Active Darcy’s Law

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.

Here, we study the cooperative effect of disorderly active flows on pressure-driven transport through porous materials of fixed obstacles (Fig. 1a).We find that, when biased due to weak external pressure gradients, activity positively enhances global drift, even in the active turbulence limit.This effect endures as the number density of obstacles is increased, resulting in undiminished kinetic energy despite increased dissipative drag-suggesting active flows autonomously fill the available porous length scales.The active auxiliary flux increases with pressure gradient and is non-monotonic with activity, possessing an optimal activity.We conclude that the flux of active fluids through porous media are described by a modified Darcy's law.
While individual swimming bacteria move with polar self-propulsion, their dipolar hydrodynamic and head-tailsymmetric steric interactions are nematic, making active nematics an appropriate minimal model [62][63][64], in agreement with experiments [5,9,65,66].To model active nematic fluids within a complex geometry, we employ active-nematic * t.shendruk@ed.ac.ukMulti-Particle Collision Dynamics (AN-MPCD [67]), a recent mesoscopic method [68].AN-MPCD simulates linearised fluctuating nematodynamics with isotropic viscosity and elasticity [69], and activity  modelled via local multi-particle force dipoles [70] The competition between elasticity and activity results in an active length scale ℓ  (Fig. S1).The corresponding continuum equations correspond to low Reynolds number flows with linearised nematic elasticity and an active stress term [71,72].Systems of motile bacterial exhibit density fluctuations [14,73], as does the particle-based AN-MPCD approach (Fig. S2), in contrast to continuum models that assume incompressibility.We focus on flow-aligning nematics with extensile activity  and a weak external forcing − = (∇)• x due to a pressure gradient ∇ down the channel x (Fig. 1a), which breaks the directional symmetry of bacterial turbulence [7].Values are reported in MPCD units [67] and unless otherwise stated,  = 0.011 and  = 0.08 ℓ  ≃ 10).All other parameters are chosen to match previous studies characterising the numerical approach [68].AN-MPCD is ideal for simulating active flows around randomly-placed offlattice obstacles with a broad distribution of voids.The active fluid and porous medium are confined within a 2D channel with heightℎ = 30 and length  = 150 with impermeable, no-slip walls with free-nematic anchoring (Fig. 1a), which models well-defined experimental set-ups [57].Finite-size effects (Fig. S3) and alternative anchoring conditions (Fig. S4) are considered in the supplementary materials.The porous medium is formed of impermeable, immobile, circular obstacles of radii  = 2 with the same boundary conditions as the walls, producing isotropic porosity  ∈ [0.67, 1.0], where  = 1.0 is an empty channel and 0.67 is likely to be impermeable [67].The random obstacles are homogeneously distributed (Fig. 1a) but overlaps are not permitted.Components of the velocity field v =   x +   ŷ are averaged temporally , longitudinally  and/or transversely , denoted as ⟨•⟩ ,, to measure flow profiles and global fluxes.
Before considering cooperative effects, we quantify the properties of the porous medium.In the limit of zero activity and large porosity ( → 1), the flow is relatively unobstructed and the mean velocity profile ⟨  ⟩ , () is parabolic (Fig. 1b; yellow).As the porosity is decreased, the greater number of obstacles slows the flow (Fig. 1a; middle  = 0.87) and broadens the profiles (Fig. 1b; pink).At low porosity, the effect of the walls is minimal compared to obstacle drag (Fig. 1a; bottom  = 0.79), producing plug-like flow (Fig. 1b    −  −1 ⟨  ⟩ , () = −/ for viscosity  and interstitial permeability  [67], indicating that anisotropic properties can be absorbed into the Brinkman terms (Fig. 1b).At the lowest porosity, systematic deviations, in the form of limited nonmonotonic shoulders in the near-wall region appear (Fig. 1b; blue), due to steric inaccessibility of obstacles.From the fits,  and corresponding pore size ℓ  = √  (or Brinkman length) are found (Fig. 1c).The resulting permeability obeys the Kozeny-Carman relationship  =  3 (1 − ) −2 with  = 0.091 ± 0.004.
