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Rheology of a concentrated suspension of spherical squirmers: monolayer in simple shear flow
- T. Ishikawa, D.R. Brumley, T.J. Pedley
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- Journal:
- Journal of Fluid Mechanics / Volume 914 / 10 May 2021
- Published online by Cambridge University Press:
- 05 March 2021, A26
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- Article
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A concentrated, vertical monolayer of identical spherical squirmers, which may be bottom heavy, and which are subjected to a linear shear flow, is modelled computationally by two different methods: Stokesian dynamics, and a lubrication-theory-based method. Inertia is negligible. The aim is to compute the effective shear viscosity and, where possible, the normal stress differences as functions of the areal fraction of spheres $\phi$, the squirming parameter $\beta$ (proportional to the ratio of a squirmer's active stresslet to its swimming speed), the ratio $Sq$ of swimming speed to a typical speed of the shear flow, the bottom-heaviness parameter $G_{bh}$, the angle $\alpha$ that the shear flow makes with the horizontal and two parameters that define the repulsive force that is required computationally to prevent the squirmers from overlapping when their distance apart is less than a critical value. The Stokesian dynamics method allows the rheological quantities to be computed for values of $\phi$ up to $0.75$; the lubrication-theory method can be used for $\phi > 0.5$. For non-bottom-heavy squirmers, which are unaffected by gravity, the effective shear viscosity is found to increase more rapidly with $\phi$ than for inert spheres, whether the squirmers are pullers ($\beta > 0$) or pushers ($\beta < 0$); it also varies with $\beta$, although not by very much. However, for bottom-heavy squirmers the behaviour for pullers and pushers as $G_{bh}$ and $\alpha$ are varied is very different, since the viscosity can fall even below that of the suspending fluid for pushers at high $G_{bh}$. The normal stress differences, which are small for inert spheres, can become very large for bottom-heavy squirmers, increasing with $\beta$, and varying dramatically as the orientation $\alpha$ of the flow is varied from 0 to ${\rm \pi} /2$. A major finding is that, despite very different assumptions, the two methods of computation give overlapping results for viscosity as a function of $\phi$ in the range $0.5 < \phi < 0.75$. This suggests that lubrication theory, based on near-field interactions alone, contains most of the relevant physics, and that taking account of interactions with more distant particles than the nearest is not essential to describe the dominant physics.
1 - Functional patterns of swimming bacteria
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- By J.O. Kessler, University of Arizona, M.A. Hoelzer, University of Alaska, T.J. Pedley, The University, Leeds, N.A. Hill, The University, Leeds
- Edited by L. Maddock, Q. Bone, J. M. V. Rayner
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- Book:
- The Mechanics and Physiology of Animal Swimming
- Published online:
- 05 March 2012
- Print publication:
- 15 September 1994, pp 3-12
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Summary
Concentrated populations of the aerobic swimming bacteria Bacillus subtilis rapidly use up the oxygen dissolved in their culture medium. As a result, oxygen diffuses in from the air interface, creating an upward concentration gradient. The organisms then swim towards the surface where they accumulate. Because this arrangement of mass density is unstable, the entire fluid culture convects. Biological and physical factors thus jointly serve to organize the population, yielding dynamics which greatly improve the transport and mixing of oxygen and the viability of the cells.
INTRODUCTION
Concentrated fluid cultures of swimming bacterial cells, such as motile strains of Bacillus subtilis, often form patterns (Kessler, 1989; Pfennig, 1962). They are easy to see when the fluid layer is shallow, and when the illumination provides adequate contrast. These patterns are fairly regular arrays of dots or stripes whose overall diameters and spacings are usually of order millimetres (Figure 1), i.e. much greater than the size of individual cells, typically micrometres. The pattern dimensions are also much greater than the average spacing between cells (∼10-3cm). What is seen are marked fluctuations in concentration n(r,t) of bacterial cells, as discussed in Appendix 1. Generally, high cell concentrations are associated with, and cause, downward motion of the suspending fluid. However, the velocity of the fluid, u(r,t), is correlated with both local and remote values of n(r,t). The relations between cell concentration, mean density of the fluid and convection are also discussed in Appendix 1.
Physical and biological factors combine to convert an originally static microbial habitat, e.g. a quiescent fluid bacterial culture in a petri dish, into a functional dynamic system.