Phys. Rev. Lett. 132, 188301 (2024)

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Active Darcy’s Law

Ryan R. Keogh, Timofey Kozhukhov, Kristian Thijssen, and Tyler N. Shendruk

Phys. Rev. Lett. 132, 188301 – Published 30 April 2024

Abstract

While bacterial swarms can exhibit active turbulence in vacant spaces, they naturally inhabit crowded environments. We numerically show that driving disorderly active fluids through porous media enhances Darcy’s law. While purely active flows average to zero flux, hybrid active/driven flows display greater drift than purely pressure-driven flows. This enhancement is nonmonotonic with activity, leading to an optimal activity to maximize flow rate. We incorporate the active contribution into an active Darcy’s law, which may serve to help understand anomalous transport of swarming in porous media.

Received 11 May 2023
Accepted 11 March 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.188301

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Bioconvection Biological fluid dynamics Collective behavior Complex fluids Flows in porous media Low Reynolds number flows Swarming Swimming

Active nematics Living matter & active matter

Theories of collective dynamics & active matter

Physics of Living SystemsFluid DynamicsPolymers & Soft MatterInterdisciplinary Physics

Authors & Affiliations

Ryan R. Keogh¹, Timofey Kozhukhov¹, Kristian Thijssen ², and Tyler N. Shendruk ^1,*

¹School of Physics and Astronomy, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom
²Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, Copenhagen, Denmark

^*t.shendruk@ed.ac.uk

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Vol. 132, Iss. 18 — 3 May 2024

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Figure 1
Purely pressure-driven flow through porous media. (a) Top: schematic of pressure-driven active nematic fluids within a obstacle-laden rectangular channel of height $h$ . (i) Flow is driven by the pressure gradient $- G$ . (ii) Extensile active nematics generate local active forces $f_{act}$ . (iii) Obstacles of radius $R$ are placed with a homogeneous probability distribution function (PDF), without overlaps with each other or the walls, creating a porous medium characterized by Brinkman pore size $ℓ_{B}$ . Middle: snapshot of passive flow ( $ζ = 0$ ) due to a pressure gradient ( $G = 0.011$ ), colored by speed at porosity $ϕ = 0.87$ . Bottom: snapshots of passive flow ( $ζ = 0$ ) due to a pressure gradient ( $G = 0.011$ ) at porosity $ϕ = 0.79$ . The color bar shows the magnitude of flow speed ranging from $0.0 \to 0.3$ in MPCD units. (b) Normalized flow profiles ${⟨ v_{x} ⟩}_{x, t}$ across the channel, fit by Eq. S(8) (solid lines). (c) Permeability $κ = ℓ_{B}^{2}$ from fits in (b) grow linearly with the Kozeny-Carman factor $ϕ^{3} (1 - ϕ)^{- 2}$ (dashed line).
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Figure 2
Global averaged flows. Red circle, purely pressure-driven ( $ζ = 0$ ; $G = 0.011$ ); Dark blue square, purely active [ $ζ = 0.08$ ( $ℓ_{ζ} ≃ 10$ ); $G = 0$ ]; and blue triangle, hybrid active/pressure-driven ( $ζ = 0.08$ ; $G = 0.011$ ). Shaded regions denote standard error. (a) Time-averaged global drift ${⟨ v_{x} ⟩}_{x, y, t}$ nondimensionalized by Poiseuille flow $U_{P}$ for $G = 0.011$ . The pressure-driven case agrees with Darcy’s law [Eq. S(9); dotted line], while the purely active case exhibits zero drift. (b) Kinetic energy density ${⟨ v^{2} ⟩}_{x, y, t}$ . Pressure-driven flow is consistent with theory (dotted red line), and hybrid flow coincides with the sum of the pure cases (dashed green line).
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Figure 3
Purely active nematic ( $ζ = 0.08$ ) through porous media. Top row: porosity $ϕ = 0.87$ ( $κ / h^{2} = 6.4 \times 10^{- 3}$ ). Bottom row: $ϕ = 0.79$ ( $κ / h^{2} = 2.3 \times 10^{- 3}$ ). (a) Snapshots of pressure-gradient-free ( $G = 0$ ) active nematic flow. The color bar represents the magnitude of flow speed in MPCD units from $0.0 \to 0.3$ . (b) Instantaneous global drift ${⟨ v_{x} ⟩}_{x, y} (t)$ for systems shown in (a) are normalized by Poiseuille flow $U_{P}$ for $G = 0.011$ . For the more dilute porous medium (top), the drift direction persists over time, whereas the denser system (bottom) exhibits more rapid fluctuations. Highlighted times: green pentagon, $t = 6 \times 10^{3}$ ; purple triangle, $10 \times 10^{2}$ ; and pink circle, $12 \times 10^{3}$ . (c) Instantaneous flow profiles ${⟨ v_{x} ⟩}_{x} (y; t)$ highlighted times in (b).
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Figure 4
Active contribution $J$ to hybridized flow. Asterisk and black square symbols denote the same datasets across panels, dashed lines represent fits, and shaded regions are the standard error. (a) Dependence of $J$ on permeability $κ$ with $ζ = 0.08$ . Dashed lines predicted from Eq. (1) with $ζ^{*}$ fit of Eq. (1) to the data in (c). (b) Linear dependence of $J$ on pressure gradient $G$ for $κ / h^{2} = 1.9 \times 10^{- 2}$ and $6.4 \times 10^{- 3}$ with $ζ = 0.08$ . Dashed lines predicted from Eq. (1) with $ζ^{*}$ fit of Eq. (1) to the data in (c). (c) Nonmonotonic dependence of $J$ on $ζ$ with a maximum at $ζ^{*} = 0.05$ for $κ / h^{2} = 1.9 \times 10^{- 2}$ ( $ϕ = 0.93$ ) with $G = 0.011$ and 0.0011. For $G = 0.011$ , at low activity, $J \sim ζ$ , while $J \sim ζ^{- 2}$ at high activity. Fit of Eq. (1) with $ζ^{*} = 0.05 \pm 0.02$ shown as a dashed yellow line. (d) Hybrid flow profiles normalized by empty channel flow $U_{p}$ compared with passive and active Darcy’s laws. Simulated flow profiles (dots) for $G = 0.011$ compared to passive [dotted lines; Eq. S(8)] and active [solid lines; Eq. (2)] Darcy’s law predictions.
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Physical Review Letters

Active Darcy’s Law

Ryan R. Keogh, Timofey Kozhukhov, Kristian Thijssen, and Tyler N. Shendruk

Phys. Rev. Lett. 132, 188301 – Published 30 April 2024

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