^{1}, Donald L. Koch

^{2}and Sean R. Fitzgibbon

^{2}

### Abstract

The linear stability of a homogeneous dilute suspension of chemotactic bacteria in a constant chemoattractant gradient is analyzed. The bacteria execute a run-and-tumble motion, typified by the species *E. coli*, wherein periods of smooth swimming (runs) are interrupted by abrupt uncorrelated changes in swimming direction (tumbles). Bacteria tumble less frequently when swimming toward regions of higher chemoattractant concentration, leading to a mean bacterial orientation and velocity in the base state. The stability of an unbounded suspension, both with and without a chemoattractant, is controlled by coupled long wavelength perturbations of the fluid velocity and bacterial orientation fields. In the former case, the most unstable perturbations have their wave vector oriented along the chemoattractant gradient. Chemotaxis reduces the critical bacteria concentration, for the onset of collective swimming, compared with that predicted by Subramanian and Koch [“Critical bacterial concentration for the onset of collective swimming,” J. Fluid Mech.632, 359 (2009)] in the absence of a chemoattractant. A part of this decrease may be attributed to the increase in the mean tumbling time in the presence of a chemoattractant gradient. A second destabilizing influence comes from the ability of the shearing motion, associated with a velocity perturbation in which the velocity and chemical gradients are aligned, to sweep prealigned bacteria into the local extensional quadrant thereby creating a stronger destabilizing active stress than in an initially isotropic suspension. The chemoattractant gradient also fundamentally alters the unstable spectrum for any finite wavenumber. In suspensions of bacteria that do not tumble, Saintillan and Shelley [“Instabilities and pattern formation in active particle suspensions: Kinetic theory and continuum simulations,” Phys. Rev. Lett.100, 178103 (2008); “Instabilities, pattern formation and mixing in active suspensions,” Phys. Fluids20, 123304 (2008)] showed that the growth rate has two real solutions (stationary modes) below a critical wavenumber at which the two solutions merge and then bifurcate to form a pair of complex conjugate solutions (propagating modes) for larger wavenumbers. The discrete spectrum terminates at a second critical wavenumber, and beyond this wavenumber, the only remaining solutions are neutrally stable waves comprising the continuous spectrum. In the presence of a chemoattractant gradient, however, the aforementioned perfect bifurcation is broken and a pair of traveling wave solutions is found for all wavenumbers. Furthermore, instead of terminating at a critical wavenumber, the solutions for the growth rate asymptote to the negative of the tumbling frequency at large wavenumbers.

This work was supported by NSF Grant No. CBET-0730579.

I. INTRODUCTION

II. EQUATIONS OF MOTION

III. LONG WAVELENGTH STABILITY ANALYSIS

IV. STABILITY ANALYSIS FOR FINITE WAVELENGTH PERTURBATIONS

A. Randomly tumbling bacteria

B. Chemotactic bacteria

V. CONCLUSIONS

### Key Topics

- Suspensions
- 50.0
- Bacteria
- 47.0
- Photon density
- 29.0
- Diffusion
- 16.0
- Probability theory
- 13.0

## Figures

The physical mechanism of mutual reinforcement of the fluid velocity and bacterial orientation perturbation fields in the absence of a chemical gradient. The solid arrows indicate the local extensional component of the imposed velocity perturbation, while the dashed arrows indicate the disturbance velocity produced by the bacterium force dipole, which reinforces the imposed flow. As the Fourier (velocity) mode grows in amplitude, the orientation distribution in the vicinity of the nodes of the velocity wave becomes increasingly peaked along the local extensional axis; this anisotropy, in turn, reinforces the velocity perturbation.

The physical mechanism of mutual reinforcement of the fluid velocity and bacterial orientation perturbation fields in the absence of a chemical gradient. The solid arrows indicate the local extensional component of the imposed velocity perturbation, while the dashed arrows indicate the disturbance velocity produced by the bacterium force dipole, which reinforces the imposed flow. As the Fourier (velocity) mode grows in amplitude, the orientation distribution in the vicinity of the nodes of the velocity wave becomes increasingly peaked along the local extensional axis; this anisotropy, in turn, reinforces the velocity perturbation.

The variation of (a measure of the critical bacterial concentration) as a function of for . The dashed line corresponds to Eq. (26) and the dotted line corresponds to Eq. (27).

The variation of (a measure of the critical bacterial concentration) as a function of for . The dashed line corresponds to Eq. (26) and the dotted line corresponds to Eq. (27).

The variation of (a measure of the critical bacterial concentration) as a function of . The solid line denotes Eq. (28) corresponding to an attractant-aligned wavevector, while the dotted-dashed line denotes Eq. (31), the nominal stability criterion.

The variation of (a measure of the critical bacterial concentration) as a function of . The solid line denotes Eq. (28) corresponding to an attractant-aligned wavevector, while the dotted-dashed line denotes Eq. (31), the nominal stability criterion.

Schematic representations of the modal structure for a suspension of random tumblers for differing values of . Note that the hatched region, which denotes the neutral continuous spectrum modes for straight swimmers, is absent for finite .

Schematic representations of the modal structure for a suspension of random tumblers for differing values of . Note that the hatched region, which denotes the neutral continuous spectrum modes for straight swimmers, is absent for finite .

The real and imaginary parts of the scaled growth rate, , as a function of for a suspension of straight swimmers. The filled and hollow symbols denote the mode 1 and mode 2 branches, respectively.

The real and imaginary parts of the scaled growth rate, , as a function of for a suspension of straight swimmers. The filled and hollow symbols denote the mode 1 and mode 2 branches, respectively.

The real and imaginary parts of the growth rate, , as a function of for a suspension of random tumblers in the absence of a chemoattractant. The filled and hollow symbols denote the mode 1 and mode 2 branches, respectively; here, .

The real and imaginary parts of the growth rate, , as a function of for a suspension of random tumblers in the absence of a chemoattractant. The filled and hollow symbols denote the mode 1 and mode 2 branches, respectively; here, .

The real and imaginary parts of the growth rate, , as a function of for chemotactic bacteria with and ; the symbols are the same as in Fig. 5. The analytical prediction for , in the limit of small , given by Eq. (73), is plotted as a dashed line.

The real and imaginary parts of the growth rate, , as a function of for chemotactic bacteria with and ; the symbols are the same as in Fig. 5. The analytical prediction for , in the limit of small , given by Eq. (73), is plotted as a dashed line.

The real and imaginary parts of as a function of for chemotactic bacteria with and ; the symbols are the same as in Fig. 5. The analytical prediction for , in the limit of small , is plotted as a dashed line.

The real and imaginary parts of as a function of for chemotactic bacteria with and ; the symbols are the same as in Fig. 5. The analytical prediction for , in the limit of small , is plotted as a dashed line.

The function , denoting the inverse of the highest (scaled) growth rate, as a function of ; here, . The highest growth rate corresponds to a stationary mode in the interval and to a propagating mode for .

The function , denoting the inverse of the highest (scaled) growth rate, as a function of ; here, . The highest growth rate corresponds to a stationary mode in the interval and to a propagating mode for .

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