In contrast, purely active turbulent flows do not drift down the channel [7].Even at lower activities, where spontaneous symmetry breaking results in unidirectional active flow, global drift averages to zero over multiple simulations [75].We focus on activities which exhibit isotropic active turbulence in an obstacle-free channel (ℎ ≫ ℓ  ) [76] (see movie 1).As  → 1, the channel height ℎ competes with the active length scale ℓ  [77] to determine the spatiotemporal structure of the active flow [72,75,78,79], which result in vortex lattices [72] recently observed as the first mode in suspensions of bacteria [11].However, as obstacles are added, the confinement length switches from ℎ to pore size ℓ  .This causes the dynamics to transition from active turbulence to local unidirectional flows (Fig. 3a; top and movie 2).There are well defined paths in which unidirectional flow might arise, though spontaneous symmetry breaking might produce a stream of unidirectional flow moving in the opposite direction along other paths, or a recirculating vortex might become trapped in a void (Fig. 3a While large porosity can allow otherwise turbulent active flows to instantaneously possess net global drift (Fig. 3a; top), lower porosity hinders instantaneous drift (Fig. 3a; bottom).In lower porosity, pore-entrapped vortices are more frequent and long-lived (Fig. 3a; bottom and movie 3).Furthermore, system-spanning streams are less likely and net fluxes become fleeting and noisier (Fig. 3b; bottom), while the instantaneous flow profiles ⟨  ⟩  (, ) approach zero across the channel (Fig. 3c; bottom).Although there are localised flows, drag on the obstacles generally dominates.
Ensemble averaged active drift is ⟨  ⟩ ,, = 0, due to the spontaneous symmetry breaking in an absence of pressure gradients (Fig. 2a).However, despite zero net global flux in the purely active systems, there is significant global kinetic energy v 2 ,, (Fig. 2b).Whereas kinetic energy drops rapidly with decreasing permeability  for pressure-driven flows (Fig. 2b; red), active systems maintain higher kinetic energy at low permeability (Fig. 2b; blue).For  = 0.08, the kinetic energy of the active flow is greater than the pressure-driven flow for  = 0.011 when /ℎ 2 ≲ 0.11.This is because active forces still generate locally coherent flow between obstacles and pores with sizes comparable to the active length scale ℓ  trap vortices, allowing localised self-sustaining recirculations.
While purely active systems have zero flux, this is not true of hybridised flows that are both active and externally biased (Fig. S5).As in the purely activity case, activity  is chosen such that active flows are in the turbulent regime without obstacles.The pressure gradient  is sufficiently high that activity acts as a perturbation to the driven case, but not so high that flow alignment is enforced.The kinetic energy density approximately corresponds to the sum of the two pure cases (Fig. 2b).At small pore sizes, the active kinetic energy of the hybrid system is close to that of the purely active flow.The hybrid flow begins to differ from the purely active case around  * /ℎ 2 ≃ 0.022.Here, the ratio of the Brinkman length and active length scale is near unity (ℓ *  /ℓ  ∼ 4.5/10).The increased kinetic energy acts as an auxiliary driving force, assisting the pressure gradient to conduct fluid through the porous space.The global flux shows an augmented drift when compared to either the purely driven or purely active cases (Fig. 2a).Though the purely active flow is disorderly, in the hybrid case local active energy injection enhances the flux.
To study the active contribution to the drift, we measure the difference  = ⟨  ⟩  ,, − ⟨  ⟩  ,, , where ⟨  ⟩  ,, is the global drift of the hybrid case and ⟨  ⟩  ,, is the purely pressuredriven case.We find  > 0 in all instances (Fig. 4), which reveals that even disorderly active flows enhance pressuredriven flux in porous systems.The active enhancement  increases linearly with  at low permeability then saturates, going as  () ∼  (ℎ/ √ ) (Fig. 4a).This is the same dependence as Darcy's law when going from a crowded channel to an obstacle-free channel (Fig. 2; red circles).Similarly at sufficiently small activity, the active contribution increases linearly with pressure gradient  ∼  (Fig. 4b), suggesting the enhancement requires a coupling to the external biasing -the stronger the pressure gradient, the more the activity can boost the flux.Pressure gradients generate directed deformations to the orientation field Q, which induce an auxiliary active forcing  act ∼ ⟨ 2  ⟩ ∼  [80].This approximation only holds in the limit  < /ℓ  , since flow-alignment eventually suppresses divergence of  and thus the active auxiliary forcing slows at larger pressure gradients.this approximation suggests the active contribution is linear with activity.Indeed, varying activity  demonstrates that the enhancement  () ∼  increases linearly at sufficiently low activities (Fig. 4c; movie 4).However, the auxiliary flux reaches a maximum value at  * = 0.05 (movie 5), then decreases (Fig. 4c), indicating there is an optimal activity for enhancing porous transport that is independent of the biasing pressure gradient.The maximum active contribution occurs when the ratio of the Brinkman and active length scales approach unity (ℓ  /ℓ *  ∼ 4.5/15).For small activities (below the optimal value  <  * ), ℓ  is larger than the characteristic pore sizedisorderly active turbulence does not occur within the pores.Active flows are unidirectional and laminar-like within the pores; hence,  ∼ .However for  >  * , active turbulence is possible within the pores and collective flows become uncorrelated even on scales smaller than ℓ  .The non-laminar uncorrelated turbulence generates additional kinetic dissipation and active vortices that are not conducive to flux (movie 6), causing the enhancement to decline as  ∼  −2 (Fig. 4c).In active nematics, much of the kinetic energy is contained with the vortices and so the additional dissipation scales with the change of enstrophy  disp ∼ − 2 [81].
