Complex and active fluids find broad applications in flows through porous materials. Nontrivial rheology can couple to porous microstructure leading to surprising flow patterns and associated transport properties in geophysical, biological, and industrial systems. Viscoelastic instabilities are highly sensitive to pore geometry and can give rise to chaotic velocity fluctuations. A number of recent studies have begun to untangle how the pore-scale geometry influences the sample-scale flow topology and the resulting dispersive transport properties of these complex systems. Beyond classical rheological properties, active colloids and swimming cells exhibit a range of unique properties, including reduced effective viscosity, collective motion, and random walks, that present novel challenges to understanding their mechanics and transport in porous media flows. This review article aims to provide a brief overview of essential, fundamental concepts followed by an in-depth summary of recent developments in this rapidly evolving field. The chosen topics are motivated by applications, and new opportunities for discovery are highlighted.

The flow of complex fluids through porous media have important natural and industrial applications in the fields of biofilm formation [1], enhanced oil recovery (EOR) [2], and groundwater remediation1 [3]. Porous media are comprised of a solid matrix material, permeated by a network of fluid-filled void spaces (pores) that are interconnected by relatively smaller throats. Relevant examples of porous media abound, including soil, porous rocks, filtration media, and biological tissues. Classically, continuum (macroscopic) descriptions of fluid transport have been sought for both Newtonian [4,5] and non-Newtonian fluid flows in porous materials. Unlike simple fluids, the mechanics of complex materials—including polymeric solutions, colloids, gels, emulsions, and more recently, active swimming cells—nonlinearly couple to fluid deformation and consequently local pore geometry due to fluid memory, solute shape, and particle-particle interactions. Despite our relatively deep current understanding of complex fluid flows in simple, model geometries, extrapolation of their stability criteria and transport properties from microscopic isolated pores to the sample scale has proven challenging for porous media. Recent innovations in experimental and numerical techniques, including microfluidics [6,7], confocal and holographic imaging [8,9], and simulation algorithms [10,11], now provide data across large domains spanning many pores along with unprecedented spatiotemporal resolution. Such novel approaches facilitate direct links between pore-scale mechanics and macroscopic transport properties. This review discusses some recent developments in this rapidly evolving field in the context of both canonical (e.g., viscoelastic fluids and colloidal suspensions) and newly emerging (e.g., active suspensions) non-Newtonian materials in porous media. The coverage of topics is strongly motivated by applications in natural and industrial flows but is not intended to be comprehensive.

Polymeric solutions encompass a range of complex fluid rheologies that are instrumental in many industrial and biological processes. Biofilms, for example, are broadly defined as surface-associated communities of one or many types of microorganisms [12]. In the case of bacteria, cells can secrete polymeric substances in the form of extracellular matrix (ECM), which mechanically behave like viscoelastic materials [13–18]. Biofilms provide protection for survival and growth in hostile environments and have implications for ecosystem dynamics and chronic infections [17,19–21]. Biofilm formation in porous media can be either desirable for applications such as microbial mining and bioremediation or detrimental to water treatment and transport [22–24]. The polymers secreted by biofilms’ microorganisms form 3D filamentous structures, called biofilm streamers. Over time, they can bridge pore structures, dramatically modify the flow through porous media, and lead to rapid clogging [25].

The flow of polymeric solutions in porous media also has implications in environmental remediation1 and EOR [2]. Improper disposal of organic compounds often leads to the contamination of groundwater, where organic compounds become trapped in the pores of reservoirs through capillary forces [26]. In a similar way, the majority of crude oil inside reservoirs remains capillary trapped even after multiple stages of the oil recovery processes [2]. Enhanced recovery techniques have the potential to vastly improve the remediation of organic compounds and the efficiency of natural resource extraction [27–30]. In both these processes, the addition of polymers to the displacing fluid enhances recovery by increasing the viscous drag on capillary trapped oil ganglia and suppressing viscous fingering [31–33]. For instance, when elastic effects dominate viscous forces, viscoelastic instabilities in the polymeric working fluids produce strong spatial and temporal velocity fluctuations, which enhance the displacement of trapped immiscible fluids [34–40] and intensify mixing and dispersion [39,41,42] inside porous media.

In the absence of inertia, such flow fluctuations are colloquially referred to as elastic turbulence due to their analogous features to inertial turbulence [43–48]. In parallel flow geometries viscoelastic flows are nonlinearly unstable [49] and often require an upstream perturbation to trigger a subcritical transition [50,51]. In flows with strong base curvature such as porous media, a linear instability is initiated through polymer stretching along curved streamlines [52] and results in a supercritical transition [53,54]. Understanding the onset and interplay of such instabilities with porous microstructure presents a fundamental problem with copious practical implications. Viscoelastic instabilities in simple isolated geometries such as cross-slot, isolated contraction, and confined cylinder lead to flow asymmetry or eddy formation [53,55,56]. The viscoelastic interaction among cylinders and contractions located in linear arrays induces multiple flow patterns, which regulate the sample scale transport of fluids and particles [57,58]. Viscoelastic instability in a 2D porous geometry promotes either stability or instability of the flow depending on the specific arrangement of the cylinders [7,59]. Whereas, elevated interaction due to higher connectivity in a 3D porous sample, viscoelastic instability induces intense velocity fluctuations, which leads to an anomalous rise of flow resistance [60].

The transport of colloidal solutions and other complex fluids through porous media has important applications in drug injection into biological tissues [61,62], contrast enhancement in specific tissues for medical imaging [63], filtration [64–66], ground water remediation [67,68], and EOR [69–71]. The surface activation and deposition of colloidal particles on a solid matrix help to mobilize trapped immiscible fluid in porous media [72–75]. For example, the deposition of colloidal particles on the surface of a porous substrate reduces its permeability resulting in an enhanced flow speed, which induces larger viscous stress and mobilizes the trapped immiscible fluid [75]. The transport of motile micro-organisms in porous media plays a critical role in contamination removal from groundwater aquifers [76,77] and agriculture [78]. Microorganisms can also be used in oil reservoirs for an alternative environmentally friendly microbial enhanced oil recovery (MEOR) [79]. MEOR can be achieved in one of the two ways whereby bacteria are injected into the pores: (i) for the purpose of developing a biofilm to plug open pores and allow subsequent drainage of remaining oil pockets or (ii) to allow bacteria-generated surfactant and polymer to reduce the oil’s surface tension and increase displacing fluid’s viscosity, again facilitating drainage. Thus, understanding how and where colloids and active cells disperse through these media determines the efficacy of recovery.

A primary experimental challenge in studying these phenomena is the need to measure inherently 3D flow structures and to peer deep within the complex microstructure of porous media. Over the past decade, the development and application of 3D imaging techniques have provided unprecedented new insights into the mechanics and transport of complex fluids in porous media with varying degrees of spatial and temporal resolution. Optical methods such as holographic imaging and micro-tomographic particle image velocimetry (PIV) have been applied to capture the flow structure of viscoelastic instabilities around obstacles and through single pores [9,80,81]. Laser scanning and spinning disk confocal microscopy have provided exquisitely high spatial resolution within 3D packed beds to gain insight into pore-pore interactions and active cell motility [60,82].

In this article, we review recent theoretical, numerical, and experimental developments in the flow of complex fluids—including polymeric fluids, colloidal solutions, and active particles—in porous media. Excellent review articles and books on the fundamentals and related topics are widely available, in particular on: complex fluids [50,83–87] and geometrical effects [88,89], elastic turbulence [90], porous media [4], colloids [65], active matter and swimmers [91–94], and applications [19,20,95–99]. The reader is pointed toward this deep body of literature for a more comprehensive treatment of these foundational concepts. Rather, in the present review in Sec. II, we begin with a brief discussion of the relevant physical flow regimes governing the mechanics and transport of complex fluids in porous media and describe a broadly applicable theoretical framework and governing equations. In Sec. III, we discuss elastic instabilities in 1D, 2D, and 3D geometries and highlight novel and promising 3D flow measurement techniques. We also discuss various proposed mechanisms for EOR by polymeric fluids and the effects of elastic instabilities on dispersive transport in ordered and disordered porous media. Section IV describes the flow of colloidal solutions in porous media and their applications. Section V addresses the role of geometry and flow in governing the transport of both dilute and concentrated suspensions of swimming micro-organisms and active matter in porous media. In Sec. VI, we conclude by summarizing the current state of this exciting field along with the experimental and theoretical challenges and opportunities for discovery that lay ahead.

In contrast to Newtonian flows through simple geometries, nontrivial rheology and pore microstructure vastly expand the space of relevant physical parameters governing the flow of complex fluids through a porous medium. In Newtonian flows, porosity and tortuosity are classically used in porous media flows to characterize the ratio of interstitial pore volume to bulk volume and the flow path length through porous media, respectively. However, these bulk-averaged measures alone are insufficient to describe the relevant geometry for non-Newtonian flows, as evidenced by recent results demonstrating that even the (dis)order of the medium can have drastic effects on viscoelastic flow stability [7]. The physical size of the voids dictates the magnitude of relevant stresses and advection time scales for flow through porous media. The local pore geometry is thus specified by the mean pore body diameter, Lb, of the voids and the diameter of the throats, Lt, which connect the voids. For example, in soil and mineral rocks, the pore and throat sizes are in the range of Lb=0.250μm and Lt=0.110μm, respectively [88,100]. The Reynolds number, Re=ρULt/η, represents the ratio of inertia to viscous forces, where ρ,U, and η are fluid density, characteristic velocity, and fluid zero-shear viscosity, respectively. Interstitial flow speeds inside crude-oil rocks, for example, caused by geothermal pressure gradients or injection of displacing fluids, are in the range of 0.01100μm/s [36,101], and the fluid viscosity in such applications can be in the range of 0.001–0.1 Pa s, corresponding to water and polymer solutions [36]. Thus, Re is in the range of Re=1011103, which implies that inertia is negligible (Table I). Although the specific physical parameters can vary significantly across diverse flows, this low Reynolds number regime Re<1 is typical of many porous media flows.

TABLE I.

Dimensionless numbers relevant to transport through porous media and their typical ranges.

Dimensionless numbersPhysical comparisonTypical range
Reynolds number, Re Inertial and viscous force 10−11 −10−3 
Weissenberg number, Wi Elastic and viscous force 10−5 −105 
Elasticity number, El Elastic and inertial force  ≫1 
Deborah number, De Polymer relaxation and advection time 10−6 −105 
Péclet number, Pe Diffusion and advection time  ≫1 
Knudsen number, Kn Polymer size and flow length scale  ≪1 
Dimensionless numbersPhysical comparisonTypical range
Reynolds number, Re Inertial and viscous force 10−11 −10−3 
Weissenberg number, Wi Elastic and viscous force 10−5 −105 
Elasticity number, El Elastic and inertial force  ≫1 
Deborah number, De Polymer relaxation and advection time 10−6 −105 
Péclet number, Pe Diffusion and advection time  ≫1 
Knudsen number, Kn Polymer size and flow length scale  ≪1 

High molecular weight polymer suspensions and emulsions are commonly found in groundwater remediation and EOR applications. Such additives impart elasticity to the solution, which is characterized by a relaxation time (λ), typically in the range λ=0.01100s [36]. The Weissenberg number (Wi) quantifies the relative importance of elastic and viscous forces in a viscoelastic flow and can be written as Wi=N1/2σ (or λU/Lt), where N1 and σ are the first normal stress difference and shear stress, respectively. Despite the small value of interstitial flow speed through porous media, the minute length scale of the flow leads to high shear rates (103103s1), and consequently, Wi in the range of Wi=105105 for EOR and groundwater remediation applications (Table I) [88]. The elasticity number (El=Wi/Re=λη/ρLt2) can also be defined to represent the ratio of elastic to inertial forces. El is independent of flow conditions and only depends on fluid properties and the geometry length scale. For viscoelastic porous media flow, elastic forces completely dominate over inertial forces due to a very small length scale (i.e., El1). Polymeric chains become stretched when they cross a throat spanning the pores due to high rates of extension, and they subsequently relax during advection inside the pore body. The Deborah number, De=λU/Lb, measures the ratio of polymer relaxation time to the advection time, and its value broadly occurs in the range 106105 for applications such as EOR and groundwater remediation (Table I). Analogous to the Re>1 regime in classical inertial turbulence, for sufficiently large De>1 (or Wi>1), viscoelastic flows are expected to become unstable and can indeed exhibit elastic turbulence [44,53,90]. For sufficiently large ReDe, a separate “inertio-elastic turbulence” regime can occur [102,103]. However, given the generally small Re incurred in most porous media applications, we restrict our discussion to purely viscoelastic phenomena here.

The flow of Newtonian materials through porous media enhances the dispersal of scalar and particulate matter suspended in the fluid, beyond the effects of diffusion alone. These effects are often exacerbated by complex fluids, for example, through mixing via elastic turbulence [45]. The relative contributions to transport by advection versus diffusion are quantified by the Péclet number, Pe=ULb/D, where D is the molecular or Brownian diffusion coefficient. In the case of active matter, the relatively large size of swimming cells (1–10 μm) means that Brownian diffusion effects are small. However, by virtue of their self propulsion, swimming cells exhibit random walk motility patterns characterized by rather large effective diffusivities that scale as DeffUs2τ, where Us is the cell swimming speed and τ is the correlation time of the random walk. For example, in the case of bacteria with Us30μm/s and τ1 s, the effective diffusivity is comparable to dissolved gasses in water, Deff900μm2/s. Size effects can greatly modify both the transport of particulates and the mechanics of polymer suspensions, for instance, if the relative length scale of the flow is comparable to the solute particle dimensions. The suspension Knudsen number (Kn) represents the ratio of the polymer radius of gyration (or particle and droplet diameter in the case of colloids and emulsions) to the characteristic physical flow length scale for the solution. For the purposes of this review, we will focus on the continuum limit (Kn1) [104].

In continuum studies, the incompressible flow of polymeric fluids is modeled by considering mass and momentum conservation,

u=0,
(1)
ρ(ut+uu)=p+(τs+τp),
(2)

where ρ, u, and p are fluid density, flow field, and pressure field, respectively. The solvent stress tensor, τs, for Newtonian solvents can be calculated as τs=ηs(u+uT), where ηs is the solvent viscosity. The extra stresses (τp) in the fluid due to polymeric chains are modeled using a variety of different constitutive equations such as Upper Convected Maxwell, Oldroyd, Finitely Extensible-Nonlinear-Elastic (FENE), and Phan–Thien–Tanner (PTT) [105,106]. Briefly, an Oldroyd-B fluid is an excellent model for highly elastic, constant viscosity (i.e., Boger [107]) fluids in shear flows [108]. However, an Oldroyd-B model gives singular extensional viscosity at a strain rate (ϵ˙=1/2λ) and fails in steady extensional flows, because it considers infinite stretching of the polymeric chains, which is nonphysical. It also does not capture the shear-thinning behavior generally exhibited by viscoelastic fluids. The FENE-P constitutive equation captures both elasticity and shear thinning behaviors of viscoelastic fluids and it also considers the finite extensibility of polymeric chains [105,106,109]. Likewise, the simplified PTT (sPTT) model considers finite extensibility of polymeric chains and gives results similar to the FENE-P model for viscometric flows [110,111]. Although these models successfully predict the flow kinematics of viscoelastic fluids in different geometries [57,112,113], the quantitative comparison between experimental data and model prediction is still challenging [114,115], because these models often fail to represent polymeric fluids at high shear rates. Current viscoelastic models also fail to estimate the dissipation and pressure correctly [116,117]. The viscoelastic models listed above consider monodisperse polymeric chains, whereas the polymeric fluids found in nature and used in the industry are often polydisperse with a broad distribution of molecular weights that significantly alter viscoelastic flows dynamics [118]. Recently, for polydisperse melts of entangled linear polymers, tube models (Rolie-Double-Poly models) have been developed [119–121]. Long-chain polymers exhibit coil-stretch hysteresis in extensional flows and require a viscoelastic model, which considers conformation-dependent drag on polymer molecules to fully capture the dynamics of long-chain solutions [122,123]. Therefore, despite these efforts and progress, there is a need to use sophisticated, higher order viscoelastic models, which can more accurately represent industrial polymeric fluids. In cases where the characteristic flow length scale is smaller than the radius of gyration of long polymer chains (or colloids or emulsions) (Kn>0.01), continuum equations are not applicable and Brownian dynamics simulations or other statistical methods are required [124].

Base flow topology strongly impacts polymer deformation and consequently affects the system scale flow dynamics of complex fluids [7]. For example, polymeric chains exhibit a coil-stretch transition in extensional flows [123], but undergo end-over-end tumbling in shear flows [125]. The flow-type parameter (Λ) characterizes local fluid deformation in mixed flows and is calculated as Λ=(|D||Ω|)/(|D|+|Ω|), where D=12(u+uT) and Ω=12(uuT) are the strain rate tensor and vorticity tensor, respectively [7,126]. The flow-type parameter is a scalar in the range 1Λ1, where Λ=1, Λ=1, and Λ=0 correspond to purely rotational, purely extensional, and shear flow, respectively. The presence of the vorticity tensor (Ω) in this simple definition of Λ can be a drawback as it depends on the rotation of the coordinate system [127]. Likewise, a flow-type parameter based solely on the strain rate tensor is also problematic for some specific flows [128]. Therefore, an alternative local and objective criterion has been proposed for the classification of flow, which considers the rate of rotation with respect to the principal axes of strain rate tensor [129,130]. This alternative, but less common, metric for flow classification is given as RD=tr(W¯2)/tr(D2), where W¯=Ωω is called the relative rate of rotation tensor and ω is the rate of rotation of tensor D [129]. The range of RD varies from RD=0 for purely extensional flow to RD=1 for viscometric flow to RD= for rigid body rotation.

