Critical bacterial concentration for the onset of collective swimming

We examine the stability of a suspension of swimming bacteria in a Newtonian medium. The bacteria execute a run-and-tumble motion, runs being periods when a bacterium on average swims in a given direction; runs are interrupted by tumbles, leading to an abrupt, albeit correlated, change in the swimming direction. An instability is predicted to occur in a suspension of ‘pushers’ (e.g. E. Coli, Bacillus subtilis, etc.), and owes its origin to the intrinsic force dipoles of such bacteria. Unlike the dipole induced in an inextensible fibre subject to an axial straining flow, the forces constituting the dipole of a pusher are directed outward along its axis. As a result, the anisotropy in the orientation distribution of bacteria due to an imposed velocity perturbation drives a disturbance velocity field that acts to reinforce the perturbation. For long wavelengths, the resulting destabilizing bacterial stress is Newtonian but with a negative viscosity. The suspension becomes unstable when the total viscosity becomes negative. In the dilute limit (nL3 ≪ 1), a linear stability analysis gives the threshold concentration for instability as (nL3)crit = ((30/Cℱ(r))(Dr L/U)(1 + 1/(6τ Dr)))/(1−(15𝒢(r)/Cℱ(r))(Dr L/U)(1 + 1/(6τ Dr))) for perfectly random tumbles; here, L and U are the length and swimming velocity of a bacterium, n is the bacterial number density, Dr characterizes the rotary diffusion during a run and τ−1 is the average tumbling frequency. The function ℱ(r) characterizes the rotation of a bacterium of aspect ratio r in an imposed linear flow; ℱ(r) = (r2 −1)/(r2 + 1) for a spheroid, and ℱ(r) ≈ 1 for a slender bacterium (r ≫ 1). The function 𝒢(r) characterizes the stabilizing viscous response arising from the resistance of a bacterium to a deforming ambient flow; 𝒢(r) = 5π/6 for a rigid spherical bacterium, and 𝒢(r)≈ π/45(ln r) for a slender bacterium. Finally, the constant C denotes the dimensionless strength of the bacterial force dipole in units of μU L2; for E. Coli, C ≈ 0.57. The threshold concentration diverges in the limit ((15𝒢(r)/Cℱ(r)) (Dr L/U)(1 + 1/(6τ Dr))) → 1. This limit defines a critical swimming speed, Ucrit = (Dr L)(15𝒢(r)/Cℱ(r))(1 + 1/(6τ Dr)). For speeds smaller than this critical value, the destabilizing bacterial stress remains subdominant and a dilute suspension of these swimmers therefore responds to long-wavelength perturbations in a manner similar to a suspension of passive rigid particles, that is, with a net enhancement in viscosity proportional to the bacterial concentration. On the other hand, the stability analysis predicts that the above threshold concentration reduces to zero in the limit Dr → 0, τ → ∞, and a suspension of non-interacting straight swimmers is therefore always unstable. It is then argued that the dominant effect of hydrodynamic interactions in a dilute suspension of such swimmers is via an interaction-driven orientation decorrelation mechanism. The latter arises from uncorrelated pair interactions in the limit nL3 ≪ 1, and for slender bacteria in particular, it takes the form of a hydrodynamic rotary diffusivity (Dhr); for E. Coli, we find Dhr = 9.4 × 10−5(nUL2). From the above expression for the threshold concentration, it may be shown that even a weakly interacting suspension of slender smooth-swimming bacteria (r ≫ 1, ℱ(r) ≈ 1, τ → ∞) will be stable provided Dhr > (C/30)(nUL2) in the limit nL3 ≪ 1. The hydrodynamic rotary diffusivity of E. Coli is, however, too small to stabilize a dilute suspension of these swimmers, and a weakly interacting suspension of E. Coli remains unstable.