The stability of a homogeneous suspension of chemotactic bacteria

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. Fluids 20, 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.