Cross-slot geometries, isolated constrictions, and cylinders are often used to model pore-scale mechanics and have provided important insights into the flow stability of viscoelastic fluids and the fluid dynamics inside porous media [88]. Polymeric flow through a perfectly symmetric cross-slot geometry at low Reynolds number leads to two different purely elastic instabilities due to the development of very high compressive normal stress upstream of the stagnation point [53,112,132]. After the first instability, the flow inside the cross-slot becomes asymmetric, but it remains time-independent [Figs. 1(a) and 1(b)] [53,112,133]. At large strain rates, the second elastic instability occurs and flow inside the cross-slot becomes time-dependent and chaotic in nature [53,112]. The optimized cross-slot channel (OSCER) [134,135], which generates an almost ideal planar homogeneous elongational flow field along its symmetry planes, exhibits an additional flow state characterized by time-dependent lateral motion of the stagnation point [Fig. 1(c)] [126]. Viscoelastic flow through a simple constriction yields a complex landscape of nonlinear flow behavior. Upstream of constrictions, persistent recirculating eddies form [57,136,137] to minimize the extensional stresses associated with polymer chain alignment [89,138–141]. Figure 1(d) shows the effect of constriction ratio and Weissenberg number on the structure of unstable eddies upstream of a sudden constriction [55]. Both the eddy size and asymmetry increase with increasing Wi as well as the constriction ratio. The presence of inertia suppresses the instabilities upstream of a constriction and the flow becomes more symmetric in the stream-wise direction [Fig. 1(e)] [131]. The flow of viscoelastic fluid around a confined cylinder at a small blockage ratio (BR) exhibits lateral flow asymmetry above a critical Weissenberg number [Fig. 1(f)] [56,113]. The blockage ratio (BR) is the ratio of cylinder diameter to channel width. At a large blockage ratio (BR>0.5), the flow resembles the behaviors of viscoelastic fluid flow through the constriction and upstream of cylinder unstable eddies form [Fig. 1(g)] [9,142–145]. Strong shear-thinning has a destabilizing effect on viscoelastic instabilities in these geometries [58,113,133], whereas the presence of inertia suppresses the instabilities [112,131]. Microfluidic experiments have revealed a variety of complex spatiotemporal flow features produced by these instabilities [7,9,51,54–56,146–158].

FIG. 1.

(a) Experiment [53] and (b) simulation [112] of purely elastic flow asymmetry in the cross-slot geometry. Images are reproduced from Arratia et al., Phys. Rev. Lett. 96, 144502 (2006). Copyright 2006, APS [53] and Poole et al., Phys. Rev. Lett. 99, 164503 (2007). Copyright 2007, APS [112], respectively. (c) Elastic instability in the optimized cross-slot (OSCER) for the flow of a monodisperse polymer solution (experiment). Image is reproduced from Haward et al., Sci. Rep. 6, 33029 (2016). Copyright 2016, Springer Nature [126]. (d) Effects of constriction ratio (β) and Wi on the structure of unstable eddies upstream of a sudden constriction (experiment). Images are reproduced from Lanzaro and Yuan, J. Nonnewton. Fluid Mech. 166, 1064–1075 (2011). Copyright 2011, Elsevier [55]. (e) Effects of Reynolds number (Re) and Wi on the structure of unstable eddies upstream of a sudden constriction (experiment). Images are reproduced from Rodd et al., J. Nonnewton. Fluid Mech. 129, 1–22 (2005). Copyright 2005, Elsevier [131]. (f) Asymmetric flow of polymeric fluid around a cylinder in a confined channel (experiment). Image is reproduced from Haward et al., J. Nonnewton. Fluid Mech. 278, 104250 (2020). Copyright 2020, Elsevier [56]. (g) Instability of polymeric fluid flow upstream of a cylinder in a confined channel at a large blockage ratio (experiment). Images are reproduced from Qin et al., J. Fluid Mech. 864, R2 (2019). Copyright 2019, Cambridge University Press [9].

FIG. 1.

(a) Experiment [53] and (b) simulation [112] of purely elastic flow asymmetry in the cross-slot geometry. Images are reproduced from Arratia et al., Phys. Rev. Lett. 96, 144502 (2006). Copyright 2006, APS [53] and Poole et al., Phys. Rev. Lett. 99, 164503 (2007). Copyright 2007, APS [112], respectively. (c) Elastic instability in the optimized cross-slot (OSCER) for the flow of a monodisperse polymer solution (experiment). Image is reproduced from Haward et al., Sci. Rep. 6, 33029 (2016). Copyright 2016, Springer Nature [126]. (d) Effects of constriction ratio (β) and Wi on the structure of unstable eddies upstream of a sudden constriction (experiment). Images are reproduced from Lanzaro and Yuan, J. Nonnewton. Fluid Mech. 166, 1064–1075 (2011). Copyright 2011, Elsevier [55]. (e) Effects of Reynolds number (Re) and Wi on the structure of unstable eddies upstream of a sudden constriction (experiment). Images are reproduced from Rodd et al., J. Nonnewton. Fluid Mech. 129, 1–22 (2005). Copyright 2005, Elsevier [131]. (f) Asymmetric flow of polymeric fluid around a cylinder in a confined channel (experiment). Image is reproduced from Haward et al., J. Nonnewton. Fluid Mech. 278, 104250 (2020). Copyright 2020, Elsevier [56]. (g) Instability of polymeric fluid flow upstream of a cylinder in a confined channel at a large blockage ratio (experiment). Images are reproduced from Qin et al., J. Fluid Mech. 864, R2 (2019). Copyright 2019, Cambridge University Press [9].

Close modal

Innovative experimental approaches, facilitated through microfluidics and advanced imaging capabilities, reveal the rich 3D structure of viscoelastic flow instabilities in otherwise 2D flows at the pore scale. Quasi-2D model pore geometries are inherently 3D, providing an additional degree of spatial freedom beyond the typical optical sectioning of the flow in the plane of the geometry, as captured by most particle velocimetry methods. Novel 3D printing fabrication methods and holographic particle tracking velocimentry have been used to understand the three-dimensional structure of flow during instabilities [9,103,159,160]. Vortices are well-known to form upstream of a confined cylinder [142,145], and while development of 3D flow structure has recognized importance in the development of viscoelastic instabilities, this 3D structure had not previously been quantified. Measurements of the 3D velocity field revealed that the instability stems from the formation of a corner vortex between the cylinder and bounding wall, which becomes unsteady at a critical Wi and can switch between the top and bottom walls of the channel [Fig. 2(a)] [9]. Similar holographic particle imaging techniques have been used to investigate the 3D structure of purely elastic cross-channel flows [80]. An elastic instability leads to symmetry breaking at the hyperbolic point of the flow [53], but the typical 2D view of these flows misses the true structure that underlies the pathway to elastic turbulence. A weak velocity component normal to the plane of the base flow emerges and distorts the flow separatrix with a structural complexity that increases with flow speed [Fig. 2(b)] [80]. In addition to holographic imaging, a stereoscopic micro-tomographic PIV technique has also been applied to capture the structure of a viscoelastic instability in a 3D contraction [Fig. 2(c)] [81]. Tomographic PIV measurements revealed a plateau region in the vortex growth with increasing Wi, which has not been reported in 2D contraction flows. Furthermore, the viscoelastic instability in a 3D contraction exhibits out-of-plane asymmetric jetting behavior [81]. Novel 3D printed glass microchannels have also been used to explore the effect of viscoelaticity on vortex formation in cross channels, which can occur at surprisingly small Re. Polymer additives reduce the vorticity and suppress the formation of the spiral vortex, which stems from an inertial instability [Fig. 2(d)] [103,161]. As the spatial and temporal resolution of such imaging and fabrication techniques improves, they will undoubtedly find broad applications in irregular 3D pore geometries and flow networks to better understand the role of porous microstructure in regulating the onset of viscoelastic instabilities.

FIG. 2.

3D pore-scale instabilities in model geometries obtained from experiments: (a) Vortex upstream of a confined cylinder switches between top and bottom walls. Images are reproduced from Qin et al., J. Fluid Mech. 864, R2 (2019). Copyright 2019, Cambridge University Press [9]. (b) Viscoelastic instability in a 3D cross-slot geometry exhibits flow asymmetry in the plane of extension as well as the plane normal to extension. Images are reproduced from Qin et al., Soft Matter 16, 6969–6974 (2020). Copyright 2020, Royal Society of Chemistry [80]. (c) Isosurfaces of velocity for viscoelastic flow through a 3D contraction. Images are reproduced from Carlson et al., J. Fluid Mech. 923, R6 (2021). Copyright 2021, Cambridge University Press [81]. (d) Fluid elasticity suppresses the vortex formation at a stagnation point in a 3D cross-slot channel. Images are reproduced from Burshtein et al., Phys. Rev. X 7, 041039 (2017). Copyright 2017, APS [103].

FIG. 2.

3D pore-scale instabilities in model geometries obtained from experiments: (a) Vortex upstream of a confined cylinder switches between top and bottom walls. Images are reproduced from Qin et al., J. Fluid Mech. 864, R2 (2019). Copyright 2019, Cambridge University Press [9]. (b) Viscoelastic instability in a 3D cross-slot geometry exhibits flow asymmetry in the plane of extension as well as the plane normal to extension. Images are reproduced from Qin et al., Soft Matter 16, 6969–6974 (2020). Copyright 2020, Royal Society of Chemistry [80]. (c) Isosurfaces of velocity for viscoelastic flow through a 3D contraction. Images are reproduced from Carlson et al., J. Fluid Mech. 923, R6 (2021). Copyright 2021, Cambridge University Press [81]. (d) Fluid elasticity suppresses the vortex formation at a stagnation point in a 3D cross-slot channel. Images are reproduced from Burshtein et al., Phys. Rev. X 7, 041039 (2017). Copyright 2017, APS [103].

Close modal

Porous media in industrial, biological, and geological applications exhibit a high degree of microstructural complexity, which can significantly alter viscoelastic flows. In 3D media, higher pore connectivity, disorder, and the presence of successive pores impact hydraulic resistance, stability, and transport in ways that are not captured by individual pore dynamics [40,162–170]. In flows through successive pores, polymers may be advected faster than they relax (De>1). Polymer stretching in subsequent throats thus leads to the accumulation of stress, which can produce spatial variations in the dominant pore-scale flow features [113,142–144,148,149]. Channels consisting of linear arrays of expansions-constrictions or cylinders are a common 1D model of porous media, which are used to study the hydrodynamic interactions among flows around adjacent obstacles and through neighboring pores [57,136,144,150,171,172]. For example, despite the well-known instability in the downstream wake of cylinder in a creeping elastic flow [173], the presence of a second cylinder in the streamwise direction results in two elastically driven transitions [150]. The first transition leads to the breaking of time-reversible symmetry due to a Hopf bifurcation, whereas the second transition corresponds to a forward bifurcation and leads to the breaking of mirror symmetry [150]. The order parameters involved in these transitions are velocity fluctuations and vorticity, respectively. After the first transition, a pair of small vortexes appears in the vicinity of the downstream cylinder and the size of the vortex increases with Wi [Fig. 3(a)] [150]. The numerical simulation of viscoelastic instability between two streamwise located cylinders exhibits tristability in the region between the confined cylinders, distinguished by two elastic transitions [Fig. 3(b)(i–iii)] [58]. The flow is steady, symmetric, and eddy-free before any elastic instability [Fig. 3(b)(i)]. After the first transition, the elastic wake produced behind the upstream cylinder [113] bifurcates into two symmetric branches, which encircle the region between the cylinders [Fig. 3(b)(v)]. The streaks characterized by high stress [Fig. 3(b)(v)] correspond to regions of high polymer stretch and act as a barrier for the flow crossing these regions, which leads to flow separation and eddy formation between the cylinders [Fig. 3(b)(ii)]. The second transition breaks the symmetry [Fig. 3(b)(vi)], which leads to a loss of the eddies and an asymmetric flow around the cylinders [Fig. 3(b)(iii)]. Viscoelastic flow past two spanwise located cylinders exhibits flow asymmetry around the cylinders [Fig. 3(c)] [174] similar to single confined cylinders [56]. However, multiple symmetry breaking events occur in the two cylinder case compared to a single confined cylinder due to the larger number of possible flow paths and multiple possible flow states [Fig. 3(c)]. The strongly nonlinear dynamics of viscoelastic fluids are exacerbated by the increasing geometrical complexity of these systems [144,175].

FIG. 3.

(a) Bistability of polymeric flow between two streamwise located cylinders (experiment). Images are reproduced from Varshney et al., Phys. Rev. Fluids 2, 051301 (2017). Copyright 2017, APS [150]. (b) Tristability of viscoelastic flow between two streamwise located cylinders (simulation). (i)–(iii) Streamlines of flow field and (iv)–(vi) trace of polymeric stress tensor. Images are reproduced from Kumar and Ardekani, Phys. Fluids 33, 074107 (2021). Copyright 2021, AIP Publishing [58]. (c) Multistability of viscoelastic flow around two cylinders located in spanwise direction (experiment). Images are reproduced from Hopkins et al., Phys. Rev. Lett. 126, 54501 (2021). Copyright 2021, APS [174].

FIG. 3.

(a) Bistability of polymeric flow between two streamwise located cylinders (experiment). Images are reproduced from Varshney et al., Phys. Rev. Fluids 2, 051301 (2017). Copyright 2017, APS [150]. (b) Tristability of viscoelastic flow between two streamwise located cylinders (simulation). (i)–(iii) Streamlines of flow field and (iv)–(vi) trace of polymeric stress tensor. Images are reproduced from Kumar and Ardekani, Phys. Fluids 33, 074107 (2021). Copyright 2021, AIP Publishing [58]. (c) Multistability of viscoelastic flow around two cylinders located in spanwise direction (experiment). Images are reproduced from Hopkins et al., Phys. Rev. Lett. 126, 54501 (2021). Copyright 2021, APS [174].

Close modal

In a similar vein to the wake interactions of cylinders, eddys form upstream of constrictions in viscoelastic flows. Unlike isolated constrictions, microfluidic investigations into linear arrays of pores and throats illustrate the presence of a surprising bistability, where each pore switches stochastically between two distinct flow structures: an eddy-dominated structure and an eddy-free structure [Fig. 4(a)] [136]. Importantly, the flow states of neighboring pores were shown to be strongly correlated. The emergence of eddy-dominated and eddy-free structures in such converging-diverging channels occur when polymers within the pore are in extended and coiled conformations, respectively, and illustrate the role of pore size in dictating relaxation between throats. Recent numerical simulations suggest that four distinct flow patterns (i.e., multi-stability) may exist inside the pores of converging-diverging channel arrays [Fig. 4(b)(i)] [57], including pores with: (1) eddy on both top and bottom, (2) eddy free, (3) eddy on bottom, and (4) eddy on top. The accumulation of stresses as the polymer chains cross successive pores create streaks characterized by high stress [Fig. 4(b)(ii)], which are closely coupled to the flow structures inside the pores [Fig. 4(b)(i)]. The streaks of polymeric stress correspond to eddy formation in different parts of the pore, and hence, to multiple flow states inside the pore above a critical Wi, including a new state characterized by eddies at the center of the pore [Fig. 4(b)(iii)]. The multistability inside the pores reduces pressure drop along the channel [Fig. 4(b)(iv)]. The eddies do not contribute to the net volumetric flow through the channel. Therefore, eddy free pores have a larger apparent width to allow larger net volumetric flow compared to pores with eddies, which leads to smaller pressure drop across the eddy free pores. Due to multistability, the pores inside the channel frequently become eddy free and hence reduce the pressure drop.

FIG. 4.

(a) Bistability of polymeric fluid flow inside the pores of a converging-diverging channel (experiment): (i) Eddy free pore and (ii) pore consisting eddies. Images are reproduced from Browne et al., J. Fluid Mech. 890, A122 (2020). Copyright 2020, Cambridge University Press [136]. (b) Multistability of the unstable flow of a polymeric fluid through the pores of a converging-diverging channel (simulation): (i) Streamlines of flow field. (ii) Trace of polymeric stress tensor inside the pores. (iii) Eddies only at the center of the pore. Top and bottom regions are eddy free. (iv) Time averaged pressure drop (Δp) across the channels at different Wi. Images are reproduced from Kumar et al., Phys. Rev. Fluids 6, 033304 (2021). Copyright 2021, APS [57].

FIG. 4.

(a) Bistability of polymeric fluid flow inside the pores of a converging-diverging channel (experiment): (i) Eddy free pore and (ii) pore consisting eddies. Images are reproduced from Browne et al., J. Fluid Mech. 890, A122 (2020). Copyright 2020, Cambridge University Press [136]. (b) Multistability of the unstable flow of a polymeric fluid through the pores of a converging-diverging channel (simulation): (i) Streamlines of flow field. (ii) Trace of polymeric stress tensor inside the pores. (iii) Eddies only at the center of the pore. Top and bottom regions are eddy free. (iv) Time averaged pressure drop (Δp) across the channels at different Wi. Images are reproduced from Kumar et al., Phys. Rev. Fluids 6, 033304 (2021). Copyright 2021, APS [57].

Close modal

Although 1D porous media analogs describe a number of fundamental phenomena of pore-pore interactions [176], a minimum of two spatial dimensions are necessary to capture the hydraulic network behavior and transport through these highly interconnected materials [167,177]. 2D microfluidic porous media analogs have illustrated how viscoelastic instability in porous media leads to a lane changing effect [167], whereby the preferential paths of the flow can fluctuate in time. Together with the formation of dead zones in the flow [148], these effects—observed in ordered porous media—are thought to be the origin of increased hydraulic resistance [177] in viscoelastic flows through porous media. Recent experimental studies explored the effect of ordered and disordered geometry on viscoelastic stability in 2D porous media, which showed that the introduction of disorder appears to promote stability [7]. While the flow through crystalline lattices of pillars readily becomes unstable and exhibits chaotic velocity fluctuations at Wi>Wicr1, as illustrated by other works. Progressively disordering the microstructure hampers temporal velocity fluctuations [Fig. 5(a)]. The onset of viscoelastic instability is thus delayed to higher Wi compared to ordered geometries [Fig. 5(b)]. The cause of the enhanced stability in disordered systems was identified as the shift of fluid deformation from extensional to shear dominated flow [125] through the formation of preferential flow paths. In contrast, the flow through the staggered hexagonal geometries investigated remained highly extensional due to persistent interactions with obstacles [Fig. 5(c)]. Additional experiments confirmed these observations and also demonstrated that specific orientations of the flow relative to obstacles with crystalline order can further stifle the transition to chaotic flow [59]. Specifically, the flow along lattice directions aligned with clear, preferential flow paths, where streamline interactions with hyperbolic points in the flow are minimal, promotes flow stability. Therefore, for geometries where the disorder of posts increases the stagnation points exposed to the viscoelastic flow, the disorder can promote instability [59,178]. These recent discoveries emphasize the importance of flow history on stability, which has been investigated theoretically for mixed shear-extensional flows [179]. Novel single molecule studies visualizing individual polymers (DNA) illustrate their dynamic stretching and relaxation as they flow through porous structures [148], and such approaches will be crucial to unraveling complex pore-scale and bulk-scale flow properties of viscoelastic materials in porous media. Taken together, these works open the door to creating new classes of engineered porous materials to control velocity fluctuations for polymer mixing, filtration, and extraction applications.

