Activity-induced clustering in model dumbbell swimmers: The role of hydrodynamic interactions

Using a fluid-particle dynamics approach, we numerically study the effects of hydrodynamic interactions on the collective dynamics of active suspensions within a simple model for bacterial motility: each microorganism is modeled as a stroke-averaged dumbbell swimmer with prescribed dipolar force pairs. Using both simulations and qualitative arguments, we show that, when the separation between swimmers is comparable to their size, the swimmers' motions are strongly affected by activity-induced hydrodynamic forces. To further understand these effects, we investigate semidilute suspensions of swimmers in the presence of thermal fluctuations. A direct comparison between simulations with and without hydrodynamic interactions shows these to enhance the dynamic clustering at a relatively small volume fraction; with our chosen model the key ingredient for this clustering behavior is hydrodynamic trapping of one swimmer by another, induced by the active forces. Furthermore, the density dependence of the motility (of both the translational and rotational motions) exhibits distinctly different behaviors with and without hydrodynamic interactions; we argue that this is linked to the clustering tendency. Our study illustrates the fact that hydrodynamic interactions not only affect kinetic pathways in active suspensions, but also cause major changes in their steady state properties.

Figure 1

Schematic of a model swimmer. Here, a swimming bacterium is represented by two spherical particles with radius $a_{\mathrm{H}}=a_{\mathrm{T}}=a$ and a “phantom” spherical particle with radius $a_{\mathrm{P}}=b$. The center-to-center distance between the head and tail particles and that between the phantom and tail particles are fixed at ${\ell }_0$ and ${\ell }_1$, respectively. The back-to-back (a) and face-to-face (b) force configurations correspond respectively to the pusher (extensile) and puller (contractile) swimming mechanisms. The (green) solid and (purple) dashed arrows respectively indicate schematically the swimming direction and the flow field around the swimmer.

Figure 2

(a) The time evolution of the velocity of an isolated swimmer. Its steady-state value for various values of $f_{\mathrm{act}}$ is shown in the inset. The results for the passive dumbbell with an unbalanced external force of the same magnitude are also plotted. (b) The velocity field $\mathit{v}(x,y,z=0)$ in the steady state at $f_{\mathrm{act}}=5$, where the axis of the swimmer is in the plane. The color bar represents the value of the velocity field scaled by the steady-state swimmer velocity. The purple arrow indicates the swimming direction. The simulation box used is ${128}^3$ and the frame size shown is ${32}^2$.

Figure 3

(a) The left-most panel represents the trajectories of the center of mass of the swimmers 1 and 2, where the swimming direction is from left to right and the initial configuration of the two swimmers is shown below this panel. The right four panels represent the trajectories of the head and tail particles of the swimmer 1 for each of the initial separations, where the positions at several times are explicitly indicated and the ratio of the horizontal scale to the vertical one is about 1/14. Here, the initial position of the tail particle of the swimmer 1 is set to the origin; that is, $\Delta x=x-R_{\mathrm{T},x}^1\left(0\right)$ and $\Delta y=y-R_{\mathrm{H},y}^1\left(0\right)$. In (b) and (c), the time dependence of the angular velocity (angles measured in radians) around the center of mass and its maximum value are shown, respectively. For $\Delta /{\ell }_0\gg 1$, the swimmers hardly change their swimming directions while traveling a distance of their own size ($\sim {\ell }_0$). However, with decreasing the separation between the swimmers, stronger distortions of the swimming directions are found: In the present case, the hydrodynamic attraction acting on the tail particle is stronger than that on the head particle, and this asymmetry is enhanced for smaller $\Delta$, which leads to faster rotation. However, as $\Delta$ decreases further ($\Delta /{\ell }_0\lesssim 1.6$), this repelling tendency gets weaker, which may be ascribed to the near-field hydrodynamic and steric effects (see the text for discussion). (d) Snapshot of two swimmers swimming side by side for $f_{\mathrm{act}}=10$ and $\Delta /{\ell }_0=1.6$ at $t\cong {\tau }_{\mathrm{a}}$. At $t=0$, they are parallel. We also show the velocity field $\mathit{v}(x,y,z=0)$, where the axes of the two swimmers lie in the plane. The color bar represents the magnitude of the fluid velocity scaled by the value of $v_0$ at $f_{\mathrm{act}}=10$ shown in the inset of Fig. 2.

Figure 4

(a) Trajectories of the head and tail particles of the two swimmers for several initial positions. Here, $\Delta =R_{\mathrm{H},y}^1-R_{\mathrm{H},y}^2$ at $t=0$. The swimmer 1 (2) is swimming from left (right) to right (left). For $\Delta /{\ell }_0=1.2$, the head particle of each swimmer touches the phantom particle of the other swimmer, and thus, due to the dynamical rule introduced in the present model, the two swimmers stop moving completely. (This would not happen for asymmetric initial conditions, nor in the presence of thermal fluctuations, or disturbance by the flow caused by other swimmers.) In (b), the velocity of the swimmer 1 along the $x$ axis, $V_{\mathrm{H},x}^1$, is plotted against $R_{\mathrm{H},x}^1$. Initially, when the two swimmers are approaching, they repel each other, because the repulsive HIs due to the incompressibility of the solvent are dominant. Then, when they are passing each other, an activity-induced hydrodynamic attraction dominates; that is, one swimmer is attracted to the streamline induced by the other swimmer, resulting in an acceleration of the swim speed. It is evident from the trajectories that such (configuration-dependent) HIs are stronger for smaller separation distances. (c) Snapshot of two swimmers passing each other for $\Delta /{\ell }_0=1.6$ and $f_{\mathrm{act}}=10$. We also show the velocity field $\mathit{v}(x,y,z=0)$, where the axes of the two swimmers lie in the plane. The color bar represents the value of the fluid velocity scaled by the value of $v_0$ at $f_{\mathrm{act}}=10$ shown in the inset of Fig. 2.

