Orientational Order and Instabilities in Suspensions of Self-Locomoting Rods

The orientational order and dynamics in suspensions of self-locomoting slender rods are investigated numerically. In agreement with previous theoretical predictions, nematic suspensions of swimming particles are found to be unstable at long wavelengths as a result of hydrodynamic fluctuations. Nevertheless, a local nematic ordering is shown to persist over short length scales and to have a significant impact on the mean swimming speed. The consequences of the large-scale orientational disorder for particle dispersion are also discussed.

Figure 1

Orientational instability in a polar nematic suspension of pushers, at an effective volume fraction of $n(L/2{)}^3=1.0$. The figure shows a region of dimensions $10\times 10\times 3$ (in units of particle length) containing 2500 particles at different stages of the instability (a)–(d).

Figure 2

(a) Typical disturbance flow field at long times in a plane cross section in the simulation of Fig. 1. (b) Mean swimming speed $⟨\mathbf{U}·\mathbf{n}⟩$ along the particle director $\mathbf{n}$, normalized by the isolated swimming speed $U_0$, as a function of the effective volume fraction $n(L/2{)}^3$.

Figure 3

Time evolution of the polar and nematic order parameters $S_1$ and $S_2$ during the instability of Fig. 1 for (a) polar and (b) apolar initial orientation distributions, in suspensions of pushers.

Figure 4

(a) Polar and nematic pair correlation functions $C_1(r)$ and $C_2(r)$ at steady state in suspensions of pushers and pullers at a volume fraction of $n(L/2{)}^3=1.0$. (b) Correlation length $l_1$, or first zero of the polar pair correlation function $C_1(r)$ in (a), as a function of volume fraction for both pushers and pullers, with and without excluded volume (EV).

Figure 5

(a) Orientational dispersion coefficient $d_r$ as a function of volume fraction in suspensions of pushers, determined from the time autocorrelation function of the director $\mathbf{n}$ (inset), with and without EV. (b) Center-of-mass dispersion coefficient $D$ in suspensions of pushers. $D$ was obtained independently using the integral of the velocity autocorrelation function (method 1) and using mean-square displacements (method 2 and inset), and is compared to the Taylor dispersion prediction $D=U_0^2/6d_r$.