^{1}and Michael D. Graham

^{2,a)}

### Abstract

Active systems, which are driven out of equilibrium, can produce long range correlations and large fluctuations that are not restricted by the fluctuation-dissipation theorem. We consider here the fluctuations and correlations in suspensions of swimming microorganisms that interact hydrodynamically. Modeling the organisms as force dipoles in Stokes flow and considering run-and-tumble and rotational diffusion models of their orientational dynamics allow derivation of closed form results for the stress fluctuations in the long-wave limit. Both of these models lead to Lorentzian distributions, in agreement with some experimental data. These fluctuations are not restricted by the fluctuation-dissipation theorem, as is explicitly verified by comparing the fluctuations with the viscosity of the suspension. In addition to the stress fluctuations in the suspension, we examine correlations between the organisms. Because of the hydrodynamic interactions, the velocities of two organisms are correlated even if the positions and orientations are uncorrelated. We develop a theory of the velocity correlations in this limit and compare with the results of computer simulations. We also formally include orientational correlations in the theory; and comparing with simulations, we are able to show that these are important even in the dilute limit and are responsible in large part for the velocity correlations. While the orientation correlations cannot as yet be predicted from this theory, by inserting the results from simulations into the theory it is possible to properly determine the form of the swimmer velocity correlations. These correlations of orientations are also the key to understanding the spatial correlations of the fluid velocity. Through simulations we show that the orientational correlations decay as *r* ^{−2} with distance—inserting this dependence into the theory leads to a logarithmic dependence of the velocity fluctuations on the size of the system.

This work was supported by the National Science Foundation, Grant No. CBET-0754573. Additional support from the Institute for Mathematics and its Applications with funds provided by the National Science Foundation is gratefully acknowledged as well. The authors also thank Jean-Luc Thiffeault and Douglas B. Weibel for many helpful discussions.

I. INTRODUCTION

II. STRESS FLUCTUATIONS AND CORRELATIONS IN A SUSPENSION OF POINT DIPOLE SWIMMERS

III. EFFECTIVE TEMPERATURE OF AN ACTIVE SUSPENSION

IV. VISCOSITY OF A DILUTE ACTIVE SUSPENSION

V. VELOCITY CORRELATIONS BETWEEN SWIMMERS

VI. FLUID DISTURBANCES

VII. CONCLUSION

### Key Topics

- Suspensions
- 56.0
- Quadrupoles
- 26.0
- Viscosity
- 17.0
- Mean field theory
- 16.0
- Tensor methods
- 13.0

## Figures

(Color online) *C _{s} _{,} _{LL} * (dark circles (blue online)) and −

*C*(light circles (red online)) where

_{s}_{,}_{NN}*r*represents the separation of the beads representing the organism bodies (as shown in the inset). The corresponding theoretical predictions using Eq. (47) are shown with the solid line (blue online) and dashed line (red online) in which spatial and orientation correlations are neglected. For comparison, we also show the orientation correlations:

*C*(dark squares (blue online)) and −

_{n}_{,}_{LL}*C*(light squares (red online)).

_{n}_{,}_{NN}(Color online) *C _{s} _{,} _{LL} * (dark circles (blue online)) and −

*C*(light circles (red online)) where

_{s}_{,}_{NN}*r*represents the separation of the beads representing the organism bodies (as shown in the inset). The corresponding theoretical predictions using Eq. (47) are shown with the solid line (blue online) and dashed line (red online) in which spatial and orientation correlations are neglected. For comparison, we also show the orientation correlations:

*C*(dark squares (blue online)) and −

_{n}_{,}_{LL}*C*(light squares (red online)).

_{n}_{,}_{NN}(Color online) *C _{s} _{,} _{LL} * (dark circles (blue online)) and −

*C*(light circles (red online)) where

_{s}_{,}_{NN}*r*represents the separation of centers including the body and flagellum (as shown in the inset). The corresponding theoretical prediction using Eq. (47) with

*q*= 0 is shown with the solid line (blue) for the longitudinal correlations. The predicted transverse correlations have the incorrect sign. For comparison, we also show the orientation correlations:

*C*(dark squares (blue online)) and −

_{n}_{,}_{LL}*C*(light squares (red online)).

_{n}_{,}_{NN}(Color online) *C _{s} _{,} _{LL} * (dark circles (blue online)) and −

*C*(light circles (red online)) where

_{s}_{,}_{NN}*r*represents the separation of centers including the body and flagellum (as shown in the inset). The corresponding theoretical prediction using Eq. (47) with

*q*= 0 is shown with the solid line (blue) for the longitudinal correlations. The predicted transverse correlations have the incorrect sign. For comparison, we also show the orientation correlations:

*C*(dark squares (blue online)) and −

_{n}_{,}_{LL}*C*(light squares (red online)).