FIG. 5.

Viscoelastic instabilities in microfluidic 2D porous media (experiment): (a) Normalized time averaged speed field with increasing disorder (β) of cylindrical obstacles comprising porous media (left to right) at Wi=0.1,4 (i). Normalized speed fluctuations for β=0 (hexagonal lattice) for increasing Wi (ii) and increasing geometric disorder for Wi=4 (iii). (b) Normalized temporal throat speed fluctuations as a function of Weissenberg number (Wi) for a range of geometric disorders. The disorder of geometry delays the onset of viscoelastic instability. (c) Ensemble-averaged flow type along tracer trajectories over one relaxation time. The flow is shear dominated for disordered geometries, whereas flow type in ordered geometries is extension dominated. Images are reproduced from Walkama et al., Phys. Rev. Lett. 124, 164501 (2020). Copyright 2020, APS [7].

FIG. 5.

Viscoelastic instabilities in microfluidic 2D porous media (experiment): (a) Normalized time averaged speed field with increasing disorder (β) of cylindrical obstacles comprising porous media (left to right) at Wi=0.1,4 (i). Normalized speed fluctuations for β=0 (hexagonal lattice) for increasing Wi (ii) and increasing geometric disorder for Wi=4 (iii). (b) Normalized temporal throat speed fluctuations as a function of Weissenberg number (Wi) for a range of geometric disorders. The disorder of geometry delays the onset of viscoelastic instability. (c) Ensemble-averaged flow type along tracer trajectories over one relaxation time. The flow is shear dominated for disordered geometries, whereas flow type in ordered geometries is extension dominated. Images are reproduced from Walkama et al., Phys. Rev. Lett. 124, 164501 (2020). Copyright 2020, APS [7].

Close modal

The pore-scale dissimilarities of flow patterns induce differences between macroscopic flow/transport properties of 2D and 3D porous media [180–182]. Conventional, quasi-2D microfluidic devices have proven invaluable in achieving controlled laboratory experiments to model complex geometries found in natural and engineered porous materials [183]. Micro-models comprised of microfluidic packed beds of spherical beads provide an excellent physical model of flow and material transport through highly tortuous 3D porous geometries [184–187]. However, direct visualization of the fluid flows has long been challenging. Refractive index matching of the beads and fluid now enable 2D or 3D optical interrogation of the flows through transparent micro-models via defocusing particle tracking velocimetry [187] or confocal microscopy [184]. Model porous media made of glass beads has been used to study viscoelastic turbulence in 3D porous media (Fig. 6) [60]. The apparent viscosity of polymeric flow in porous media abruptly increases above a critical flow rate and the onset of increase of flow resistance coincides with the onset of flow fluctuations inside the pores [Fig. 6(b)] [39,60]. Unstable flow fluctuations induced by elastic turbulence inside the pores augment energy dissipation, which enhances flow resistance. Flow fluctuations inside the pores increase as the flow rate (i.e., Wi) increases [Figs. 6(c)6(f)]. The enhancement of apparent viscosity of viscoelastic flow through porous media has also been reported in numerical studies [162,188]. Spatial correlations of velocity and pore-space are nearly identical in the creeping flow regime (Fig. 7) [189], and it will be interesting to see if this feature also holds for viscoelastic fluids.

FIG. 6.

Viscoelastic instability in 3D porous media (experiment) [60]: (a) Transparent 3D model of porous media made of randomly packed borosilicate glass beads with an index of refraction matched fluid. (b) Apparent viscosity of polymeric flow through the porous medium increases due to elastic instability. (c) and (d) Time sequence of the flow field in a single pore at a low flow rate. (e) and (f) Time sequence of the flow field in the same pore at a high flow rate. Images are reproduced from Browne et al., Sci. Adv. 7, eabj2619 (2021). Copyright 2021, American Association for the Advancement of Science (AAAS) [60].

FIG. 6.

Viscoelastic instability in 3D porous media (experiment) [60]: (a) Transparent 3D model of porous media made of randomly packed borosilicate glass beads with an index of refraction matched fluid. (b) Apparent viscosity of polymeric flow through the porous medium increases due to elastic instability. (c) and (d) Time sequence of the flow field in a single pore at a low flow rate. (e) and (f) Time sequence of the flow field in the same pore at a high flow rate. Images are reproduced from Browne et al., Sci. Adv. 7, eabj2619 (2021). Copyright 2021, American Association for the Advancement of Science (AAAS) [60].

Close modal
FIG. 7.

Pore-scale statistics of Newtonian porous media flow (simulation) [189]: Three-dimensional periodic assemblies of (a) monodisperse hard-spheres, (b) polydisperse hard-spheres, and (c) monodisperse overlapping spheres. Spatial correlation in velocity fluctuations (Cuu) and two-point correlation function calculated on pore-space (S2) for (d) monodisperse hard-spheres, (e) polydisperse hard-spheres, and (f) monodisperse overlapping spheres. Images are reproduced from Aramideh et al., Phys. Rev. E 98, 013104 (2018). Copyright 2018, APS [189].

FIG. 7.

Pore-scale statistics of Newtonian porous media flow (simulation) [189]: Three-dimensional periodic assemblies of (a) monodisperse hard-spheres, (b) polydisperse hard-spheres, and (c) monodisperse overlapping spheres. Spatial correlation in velocity fluctuations (Cuu) and two-point correlation function calculated on pore-space (S2) for (d) monodisperse hard-spheres, (e) polydisperse hard-spheres, and (f) monodisperse overlapping spheres. Images are reproduced from Aramideh et al., Phys. Rev. E 98, 013104 (2018). Copyright 2018, APS [189].

Close modal

Spatiotemporal velocity fluctuations stemming from viscoelastic instabilities enhance the transport and mixing of particulate and multiphase flows in porous media. Despite the successful application of polymer solutions in oil recovery [2], the precise mechanism of oil recovery enhancement by viscoelastic fluids [95] has yet to be identified. It has been hypothesized that the normal stresses created between oil and displacing fluid due to the elastic properties of polymeric fluid pull the oil ganglion out of dead end pores and enhance oil recovery [190,191]. It has also been proposed that normal stresses stabilize the flow of an oil column (thread) and improve displacement efficiency [192,193]. In some experimental studies [39,40,194], elastic turbulence has been identified as the mechanism of EOR [Figs. 8(a) and 8(b)]. Velocity and pressure fluctuations induced by elastic turbulence depin trapped oil ganglia and also disrupt larger ganglia into smaller droplets, which are more easily displaced [Fig. 8(a)] [39]. The shear-thickening behavior of polymeric flows through porous media due to elastic turbulence [39,60] is also considered a mechanism of EOR [183,195,196]. Conversely, in recent experiments, it has been reported that the retention of polymers in porous media enhances oil recovery [Fig. 8(c)] [74]. The retention of polymers induces large and highly heterogeneous changes in local fluid velocities, which create viscous forces strong enough to overcome the capillary forces [74].

FIG. 8.

(a) The remaining oil ganglia in a microfluidic porous medium (experiment) after the flow of (i) weakly elastic Xanthan gum solution and (iii) highly elastic HPAM solution. Integrated differences of successive captured images from video sequences of (ii) Xanthan gum flow and (iv) HPAM solution flow. Images are reproduced from Clarke et al., Soft Matter 11, 3536–3541 (2015). Copyright 2015, Royal Society of Chemistry [39]. (b) Remaining oil saturation in pillared microfluidic porous media (experiment) for the flow of different displacing fluids as a function of the capillary number. Image is reproduced from De et al., J. Colloid Interface Sci. 510, 262–271 (2018). Copyright 2018, Elsevier [40]. (c) Confocal images of crude oil (purple) and displacing fluid (black) during polymer flooding (experiment) in a capillary packed bed porous medium after (i) initial water flow, (ii) polymer flow, and (iii) chase water flow (i.e., additional flow of displacing water after polymer flow). Images are reproduced from Parsa et al., Phys. Rev. Fluids 5, 022001 (2020). Copyright 2020, APS [74].

FIG. 8.

(a) The remaining oil ganglia in a microfluidic porous medium (experiment) after the flow of (i) weakly elastic Xanthan gum solution and (iii) highly elastic HPAM solution. Integrated differences of successive captured images from video sequences of (ii) Xanthan gum flow and (iv) HPAM solution flow. Images are reproduced from Clarke et al., Soft Matter 11, 3536–3541 (2015). Copyright 2015, Royal Society of Chemistry [39]. (b) Remaining oil saturation in pillared microfluidic porous media (experiment) for the flow of different displacing fluids as a function of the capillary number. Image is reproduced from De et al., J. Colloid Interface Sci. 510, 262–271 (2018). Copyright 2018, Elsevier [40]. (c) Confocal images of crude oil (purple) and displacing fluid (black) during polymer flooding (experiment) in a capillary packed bed porous medium after (i) initial water flow, (ii) polymer flow, and (iii) chase water flow (i.e., additional flow of displacing water after polymer flow). Images are reproduced from Parsa et al., Phys. Rev. Fluids 5, 022001 (2020). Copyright 2020, APS [74].

Close modal

The transport of submicrometer particles is central to the efficacy of EOR [197,198]. Likewise, biological processes, such as drug delivery [62,199,200], the transport of gametes and embryos in the reproductive track [91], and the trapping of inhaled dust particles in the airways of the lungs [201], also rely on the transport of particles through viscoelastic fluid in a porous substrate. Pore-scale instabilities of polymeric fluid flow in porous media affect the macroscopic transport properties. The elastic stresses in the flow of polymeric fluid through ordered porous media enhance the velocity fluctuations [7,167,202], which break the flow periodicity and enhance the transverse dispersion [Fig. 9(a)] [41]. In disordered porous media at low porosity (ϕ0.4), numerical and experimental studies have independently shown that the dispersion of small particles does not depend on the fluid rheology due to the randomness of the geometry [Fig. 9(b)] [162,168]. Conversely, an experimental study reports that polymeric stresses enhance the transverse dispersion in disordered porous media at high porosity (ϕ0.8) [Fig. 9(c)] [203]. At low porosity (high confinement), fluid is highly deformed and hydrodynamic interactions between nearby pore surfaces become increasingly important [204,205]. In 3D porous geometries, it is expected that the higher connectivity and elevated disorder affect the elastic instabilities as well as dispersion tensor. An important future direction is to expand the above mentioned studies to 3D geometries.

FIG. 9.

(a) Enhancement of transverse dispersion in an ordered microfluidic porous medium due to viscoelastic instability (experiment). Image is reproduced from Scholz et al., Europhys. Lett. 107, 54003 (2014). Copyright 2014, Europhysics Letters Association (EPLA) [41]. (b) Numerical simulations show that viscoelastic fluid rheology does not affect transverse or longitudinal dispersion in disordered porous media at porosity ϕ=0.4. Image is reproduced from Aramideh et al., J. Nonnewton. Fluid Mech. 268, 75–80 (2019). Copyright 2019, Elsevier [162]. (c) Slight enhancement of transverse dispersion observed in disordered microfluidic porous media at porosity ϕ=0.8 (experiment). Image is reproduced from Jacob et al., Phys. Rev. E 96, 022610 (2017). Copyright 2017, APS [203].

FIG. 9.

(a) Enhancement of transverse dispersion in an ordered microfluidic porous medium due to viscoelastic instability (experiment). Image is reproduced from Scholz et al., Europhys. Lett. 107, 54003 (2014). Copyright 2014, Europhysics Letters Association (EPLA) [41]. (b) Numerical simulations show that viscoelastic fluid rheology does not affect transverse or longitudinal dispersion in disordered porous media at porosity ϕ=0.4. Image is reproduced from Aramideh et al., J. Nonnewton. Fluid Mech. 268, 75–80 (2019). Copyright 2019, Elsevier [162]. (c) Slight enhancement of transverse dispersion observed in disordered microfluidic porous media at porosity ϕ=0.8 (experiment). Image is reproduced from Jacob et al., Phys. Rev. E 96, 022610 (2017). Copyright 2017, APS [203].

Close modal

The flow of colloidal particles in porous media also play a significant role in efforts to mobilize capillary trapped immiscible fluid for groundwater remediation [73,206] as well as EOR applications [207–210]. There are several mechanisms through which colloidal particles mobilize trapped immiscible fluid in porous media: (i) Chemically active colloidal particles accumulate at the interface of immiscible fluids and reduce surface tension through their surface chemical activities, which weaken capillary forces and help to mobilize trapped fluids [69,72,73]. The interfacial tension and contact angle of the aqueous phase decrease as the concentration of nano-particles in displacing fluid increases [69]. However, it does not induce additional oil recovery at low to medium permeability due to the blocking of the pore network by the retention of particles, for instance, on sandstone surfaces. (ii) The presence of inactive nanoparticles in the displacing fluid induces surface slip, which enhances flow speed (or reduces the injection pressure) in porous media [Fig. 10(a)] [211]. The enhanced flow speed leads to larger viscous stress, which can mobilize capillary trapped immiscible fluids. The characteristic slip length depends on nano-particle morphology; however, it is independent of particle concentration and shear rate [211]. (iii) The self-structuring of nanoparticles inside the wedge film between an oil droplet and the solid matrix surface creates a structural disjoining pressure, which enhances the spreading and wetting of injecting fluid on the solid surface [Fig. 10(b)] [212]. The spreading of injecting fluid increases with particle concentration. During injection, the wedge film deforms into a thin-film parallel to the solid surface at a threshold concentration, where oil is separated from the solid surface by a thin equilibrium film of injecting fluid containing nanoparticles. The enhancement of wettability has the potential to weaken the capillary force responsible for trapping oil and subsequently to mobilize trapped oil droplets. (iv) The deposition of colloidal particles on the solid surfaces of porous media also mobilizes capillary trapped oil by enhancing the viscous stresses on oil ganglia [Fig. 10(c)] [75]. In this scenario, colloidal particles reduce the permeability of porous media [213–215], which enhances the flow speed and ultimately the viscous stress on the trapped fluid [75]. The mobilization of trapped immiscible fluid increases with colloid concentration as it depends on the extent of particle deposition [216].

FIG. 10.

Mechanisms of fluid mobilization in porous media through colloidal particles: (a) The enhancement of flow speed with slip length due to slip flow induced by nanoparticles (experiment). Image is reproduced from Yu et al., Sci. Rep. 5, 8702 (2015). Copyright 2015, Springer Nature [211]. (b) Enhanced spreading of fluid due to ordering of nanoparticles in the oil-fluid-solid three-phase contact region (experiment and theory). Images are reproduced from Kondiparty et al., Langmuir 27, 3324–3335 (2011). Copyright 2011, American Chemical Society [212]. (c) Deposition of colloidal particles in porous media mobilizes immiscible fluid (experiment). (i) Residual oil saturation before and after colloidal particle injection. Image of porous medium containing trapped oil (dark regions) before (ii) and after (iii) the injection of colloidal particles. Images are reproduced from Schneider et al., Phys. Rev. Fluids 6, 014001 (2021). Copyright 2021, APS [75]. (d) Deposition profile of colloidal particles (experiment) at high (i) and low (iii) injection pressure [217]. Laterally averaged amount of deposited particles along the flow direction at high (ii) and low (iv) injection pressure at different injected pore volumes. Images are reproduced from Bizmark et al., Sci. Adv. 6, eabc2530 (2020). Copyright 2020, American Association for the Advancement of Science (AAAS) [217].

FIG. 10.

Mechanisms of fluid mobilization in porous media through colloidal particles: (a) The enhancement of flow speed with slip length due to slip flow induced by nanoparticles (experiment). Image is reproduced from Yu et al., Sci. Rep. 5, 8702 (2015). Copyright 2015, Springer Nature [211]. (b) Enhanced spreading of fluid due to ordering of nanoparticles in the oil-fluid-solid three-phase contact region (experiment and theory). Images are reproduced from Kondiparty et al., Langmuir 27, 3324–3335 (2011). Copyright 2011, American Chemical Society [212]. (c) Deposition of colloidal particles in porous media mobilizes immiscible fluid (experiment). (i) Residual oil saturation before and after colloidal particle injection. Image of porous medium containing trapped oil (dark regions) before (ii) and after (iii) the injection of colloidal particles. Images are reproduced from Schneider et al., Phys. Rev. Fluids 6, 014001 (2021). Copyright 2021, APS [75]. (d) Deposition profile of colloidal particles (experiment) at high (i) and low (iii) injection pressure [217]. Laterally averaged amount of deposited particles along the flow direction at high (ii) and low (iv) injection pressure at different injected pore volumes. Images are reproduced from Bizmark et al., Sci. Adv. 6, eabc2530 (2020). Copyright 2020, American Association for the Advancement of Science (AAAS) [217].

