Effective viscosity of bacterial suspensions: a three-dimensional PDE model with stochastic torque

B. M. Haines; Igor S. Aranson; Leonid Berlyand; Dmitry A. Karpeev

doi:10.3934/cpaa.2012.11.19

Article Contents

2012, Volume 11, Issue 1: 19-46. Doi: 10.3934/cpaa.2012.11.19

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Effective viscosity of bacterial suspensions: a three-dimensional PDE model with stochastic torque

1.
Department of Mathematics, Penn State University, McAllister Bldg., University Park, PA 16802
2.
Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439
3.
Department of Mathematics and Materials Research Institute, Penn State University, University Park, PA 16802
4.
Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439

Received: December 2009

Revised: March 2010

Published online: September 2011

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Abstract

We present a PDE model for dilute suspensions of swimming bacteria in a three-dimensional Stokesian fluid. This model is used to calculate the statistically-stationary bulk deviatoric stress and effective viscosity of the suspension from the microscopic details of the interaction of an elongated body with the background flow. A bacterium is modeled as an impenetrable prolate spheroid with self-propulsion provided by a point force, which appears in the model as an inhomogeneous delta function in the PDE. The bacterium is also subject to a stochastic torque in order to model tumbling (random reorientation). Due to a bacterium's asymmetric shape, interactions with prescribed generic planar background flows, such as a pure straining or planar shear flow, cause the bacterium to preferentially align in certain directions. Due to the stochastic torque, the steady-state distribution of orientations is unique for a given background flow. Under this distribution of orientations, self-propulsion produces a reduction in the effective viscosity. For sufficiently weak background flows, the effect of self-propulsion on the effective viscosity dominates all other contributions, leading to an effective viscosity of the suspension that is lower than the viscosity of the ambient fluid. This is in qualitative agreement with recent experiments on suspensions of Bacillus subtilis.

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Mathematics Subject Classification: Primary: 92B99, 76A05, 35Q35.

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