Chaos and mixing in self-propelled droplets

We consider self-propelled droplets that are driven by internal flow. Tracer particles, which are advected by the flow, in general follow chaotic trajectories, even though the motion of the autonomous swimmer is completely regular. The flow is mixing, and for Péclet and Batchelor numbers, which are realized, e.g., in eukaryotic cells, advective mixing can substantially accelerate and even dominate transport by diffusion.

Figure 1

(a) Trajectories of tracer particle inside a droplet, which moves with constant linear velocity and without rotation. Initial conditions are $\mathbf{r}\left(0\right)=\pm k\times {10}^{-1}\widehat{\mathbf{x}}$ with $k=1,\cdots 6.$ (b) A 3D graph of the cycles for $\widehat{\mathit{\omega }}$ tilted with respect to $\widehat{\mathbf{z}}$ by $\theta =0.3\pi$ (see also Fig. 2).

Figure 2

Poincaré section (x-z plane) of time-independent flow with $\widehat{\mathbf{n}}=\widehat{\mathbf{z}}$ and $\widehat{\mathit{\omega }}$ tilted as indicated in the figure for different strength of the rotational flow, $a_r=1.1$ (a), 2 (b), 3 (c), 9 (d); initial conditions as used in Fig. 1, with two additional trajectories starting on the $z$ axis at $0.1·\widehat{\mathbf{z}}$ and $0.3·\widehat{\mathbf{z}}$.

Figure 3

Stroboscopic views for a flow due to time-dependent $\widehat{\mathbf{n}}$ (see text); tilt angles $\mathrm{\Delta }\theta =k\pi /180$ with $k=4$ (a), 10 (b), 20 (c), and 30 (d) for same initial conditions as in Fig. 2.

Figure 4

Temporal development of an initial distribution of $5\times {10}^3$ particles distributed at random within the solid square centered at $\mathbf{r}=(0.6,0,0.6)$ for 3 (a), 4 (b), and 5 (c) periods of $\theta \left(t\right)$, with $\mathrm{\Delta }\theta =0.2\pi$. (d) Centered at $\mathbf{r}=(-0.6,0,0.5)$ for $\mathrm{\Delta }\theta =0.12$ after 30 periods of $\theta \left(t\right)$.

Figure 5

Positions of $5\times {10}^3$ particles initially distributed randomly within a small solid cube centered at ${\mathbf{r}}_0=(0.6,0.1,0.5)$ after 6 periods of oscillations of $\widehat{\mathbf{n}}\left(t\right)$ with $\mathrm{\Delta }\theta =0.2\pi$ as in Figs. 4. (a) Without rotational flow and (b) with rotational flow of strength $a_r=1$.

Figure 6

Coarse-grained Kullback-Leibler entropy at $t=24\pi$ versus $\mathrm{\Delta }\theta$ for the initial distribution shown in Fig. 4; insets: temporal development as in Fig. 4 for three selected values (a), (b), and (c) of $\mathrm{\Delta }\theta$ and 12 periods of $\theta \left(t\right)$.