Close modal

There is a significant difference between the mechanisms of the enhancement of viscous stresses on oil ganglia by the retention of polymers [74] and colloidal particles [75]. In contrast to the study of colloidal deposition [75], the reduction of permeability of porous media due to the retention of polymers itself was not sufficient to displace oil ganglia in the experiment with polymers [74]. Instead, highly heterogeneous changes in local fluid velocities due to polymer deposition lead to large fluctuations in velocities, which are essential to overcome the capillary force [74]. Thus, the deposition of colloidal particles in a porous matrix is beneficial in EOR. The deposition profile of colloidal particles in porous media depends on the injection pressure [Fig. 10(d)] [217]. At a high injection pressure, deposition and erosion of colloidal particles at solid surfaces of the porous matrix occur continuously due to the large hydrodynamic stresses, which lead to an extended distribution of deposited colloidal particles throughout the domain. In contrast at a low injection pressure, the erosion of deposited particles is highly suppressed, which facilitates local deposition of particles close to the inlet. In model systems, the pore-scale distribution of deposited particles depends on the charge on the particles due to the inherent charge on the glass beads of packed bed porous media. However, the macroscopic distribution does not depend on particles’ charges and is tuned by imposed pressure [217].

The transport of active particles, especially microorganisms, through porous media is relevant to bacterial infections in bodily tissues [218–222] and sperm cell transport through the complex topology of the female reproductive tract during fertilization [223,224]. In soils, swimming cell dispersal regulates environmental bioremediation [225–227] and agriculture applications [78,228,229], as well as the deposition, formation, and growth of biofilms [21] in groundwater flows. Unlike passive colloidal particles, self propulsion significantly changes the nature of swimming cell transport in porous media due to physical interactions with fluid flows and solid surfaces. Multi-flagellated bacteria and other microbes exhibit random walk motility patterns for swimming in bulk viscous fluids. In the case of the model bacterium Escherichia coli, cells swim with roughly constant speed (Us) in straight trajectories (i.e., “runs”), which are disrupted by flagellar-induced random reorientations (i.e., “tumbles”) every τ1 s [230]. These random walks are often characterized by an effective diffusion DeffUs2τ. Solid surfaces are known to strongly impact cell motility, leading to surface attraction and scattering [231–233]. Solid obstructions can thus impede microbial runs [Fig. 11(a)] [234–239], and in porous environments, if the pore body scale is smaller than the run length Lb/Usτ<1, cell transport properties will be affected. To account for this effect, a Bosanquet model for random walks of gases in porous media was adapted for run-and-tumble bacteria [225] in relatively porous environments and has been investigated using quasi-2D microfluidic porous media models and simulations. In contrast, many microbes thrive in 3D environments having a typically low porosity. Microfluidic packed bed models allow direct 3D visualization of bacterial motility in transparent porous media [82]. Based on such experiments, a new transport mechanism for multi-flagellated bacteria has been proposed, in which cells become intermittently trapped inside the pore body [82,240]. When trapped, bacteria frequently reorient within a single pore until randomly escaping through a throat. When one or more pores are consecutively transited, cells exhibit a long duration hop, before becoming trapped again [Fig. 11(b)(i)]. Bacteria generally exhibit a faster swimming speed in hopping states with relatively slower motility in trapping states [Fig. 11(b)(ii)]. The small reorientation angles of cells in the hopping state indicate that their motion occurs along a directed path, whereas large reorientation angles are observed in trapped states and accompanied by several successive cell reorientations [Fig. 11(b)(iii)].

FIG. 11.

(a) Schematic of run (i and iv) and tumble (ii–iv) of bacteria in porous media. Images are reproduced from Licata et al., Biophys. J. 110, 247–257 (2016). Copyright 2016, Elsevier [236]. (b) Hopping and trapping mechanism of bacteria motion in porous media (experiment) [82]: (i) The trajectory of a single bacterium measured by confocal microscopy showing long correlated jumps (hopping) and stalled regions (trapping) in porous media. Instantaneous speed (ii) and reorientation angle of a cell during trapping and hopping in the porous media. Images are reproduced from Bhattacharjee et al., Nat. Commun. 10, 2075 (2019). Copyright 2019, Springer Nature [82].

FIG. 11.

(a) Schematic of run (i and iv) and tumble (ii–iv) of bacteria in porous media. Images are reproduced from Licata et al., Biophys. J. 110, 247–257 (2016). Copyright 2016, Elsevier [236]. (b) Hopping and trapping mechanism of bacteria motion in porous media (experiment) [82]: (i) The trajectory of a single bacterium measured by confocal microscopy showing long correlated jumps (hopping) and stalled regions (trapping) in porous media. Instantaneous speed (ii) and reorientation angle of a cell during trapping and hopping in the porous media. Images are reproduced from Bhattacharjee et al., Nat. Commun. 10, 2075 (2019). Copyright 2019, Springer Nature [82].

Close modal

Fluid flows through confined geometries modify cell swimming dynamics and lead to heterogeneous cell concentration and augmented dispersion. In flows through straight ducts, the trajectories of self-propelled particles are modified by local fluid advection and rotation. Velocity gradients cause spherical particles to rotate due to the local vorticity, leading to periodic orbits [15,241]. The elongated shapes of most bacteria complicate the cell trajectories caused by their rotational Jeffery orbits, which stem from extensional flow components in shear [242,243]. The coupling of cell motility and shape with flow leads to bacterial trapping and the suppression of transport in regions of high shear near channel walls [242]. In nontrivial 2D flows through porous media, where mixed velocity gradients are common, recent experiments and simulations have illustrated that the Lagrangian stretching history of the flow is a strong indicator of bacterial accumulation [244]. The reorientation of cells due to hydrodynamic gradients creates filamentous density patterns, which depend on the incident angle of flow relative to an ordered porous medium and are highly correlated with the topology of the Lagrangian stretching field [Fig. 12(a)]. Enhanced longitudinal dispersion [244] and front spreading [245] have also been reported in microfluidic experiments. These effects appear to be absent for isotropic particle shapes, where a muted dispersion is observed in simulations [246]. The landscape of these transport properties is further complicated by cell-surface interactions [246], which can lead to enhanced active particle retention [245]. A unique feature of biological micro-swimmers is that they often exhibit directed motility (“-taxis”) toward a mechanical, chemical, or optical stimulus, relative to the ambient flow. Chemotaxis of micro-swimmers across a straight channel can be stifled by strong shear [224,242]. In 3D porous media flows, chemotaxis has been shown to dramatically increase the retention of bacteria [227]. 2D microfluidic experiments indicate that this phenomenon and the associated dispersion are likely due to chemotaxis in slow flowing regions in disordered media [Fig. 12(b)] [247]. Magnetotactic bacteria—which orient along Earth’s magnetic field—also exhibit highly nonlinear transport properties in microfluidic porous media when directed to swim upstream against a flow [248]. These results have obvious implications for not only biochemical cycling, remineralization, and bioremediation [249], but also drug delivery in the human body [250]. However, the astounding complexity of transport in these system—due to the interplay between flow, pore structure, cell shape and motility, surface interactions, and -taxis effects—poses significant experimental and theoretical challenges to advance potential applications.

FIG. 12.

(a) Filamentous density patterns of cells in porous media flow (experiment) [244]: (i)–(iv) Cell densities for different flow incident angles and shear rates and (v)–(vi) Lagrangian stretching field for different incident angles of flow. Images are reproduced from Dehkharghani et al., Proc. Natl. Acad. Sci. U.S.A. 116, 11119–11124 (2019). Copyright 2019, National Academy of Sciences [244]. (b) Chemotaxis of cells in slow flow regions of disordered porous media (experiment) [247]: (i) The residence time of the invading chemical gradients, (ii) snapshot of bacterial spatial distribution during a chemorepellent injection, and (iii) snapshot of bacterial spatial distribution during a chemoattractant injection. Bacteria chemotax toward low-velocity regions during chemorepellent injection and out of them during chemoattractant injection. Images are reproduced from de Anna et al., Nat. Phys. 17, 68–73 (2021). Copyright 2021, Springer Nature [247].

FIG. 12.

(a) Filamentous density patterns of cells in porous media flow (experiment) [244]: (i)–(iv) Cell densities for different flow incident angles and shear rates and (v)–(vi) Lagrangian stretching field for different incident angles of flow. Images are reproduced from Dehkharghani et al., Proc. Natl. Acad. Sci. U.S.A. 116, 11119–11124 (2019). Copyright 2019, National Academy of Sciences [244]. (b) Chemotaxis of cells in slow flow regions of disordered porous media (experiment) [247]: (i) The residence time of the invading chemical gradients, (ii) snapshot of bacterial spatial distribution during a chemorepellent injection, and (iii) snapshot of bacterial spatial distribution during a chemoattractant injection. Bacteria chemotax toward low-velocity regions during chemorepellent injection and out of them during chemoattractant injection. Images are reproduced from de Anna et al., Nat. Phys. 17, 68–73 (2021). Copyright 2021, Springer Nature [247].

Close modal

Concentrated suspensions of self-propelled particles, including bacteria and other microorganisms, exhibit striking collective motility patterns [251] characterized by large scale correlated motion that is an order of magnitude beyond the size of individual cells [252]. Stemming from steric and hydrodyamic cell-cell interactions, such motility patterns lead to enhanced diffusive transport of suspended material [253] and novel rheological properties [94]. As the concentration increases (109 ml1), the shear viscosity of the suspension decreases [254] and can reach a superfluid-like behavior [94]. In the bulk, active fluids exhibit distinct topological structure [92,93] with turbulence-like dynamics and correlation times of around 1 s [255,256]. In contrast, geometrical confinement at spatial scales comparable to the correlation length of the collective motility (50μm) stabilizes the cell motion. Experiments have shown that under circular confinement vortical collective flow structures emerge [257] [Fig. 13(a)], whereas confinement in a parallel channel leads to rectified self-pumping of the suspension below a critical length scale [258] [Fig. 13(b)]. Cell interactions with solid boundaries were shown to regulate such behaviors [16,259]. Building up from the individual cell perspective, numerical studies of active run-and-tumble particles in 2D porous media predict an initial increase in particle transport with increasing run length before reaching a maximum and declining at larger run lengths [260]. The reduction of transport at large run lengths occurs due to the formation of clusters, which get clogged in the pore space. Activity induced self-clustering of highly active particles also causes intermittent transport, where the motion occurs with avalanche-like statistics [261]. Experiments in microfluidic lattices demonstrated that collective motility can be coupled across pores leading to large-scale pattern formation [262] [Fig. 13(c)]. In this work, communication between bacterial vortices isolated in the pores of an ordered lattice structure is regulated by the throat geometry, where the resulting statistics are reminiscent of the Ising model. Imposing an external flow upon dense bacterial suspensions in a confined channel leads to intermittent dynamics, which switches between parabolic and plug-like flow due to the collective motility of the cells [263]. Despite twenty years of innovative research, the collective dynamics of dense cell suspensions are still not fully understood. Interactions between the novel rheology of these active materials with boundaries and external flows are still in their infancy, and there exists rich potential for innovation especially in porous media, where pore geometry and long range order/disorder can impact system dynamics. The study of collective transport and cluster formation under strong confinement in the context of recently discovered hopping-and-trapping motility will be valuable for elucidating bacterial transport processes for ecological and remediation applications in porous soil and rock. Furthermore, engineered porous metamaterials hold the potential to tailor collective transport in these systems, including the possibility of biologically rooted computing and autonomous lab-on-a-chip devices [264].

FIG. 13.

(a) Vortical collective motion of micro-swimmers in circular confinement (experiment). Images are reproduced from Wioland et al., Phys. Rev. Lett. 110, 268102 (2013). Copyright 2013, APS [257]. (b) Collective motion of micro-swimmers in rectangular channels of different width (experiment). Images are reproduced from Wioland et al., New J. Phys. 18, 075002 (2016). Copyright 2016, IOP Publishing Ltd and Deutsche Physikalische Gesellschaft [258]. (c) Collective motion of micro-swimmers in microfluidic porous geometries (experiment) [262]: Vortex spin in neighboring pores for (i) anticorrelated and (iv) correlated patterns. Flow fields in pores for (ii) anticorrelated and (v) correlated patterns. Schematic of bacterial circulation in the vicinity of gap for (iii) anticorrelated and (vi) correlated patterns. Images are reproduced from Wioland et al., Nat. Phys. 12, 341–345 (2016). Copyright 2016, Springer Nature [262].

FIG. 13.

(a) Vortical collective motion of micro-swimmers in circular confinement (experiment). Images are reproduced from Wioland et al., Phys. Rev. Lett. 110, 268102 (2013). Copyright 2013, APS [257]. (b) Collective motion of micro-swimmers in rectangular channels of different width (experiment). Images are reproduced from Wioland et al., New J. Phys. 18, 075002 (2016). Copyright 2016, IOP Publishing Ltd and Deutsche Physikalische Gesellschaft [258]. (c) Collective motion of micro-swimmers in microfluidic porous geometries (experiment) [262]: Vortex spin in neighboring pores for (i) anticorrelated and (iv) correlated patterns. Flow fields in pores for (ii) anticorrelated and (v) correlated patterns. Schematic of bacterial circulation in the vicinity of gap for (iii) anticorrelated and (vi) correlated patterns. Images are reproduced from Wioland et al., Nat. Phys. 12, 341–345 (2016). Copyright 2016, Springer Nature [262].

Close modal

The study of micro-organisms in porous media is also relevant for EOR. The use of microbes for EOR is very promising and attractive due to environment compatibility, low cost, and potentially high efficiency [79,96–98]. In MEOR, suitable micro-organisms along with nutrients are injected into the oil reservoirs, where the by-products of these micro-organisms help to mobilize capillary trapped oil ganglia [265–267]. There are a large number of metabolic products produced by micro-organisms such as biosurfactants [99,267,268], biopolymers [269,270], biomass, solvents, acids, and gases, which are useful in the EOR. These products mobilize trapped oil drop through different mechanisms [79,97]. For example, the biosurfactants mobilize trapped oil ganglia by reducing the capillary forces and suppressing the viscous fingering [32,33], whereas biopolymers mobilize the trapped phase by increasing viscous stresses and inducing velocity and pressure fluctuations [31,39,74]. The transport and distribution of micro-organisms in reservoirs are expected to be different from those in simple porous media due to the presence of interfaces and surfactants [271–274] or the presence of polymers and extracellular polymeric matrix [16,275,276].

The coupling of complex rheology and particle activity to porous microstructure yields anomalous and unexpected transport phenomena in comparison to their established Newtonian and passive counterparts, respectively. While a bevy of applications—including EOR, environmental remediation, bacterial infection, agriculture, and microbial mining—exploit their novel properties, our fundamental understanding of these systems is still evolving [2,22].1 Viscoelastic flows through confined geometries induce large elastic stresses, which lead to purely elastic instabilities [57,58]. Extrapolating the dynamics and transport properties of porous media flows from observations of individual pores remains challenging, largely due to the potential for pore-pore coupling. The onset of flow instability is strongly dependent on the sample scale geometry, and the resulting velocity fluctuations cause an anomalous rise of apparent fluid resistance and enhanced dispersion [7,60]. The transverse dispersion of passive tracers in ordered porous media increases due to viscoelastic instability [41]; however, a broader consensus on the role of disorder and porosity in regulating dispersion viscoelastic flow through porous media has yet to emerge [162,203]. Sophisticated experiments have revealed the rich 3D dynamics of viscoelastic instabilities [9,80,103], but numerical simulations have been mainly restricted to 2D geometries due to the enormous computational cost. The quantitative comparison between experiments and simulations of viscoelastic flows is still challenging due to the lack of robustness of available viscoelastic models [114,115]. There exists a major need for higher order models, which can accurately represent realistic industrial and naturally occurring polymeric fluids. The polydispersity of polymeric solutions used industrial, natural, and model flows creates challenges for the direct comparison between experiments and simulations, as most simulations of viscoelastic instabilities are limited to models of monodisperse polymeric fluids. Experiments using polymeric solutions of precise molecular weight distributions will help to bridge the gap between simulations and experiments and facilitate the development of improved rheological models. The injection of colloidal suspensions has also seen successful application in EOR and groundwater remediation [206,209]. Several distinct mechanisms for the mobilization of immiscible, trapped fluids in porous media by colloidal solutions have been suggested. The extent to which these mechanism work in conjunction with one another or with polymeric fluids remains to be determined, as well as consideration for potentially harmful impacts on the environment. Swimming microorganisms are widely known to play important roles in natural processes, and harnessing their novel transport properties for engineering applications (e.g., MEOR) promises an environment-friendly, low cost, and potentially highly efficient alternative to chemical-based remediation and recovery of trapped resources [79,99]. However, our understanding of the effects of external flows, interfaces, collective motility, and microstructure geometry on the transport of microbial suspensions in porous media is still in an early stage. In particular, understanding the effects of confinement and geometry on the rheological properties of active suspensions [94] will help us to develop improved models of their transport in porous media. The sheer complexity of these systems adds to both their challenging nature and their potential for new discoveries. Despite the presence of multiple phases and components in the porous media flows [277,278] relevant in environmental and industrial applications, the studies of complex and active fluids through porous media have mainly focused on the transport of a single phase and component. A deeper fundamental understanding of the flow and transport processes in non-Newtonian and active fluids in porous media in the presence of multiple phases and components will pave the way for novel solutions to a host of environmental and industrial hurdles.

The authors acknowledge financial support from the National Science Foundation (NSF) including Grant Nos. CBET-1700961 (to A.M.A.), CAREER-1554095 (to J.S.G.), and CBET-1701392 (to J.S.G.). M.K. also acknowledges financial support from the Ross graduate fellowship from Purdue College of Engineering.