Figure 5

The radial distribution function $g_{m{m}^{\prime }}\left(r\right)$ where $\left(m,m^{\prime }\right)=\left(H,T\right)$ with (a) and without (b) HIs.

Figure 6

The structure factor $S\left(k\right)$ for various values of $f_{\mathrm{act}}$. With increasing $f_{\mathrm{act}}$, $S\left(k\right)$ grows at small $k$. For $f_{\mathrm{act}}\ne 0$, $S\left(k\right)$ for $2ak\lesssim 1$ can be described by the Ornstein-Zernike form, which is represented by the black solid curves in the cases with and without HIs, respectively. The results with neither the dynamical rule nor HIs are also shown.

Figure 7

The pair distribution function $g_{m{m}^{\prime }}(r,\theta )$ defined as $g_{m{m}^{\prime }}(r,\theta )=\frac{L^3}{4\pi N^2{r}^2}{\sum }_{\alpha \ne {\alpha }^{\prime }}\delta [{cos}^{-1}({\widehat{\mathit{n}}}_{\alpha }·{\widehat{\mathit{n}}}_{\alpha }^{\prime })-\theta ]\delta \left(\right|{\mathit{R}}_m^{\alpha }-{\mathit{R}}_{m^{\prime }}^{{\alpha }^{\prime }}|-r)$ in the steady state at $f_{\mathrm{act}}=14$ with (a) and without (b) HIs. The purple, green, and yellow curves are the contours corresponding to 1/6, 3/6, and 5/6 of the maximum value of $g_{m{m}^{\prime }}(r,\theta )$, respectively.

Figure 8

The typical time evolution of the cluster configuration in the steady state at $f_{\mathrm{act}}=14$ with (a) and without (b) HIs. Here, colored dumbbells represent the swimmers that compose a cluster at $\Delta t=0$. Different colors represent different numbers of swimmers in a cluster ($N_{\mathrm{c}}$). The swimmers which are isolated at $\Delta t=0$ are shown as white semitransparent dumbbells, and the phantom particles are not shown here. The simulation box is ${128}^3={\left(40a\right)}^3$ with periodic boundary condition. (c) The probability distribution for a given cluster have $N_{\mathrm{c}}$ swimmers with and without HIs.

Figure 9

The average separation of a pair of swimmers as a function of its initial value and elapsed time with and without HIs at $f_{\mathrm{act}}=14$. Here, the separation is defined as $\Delta R_{\alpha ,{\alpha }^{\prime }}\left(t\right)=|{\mathit{R}}_{\mathrm{C}}^{\alpha }\left(t\right)-{\mathit{R}}_{\mathrm{C}}^{{\alpha }^{\prime }}\left(t\right)|$, where ${\mathit{R}}_{\mathrm{C}}^{\alpha }=({\mathit{R}}_{\mathrm{H}}^{\alpha }+{\mathit{R}}_{\mathrm{T}}^{\alpha })/2$ is the center of mass of the $\alpha \mathrm{th}$ swimmer. The colors represent the scaled value of $\Delta R_{\alpha ,{\alpha }^{\prime }}\left(t\right)$. With HIs, the initially adjacent pairs [$\Delta R_{\alpha ,{\alpha }^{\prime }}(\Delta t=0)\lesssim {\ell }_0(=2.5a)$] stay close to each other for a longer time than ${\tau }_{\mathrm{a}}$, although without HIs, such swimmers separate themselves much quicker.

Figure 10

The average swim speed (a) and the rotational relaxation time (b) for various volume fraction $\Psi$ at $f_{\mathrm{act}}=14$. The red and green curves are for the cases with and without HIs, respectively.

Figure 11

The structure factor with (a) and without (b) HIs for various $\Psi$ at $f_{\mathrm{act}}=14$.

Figure 12

Schematic of the repulsive forces due to the additional interactions.

Figure 13

The average swim speed (a) and the rotational relaxation time (b) for various volume fraction $\Psi$ at $f_{\mathrm{act}}=14$. The red and green curves are for the cases with and without HIs, respectively. These curves exhibit similar behaviors to those with the dynamical rule shown in Fig. 10 but with an overall shift to higher $\Psi$ (due to the absence of the rule-induced slowing down).

Figure 14

The structure factor with (a) and without (b) HIs for various $\Psi$ at $f_{\mathrm{act}}=14$. With HIs, within the range of $\Psi$ investigated here, the structure factor at small $k(=L/2\pi )$ increases with increasing $\Psi$.