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Instability regimes in flowing suspensions of swimming micro-organisms

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10.1063/1.3529411

### Abstract

The effects of an external shear flow on the dynamics and pattern formation in a dilute suspension of swimming micro-organisms are investigated using a linear stability analysis and three-dimensional numerical simulations, based on the kinetic model previously developed by [D. Saintillan and M. J. Shelley, Phys. Fluids20, 123304 (2008)]. The external shear flow is found to damp the instabilities that occur in these suspensions by controlling the orientation of the particles. We demonstrate in our simulations that the rate of damping is direction-dependent: it is fastest in the flow direction, but slowest in the direction perpendicular to the shear plane. As a result, transitions from three- to two- to one-dimensional instabilities are observed to occur as shear rate increases, and above a certain shear rate the instabilities altogether disappear. The density patterns and complex flows that arise at long time in the suspensions are also analyzed from the numerical simulations using standard techniques from the literature on turbulent flows. The imposed shear flow is found to have an effect on both density patterns and flow structures, which typically align with the extensional axis of the external flow. The disturbance flows in the simulations are shown to exhibit similarities with turbulent flows, and in particular two of the seemingly universal characteristics of turbulent flows also occur, namely: (i) the bias of plots toward the second and fourth quadrants, corresponding to stable focus/stretching and unstable node/saddle/saddle flow topologies, respectively, and (ii) the alignment of the vorticity vector with the intermediate strain-rate eigenvector. However, the flows described herein also significantly differ from turbulent flows owing to the strong predominance of large scales, as exemplified by the very rapid decay of the kinetic energy spectrum, an effect further enhanced after the transitions to two- and one-dimensional instabilities.

© 2011 American Institute of Physics

Received 31 August 2010
Accepted 28 November 2010
Published online 06 January 2011

Acknowledgments: The authors thank Michael Shelley (Courant Institute) for useful conversations on this work, and gratefully acknowledge funding from NSF Grant No. DMS-0920931-ARRA. Computer resources were provided by the National Center for Supercomputing Applications (NCSA) at the University of Illinois.

Article outline:

I. INTRODUCTION

II. KINETIC MODEL

A. Governing equations

B. Nondimensionalization

III. LINEAR STABILITY ANALYSIS

A. Eigenvalue problem

B. Results

IV. NUMERICAL SIMULATIONS

A. Simulation method

B. Flow structures and velocity field characterization

1. Density fluctuations

2. Velocity fields

3. Vortical structures

4. *plots*

C. Joint probability distribution functions

D. Alignment with rate-of-strain eigenvectors

E. Autocorrelation functions

F. Time dynamics

V. SUMMARY

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2011-01-06

2014-04-25

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