1.
Carrel
,
M.
,
V. L.
Morales
,
M. A.
Beltran
,
N.
Derlon
,
R.
Kaufmann
,
E.
Morgenroth
, and
M.
Holzner
, “
Biofilms in 3D porous media: Delineating the influence of the pore network geometry, flow and mass transfer on biofilm development
,”
Water Res.
134
,
280
291
(
2018
).
2.
Sorbie
,
K. S.
,
Polymer-Improved Oil Recovery
(
Springer Science & Business Media
,
New York
,
2013
).
3.
Smith
,
M. M.
,
J. A.
Silva
,
J.
Munakata-Marr
, and
J. E.
Mccray
, “
Compatibility of polymers and chemical oxidants for enhanced groundwater remediation
,”
Environ. Sci. Technol.
42
,
9296
9301
(
2008
).
4.
Bear
,
J.
, Introduction to Modeling of Transport Phenomena in Porous Media, Theory and Applications of Transport in Porous Media v. 4 (Kluwer Academic, Dordrecht, 1991).
5.
Das
,
M. K.
, Modeling Transport Phenomena in Porous Media with Applications, 1st ed., Mechanical Engineering Series (Springer International, Cham, 2018).
6.
Krummel
,
A. T.
,
S. S.
Datta
,
S.
Münster
, and
D. A.
Weitz
, “
Visualizing multiphase flow and trapped fluid configurations in a model three-dimensional porous medium
,”
AIChE J.
59
,
1022
1029
(
2013
).
7.
Walkama
,
D. M.
,
N.
Waisbord
, and
J. S.
Guasto
, “
Disorder suppresses chaos in viscoelastic flows
,”
Phys. Rev. Lett.
124
,
164501
(
2020
).
8.
Datta
,
S. S.
,
H.
Chiang
,
T. S.
Ramakrishnan
, and
D. A.
Weitz
, “
Spatial fluctuations of fluid velocities in flow through a three-dimensional porous medium
,”
Phys. Rev. Lett.
111
,
064501
(
2013
).
9.
Qin
,
B.
,
P. F.
Salipante
,
S. D.
Hudson
, and
P. E.
Arratia
, “
Upstream vortex and elastic wave in the viscoelastic flow around a confined cylinder
,”
J. Fluid Mech.
864
,
R2
(
2019
).
10.
Fattal
,
R.
, and
R.
Kupferman
, “
Constitutive laws for the matrix-logarithm of the conformation tensor
,”
J. Nonnewton. Fluid Mech.
123
,
281
285
(
2004
).
11.
Pimenta
,
F.
, and
M. A.
Alves
, “
Stabilization of an open-source finite-volume solver for viscoelastic fluid flows
,”
J. Nonnewton. Fluid Mech.
239
,
85
104
(
2017
).
12.
Hall-Stoodley
,
L.
, and
P.
Stoodley
, “
Evolving concepts in biofilm infections
,”
Cell. Microbiol.
11
,
1034
1043
(
2009
).
13.
Klapper
,
I.
,
C. J.
Rupp
,
R.
Cargo
,
B.
Purvedorj
, and
P.
Stoodley
, “
Viscoelastic fluid description of bacterial biofilm material properties
,”
Biotechnol. Bioeng.
80
,
289
296
(
2002
).
14.
Fabbri
,
S.
,
D.
Johnston
,
A.
Rmaile
,
B.
Gottenbos
,
M.
De Jager
,
M.
Aspiras
,
E.
Starke
,
M.
Ward
, and
P.
Stoodley
, “
Streptococcus mutans biofilm transient viscoelastic fluid behaviour during high-velocity microsprays
,”
J. Mech. Behav. Biomed. Mater.
59
,
197
206
(
2016
).
15.
Ardekani
,
A. M.
, and
E.
Gore
, “
Emergence of a limit cycle for swimming microorganisms in a vortical flow of a viscoelastic fluid
,”
Phys. Rev. E
85
,
056309
(
2012
).
16.
Karimi
,
A.
,
D.
Karig
,
A.
Kumar
, and
A. M.
Ardekani
, “
Interplay of physical mechanisms and biofilm processes: Review of microfluidic methods
,”
Lab Chip
15
,
23
42
(
2015
).
17.
Mello
,
T. P.
,
A. C.
Aor
,
D. S.
Gonçalves
,
S. H.
Seabra
,
M. H.
Branquinha
, and
A. L. S.
Santos
, “
Assessment of biofilm formation by Scedosporium apiospermum, S. aurantiacum, S. minutisporum and Lomentospora prolificans
,”
Biofouling
32
,
737
749
(
2016
).
18.
dos Santos
,
A. L. S.
,
A. C. M.
Galdino
,
T. P.
de Mello
,
L. D. S.
Ramos
,
M. H.
Branquinha
,
A. M.
Bolognese
,
J.
Columbano Neto
, and
M.
Roudbary
, “
What are the advantages of living in a community? A microbial biofilm perspective!
,”
Mem. Inst. Oswaldo Cruz
113
,
e180212
(
2018
).
19.
Koo
,
H.
,
R. N.
Allan
,
R. P.
Howlin
,
P.
Stoodley
, and
L.
Hall-Stoodley
, “
Targeting microbial biofilms: Current and prospective therapeutic strategies
,”
Nat. Rev. Microbiol.
15
,
740
755
(
2017
).
20.
Hall-Stoodley
,
L.
,
J. W.
Costerton
, and
P.
Stoodley
, “
Bacterial biofilms: From the natural environment to infectious diseases
,”
Nat. Rev. Microbiol.
2
,
95
108
(
2004
).
21.
Coyte
,
K. Z.
,
H.
Tabuteau
,
E. A.
Gaffney
,
K. R.
Foster
, and
W. M.
Durham
, “
Microbial competition in porous environments can select against rapid biofilm growth
,”
Proc. Natl. Acad. Sci. U.S.A.
114
,
E161
E170
(
2017
).
22.
Bouwer
,
E. J.
,
H. H.
Rijnaarts
,
A. B.
Cunningham
, and
R.
Gerlach
,
Biofilms in Porous Media
(
Wiley
,
New York, NY
,
2000
).
23.
Sharp
,
R.
,
P.
Stoodley
,
M.
Adgie
,
R.
Gerlach
, and
A.
Cunningham
, “
Visualization and characterization of dynamic patterns of flow, growth and activity of biofilms growing in porous media
,”
Water Sci. Technol.
52
,
85
90
(
2005
).
24.
Stoodley
,
P.
,
I.
Dodds
,
D.
De Beer
,
H. L.
Scott
, and
J. D.
Boyle
, “
Flowing biofilms as a transport mechanism for biomass through porous media under laminar and turbulent conditions in a laboratory reactor system
,”
Biofouling
21
,
161
168
(
2005
).
25.
Drescher
,
K.
,
Y.
Shen
,
B. L.
Bassler
, and
H. A.
Stone
, “
Biofilm streamers cause catastrophic disruption of flow with consequences for environmental and medical systems
,”
Proc. Natl. Acad. Sci. U.S.A.
110
,
4345
4350
(
2013
).
26.
Mackay
,
D. M.
,
P. V.
Roberts
, and
J. A.
Cherry
, “
Transport of organic contaminants in groundwater
,”
Environ. Sci. Technol.
19
,
384
392
(
1985
).
27.
Kokal
,
S.
, and
A.
Al-Kaabi
, “
Enhanced oil recovery: Challenges and opportunities
,”
Global Energy Solutions
64
,
64
69
(
2010
).
28.
Naik
,
P.
,
P.
Pandita
,
S.
Aramideh
,
I.
Bilionis
, and
A. M.
Ardekani
, “
Bayesian model calibration and optimization of surfactant-polymer flooding
,”
Comput. Geosci.
23
,
981
996
(
2019
).
29.
Naik
,
P.
,
S.
Aramideh
, and
A. M.
Ardekani
, “
History matching of surfactant-polymer flooding using polynomial chaos expansion
,”
J. Pet. Sci. Eng.
173
,
1438
1452
(
2019
).
30.
Aramideh
,
S.
,
R.
Borgohain
,
P. K.
Naik
,
C. T.
Johnston
,
P. P.
Vlachos
, and
A. M.
Ardekani
, “
Multi-objective history matching of surfactant-polymer flooding
,”
Fuel
228
,
418
428
(
2018
).
31.
Delamaide
,
E.
,
A.
Zaitoun
,
G.
Renard
, and
R.
Tabary
, “
Pelican lake field: First successful application of polymer flooding in a heavy-oil reservoir
,”
SPE Reservoir Eval. Eng.
17
,
340
354
(
2014
).
32.
Sheng
,
J. J.
, “
A comprehensive review of alkaline-surfactant-polymer (ASP) flooding
,”
Asia-Pac. J. Chem. Eng.
9
,
471
489
(
2014
).
33.
Aramideh
,
S.
,
P. P.
Vlachos
, and
A. M.
Ardekani
, “
Unstable displacement of non-aqueous phase liquids with surfactant and polymer
,”
Transp. Porous Media
126
,
455
474
(
2019
).
34.
Howe
,
A. M.
,
A.
Clarke
, and
D.
Giernalczyk
, “
Flow of concentrated viscoelastic polymer solutions in porous media: Effect of MW and concentration on elastic turbulence onset in various geometries
,”
Soft Matter
11
,
6419
6431
(
2015
).
35.
Mitchell
,
J.
,
K.
Lyons
,
A. M.
Howe
, and
A.
Clarke
, “
Viscoelastic polymer flows and elastic turbulence in three-dimensional porous structures
,”
Soft Matter
12
,
460
468
(
2016
).
36.
Clarke
,
A.
,
A. M.
Howe
,
J.
Mitchell
,
J.
Staniland
, and
L. A.
Hawkes
, “
How viscoelastic-polymer flooding enhances displacement efficiency
,”
SPE J.
21
,
0675
0687
(
2016
).
37.
Datta
,
S. S.
,
J. B.
Dupin
, and
D. A.
Weitz
, “
Fluid breakup during simultaneous two-phase flow through a three-dimensional porous medium
,”
Phys. Fluids
26
,
062004
(
2014
).
38.
Skauge
,
A.
,
N.
Zamani
,
J.
Gausdal Jacobsen
,
B. S.
Shiram
,
B.
Al-Shakry
, and
T.
Skauge
, “
Polymer flow in porous media: Relevance to enhanced oil recovery
,”
Colloids Interfaces
2
(3),
27
(
2018
).
39.
Clarke
,
A.
,
A. M.
Howe
,
J.
Mitchell
,
J.
Staniland
,
L.
Hawkes
, and
K.
Leeper
, “
Mechanism of anomalously increased oil displacement with aqueous viscoelastic polymer solutions
,”
Soft Matter
11
,
3536
3541
(
2015
).
40.
De
,
S.
,
P.
Krishnan
,
J.
van der Schaaf
,
J.
Kuipers
,
E.
Peters
, and
J.
Padding
, “
Viscoelastic effects on residual oil distribution in flows through pillared microchannels
,”
J. Colloid Interface Sci.
510
,
262
271
(
2018
).
41.
Scholz
,
C.
,
F.
Wirner
,
J. R.
Gomez-Solano
, and
C.
Bechinger
, “
Enhanced dispersion by elastic turbulence in porous media
,”
Europhys. Lett.
107
,
54003
(
2014
).
42.
Groisman
,
A.
, and
V.
Steinberg
, “
Efficient mixing at low Reynolds numbers using polymer additives
,”
Nature
410
,
905
908
(
2001
).
43.
Weissenberg
,
K.
, “
A continuum theory of rheological phenomena
,”
Nature
159
,
310
311
(
1947
).
44.
Groisman
,
A.
, and
V.
Steinberg
, “
Elastic turbulence in a polymer solution flow
,”
Nature
405
,
53
55
(
2000
).
45.
Groisman
,
A.
, and
V.
Steinberg
, “
Efficient mixing at low Reynolds numbers using polymer additives
,”
Nature
410
,
905
908
(
2001
).
46.
Groisman
,
A.
, and
V.
Steinberg
, “
Elastic turbulence in curvilinear flows of polymer solutions
,”
New J. Phys.
6
,
29
(
2004
).
47.
Avgousti
,
M.
, and
A. N.
Beris
, “
Non-axisymmetric modes in viscoelastic Taylor-Couette flow
,”
J. Nonnewton. Fluid Mech.
50
,
225
251
(
1993
).
48.
Sureshkumar
,
R.
,
A. N.
Beris
, and
M.
Avgousti
, “
Non-axisymmetric subcritical bifurcations in viscoelastic Taylor-Couette flow
,”
Proc. R. Soc. London, Ser. A
447
,
135
153
(
1994
).
49.
Morozov
,
A. N.
, and
W.
van Saarloos
, “
An introductory essay on subcritical instabilities and the transition to turbulence in visco-elastic parallel shear flows
,”
Phys. Rep.
447
,
112
143
(
2007
).
50.
Bonn
,
D.
,
M. M.
Denn
,
L.
Berthier
,
T.
Divoux
, and
S.
Manneville
, “
Yield stress materials in soft condensed matter
,”
Rev. Mod. Phys.
89
,
035005
(
2017
).
51.
Pan
,
L.
,
A.
Morozov
,
C.
Wagner
, and
P.
Arratia
, “
Nonlinear elastic instability in channel flows at low Reynolds numbers
,”
Phys. Rev. Lett.
110
,
174502
(
2013
).
52.
Pakdel
,
P.
, and
G. H.
McKinley
, “
Elastic instability and curved streamlines
,”
Phys. Rev. Lett.
77
,
2459
2462
(
1996
).
53.
Arratia
,
P. E.
,
C. C.
Thomas
,
J.
Diorio
, and
J. P.
Gollub
, “
Elastic instabilities of polymer solutions in cross-channel flow
,”
Phys. Rev. Lett.
96
,
144502
(
2006
).
54.
Zilz
,
J.
,
R.
Poole
,
M.
Alves
,
D.
Bartolo
,
B.
Levaché
, and
A.
Lindner
, “
Geometric scaling of a purely elastic flow instability in serpentine channels
,”
J. Fluid Mech.
712
,
203
218
(
2012
).
55.
Lanzaro
,
A.
, and
X. F.
Yuan
, “
Effects of contraction ratio on non-linear dynamics of semi-dilute, highly polydisperse PAAm solutions in microfluidics
,”
J. Nonnewton. Fluid Mech.
166
,
1064
1075
(
2011
).
56.
Haward
,
S. J.
,
C. C.
Hopkins
, and
A. Q.
Shen
, “
Asymmetric flow of polymer solutions around microfluidic cylinders: Interaction between shear-thinning and viscoelasticity
,”
J. Nonnewton. Fluid Mech.
278
,
104250
(
2020
).
57.
Kumar
,
M.
,
S.
Aramideh
,
C. A.
Browne
,
S. S.
Datta
, and
A. M.
Ardekani
, “
Numerical investigation of multistability in the unstable flow of a polymer solution through porous media
,”
Phys. Rev. Fluids
6
,
033304
(
2021
).
58.
Kumar
,
M.
, and
A. M.
Ardekani
, “
Elastic instabilities between two cylinders confined in a channel
,”
Phys. Fluids
33
,
074107
(
2021
).
59.
Haward
,
S. J.
,
C. C.
Hopkins
, and
A. Q.
Shen
, “
Stagnation points control chaotic fluctuations in viscoelastic porous media flow
,”
Proc. Natl. Acad. Sci. U.S.A.
118
,
e2111651118
(
2021
).
60.
Browne
,
C. A.
, and
S. S.
Datta
, “
Elastic turbulence generates anomalous flow resistance in porous media
,”
Sci. Adv.
7
,
eabj2619
(
2021
).
61.
Jain
,
R. K.
, “
Delivery of novel therapeutic agents in tumors: Physiological barriers and strategies
,”
JNCI, J. Natl. Cancer Inst.
81
,
570
576
(
1990
).
62.
Rahimi
,
E.
,
S.
Aramideh
,
D.
Han
,
H.
Gomez
, and
A. M.
Ardekani
, “
Transport and lymphatic uptake of monoclonal antibodies after subcutaneous injection
,”
Microvasc. Res.
139
,
104228
(
2021
).
63.
Alazraki
,
N. P.
,
D.
Eshima
,
L. A.
Eshima
,
S. C.
Herda
,
D. R.
Murray
,
J. P.
Vansant
, and
A. T.
Taylor
, “
Lymphoscintigraphy, the sentinel node concept, and the intraoperative gamma probe in melanoma, breast cancer, and other potential cancers
,”
Semin. Nucl. Med.
27
,
55
67
(
1997
).
64.
Ersahin
,
M. E.
,
H.
Ozgun
,
R. K.
Dereli
,
I.
Ozturk
,
K.
Roest
, and
J. B.
van Lier
, “
A review on dynamic membrane filtration: Materials, applications and future perspectives
,”
Bioresour. Technol.
122
,
196
206
(
2012
).
65.
Molnar
,
I. L.
,
E.
Pensini
,
M. A.
Asad
,
C. A.
Mitchell
,
L. C.
Nitsche
,
L. J.
Pyrak-Nolte
,
G. L.
Miño
, and
M. M.
Krol
, “
Colloid transport in porous media: A review of classical mechanisms and emerging topics
,”
Transp. Porous Media
130
,
129
156
(
2019
).
66.
Song
,
W.
,
H.
Joshi
,
R.
Chowdhury
,
J. S.
Najem
,
Y.-X.
Shen
,
C.
Lang
,
C. B.
Henderson
,
Y.-M.
Tu
,
M.
Farell
,
M. E.
Pitz
,
C. D.
Maranas
,
P. S.
Cremer
,
R. J.
Hickey
,
S. A.
Sarles
,
J.-L.
Hou
,
A.
Aksimentiev
, and
M.
Kumar
, “
Artificial water channels enable fast and selective water permeation through water-wire networks
,”
Nat. Nanotechnol.
15
,
73
79
(
2020
).
67.
Phenrat
,
T.
,
N.
Saleh
,
K.
Sirk
,
R. D.
Tilton
, and
G. V.
Lowry
, “
Aggregation and sedimentation of aqueous nanoscale zerovalent iron dispersions
,”
Environ. Sci. Technol.
41
,
284
290
(
2007
).
68.
Phenrat
,
T.
,
F.
Fagerlund
,
T.
Illangasekare
,
G. V.
Lowry
, and
R. D.
Tilton
, “