Having obtained the active auxiliary contribution as a function of its dependencies, the full effect of local activity on the global flux is where 2 * is the optimal activity at which the active length scale ℓ  matches the Brinkman length ℓ  .The active contribution only depends directly on the Darcy velocity and dimensionless ratios.Equation 1 is linear in pressure gradient for ℓ  / < 1, nonlinear-but-monotonic with permeability and non-monotonic in activity, with a maximum where ℓ  = ℓ  .The finite size of the channel enters through the factor of  [67].The active contribution in Eq. 1 is fit as a function of activity for a single pressure gradient (Fig. 4c;  = 0.011), with  * the only fitting parameter since  D and  are given by the permeability from Fig. 1.Having fit  * for a single system, the predicted enhancement as a function of  and  is seen to agree with all other simulations without any further fitting (Fig. 4a-b).The fit is accurate for activities above the pressure-dominated limit (ℓ  / > 1), where and flowalignment suppresses active forcing.Hence, activity enhances transport in porous channels via an effective active Darcy velocity which recasts the Brinkman equation and channel-averaged drift into active Brinkman and drift equations by substitution of  D →  AD .This prediction captures the enhancement of the flow profiles across the channel without additional fitting (Fig. 4d).
Here, we studied the auxiliary contribution of collective bacterial motion on fluid transport through porous media.Our results show that activity enhances transport of pressure-driven fluids, even in the limit of bacterial turbulence.The pressure gradient breaks the symmetry of disorderly active flows, which enhances the total transport properties.Optimal activity for maximising the flux arises from competition between the characteristic active and porous length scales.By measuring the active contribution as a function of permeability, pressure gradient and activity, we discover an active version of Darcy's law.While the presented results are of active fluids and porous media confined within 2D channels, simulations of larger systems with periodic boundary conditions on all sides (movies 7-8) confirm the proposed active Darcy's law (Fig. S6).Through  * , the active Darcy equation can be written as This expands on recent work studying driven active fluids in empty channels [21,80] and previous description of activity lowering the apparent viscosity [82][83][84][85][86] to complex disordered environments.While the active Darcy's law presented here could be interpreted as reducing the apparent viscosity, raising the effective permeability or augmenting the pressure gradient, none of these interpretations make clear the physical mechanism of weak pressure gradients biasing the local force densities ∼ /ℓ  within pores to point in the same direction, nor do they account for an optimal value for active auxiliary forcing when the active length scale is comparable to pore size.
Our results indicate that ℓ  /ℓ  ∼ 1 is precisely the condition to maximise the active enhancement to the drift.Active transport enhancement could be utilised in future research as a framework to understand anomalous transport of nutrients by soil bacteria, as well as provide an approach for controlling active flows akin to continuous friction [93][94][95].Previous work on dilute bacteria dynamics has highlighted the role of transient trapping [30,52] and escape from cavities [96], and deadend pores [97], and future work considering dense collective dynamics within random networks of cavities and dead-ends rather than obstacles may be fruitful.While we simulated immobile obstacles, future studies could consider how collective flows in turn modify deformable surroundings.Active clogging, erosion and infiltration may reveal much about the role of motile microbes as microecosystem engineers [98].

FIG. 1 .FIG. 2 .
FIG. 1. Purely pressure-driven flow through porous media.a) Top: Schematic of pressure-driven active nematic fluids within a obstacleladen rectangular channel of height ℎ. i) Flow is driven by the pressure gradient −.ii) Extensile active nematics generate local active forces f act .iii) Obstacles of radius  are placed with a homogeneous probability distribution function (PDF), without overlaps with each other or the walls, creating a porous medium characterised by Brinkman pore size ℓ  .Middle: Snapshot of passive flow ( = 0) due to a pressure gradient ( = 0.011), coloured by speed at porosity  = 0.87.Bottom: Snapshots of passive flow ( = 0) due to a pressure gradient ( = 0.011) at porosity  = 0.79.The colour bar shows the magnitude of flow speed ranging from 0.0 → 0.3 in MPCD units.b) Normalised flow profiles ⟨  ⟩ , across the channel, fit by Eq.S(8) (solid lines).c) Permeability  = ℓ 2  from fits in (b) grow linearly with the Kozeny-Carman factor  3 (1 − ) −2 (dashed line).