Vorticity-induced surfing and trapping in porous media

TL;DR

This work addresses how active propulsion interacts with highly heterogeneous flows in dense porous media to govern cross-channel transport. It combines an active Brownian particle model with Faxén-type hydrodynamic couplings to a flow field solved by finite-element Stokes calculations and a Voronoi-lubrication flow-network, enabling detailed exit-time analysis. The authors find that swimming slows transport and creates universal long-time tails in exit-time distributions with exponent $\alpha \approx 3/2$, collapsing across packing fractions and motility parameters via scaling with ${\rm Pe}^s$ and ${\rm Pe}^f$ and revealing a surf-and-trap dynamic akin to diffusion in a comb-like geometry. A key mechanism is vorticity-induced trapping in the flow backbone, where orientation changes cause prolonged entrapment, with significant implications for biofilm formation and microrobot design in heterogeneous environments.

Abstract

Microorganisms often encounter strong confinement and complex hydrodynamic flows while navigating their habitats. Combining finite-element methods and stochastic simulations, we study the interplay of active transport and heterogeneous flows in dense porous channels. We find that swimming always slows down the traversal of agents across the channel, giving rise to robust power-law tails of their exit-time distributions. These exit-time distributions collapse onto a universal master curve with a scaling exponent of $\approx 3/2$ across a wide range of packing fractions and motility parameters, which can be rationalized by a scaling relation. We further identify a new motility pattern where agents alternate between surfing along fast streams and extended trapping phases, the latter determining the power-law exponent. Unexpectedly, trapping occurs in the flow backbone itself -- not only at obstacle boundaries -- due to vorticity-induced reorientation in the highly-heterogeneous fluid environment. These findings provide a fundamentally new active transport mechanism with direct implications for biofilm clogging and the design of novel microrobots capable of operating in heterogeneous media.

Figures (4)

• Figure 1: Model set-up. The active agent self-propels along its instantaneous orientation $\boldsymbol{e}$ at velocity $v$ and rotates due to rotational Brownian motion with diffusivity $D_{\rm rot}$. It is also advected and rotated by the velocity and vorticity fields, $\boldsymbol{u}$ and $\omega$, respectively. Simulation snapshots of swimmer positions at two consecutive times, $t_1=0.1/D_{\rm rot}$ and $t_2= 1.5/D_{\rm rot}$, are depicted by dark gray and black dots, respectively. The gray discs are the obstacles and the externally-applied flow is shown in the background. The color bar corresponds to the magnitude of the velocity field $|\boldsymbol{u}|$ and the white arrow indicates the direction of the applied flow. For illustration purpose we only show a slice of the porous channel. Here, the packing fraction and Péclet numbers are $\phi = 0.51$, $\mathrm{Pe}^f = 40$, and $\mathrm{Pe}^s = 1$, respectively, and $a$ denotes the obstacle radius.
• Figure 2: Power-law behavior of the exit-time distributions of active particles. a. Exit-time distribution $\varphi_{\rm exit}(\tau)$ for different swimming Péclet numbers. Here, the packing fraction and flow strength are $\phi=0.51$ and ${\mathrm{Pe}}^f=40$, respectively. The solid line represents the exit-time distribution of the streamlines of the fluid flow. b. Data collapse for $\mathrm{Pe}^s \gtrsim \mathrm{Pe}^f$ by rescaling the $x-$axis with ${\rm Pe}^s$. c. Data collapse for various packing fractions $\phi=0.34, 0.51, 0.58$ with their respective flow Péclet number $\mathrm{Pe}^f = 70,40,20$ and varying higher activity ${\rm Pe}^s$. The $x-$axis is rescaled by ${\rm Pe}^f/{\rm {Pe}^s}$. ( Inset) Sketch of the proposed 'comb potential' analog. The black lines in b, c indicate the power-law behavior at long times $\propto \tau^{-3/2}$. The dashed black line is added as guide to the eye. We denote by $D_{\rm rot}$ the rotational diffusivity.
• Figure 3: Surf-and-trap motility. a. Exemplary trajectory of a swimmer. Here, the activity is $\mathrm{Pe}^s=80$ and the agent's exit time is $\tau = 100/D_{\rm rot}$. The trajectory is colored according to the instantaneous particle velocity $U$. The color bar for the flow velocity is shown in Fig. \ref{['fig:method']}. b. Trap- and surf-time distributions, $\varphi_T(\tau_T)$ and $\varphi_S(\tau_S)$, for different activities ${\rm Pe}^s$. ( Inset) Exemplary displacements $\Delta r(t)$ as a function of time. Here, the packing fraction and flow Péclet number are $\phi = 0.51$ and $\mathrm{Pe}^f = 40$, respectively. We denote by $D_{\rm rot}$ the rotational diffusivity and $a$ is the obstacle radius.
• Figure 4: Vorticity-induced trapping. a. Positions of active agents (black dots) during their trapping phases shown on top of the underlying flow network and porous geometry. b. Trajectory and swimming orientation (arrows) of an active agent in the marked zone with (left) fluid velocity and (right) fluid vorticity in the background. c. Rescaled exit-time distributions of swimmers with and without the effect of vorticity-induced reorientation. ( Inset) Trajectory of a swimmer without orientation-vorticity coupling. d. Trajectories of fastest $10\%$ passive tracers (left), active agents with (middle) and without (right) orientation-vorticity coupling. Here, the flow Péclet number and packing fraction are $\mathrm{Pe}^f = 80$ and $\phi = 0.51$, respectively. Unless otherwise stated, the activity is $\mathrm{Pe}^s = 80$. We denote by $D_{\rm rot}$ the rotational diffusivity. The color bar for the flow velocity in a-d is shown in Fig. \ref{['fig:method']}.