Polymer-modified Fe0 nanoparticles target entrapped NAPL in two dimensional porous media: Effect of particle concentration, NAPL saturation, and injection strategy
,”
Environ. Sci. Technol.
45
,
6102
6109
(
2011
).
69.
Hendraningrat
,
L.
,
S.
Li
, and
O.
Torsæter
, “
A coreflood investigation of nanofluid enhanced oil recovery
,”
J. Pet. Sci. Eng.
111
,
128
138
(
2013
).
70.
Hussain
,
F.
,
A.
Zeinijahromi
,
P.
Bedrikovetsky
,
A.
Badalyan
,
T.
Carageorgos
, and
Y.
Cinar
, “
An experimental study of improved oil recovery through fines-assisted waterflooding
,”
J. Pet. Sci. Eng.
109
,
187
197
(
2013
).
71.
Zhang
,
H.
,
A.
Nikolov
, and
D.
Wasan
, “
Enhanced oil recovery (EOR) using nanoparticle dispersions: Underlying mechanism and imbibition experiments
,”
Energy Fuels
28
,
3002
3009
(
2014
).
72.
Franzetti
,
A.
,
P.
Caredda
,
C.
Ruggeri
,
P. L.
Colla
,
E.
Tamburini
,
M.
Papacchini
, and
G.
Bestetti
, “
Potential applications of surface active compounds by Gordonia sp. strain BS29 in soil remediation technologies
,”
Chemosphere
75
,
801
807
(
2009
).
73.
Roy
,
S. B.
, and
D. A.
Dzombak
, “
Chemical factors influencing colloid-facilitated transport of contaminants in porous media
,”
Environ. Sci. Technol.
31
,
656
664
(
1997
).
74.
Parsa
,
S.
,
E.
Santanach-Carreras
,
L.
Xiao
, and
D. A.
Weitz
, “
Origin of anomalous polymer-induced fluid displacement in porous media
,”
Phys. Rev. Fluids
5
,
022001
(
2020
).
75.
Schneider
,
J.
,
R. D.
Priestley
, and
S. S.
Datta
, “
Using colloidal deposition to mobilize immiscible fluids from porous media
,”
Phys. Rev. Fluids
6
,
014001
(
2021
).
76.
Fontes
,
D. E.
,
A. L.
Mills
,
G. M.
Hornberger
, and
J. S.
Herman
, “
Physical and chemical factors influencing transport of microorganisms through porous media
,”
Appl. Environ. Microbiol.
57
,
2473
2481
(
1991
).
77.
Yavuz Corapcioglu
,
M.
, and
A.
Haridas
, “
Transport and fate of microorganisms in porous media: A theoretical investigation
,”
J. Hydrol.
72
,
149
169
(
1984
).
78.
Watt
,
M.
,
J. A.
Kirkegaard
, and
J. B.
Passioura
, “
Rhizosphere biology and crop productivity—A review
,”
Soil Res.
44
,
299
317
(
2006
).
79.
Nikolova
,
C.
, and
T.
Gutierrez
, “
Use of microorganisms in the recovery of oil from recalcitrant oil reservoirs: Current state of knowledge, technological advances and future perspectives
,”
Front. Microbiol.
10
,
2996
(
2020
).
80.
Qin
,
B.
,
R.
Ran
,
P. F.
Salipante
,
S. D.
Hudson
, and
P. E.
Arratia
, “
Three-dimensional structures and symmetry breaking in viscoelastic cross-channel flow
,”
Soft Matter
16
,
6969
6974
(
2020
).
81.
Carlson
,
D. W.
,
A. Q.
Shen
, and
S. J.
Haward
, “
Microtomographic particle image velocimetry measurements of viscoelastic instabilities in a three-dimensional microcontraction
,”
J. Fluid Mech.
923
,
R6
(
2021
).
82.
Bhattacharjee
,
T.
, and
S. S.
Datta
, “
Bacterial hopping and trapping in porous media
,”
Nat. Commun.
10
,
2075
(
2019
).
83.
Denn
,
M. M.
, “
Issues in viscoelastic fluid mechanics
,”
Annu. Rev. Fluid Mech.
22
,
13
32
(
1990
).
84.
Datta
,
S. S.
,
A. M.
Ardekani
,
P. E.
Arratia
,
A. N.
Beris
,
I.
Bischofberger
,
J. G.
Eggers
,
J. E.
López-Aguilar
,
S. M.
Fielding
,
A.
Frishman
,
M. D.
Graham
,
J. S.
Guasto
,
S. J.
Haward
,
S.
Hormozi
,
G. H.
McKinley
,
R. J.
Poole
,
A.
Morozov
,
V.
Shankar
,
E. S. G.
Shaqfeh
,
A. Q.
Shen
,
H.
Stark
,
V.
Steinberg
,
G.
Subramanian
, and
H. A.
Stone
, “Perspectives on viscoelastic flow instabilities and elastic turbulence,” arXiv:2108.09841 (2021).
85.
Coussot
,
P.
, “
Yield stress fluid flows: A review of experimental data
,”
J. Nonnewton. Fluid Mech.
211
,
31
49
(
2014
).
86.
Morris
,
J. F.
, “
Shear thickening of concentrated suspensions: Recent developments and relation to other phenomena
,”
Annu. Rev. Fluid Mech.
52
,
121
144
(
2020
).
87.
Alves
,
M.
,
P.
Oliveira
, and
F.
Pinho
, “
Numerical methods for viscoelastic fluid flows
,”
Annu. Rev. Fluid Mech.
53
,
509
541
(
2021
).
88.
Browne
,
C. A.
,
A.
Shih
, and
S. S.
Datta
, “
Pore-scale flow characterization of polymer solutions in microfluidic porous media
,”
Small
16
,
1903944
(
2020
).
89.
Boger
,
D. V.
, “
Viscoelastic flows through contractions
,”
Annu. Rev. Fluid Mech.
19
,
157
182
(
1987
).
90.
Steinberg
,
V.
, “
Elastic turbulence: An experimental view on inertialess random flow
,”
Annu. Rev. Fluid Mech.
53
,
27
58
(
2021
).
91.
Fauci
,
L. J.
, and
R.
Dillon
, “
Biofluidmechanics of reproduction
,”
Annu. Rev. Fluid Mech.
38
,
371
394
(
2006
).
92.
Marchetti
,
M. C.
,
J. F.
Joanny
,
S.
Ramaswamy
,
T. B.
Liverpool
,
J.
Prost
,
M.
Rao
, and
R. A.
Simha
, “
Hydrodynamics of soft active matter
,”
Rev. Mod. Phys.
85
,
1143
1189
(
2013
).
93.
Bechinger
,
C.
,
R.
Di Leonardo
,
H.
Löwen
,
C.
Reichhardt
,
G.
Volpe
, and
G.
Volpe
, “
Active particles in complex and crowded environments
,”
Rev. Mod. Phys.
88
,
045006
(
2016
).
94.
Saintillan
,
D.
, “
Rheology of active fluids
,”
Annu. Rev. Fluid Mech.
50
,
563
592
(
2018
).
95.
Wei
,
B.
,
L.
Romero-Zerón
, and
D.
Rodrigue
, “
Oil displacement mechanisms of viscoelastic polymers in enhanced oil recovery (EOR): A review
,”
J. Pet. Explor. Prod. Technol.
4
,
113
121
(
2014
).
96.
Lazar
,
I.
,
I. G.
Petrisor
, and
T. F.
Yen
, “
Microbial enhanced oil recovery (MEOR)
,”
Pet. Sci. Technol.
25
,
1353
1366
(
2007
).
97.
Sen
,
R.
, “
Biotechnology in petroleum recovery: The microbial EOR
,”
Prog. Energy Combust. Sci.
34
,
714
724
(
2008
).
98.
Shibulal
,
B.
,
S. N.
Al-Bahry
,
Y. M.
Al-Wahaibi
,
A. E.
Elshafie
,
A. S.
Al-Bemani
, and
S. J.
Joshi
, “
Microbial enhanced heavy oil recovery by the aid of inhabitant spore-forming bacteria: An insight review
,”
The Scientific World Journal
2014
,
1
12
(
2014
).
99.
Banat
,
I.
, “
Biosurfactants production and possible uses in microbial enhanced oil recovery and oil pollution remediation: A review
,”
Bioresour. Technol.
51
,
1
12
(
1995
).
100.
Blunt
,
M. J.
,
B.
Bijeljic
,
H.
Dong
,
O.
Gharbi
,
S.
Iglauer
,
P.
Mostaghimi
,
A.
Paluszny
, and
C.
Pentland
, “
Pore-scale imaging and modelling
,”
Adv. Water Resour.
51
,
197
216
(
2013
).
101.
Muggeridge
,
A.
,
A.
Cockin
,
K.
Webb
,
H.
Frampton
,
I.
Collins
,
T.
Moulds
, and
P.
Salino
, “
Recovery rates, enhanced oil recovery and technological limits
,”
Philos. Trans. R. Soc., A
372
,
20120320
(
2014
).
102.
Samanta
,
D.
,
Y.
Dubief
,
M.
Holzner
,
C.
Schafer
,
A. N.
Morozov
,
C.
Wagner
, and
B.
Hof
, “
Elasto-inertial turbulence
,”
Proc. Natl. Acad. Sci. U.S.A.
110
,
10557
10562
(
2013
).
103.
Burshtein
,
N.
,
K.
Zografos
,
A. Q.
Shen
,
R. J.
Poole
, and
S. J.
Haward
, “
Inertioelastic flow instability at a stagnation point
,”
Phys. Rev. X
7
,
041039
(
2017
).
104.
Li
,
G.
,
E.
Lauga
, and
A. M.
Ardekani
, “
Microswimming in viscoelastic fluids
,”
J. Nonnewton. Fluid Mech.
297
,
104655
(
2021
).
105.
Bird
,
R.
,
R.
Armstrong
, and
O.
Hassager
,
Dynamics of Polymeric Liquids. Vol. 1: Fluid Mechanics
, 2nd ed. (
John Wiley and Sons Inc.
,
New York
,
1987
).
106.
Bird
,
R. B.
,
C. F.
Curtiss
,
R. C.
Armstrong
, and
O.
Hassager
,
Dynamics of Polymeric Liquids, Vol. 2: Kinetic Theory
, 2nd ed. (
Wiley
,
New York
,
1987
).
107.
Boger
,
D.
, “
A highly elastic constant-viscosity fluid
,”
J. Nonnewton. Fluid Mech.
3
,
87
91
(
1977
).
108.
Oldroyd
,
J. G.
, “
On the formulation of rheological equations of state
,”
Proc. R. Soc. London, Ser. A
200
,
523
541
(
1950
).
109.
Bird
,
R. B.
,
P. J.
Dotson
, and
N. L.
Johnson
, “
Polymer solution rheology based on a finitely extensible bead-spring chain model
,”
J. Nonnewton. Fluid Mech.
7
,
213
235
(
1980
).
110.
Poole
,
R.
,
M.
Davoodi
, and
K.
Zografos
, “
On the similarities between the simplified Phan-Thien Tanner (sPTT) and FENE-P models
,”
Br. Soc. Rheol. Rheol. Bull.
60
,
29
34
(
2019
).
111.
Davoodi
,
M.
,
A. F.
Domingues
, and
R. J.
Poole
, “
Control of a purely elastic symmetry-breaking flow instability in cross-slot geometries
,”
J. Fluid Mech.
881
,
1123
1157
(
2019
).
112.
Poole
,
R. J.
,
M. A.
Alves
, and
P. J.
Oliveira
, “
Purely elastic flow asymmetries
,”
Phys. Rev. Lett.
99
,
164503
(
2007
).
113.
Varchanis
,
S.
,
C. C.
Hopkins
,
A. Q.
Shen
,
J.
Tsamopoulos
, and
S. J.
Haward
, “
Asymmetric flows of complex fluids past confined cylinders: A comprehensive numerical study with experimental validation
,”
Phys. Fluids
32
,
053103
(
2020
).
114.
McKinley
,
G. H.
, “Steady and transient motion of spherical particles in viscoelastic liquids,” in Transport Processes in Bubbles, Drops and Particles (Taylor and Francis, London, 2001), p. 338.
115.
Wagner
,
C.
,
Y.
Amarouchene
,
D.
Bonn
, and
J.
Eggers
, “
Droplet detachment and satellite bead formation in viscoelastic fluids
,”
Phys. Rev. Lett.
95
,
164504
(
2005
).
116.
Rothstein
,
J. P.
, and
G. H.
McKinley
, “
Extensional flow of a polystyrene Boger fluid through a 4:1:4 axisymmetric contraction/expansion
,”
J. Nonnewton. Fluid Mech.
86
,
61
88
(
1999
).
117.
Rothstein
,
J. P.
, and
G. H.
McKinley
, “
The axisymmetric contraction–expansion: The role of extensional rheology on vortex growth dynamics and the enhanced pressure drop
,”
J. Nonnewton. Fluid Mech.
98
,
33
63
(
2001
).
118.
Hassell
,
D.
,
J.
Embery
,
T.
McLeish
, and
M.
Mackley
, “
An experimental evaluation of the formation of an instability in monodisperse and polydisperse polystyrenes
,”
J. Nonnewton. Fluid Mech.
157
,
1
14
(
2009
).
119.
Likhtman
,
A. E.
, and
R. S.
Graham
, “
Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation
,”
J. Nonnewton. Fluid Mech.
114
,
1
12
(
2003
).
120.
Read
,
D. J.
,
K.
Jagannathan
,
S. K.
Sukumaran
, and
D.
Auhl
, “
A full-chain constitutive model for bidisperse blends of linear polymers
,”
J. Rheol.
56
,
823
873
(
2012
).
121.
Boudara
,
V. A. H.
,
J. D.
Peterson
,
L. G.
Leal
, and
D. J.
Read
, “
Nonlinear rheology of polydisperse blends of entangled linear polymers: Rolie-Double-Poly models
,”
J. Rheol.
63
,
71
91
(
2019
).
122.
Fuller
,
G.
, and
L.
Leal
, “
The effects of conformation-dependent friction and internal viscosity on the dynamics of the nonlinear dumbbell model for a dilute polymer solution
,”
J. Nonnewton. Fluid Mech.
8
,
271
310
(
1981
).
123.
Hsieh
,
C.-C.
, and
R. G.
Larson
, “
Prediction of coil-stretch hysteresis for dilute polystyrene molecules in extensional flow
,”
J. Rheol.
49
,
1081
1089
(
2005
).
124.
Zhang
,
Y.
,
G.
Li
, and
A. M.
Ardekani
, “
Reduced viscosity for flagella moving in a solution of long polymer chains
,”
Phys. Rev. Fluids
3
,
023101
(
2018
).
125.
Smith
,
D. E.
, “
Single-polymer dynamics in steady shear flow
,”
Science
283
,
1724
1727
(
1999
).
126.
Haward
,
S. J.
,
G. H.
Mckinley
, and
A. Q.
Shen
, “
Elastic instabilities in planar elongational flow of monodisperse polymer solutions
,”
Sci. Rep.
6
,
33029
(
2016
).
127.
Huilgol
,
R. R.
, “
On a characterization of simple extensional flows
,”
Rheol. Acta
14
,
48
50
(
1975
).
128.
Astarita
,
G.
, “
Quasi–Newtonian constitutive equations exhibiting flow-type sensitivity
,”
J. Rheol.
35
,
687
689
(
1991
).
129.
Astarita
,
G.
, “
Objective and generally applicable criteria for flow classification
,”
J. Nonnewton. Fluid Mech.
6
,
69
76
(
1979
).
130.
Thompson
,
R. L.
, and
P. R.
Souza Mendes
, “
Persistence of straining and flow classification
,”
Int. J. Eng. Sci.
43
,
79
105
(
2005
).
131.
Rodd
,
L. E.
,
T. P.
Scott
,
D. V.
Boger
,
J. J.
Cooper-White
, and
G. H.
McKinley
, “
The inertio-elastic planar entry flow of low-viscosity elastic fluids in micro-fabricated geometries
,”
J. Nonnewton. Fluid Mech.
129
,
1
22
(
2005
).
132.
Sousa
,
P. C.
,
F. T.
Pinho
,
M. S. N.
Oliveira
, and
M. A.
Alves
, “
Purely elastic flow instabilities in microscale cross-slot devices
,”
Soft Matter
11
,
8856
8862
(
2015
).
133.
Rocha
,
G. N.
,
R. J.
Poole
,
M. A.
Alves
, and
P. J.
Oliveira
, “
On extensibility effects in the cross-slot flow bifurcation
,”
J. Nonnewton. Fluid Mech.
156
,
58
69
(
2009
).
134.
Haward
,
S. J.
,
M. S. N.
Oliveira
,
M. A.
Alves
, and
G. H.
McKinley
, “
Optimized cross-slot flow geometry for microfluidic extensional rheometry
,”
Phys. Rev. Lett.
109
,
128301
(
2012
).
135.
Haward
,
S. J.
, and
G. H.
McKinley
, “
Instabilities in stagnation point flows of polymer solutions
,”
Phys. Fluids
25
,
083104
(
2013
).
136.
Browne
,
C. A.
,
A.
Shih
, and
S. S.
Datta
, “
Bistability in the unstable flow of polymer solutions through pore constriction arrays
,”
J. Fluid Mech.
890
,
A122
(
2020
).
137.
Ekanem
,
E. M.
,
S.
Berg
,
S.
De
,
A.
Fadili
,
T.
Bultreys
,
M.
Rücker
,
J.
Southwick
,
J.
Crawshaw
, and
P. F.
Luckham
, “
Signature of elastic turbulence of viscoelastic fluid flow in a single pore throat
,”
Phys. Rev. E
101
,
42605
(
2020
).
138.
Batchelor
,
G.
, “
The stress generated in a non-dilute suspension of elongated particles by pure straining motion
,”
J. Fluid Mech.
46
,
813
829
(
1971
).
139.
Mongruel
,
A.
, and
M.
Cloitre
, “
Extensional flow of semidilute suspensions of rod-like particles through an orifice
,”
Phys. Fluids
7
,
2546
2552
(
1995
).
140.
Mongruel
,
A.
, and
M.
Cloitre
, “
Axisymmetric orifice flow for measuring the elongational viscosity of semi-rigid polymer solutions
,”
J. Nonnewton. Fluid Mech.
110
,
27
43
(
2003
).
141.
Rodd
,
L.
,
J.
