Hydrodynamics of self-propelled hard rods

Motivated by recent simulations and by experiments on aggregation of gliding bacteria, we study a model of the collective dynamics of self-propelled hard rods on a substrate in two dimensions. The rods have finite size, interact via excluded volume, and their dynamics is overdamped by the interaction with the substrate. Starting from a microscopic model with nonthermal noise sources, a continuum description of the system is derived. The hydrodynamic equations are then used to characterize the possible steady states of the systems and their stability as a function of the particles packing fraction and the speed of self-propulsion.

This article appears in the following collection:

Figure 1

(Color online) Geometry of overlap between two rods of length $\ell$. Here $\mathbf{\xi }={\mathbf{r}}_2-{\mathbf{r}}_1$ is the separation of the centers of mass of the two rods, $s_1$ and $s_2$ parametrize the position along the rod relative to the center of mass, with $-\ell ∕2⩽s_i⩽\ell ∕2$, for $i=1,2$. Finally, ${\widehat{\mathbf{u}}}_i$ are the unit vectors directed along the rods’ axes.

Figure 2

(Color online) A description of the crossover from diffusive to propagating density fluctuations in the isotropic state. The solid line (red online) is the boundary $v_0\left(k\right)$ given in Eq. (36) for $\frac{D_p-D}{\mathcal{l}{D}_R}=1$. Velocities on the vertical axis are measured in units of $\mathcal{l}{D}_R$, where $\mathcal{l}$ is the length of the rods. The magnitude of the wave vector $k$ is measured in units of ${\mathcal{l}}^{-1}$. The boundary $v_0\left(k\right)$ has a minimum at $v_{c0}$ below which the density fluctuations are always diffusive. For $v_0>v_{c0}$, the system can support propagating sound waves when $k_{c1}\left(v_0\right)⩽k⩽k_{c2}\left(v_0\right)$, with $k_{ci}\left(v_0\right)$ given in Eq. (37).

Figure 3

(Color online) The boundary $v_c(\varphi ,S)$ of the stability of the homogeneous nematic state is shown as a function of the angle $\varphi$ between the wave vector and the direction of nematic order for $S=1$. The nematic state is always stable for all $\varphi$ when $v_0<v_c\left(S\right)$. For a fixed value of $v_0>v_c\left(S\right)$ the nematic state is unstable for ${\varphi }_{c1}\left(S\right)⩽\varphi ⩽{\varphi }_{c2}\left(S\right)$. The behavior of $v_c\left(S\right)$ with $S$ is shown in the inset. The critical speed $v_c\left(S\right)$ diverges as $S\to 0$.

Figure 4

(Color online) The boundary $v_c(\varphi ,S)$ above which the homogeneous nematic state is linearly unstable is shown as a function of the angle $\varphi$ between the wave vector and the direction of nematic order for three values of $S$. The horizontal line at a fixed $v_0$ intersects each curve at two values of $\varphi$, ${\varphi }_{c1}\left(S\right)<{\varphi }_{c2}\left(S\right)$. The nematic state is unstable for ${\varphi }_{c1}\left(S\right)⩽\varphi ⩽{\varphi }_{c2}\left(S\right)$. The figure illustrates that the range of angles $\varphi$ for which the system becomes unstable decreases as the degree of order in the system decreases.