_{n}_{,}_{NN}(Color online) *C _{s} _{,} _{LL} * (dark circles (blue online)) and −

*C*(light circles (red online)) where

_{s}_{,}_{NN}*r*represents the separation of centers including the body and flagellum. The lines represent Eq. (52) in which

*q*= 0 and

**C**

*and*

_{n}**C**

*are both measured in our simulations independently of the swimmer correlations.*

_{f}(Color online) *C _{s} _{,} _{LL} * (dark circles (blue online)) and −

*C*(light circles (red online)) where

_{s}_{,}_{NN}*r*represents the separation of centers including the body and flagellum. The lines represent Eq. (52) in which

*q*= 0 and

**C**

*and*

_{n}**C**

*are both measured in our simulations independently of the swimmer correlations.*

_{f}(Color online) *C _{s} _{,} _{LL} * (dark circles (blue online)) and −

*C*(light circles (red online)) where

_{s}_{,}_{NN}*r*represents the separation of the beads representing the organism bodies. The corresponding theoretical predictions using Eq. (52) are shown with the solid lines.

(Color online) *C _{s} _{,} _{LL} * (dark circles (blue online)) and −

*C*(light circles (red online)) where

_{s}_{,}_{NN}*r*represents the separation of the beads representing the organism bodies. The corresponding theoretical predictions using Eq. (52) are shown with the solid lines.

(Color online) *C _{s} _{,} _{LL} * (dark circles (blue online)) and −

*C*(light circles (red online)) where

_{s}_{,}_{NN}*r*represents the separation of the beads representing the flagella. The corresponding theoretical predictions using Eq. (52) are shown with the solid lines.

(Color online) *C _{s} _{,} _{LL} * (dark circles (blue online)) and −

*C*(light circles (red online)) where

_{s}_{,}_{NN}*r*represents the separation of the beads representing the flagella. The corresponding theoretical predictions using Eq. (52) are shown with the solid lines.

(Color online) Illustration of the key variables in determining how correlations between organisms lead to fluid correlations. The two organisms with orientations **n** _{1} and **n** _{2} are separated by a vector **s**. The two fluid elements for which the velocity correlation is calculated are separated by a **r**.

(Color online) Illustration of the key variables in determining how correlations between organisms lead to fluid correlations. The two organisms with orientations **n** _{1} and **n** _{2} are separated by a vector **s**. The two fluid elements for which the velocity correlation is calculated are separated by a **r**.

(Color online) Measured correlation functions *A* (circles) and *C* (squares) as a function of *s* from Eqs. (80) and (81) for pushers (dark (blue online)) and pullers (light (red online)) at *ϕ _{e} * = 10

^{−2}. Because

*A*for pushers and

*C*for pullers are negative over most of the range, −

*A*and −

*C*are plotted for those curves. The dashed line is a power law of

*s*

^{−2}for reference.

(Color online) Measured correlation functions *A* (circles) and *C* (squares) as a function of *s* from Eqs. (80) and (81) for pushers (dark (blue online)) and pullers (light (red online)) at *ϕ _{e} * = 10

^{−2}. Because

*A*for pushers and

*C*for pullers are negative over most of the range, −

*A*and −

*C*are plotted for those curves. The dashed line is a power law of

*s*

^{−2}for reference.

(Color online) Measured correlation function *I* (triangles) as a function of *s* from Eq. (82) for pushers (dark (blue online)) and pullers (light (red online)) at *ϕ _{e} * = 10

^{−2}. Because

*I*for pullers is negative over most of the range, −

*I*is plotted. The dashed line is a power law of

*s*

^{−2}for reference.

(Color online) Measured correlation function *I* (triangles) as a function of *s* from Eq. (82) for pushers (dark (blue online)) and pullers (light (red online)) at *ϕ _{e} * = 10

^{−2}. Because

*I*for pullers is negative over most of the range, −

*I*is plotted. The dashed line is a power law of

*s*

^{−2}for reference.

Dependence of the mean-squared velocity of fluid elements on system size *L* for pushers at an effective volume fraction of *ϕ _{e} * = 10

^{−1}.

Dependence of the mean-squared velocity of fluid elements on system size *L* for pushers at an effective volume fraction of *ϕ _{e} * = 10

^{−1}.

(Color online) Comparison of spatial fluid correlations for suspensions of pushers at *ϕ _{e} * = 10

^{−1}at different system sizes to the theory based on orientational correlations of the organisms. The symbols correspond to the simulations with number of organisms equal to

*N*= 400 (circles),

*N*= 800 (squares),

*N*= 1600 (diamonds),

*N*= 3200 (stars), and

*N*= 6400 (×). The corresponding theory from Eq. (90) is shown by the lines.

(Color online) Comparison of spatial fluid correlations for suspensions of pushers at *ϕ _{e} * = 10

^{−1}at different system sizes to the theory based on orientational correlations of the organisms. The symbols correspond to the simulations with number of organisms equal to

*N*= 400 (circles),

*N*= 800 (squares),

*N*= 1600 (diamonds),

*N*= 3200 (stars), and

*N*= 6400 (×). The corresponding theory from Eq. (90) is shown by the lines.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content