Cooper-White
,
D.
Boger
, and
G.
McKinley
, “
Role of the elasticity number in the entry flow of dilute polymer solutions in micro-fabricated contraction geometries
,”
J. Nonnewton. Fluid Mech.
143
,
170
191
(
2007
).
142.
Kenney
,
S.
,
K.
Poper
,
G.
Chapagain
, and
G. F.
Christopher
, “
Large Deborah number flows around confined microfluidic cylinders
,”
Rheol. Acta
52
,
485
497
(
2013
).
143.
Shi
,
X.
,
S.
Kenney
,
G.
Chapagain
, and
G. F.
Christopher
, “
Mechanisms of onset for moderate mach number instabilities of viscoelastic flows around confined cylinders
,”
Rheol. Acta
54
,
805
815
(
2015
).
144.
Shi
,
X.
, and
G. F.
Christopher
, “
Growth of viscoelastic instabilities around linear cylinder arrays
,”
Phys. Fluids
28
,
124102
(
2016
).
145.
Zhao
,
Y.
,
A. Q.
Shen
, and
S. J.
Haward
, “
Flow of wormlike micellar solutions around confined microfluidic cylinders
,”
Soft Matter
12
,
8666
8681
(
2016
).
146.
Qin
,
B.
,
P. F.
Salipante
,
S. D.
Hudson
, and
P. E.
Arratia
, “
Flow resistance and structures in viscoelastic channel flows at low re
,”
Phys. Rev. Lett.
123
,
194501
(
2019
).
147.
Qin
,
B.
, and
P. E.
Arratia
, “
Characterizing elastic turbulence in channel flows at low Reynolds number
,”
Phys. Rev. Fluids
2
,
083302
(
2017
).
148.
Kawale
,
D.
,
E.
Marques
,
P. L.
Zitha
,
M. T.
Kreutzer
,
W. R.
Rossen
, and
P. E.
Boukany
, “
Elastic instabilities during the flow of hydrolyzed polyacrylamide solution in porous media: Effect of pore-shape and salt
,”
Soft Matter
13
,
765
775
(
2017
).
149.
Kawale
,
D.
,
G.
Bouwman
,
S.
Sachdev
,
P. L.
Zitha
,
M. T.
Kreutzer
,
W. R.
Rossen
, and
P. E.
Boukany
, “
Polymer conformation during flow in porous media
,”
Soft Matter
13
,
8745
8755
(
2017
).
150.
Varshney
,
A.
, and
V.
Steinberg
, “
Elastic wake instabilities in a creeping flow between two obstacles
,”
Phys. Rev. Fluids
2
,
051301
(
2017
).
151.
Lam
,
Y.
,
H.
Gan
,
N.-T.
Nguyen
, and
H.
Lie
, “
Micromixer based on viscoelastic flow instability at low Reynolds number
,”
Biomicrofluidics
3
,
014106
(
2009
).
152.
Teclemariam
,
N. P.
,
V. A.
Beck
,
E. S.
Shaqfeh
, and
S. J.
Muller
, “
Dynamics of DNA polymers in post arrays: Comparison of single molecule experiments and simulations
,”
Macromolecules
40
,
3848
3859
(
2007
).
153.
Galindo-Rosales
,
F. J.
,
L.
Campo-Deaño
,
F.
Pinho
,
E.
Van Bokhorst
,
P.
Hamersma
,
M. S.
Oliveira
, and
M.
Alves
, “
Microfluidic systems for the analysis of viscoelastic fluid flow phenomena in porous media
,”
Microfluid. Nanofluid.
12
,
485
498
(
2012
).
154.
Ribeiro
,
V.
,
P.
Coelho
,
F.
Pinho
, and
M.
Alves
, “
Viscoelastic fluid flow past a confined cylinder: Three-dimensional effects and stability
,”
Chem. Eng. Sci.
111
,
364
380
(
2014
).
155.
Lanzaro
,
A.
, and
X.-F.
Yuan
, “
A quantitative analysis of spatial extensional rate distribution in nonlinear viscoelastic flows
,”
J. Nonnewton. Fluid Mech.
207
,
32
41
(
2014
).
156.
Lanzaro
,
A.
,
Z.
Li
, and
X.-F.
Yuan
, “
Quantitative characterization of high molecular weight polymer solutions in microfluidic hyperbolic contraction flow
,”
Microfluid. Nanofluid.
18
,
819
828
(
2015
).
157.
Lanzaro
,
A.
,
D.
Corbett
, and
X.-F.
Yuan
, “
Non-linear dynamics of semi-dilute paam solutions in a microfluidic 3D cross-slot flow geometry
,”
J. Nonnewton. Fluid Mech.
242
,
57
65
(
2017
).
158.
Haward
,
S. J.
,
N.
Kitajima
,
K.
Toda-Peters
,
T.
Takahashi
, and
A. Q.
Shen
, “
Flow of wormlike micellar solutions around microfluidic cylinders with high aspect ratio and low blockage ratio
,”
Soft Matter
15
,
1927
1941
(
2019
).
159.
Haward
,
S. J.
,
K.
Toda-Peters
, and
A. Q.
Shen
, “
Steady viscoelastic flow around high-aspect-ratio, low-blockage-ratio microfluidic cylinders
,”
J. Nonnewton. Fluid Mech.
254
,
23
35
(
2018
).
160.
Chan
,
S. T.
,
S. J.
Haward
, and
A. Q.
Shen
, “
Microscopic investigation of vortex breakdown in a dividing T-junction flow
,”
Phys. Rev. Fluids
3
,
072201
(
2018
).
161.
Zografos
,
K.
,
N.
Burshtein
,
A. Q.
Shen
,
S. J.
Haward
, and
R. J.
Poole
, “
Elastic modifications of an inertial instability in a 3D cross-slot
,”
J. Nonnewton. Fluid Mech.
262
,
12
24
(
2018
).
162.
Aramideh
,
S.
,
P. P.
Vlachos
, and
A. M.
Ardekani
, “
Nanoparticle dispersion in porous media in viscoelastic polymer solutions
,”
J. Nonnewton. Fluid Mech.
268
,
75
80
(
2019
).
163.
De
,
S.
,
S.
Koesen
,
R.
Maitri
,
M.
Golombok
,
J.
Padding
, and
J.
van Santvoort
, “
Flow of viscoelastic surfactants through porous media
,”
AIChE J.
64
,
773
781
(
2018
).
164.
De
,
S.
,
J.
Kuipers
,
E.
Peters
, and
J.
Padding
, “
Viscoelastic flow simulations in random porous media
,”
J. Nonnewton. Fluid Mech.
248
,
50
61
(
2017
).
165.
De
,
S.
,
J.
Kuipers
,
E.
Peters
, and
J.
Padding
, “
Viscoelastic flow simulations in model porous media
,”
Phys. Rev. Fluids
2
,
053303
(
2017
).
166.
Zami-Pierre
,
F.
,
R.
De Loubens
,
M.
Quintard
, and
Y.
Davit
, “
Transition in the flow of power-law fluids through isotropic porous media
,”
Phys. Rev. Lett.
117
,
074502
(
2016
).
167.
De
,
S.
,
J.
van der Schaaf
,
N. G.
Deen
,
J. A. M.
Kuipers
,
E. A. J. F.
Peters
, and
J. T.
Padding
, “
Lane change in flows through pillared microchannels
,”
Phys. Fluids
29
,
113102
(
2017
).
168.
Babayekhorasani
,
F.
,
D. E.
Dunstan
,
R.
Krishnamoorti
, and
J. C.
Conrad
, “
Nanoparticle dispersion in disordered porous media with and without polymer additives
,”
Soft Matter
12
,
5676
5683
(
2016
).
169.
Odell
,
J. A.
, and
S. J.
Haward
, “
Viscosity enhancement in non-Newtonian flow of dilute aqueous polymer solutions through crystallographic and random porous media
,”
Rheol. Acta
45
,
853
863
(
2006
).
170.
Haward
,
S. J.
, and
J. A.
Odell
, “
Viscosity enhancement in non-Newtonian flow of dilute polymer solutions through crystallographic porous media
,”
Rheol. Acta
42
,
516
526
(
2003
).
171.
Deiber
,
J. A.
, and
W. R.
Schowalter
, “
Modeling the flow of viscoelastic fluids through porous media
,”
AIChE J.
27
,
912
920
(
1981
).
172.
Blunt
,
M. J.
,
Multiphase Flow in Permeable Media: A Pore-Scale Perspective
(
Cambridge University Press
,
Cambridge, England
,
2017
).
173.
McKinley
,
G. H.
,
R. C.
Armstrong
, and
R. A.
Brown
, “
The wake instability in viscoelastic flow past confined circular cylinders
,”
Philos. Trans. R. Soc., A
344
,
265
304
(
1993
).
174.
Hopkins
,
C. C.
,
S. J.
Haward
, and
A. Q.
Shen
, “
Tristability in viscoelastic flow past side-by-side microcylinders
,”
Phys. Rev. Lett.
126
,
54501
(
2021
).
175.
Grilli
,
M.
,
A.
Vázquez-Quesada
, and
M.
Ellero
, “
Transition to turbulence and mixing in a viscoelastic fluid flowing inside a channel with a periodic array of cylindrical obstacles
,”
Phys. Rev. Lett.
110
,
174501
(
2013
).
176.
Arora
,
K.
,
R.
Sureshkumar
, and
B.
Khomami
, “
Experimental investigation of purely elastic instabilities in periodic flows
,”
J. Nonnewton. Fluid Mech.
108
,
209
226
(
2002
).
177.
Müller
,
M.
,
J.
Vorwerk
, and
P. O.
Brunn
, “
Optical studies of local flow behaviour of a non-Newtonian fluid inside a porous medium
,”
Rheol. Acta
37
,
189
194
(
1998
).
178.
Chauhan
,
A.
,
S.
Gupta
, and
C.
Sasmal
, “Effect of geometric disorder on chaotic viscoelastic porous media flows,” arXiv:2107.10617 [physics.flu-dyn] (2021).
179.
Wagner
,
C. E.
, and
G. H.
McKinley
, “
The importance of flow history in mixed shear and extensional flows
,”
J. Nonnewton. Fluid Mech.
233
,
133
145
(
2015
).
180.
Maier
,
R. S.
,
D. M.
Kroll
,
R. S.
Bernard
,
S. E.
Howington
,
J. F.
Peters
, and
H. T.
Davis
, “
Pore-scale simulation of dispersion
,”
Phys. Fluids
12
,
2065
2079
(
2000
).
181.
De
,
S.
,
S. P.
Koesen
,
R. V.
Maitri
,
M.
Golombok
,
J. T.
Padding
, and
J. F. M.
van Santvoort
, “
Flow of viscoelastic surfactants through porous media
,”
AIChE J.
64
,
773
781
(
2018
).
182.
Marafini
,
E.
,
M.
La Rocca
,
A.
Fiori
,
I.
Battiato
, and
P.
Prestininzi
, “
Suitability of 2D modelling to evaluate flow properties in 3D porous media
,”
Transp. Porous Media
134
,
315
329
(
2020
).
183.
Nilsson
,
M. A.
,
R.
Kulkarni
,
L.
Gerberich
,
R.
Hammond
,
R.
Singh
,
E.
Baumhoff
, and
J. P.
Rothstein
, “
Effect of fluid rheology on enhanced oil recovery in a microfluidic sandstone device
,”
J. Nonnewton. Fluid Mech.
202
,
112
119
(
2013
).
184.
Anbari
,
A.
,
H. T.
Chien
,
S. S.
Datta
,
W.
Deng
,
D. A.
Weitz
, and
J.
Fan
, “
Microfluidic model porous media: Fabrication and applications
,”
Small
14
,
1703575
(
2018
).
185.
Datta
,
S. S.
,
T. S.
Ramakrishnan
, and
D. A.
Weitz
, “
Mobilization of a trapped non-wetting fluid from a three-dimensional porous medium
,”
Phys. Fluids
26
,
022002
(
2014
).
186.
do Nascimento
,
D. F.
,
J. R.
Vimieiro Junior
,
S.
Paciornik
, and
M. S.
Carvalho
, “
Pore scale visualization of drainage in 3D porous media by confocal microscopy
,”
Sci. Rep.
9
,
12333
(
2019
).
187.
Guo
,
T.
,
A. M.
Ardekani
, and
P. P.
Vlachos
, “
Microscale, scanning defocusing volumetric particle-tracking velocimetry
,”
Exp. Fluids
60
,
89
(
2019
).
188.
De
,
S.
,
J.
Kuipers
,
E.
Peters
, and
J.
Padding
, “
Viscoelastic flow simulations in random porous media
,”
J. Nonnewton. Fluid Mech.
248
,
50
61
(
2017
).
189.
Aramideh
,
S.
,
P. P.
Vlachos
, and
A. M.
Ardekani
, “
Pore-scale statistics of flow and transport through porous media
,”
Phys. Rev. E
98
,
013104
(
2018
).
190.
Wang
,
D.
,
G.
Wang
,
W.
Wu
,
H.
Xia
, and
H.
Yin
, “The influence of viscoelasticity on displacement efficiency–From micro to macro scale,” in SPE Annual Technical Conference and Exhibition (Society of Petroleum Engineers, Anaheim, CA, 2007).
191.
Xia
,
H.
,
D.
Wang
,
G.
Wang
, and
J.
Wu
, “
Effect of polymer solution viscoelasticity on residual oil
,”
Pet. Sci. Technol.
26
,
398
412
(
2008
).
192.
Wang
,
D.
,
H.
Xia
,
Z.
Liu
, and
Q.
Yang
, “Study of the mechanism of polymer solution with visco-elastic behavior increasing microscopic oil displacement efficiency and the forming of steady ‘oil thread’ flow channels,” in SPE Asia Pacific Oil and Gas Conference and Exhibition (Society of Petroleum Engineers, Jakarta, Indonesia, 2001).
193.
Wang
,
D.
,
J.
Cheng
,
H.
Xia
,
Q.
Li
, and
J.
Shi
, “Viscous-elastic fluids can mobilize oil remaining after water-flood by force parallel to the oil-water interface,” in SPE Asia Pacific Improved Oil Recovery Conference (Society of Petroleum Engineers, Kuala Lumpur, Malaysia, 2001).
194.
Shakeri
,
P.
,
M.
Jung
, and
R.
Seemann
, “
Effect of elastic instability on mobilization of capillary entrapments
,”
Phys. Fluids
33
,
113102
(
2021
).
195.
Jones
,
W. M.
, “
Polymer additives in reservoir flooding for oil recovery: Shear thinning or shear thickening?
,”
J. Phys. D: Appl. Phys.
13
,
L87
L88
(
1980
).
196.
Delshad
,
M.
,
D. H.
Kim
,
O. A.
Magbagbeola
,
C.
Huh
,
G. A.
Pope
, and
F.
Tarahhom
, “Mechanistic interpretation and utilization of viscoelastic behavior of polymer solutions for improved polymer-flood efficiency,” in SPE Symposium on Improved Oil Recovery (Society of Petroleum Engineers, Tulsa, OK, 2008).
197.
Upchurch
,
E.
, and
E.
Meiburg
, “
Miscible porous media displacements driven by non-vertical injection wells
,”
J. Fluid Mech.
607
,
289
312
(
2008
).
198.
Holtzman
,
R.
, “
Effects of pore-scale disorder on fluid displacement in partially-wettable porous media
,”
Sci. Rep.
6
,
36221
(
2016
).
199.
Tang
,
R.
,
C. S.
Kim
,
D. J.
Solfiell
,
S.
Rana
,
R.
Mout
,
E. M.
Velázquez-Delgado
,
A.
Chompoosor
,
Y.
Jeong
,
B.
Yan
,
Z. J.
Zhu
,
C.
Kim
,
J. A.
Hardy
, and
V. M.
Rotello
, “
Direct delivery of functional proteins and enzymes to the cytosol using nanoparticle-stabilized nanocapsules
,”
ACS Nano
7
,
6667
6673
(
2013
).
200.
Parodi
,
A.
,
S. G.
Haddix
,
N.
Taghipour
,
S.
Scaria
,
F.
Taraballi
,
A.
Cevenini
,
I. K.
Yazdi
,
C.
Corbo
,
R.
Palomba
,
S. Z.
Khaled
,
J. O.
Martinez
,
B. S.
Brown
,
L.
Isenhart
, and
E.
Tasciotti
, “
Bromelain surface modification increases the diffusion of silica nanoparticles in the tumor extracellular matrix
,”
ACS Nano
8
,
9874
9883
(
2014
).
201.
Smith
,
D. J.
,
E. A.
Gaffney
, and
J. R.
Blake
, “
Modelling mucociliary clearance
,”
Resp. Physiol. Neurobiol.
163
,
178
188
(
2008
).
202.
Bodiguel
,
H.
,
J.
Beaumont
,
A.
Machado
,
L.
Martinie
,
H.
Kellay
, and
A.
Colin
, “
Flow enhancement due to elastic turbulence in channel flows of shear thinning fluids
,”
Phys. Rev. Lett.
114
,
028302
(
2015
).
203.
Jacob
,
J. D.
,
R.
Krishnamoorti
, and
J. C.
Conrad
, “
Particle dispersion in porous media: Differentiating effects of geometry and fluid rheology
,”
Phys. Rev. E
96
,
022610
(
2017
).
204.
Frechette
,
J.
, and
G.
Drazer
, “
Directional locking and deterministic separation in periodic arrays
,”
J. Fluid Mech.
627
,
379
401
(
2009
).
205.
Huang
,
L. R.
,
E. C.
Cox
,
R. H.
Austin
, and
J. C.
Sturm
, “
Continuous particle separation through deterministic lateral displacement
,”
Science
304
,
987
990
(
2004
).
206.
Corapcioglu
,
M. Y.
, and
S.
Jiang
, “
Colloid-facilitated groundwater contaminant transport
,”
Water Resour. Res.
29
,
2215
2226
(
1993
).
207.
Wei
,
B.
,
Q.
Li
,
F.
Jin
,
H.
Li
, and
C.
Wang
, “
The potential of a novel nanofluid in enhancing oil recovery
,”
Energy Fuels
30
,
2882
2891
(
2016
).
208.
Li
,
R.
