Activity-induced asymmetric dispersion in confined channels with constriction

Microorganisms, such as Escherichia coli, are known to display upstream behavior and respond rheotactically to shear flows. In particular, E. coli suspensions have been shown to display strong sensitivity to spatial constrictions, leading to an anomalous densification past the constriction for incoming fluid velocities comparable to the microoganism's self-propulsion speed. We introduce a Brownian dynamics model for ellipsoidal self-propelling particles in a confined channel subject to a constriction. The model allows us to identify the relevant parameters that characterize the relevant dynamical regimes of the accumulation of the active particles at the constriction, and clarify the mechanisms underlying the experimental observations. We find that particles are trapped in butterfly-like attractors in front of the constriction, which is the origin of the symmetry breaking in the emerging density profiles of active particles passing the constriction. In addition, the probability of trapping and thus the strength of asymmetry are affected by size of the particles and geometry of the channel, as well as the ratio of fluid velocity to propulsion speed.

Figure 1

(a) Schematic of the system used in this study, displaying the relevant geometry and parameters. (b) Velocity field and streamlines in the channel of $L_x=4\mathrm{mm}$, $L_y=200\mu \mathrm{m}$, $L_c=40\mu \mathrm{m}$, incoming velocity $u_{\mathrm{in}}=5\mu {\mathrm{m}\mathrm{s}}^{-1}$ and average velocity in the constriction ${\langle u_x\rangle }_c=24.35\mu \mathrm{m}{\text{s}}^{-1}$.

Figure 2

Trajectories of 50 ellipsoidal particles of aspect ratio $\lambda =6$ and ${\text{Pe}}_s=45.7$ ($v_s=20\mu \mathrm{m}{\text{s}}^{-1}$) within a time interval of $t\sim 30{\tau }_p$. Panels (a)–(d) display the trajectories at four different relative strengths of incoming flow, ${\tilde{u}}_{\mathrm{in}}=0,1/4,1/2,1$, corresponding to incoming velocities $u_{\mathrm{in}}=0,5,10,20\mu \mathrm{m}{\text{s}}^{-1}$, respectively. In (a) and (d), particles' trajectories distribute uniformly in the right and left sides of the channel, while for intermediate ${\text{Pe}}_f$ (b, c), the trajectories are more concentrated on the right side of the constriction, resulting in asymmetric density profiles.

Figure 3

Time evolution of $p\left(x\right)$ for $N=4\times {10}^4$ ellipsoidal particles of aspect ratio $\lambda =6$ in the same channel as described in Fig 1, averaged over 10 realizations. Starting from a uniformly distributed state (gray dotted line), the PDF evolves to an asymmetric distribution, for $\widehat{t}\gtrsim 200$, with a sharp long-ranged peak just on the right side of the constriction (orange solid line).

Figure 4

SB as a function of relative incoming flow strength, ${\tilde{u}}_{\mathrm{in}}$, for SPPs of aspect ratio $\lambda =6$. Symbols show simulated results; the line is a guide to the eye. Insets show dimensionless PDF ($p\left(x\right)/p_0$) for three special cases, corresponding to ${\tilde{u}}_{\mathrm{in}}=1/10,1/4,3/4$.

Figure 5

SB as a function of ${\tilde{u}}_{\mathrm{in}}$ for ellipsoidal particles of various aspect ratios and fixed semiminor axis $b=0.5\mu \mathrm{m}$. In the insets, critical relative incoming flow strength, ${\tilde{u}}_{\mathrm{in}}^{*}$ is plotted as a function of $\lambda$, indicating that the peak position increases linearly with particle asymmetry.

Figure 6

SB (a) as a function of SPP size and asymmetry for ${\tilde{u}}_{\mathrm{in}}=1/4$, ${\text{Pe}}_s=45.7$ and (b) as a function of SPPs' and fluid's Péclet numbers for a fixed particle shape, $\lambda =6$. The rest of the parameters are the same as in Fig. 1.

Figure 7

Symmetry breaking versus relative incoming flow strength for various confinement parameters ($lp/L_y={\tilde{L}}_y^{-1}$). Inset shows how the maximum value of the SB profile decreases by decreasing confinement parameter (or increasing ${\tilde{L}}_y$).

Figure 8

Streamlines of passive particles ($v_s=0$) moving in the channel of width ${\tilde{L}}_y=1.46,{\tilde{L}}_c=0.29$, for incoming fluid velocity $u_{\mathrm{in}}=0,5\mu \mathrm{m}{\text{s}}^{-1}$. The velocity vector field, $\mathbf{v}(x,y)$, is the local average velocity of particles found in surface element $\delta x\delta y$ around each point, $(x,y)$, in the channel. The color of streamlines changes by the absolute value of average velocity from zero (blue) to its maximum value (red) as shown in the color bar. The triangles on the streamlines show the direction of average velocity field.

Figure 9

Streamlines of SPPs of aspect ratio $\lambda =6$ moving in the same channel as described in Fig. 8 for ${\tilde{u}}_{\mathrm{in}}=0,1/10,1/4,3/10,3/4$. Vertical and horizontal axes are rescaled by ${ł}_p$.

Figure 10

Typical trajectories of SPPs ($\lambda =6$) moving upstream in the same channel as described in Fig. 8, for ${\text{Pe}}_f=0.3$, corresponding to maximum symmetry breaking. Starting from the right side of the channel (filled circles), particles experience an up-and-down motion between the walls and finally are trapped in a butterfly-like trajectory in front of the constriction. Arrows show the direction of particles' velocity at the ending point of their trajectories.

Figure 11

Mean-square horizontal displacement (MSD) divided by time for (a) ellipsoidal particles of aspect ratio $\lambda =6$ for different incoming flow ${\tilde{u}}_{\mathrm{in}}$ and (b) spherical particles of different ${\tilde{L}}_y$ (corresponding to radii $a=1,1.2,1.3,1.5,2.0\mu \mathrm{m}$, respectively) and ${\tilde{u}}_{\mathrm{in}}=1/4$.