Collective Entrainment and Confinement Amplify Transport by Schooling Microswimmers

Microswimmers can serve as cargo carriers that move deep inside complex flow networks. When a school collectively entrains the surrounding fluid, their transport capacity can be enhanced. This effect is quantified with good agreement between experiments with self-propelled droplets and a confined Brinkman squirmer model. The volume of liquid entrained can be much larger than the droplet itself, amplifying the effective cargo capacity over an order of magnitude, even for dilute schools. Hence, biological and engineered swimmers can efficiently transport materials into confined environments.

Figure 1

Fluid transport by a self-propelled droplet. (a) Schematic of experimental setup. Cell bottom glass, top PDMS, $H=52\mu \mathrm{m}$, droplet radius $R=24\mu \mathrm{m}$. (b) Diagram of an entrainment event. Because of the flow generated by the droplet, the tracer is displaced by $\mathbf{\Delta }(a,b,\lambda )$ as a function of the impact parameters $a$ and $b$, and the swimming path length $\lambda$. (c) Movie screenshot showing particle trajectories due to an active droplet swimming along a straight line (see Video 1). (d) Tangential flow velocity at the surface of the droplet, in the comoving frame of reference. Data points are PIV measurements close to the interface, for four different surfactant concentrations. Lines are fits to the first two modes of the squirmer model (Eq. (1)). Legend: resulting dipole coefficient $\beta =B_2/B_1$ for each TTAB concentration.

Figure 2

Flows $\mathit{u}(\mathit{r})$ generated by an active droplet. (a),(b) Comparison of the flow fields between (top half) experiments and (bottom half) the squirmer model in a Brinkman fluid (Supplemental Material, Eq. S13). Flows are shown at the midplane of the microfluidic chamber. Streamlines are shown (a) in the laboratory frame, and (b) in the frame comoving with the droplet. Background colors denote (a) the longitudinal flow $u_x$ and (b) the transverse flow $u_y$, scaled with the swimming speed. Note that the transverse flow $u_y$ changes sign across the $x$ axis. (c),(d) Decay of the longitudinal flow strength in the laboratory frame along the (c) $x$ axis and (d) $y$ axis. Points show experiments, and the solid line is the corresponding squirmer model in a Brinkman fluid. The dashed line shows the squirmer model in an unconfined Stokesian fluid, which underestimates the flow strengths.

Figure 3

Entrainment of particles by a confined self-propelled droplet. (a) Fluid transport by a squirmer in a Brinkman fluid. Shown is the deformation of an initially uniform rectangular grid of tracer particles, after a droplet swims from $x/R=-5$ to 5. Colored trajectories show the motion of particles starting at different positions along the $y$ axis. (b) Experimental trajectories, with their initial positions aligned along the $y$ axis. The initial positions are marked with a (Circle), and the final positions are marked with a (Cross mark). (c) Comparison. (Blue cross mark) show the experimental entrainment along $x$ as a function of lateral distance $y$, (Pink plus signs) show simulated trajectories using the same initial positions using the Brinkman model with Brownian motion, the solid blue line shows the Brinkman theory without noise, and the dashed gray line shows the squirmer model in an unconfined Stokesian fluid. Far away the particles are displaced backward, but nearby they are entrained forward, along the swimming direction.

Figure 4

Entrainment by a school of microswimmers. (a) Simulation snapshot, see Video 3. Self-propelled droplets (black) move collectively along the positive $x$ direction, transporting particles (colors) that initially started at $x=0$. (b) Displacement of the particles along the swimming direction as a function of time, $x_T(t)$. Individual trajectories (gray lines) show jumps and drift events, which on average (black line) lead to the entrainment velocity $U_{\mathrm{ent}}$ that is predicted analytically (red dashed line) by Eq. (5). (c) Diagram of the entrainment volume (blue shaded area) that is dragged behind an active droplet, in the comoving reference frame. The gray lines depict stream lines, the purple area is the swimmer wake, and the green area is the auxiliary volume. (d),(e) Stirring by active droplets in a Brinkman fluid. Shown is the integrand $a{R}^{-3}{\mathrm{\Delta }}^2(a,b,\lambda )$ from Eq. (7), measured experimentally and computed numerically.