,
P.
Jiang
,
C.
Gao
,
F.
Huang
,
R.
Xu
, and
X.
Chen
, “
Experimental investigation of silica-based nanofluid enhanced oil recovery: The effect of wettability alteration
,”
Energy Fuels
31
,
188
197
(
2017
).
209.
Suleimanov
,
B.
,
F.
Ismailov
, and
E.
Veliyev
, “
Nanofluid for enhanced oil recovery
,”
J. Pet. Sci. Eng.
78
,
431
437
(
2011
).
210.
Roustaei
,
A.
, and
H.
Bagherzadeh
, “
Experimental investigation of SiO2 nanoparticles on enhanced oil recovery of carbonate reservoirs
,”
J. Pet. Explor. Prod. Technol.
5
,
27
33
(
2015
).
211.
Yu
,
H.
,
Y.
He
,
P.
Li
,
S.
Li
,
T.
Zhang
,
E.
Rodriguez-Pin
,
S.
Du
,
C.
Wang
,
S.
Cheng
,
C. W.
Bielawski
,
S. L.
Bryant
, and
C.
Huh
, “
Flow enhancement of water-based nanoparticle dispersion through microscale sedimentary rocks
,”
Sci. Rep.
5
,
8702
(
2015
).
212.
Kondiparty
,
K.
,
A.
Nikolov
,
S.
Wu
, and
D.
Wasan
, “
Wetting and spreading of nanofluids on solid surfaces driven by the structural disjoining pressure: Statics analysis and experiments
,”
Langmuir
27
,
3324
3335
(
2011
).
213.
Chen
,
C.
,
A. I.
Packman
, and
J.-F.
Gaillard
, “
Pore-scale analysis of permeability reduction resulting from colloid deposition
,”
Geophys. Res. Lett.
35
,
L07404
(
2008
).
214.
Civan
,
F.
, “
Non-isothermal permeability impairment by fines migration and deposition in porous media including dispersive transport
,”
Transp. Porous Media
85
,
233
258
(
2010
).
215.
Wiesner
,
M. R.
,
M. C.
Grant
, and
S. R.
Hutchins
, “
Reduced permeability in groundwater remediation systems: Role of mobilized colloids and injected chemicals
,”
Environ. Sci. Technol.
30
,
3184
3191
(
1996
).
216.
Zhang
,
W.
,
V. L.
Morales
,
M. E.
Cakmak
,
A. E.
Salvucci
,
L. D.
Geohring
,
A. G.
Hay
,
J.-Y.
Parlange
, and
T. S.
Steenhuis
, “
Colloid transport and retention in unsaturated porous media: Effect of colloid input concentration
,”
Environ. Sci. Technol.
44
,
4965
4972
(
2010
).
217.
Bizmark
,
N.
,
J.
Schneider
,
R. D.
Priestley
, and
S. S.
Datta
, “
Multiscale dynamics of colloidal deposition and erosion in porous media
,”
Sci. Adv.
6
,
eabc2530
(
2020
).
218.
Balzan
,
S.
,
C.
de Almeida Quadros
,
R.
de Cleva
,
B.
Zilberstein
, and
I.
Cecconello
, “
Bacterial translocation: Overview of mechanisms and clinical impact
,”
J. Gastroenterol. Hepatol.
22
,
464
471
(
2007
).
219.
Chaban
,
B.
,
H. V.
Hughes
, and
M.
Beeby
, “
The flagellum in bacterial pathogens: For motility and a whole lot more
,”
Semin. Cell Dev. Biol.
46
,
91
103
(
2015
).
220.
Ribet
,
D.
, and
P.
Cossart
, “
How bacterial pathogens colonize their hosts and invade deeper tissues
,”
Microbes Infect.
17
,
173
183
(
2015
).
221.
Harman
,
M. W.
, “
The heterogeneous motility of the Lyme disease spirochete in gelatin mimics dissemination through tissue
,”
Biophys. J.
102
,
151a
152a
(
2012
).
222.
Siitonen
,
A.
, and
M.
Nurminen
, “
Bacterial motility is a colonization factor in experimental urinary tract infection
,”
Infect. Immun.
60
,
3918
3920
(
1992
).
223.
Kumar
,
M.
,
D. M.
Walkama
,
J. S.
Guasto
, and
A. M.
Ardekani
, “
Flow-induced buckling dynamics of sperm flagella
,”
Phys. Rev. E
100
,
063107
(
2019
).
224.
Kumar
,
M.
, and
A. M.
Ardekani
, “
Effect of external shear flow on sperm motility
,”
Soft Matter
15
,
6269
6277
(
2019
).
225.
Ford
,
R. M.
, and
R. W.
Harvey
, “
Role of chemotaxis in the transport of bacteria through saturated porous media
,”
Adv. Water Resour.
30
,
1608
1617
(
2007
).
226.
Adadevoh
,
J. S. T.
,
S.
Triolo
,
C. A.
Ramsburg
, and
R. M.
Ford
, “
Chemotaxis increases the residence time of bacteria in granular media containing distributed contaminant sources
,”
Environ. Sci. Technol.
50
,
181
187
(
2016
).
227.
Adadevoh
,
J. S. T.
,
C. A.
Ramsburg
, and
R. M.
Ford
, “
Chemotaxis increases the retention of bacteria in porous media with residual NAPL entrapment
,”
Environ. Sci. Technol.
52
,
7289
7295
(
2018
).
228.
Dechesne
,
A.
,
G.
Wang
,
G.
Gulez
,
D.
Or
, and
B. F.
Smets
, “
Hydration-controlled bacterial motility and dispersal on surfaces
,”
Proc. Natl. Acad. Sci. U.S.A.
107
,
14369
14372
(
2010
).
229.
Turnbull
,
G. A.
,
J. W.
Morgan
,
J. M.
Whipps
, and
J. R.
Saunders
, “
The role of bacterial motility in the survival and spread of Pseudomonas fluorescens in soil and in the attachment and colonisation of wheat roots
,”
FEMS Microbiol. Ecol.
36
,
21
31
(
2001
).
230.
Berg
,
H. C.
,
Random Walks in Biology
(
Princeton University
,
Princeton, NJ
,
1993
), expanded edition.
231.
Rothschild
,
L.
, “
Non-random distribution of bull spermatozoa in a drop of sperm suspension
,”
Nature
198
,
1221
1222
(
1963
).
232.
Sipos
,
O.
,
K.
Nagy
,
R.
Di Leonardo
, and
P.
Galajda
, “
Hydrodynamic trapping of swimming bacteria by convex walls
,”
Phys. Rev. Lett.
114
,
258104
(
2015
).
233.
Kantsler
,
V.
,
J.
Dunkel
,
M.
Polin
, and
R. E.
Goldstein
, “
Ciliary contact interactions dominate surface scattering of swimming eukaryotes
,”
Proc. Natl. Acad. Sci. U.S.A.
110
,
1187
1192
(
2013
).
234.
Lauffenburger
,
D.
,
C.
Kennedy
, and
R.
Aris
, “
Traveling bands of chemotactic bacteria in the context of population growth
,”
Bull. Math. Biol.
46
,
19
40
(
1984
).
235.
Hilpert
,
M.
, “
Lattice-Boltzmann model for bacterial chemotaxis
,”
J. Math. Biol.
51
,
302
332
(
2005
).
236.
Licata
,
N. A.
,
B.
Mohari
,
C.
Fuqua
, and
S.
Setayeshgar
, “
Diffusion of bacterial cells in porous media
,”
Biophys. J.
110
,
247
257
(
2016
).
237.
Croze
,
O. A.
,
G. P.
Ferguson
,
M. E.
Cates
, and
W. C.
Poon
, “
Migration of chemotactic bacteria in soft agar: Role of gel concentration
,”
Biophys. J.
101
,
525
534
(
2011
).
238.
Duffy
,
K.
,
P.
Cummings
, and
R.
Ford
, “
Random walk calculations for bacterial migration in porous media
,”
Biophys. J.
68
,
800
806
(
1995
).
239.
Brun-Cosme-Bruny
,
M.
,
E.
Bertin
,
B.
Coasne
,
P.
Peyla
, and
S.
Rafaï
, “
Effective diffusivity of microswimmers in a crowded environment
,”
J. Chem. Phys.
150
,
104901
(
2019
).
240.
Bhattacharjee
,
T.
, and
S. S.
Datta
, “
Confinement and activity regulate bacterial motion in porous media
,”
Soft Matter
15
,
9920
9930
(
2019
).
241.
Zöttl
,
A.
, and
H.
Stark
, “
Nonlinear dynamics of a microswimmer in Poiseuille flow
,”
Phys. Rev. Lett.
108
,
218104
(
2012
).
242.
Rusconi
,
R.
,
J. S.
Guasto
, and
R.
Stocker
, “
Bacterial transport suppressed by fluid shear
,”
Nat. Phys.
10
,
212
217
(
2014
).
243.
Zöttl
,
A.
, and
H.
Stark
, “
Periodic and quasiperiodic motion of an elongated microswimmer in Poiseuille flow
,”
Eur. Phys. J. E
36
,
4
(
2013
).
244.
Dehkharghani
,
A.
,
N.
Waisbord
,
J.
Dunkel
, and
J. S.
Guasto
, “
Bacterial scattering in microfluidic crystal flows reveals giant active Taylor–Aris dispersion
,”
Proc. Natl. Acad. Sci. U.S.A.
116
,
11119
11124
(
2019
).
245.
Creppy
,
A.
,
E.
Clément
,
C.
Douarche
,
M. V.
D’Angelo
, and
H.
Auradou
, “
Effect of motility on the transport of bacteria populations through a porous medium
,”
Phys. Rev. Fluids
4
,
013102
(
2019
).
246.
Alonso-Matilla
,
R.
,
B.
Chakrabarti
, and
D.
Saintillan
, “
Transport and dispersion of active particles in periodic porous media
,”
Phys. Rev. Fluids
4
,
043101
(
2019
).
247.
de Anna
,
P.
,
A. A.
Pahlavan
,
Y.
Yawata
,
R.
Stocker
, and
R.
Juanes
, “
Chemotaxis under flow disorder shapes microbial dispersion in porous media
,”
Nat. Phys.
17
,
68
73
(
2021
).
248.
Waisbord
,
N.
,
A.
Dehkharghani
, and
J. S.
Guasto
, “
Fluidic bacterial diodes rectify magnetotactic cell motility in porous environments
,”
Nat. Commun.
12
,
5949
(
2021
).
249.
McClain
,
M. E.
,
E. W.
Boyer
,
C. L.
Dent
,
S. E.
Gergel
,
N. B.
Grimm
,
P. M.
Groffman
,
S. C.
Hart
,
J. W.
Harvey
,
C. A.
Johnston
,
E.
Mayorga
,
W. H.
McDowell
, and
G.
Pinay
, “
Biogeochemical hot spots and hot moments at the interface of terrestrial and aquatic ecosystems
,”
Ecosystems
6
,
301
312
(
2003
).
250.
Felfoul
,
O.
,
M.
Mohammadi
,
S.
Taherkhani
,
D.
De Lanauze
,
Y.
Zhong Xu
,
D.
Loghin
,
S.
Essa
,
S.
Jancik
,
D.
Houle
,
M.
Lafleur
,
L.
Gaboury
,
M.
Tabrizian
,
N.
Kaou
,
M.
Atkin
,
T.
Vuong
,
G.
Batist
,
N.
Beauchemin
,
D.
Radzioch
, and
S.
Martel
, “
Magneto-aerotactic bacteria deliver drug-containing nanoliposomes to tumour hypoxic regions
,”
Nat. Nanotechnol.
11
,
941
947
(
2016
).
251.
Dombrowski
,
C.
,
L.
Cisneros
,
S.
Chatkaew
,
R. E.
Goldstein
, and
J. O.
Kessler
, “
Self-concentration and large-scale coherence in bacterial dynamics
,”
Phys. Rev. Lett.
93
,
098103
(
2004
).
252.
Sokolov
,
A.
, and
I. S.
Aranson
, “
Reduction of viscosity in suspension of swimming bacteria
,”
Phys. Rev. Lett.
103
,
148101
(
2009
).
253.
Wu
,
X.-L.
, and
A.
Libchaber
, “
Particle diffusion in a quasi-two-dimensional bacterial bath
,”
Phys. Rev. Lett.
84
,
3017
3020
(
2000
).
254.
Sokolov
,
A.
,
I. S.
Aranson
,
J. O.
Kessler
, and
R. E.
Goldstein
, “
Concentration dependence of the collective dynamics of swimming bacteria
,”
Phys. Rev. Lett.
98
,
158102
(
2007
).
255.
Dunkel
,
J.
,
S.
Heidenreich
,
K.
Drescher
,
H. H.
Wensink
,
M.
Bär
, and
R. E.
Goldstein
, “
Fluid dynamics of bacterial turbulence
,”
Phys. Rev. Lett.
110
,
228102
(
2013
).
256.
Li
,
G.
, and
A. M.
Ardekani
, “
Collective motion of microorganisms in a viscoelastic fluid
,”
Phys. Rev. Lett.
117
,
118001
(
2016
).
257.
Wioland
,
H.
,
F. G.
Woodhouse
,
J.
Dunkel
,
J. O.
Kessler
, and
R. E.
Goldstein
, “
Confinement stabilizes a bacterial suspension into a spiral vortex
,”
Phys. Rev. Lett.
110
,
268102
(
2013
).
258.
Wioland
,
H.
,
E.
Lushi
, and
R. E.
Goldstein
, “
Directed collective motion of bacteria under channel confinement
,”
New J. Phys.
18
,
075002
(
2016
).
259.
Li
,
G.-J.
, and
A. M.
Ardekani
, “
Hydrodynamic interaction of microswimmers near a wall
,”
Phys. Rev. E
90
,
013010
(
2014
).
260.
Reichhardt
,
C.
, and
C. J.
Olson Reichhardt
, “
Active matter transport and jamming on disordered landscapes
,”
Phys. Rev. E
90
,
012701
(
2014
).
261.
Reichhardt
,
C. J. O.
, and
C.
Reichhardt
, “
Avalanche dynamics for active matter in heterogeneous media
,”
New J. Phys.
20
,
025002
(
2018
).
262.
Wioland
,
H.
,
F. G.
Woodhouse
,
J.
Dunkel
, and
R. E.
Goldstein
, “
Ferromagnetic and antiferromagnetic order in bacterial vortex lattices
,”
Nat. Phys.
12
,
341
345
(
2016
).
263.
Secchi
,
E.
,
R.
Rusconi
,
S.
Buzzaccaro
,
M. M.
Salek
,
S.
Smriga
,
R.
Piazza
, and
R.
Stocker
, “
Intermittent turbulence in flowing bacterial suspensions
,”
J. R. Soc. Interface
13
,
20160175
(
2016
).
264.
Woodhouse
,
F. G.
, and
J.
Dunkel
, “
Active matter logic for autonomous microfluidics
,”
Nat. Commun.
8
,
15169
(
2017
).
265.
Gao
,
C. H.
, and
A.
Zekri
, “
Applications of microbial-enhanced oil recovery technology in the past decade
,”
Energy Sources, Part A
33
,
972
989
(
2011
).
266.
Voordouw
,
G.
, “
Production-related petroleum microbiology: Progress and prospects
,”
Curr. Opin. Biotechnol.
22
,
401
405
(
2011
).
267.
Marchant
,
R.
, and
I. M.
Banat
, “
Microbial biosurfactants: Challenges and opportunities for future exploitation
,”
Trends Biotechnol.
30
,
558
565
(
2012
).
268.
Joshi
,
G. S.
,
I. M.
Banat
, and
S. J.
Joshi
, “
Biosurfactants: Production and potential applications in microbial enhanced oil recovery (MEOR)
,”
Biocatal. Agric. Biotechnol.
14
,
23
32
(
2018
).
269.
Nazina
,
T.
,
D. S.
Sokolova
,
A.
Grigor’yan
,
Y.-F.
Xue
,
S.
Belyaev
, and
M.
Ivanov
, “
Production of oil-releasing compounds by microorganisms from the Daqing Oil Field, China
,”
Microbiology
72
,
173
178
(
2003
).
270.
Ramsay
,
J. A.
,
D.
Cooper
, and
R. J.
Neufeld
, “
Effects of oil reservoir conditions on the production of water-insoluble Levan by Bacillus licheniformis
,”
Geomicrobiol. J.
7
,
155
165
(
1989
).
271.
Shaik
,
V. A.
, and
A. M.
Ardekani
, “
Swimming sheet near a plane surfactant-laden interface
,”
Phys. Rev. E
99
,
033101
(
2019
).
272.
Desai
,
N.
, and
A. M.
Ardekani
, “
Biofilms at interfaces: Microbial distribution in floating films
,”
Soft Matter
16
,
1731
1750
(
2020
).
273.
Desai
,
N.
,
V. A.
Shaik
, and
A. M.
Ardekani
, “
Hydrodynamics-mediated trapping of micro-swimmers near drops
,”
Soft Matter
14
,
264
278
(
2018
).
274.
Shaik
,
V. A.
, and
A. M.
Ardekani
, “
Point force singularities outside a drop covered with an incompressible surfactant: Image systems and their applications
,”
Phys. Rev. Fluids
2
,
113606
(
2017
).
275.
Li
,
G. J.
,
A.
Karimi
, and
A. M.
Ardekani
, “
Effect of solid boundaries on swimming dynamics of microorganisms in a viscoelastic fluid
,”
Rheol. Acta
53
,
911
926
(
2014
).
276.
Li
,
G.
, and
A. M.
Ardekani
, “
Near wall motion of undulatory swimmers in non-Newtonian fluids
,”
Eur. J. Comput. Mech.
26
,
44
60
(
2017
).
277.
Huang
,
Z.
,
G.
Lin
, and
A. M.
Ardekani
, “
A consistent and conservative model and its scheme for N-phase-M-component incompressible flows
,”
J. Comput. Phys.
434
,
110229
(
2021
).
278.
Huang
,
Z.
,
G.
Lin
, and
A. M.
Ardekani
, “
A consistent and conservative volume distribution algorithm and its applications to multiphase flows using phase-field models
,”
Int. J. Multiphase Flow
142
,
103727
(
2021
).
1

Roote, D., Technology status report: In situ flushing, Ground Water Remediation Technology Analysis Center, 1998, http://www.gwrtac.org.