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Stirring by squirmers

Published online by Cambridge University Press:  01 February 2011

ZHI LIN ,
JEAN-LUC THIFFEAULT  and
STEPHEN CHILDRESS
ZHI LIN
Affiliation:
Institute for Mathematics and Applications, University of Minnesota – Twin Cities, 207 Church Street SE, Minneapolis, MN 55455, USA
JEAN-LUC THIFFEAULT*
Affiliation:
Institute for Mathematics and Applications, University of Minnesota – Twin Cities, 207 Church Street SE, Minneapolis, MN 55455, USA Department of Mathematics, University of Wisconsin – Madison, 480 Lincoln Drive, Madison, WI 53706, USA
STEPHEN CHILDRESS
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA
*
†Email address for correspondence: [email protected]
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Abstract

We analyse a simple ‘Stokesian squirmer’ model for the enhanced mixing due to swimming micro-organisms. The model is based on a calculation of Thiffeault & Childress (Phys. Lett. A, vol. 374, 2010, p. 3487), where fluid particle displacements due to inviscid swimmers are added to produce an effective diffusivity. Here we show that, for the viscous case, the swimmers cannot be assumed to swim an infinite distance, even though their total mass displacement is finite. Instead, the largest contributions to particle displacement, and hence to mixing, arise from random changes of direction of swimming and are dominated by the far-field stresslet term in our simple model. We validate the results by numerical simulation. We also calculate non-zero Reynolds number corrections to the effective diffusivity. Finally, we show that displacements due to randomly swimming squirmers exhibit probability distribution functions with exponential tails and a short-time superdiffusive regime, as found previously by several authors. In our case, the exponential tails are due to ‘sticking’ near the stagnation points on the squirmer's surface.

JFM classification

Mixing: Mixing Biological Fluid Dynamics: Micro-organism dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 669 , 25 February 2011 , pp. 167 - 177
Copyright
Copyright © Cambridge University Press 2011

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References

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