation exactly for epsilon = 0 (lambda arbitrary) and also obtain the first-order solution for small epsilon. Using experimental data on Vc and D obtained with the swimming alga, Chlamydomonas nivalis, in the absence of bulk flow, the epsilon = 0 results can be used to estimate the value of lambda for that species (lambda approximately 2.2; Dr approximately 0.13 s-1). The continuum model for small epsilon is then used to reanalyse the instability of a uniform suspension, previously investigated by Pedley, Hill & Kessler (1988). The only qualitatively different result is that there no longer seem to be circumstances in which disturbances with a non-zero vertical wavenumber are more unstable than purely horizontal disturbances. On the way, it is demonstrated that the only significant contribution to sigma, other than the basic Newtonian stress, is that derived from the stresslets associated with the cells' intrinsic swimming motions.