Multi-scale permeability of deformable fibrous porous media

L. Bergamasco a, S. Izquierdo a,n, I. Pagonabarraga b, N. Fueyo c

a ITAINNOVA – Instituto Tecnológico de Aragón – María de Luna 8, 50018 Zaragoza, Spainb Departament de Física Fonamental, Universitat de Barcelona, Martí i Franqués 1, 08028 Barcelona, Spainc Grupo de Fluidodinámica Numérica, Universidad de Zaragoza – María de Luna 3, 50018 Zaragoza, Spain

H I G H L I G H T S

� A numerical model for fluid-induced topology in fibrous porous media is proposed.� Topology is defined by clustering, computed as convolution of fiber distribution.� 2D clustering locally depends on porosity, shear rate and out-of-plane forces.� Permeability may decrease one order of magnitude due to flow-induced variations.� A hybrid FV/LB CFD method is developed to solve the multiscale fluid flow problem.

a r t i c l e i n f o

Article history:Received 20 August 2014Received in revised form11 November 2014Accepted 15 November 2014Available online 22 December 2014

Keywords:Resin transfer moulding (RTM)Transport propertiesFabrics/textilesFibersMultiscale modeling

a b s t r a c t

The contribution of fiber dynamics and clustering to the effective permeability in hierarchical fibrousmedia is poorly understood, due to the complex fluid–structure interactions taking place across fiber,yarn and textile scales. In this work, a two-dimensional model for fiber deformation subject to out-of-plane movement restrictions is derived for creeping flow conditions by analogy with non-Browniansuspensions of particles with confining potentials. This leads to a homogeneous Fokker–Planck equationin a phase space of fiber configurations, for the probability density function of the fiber displacements. Afiber clustering criterion is then defined using autoconvolution functions of the local probabilitydensities, which yields the local change in fiber-scale permeability according to a topological descriptionof the porous media instead of the typical geometric description. The resulting multi-scale hydro-dynamic system is numerically solved by a coupled method, where the Stokes flow at yarn-scale issolved with a finite volume method and the mesoscopic model that recovers information from the fiber-scale is solved by a lattice Boltzmann method. The fiber-scale permeability is characterized in terms ofporosity, dimensionless shear rate and dimensionless out-of-plane forces. When assessed in terms of areduced viscosity related to Brinkman's closure for porous media, the mesoscopic model shows thatdeformable fibrous porous media qualitatively behave like dense particle suspensions. For low volumefractions a non-Newtonian reduced viscosity exhibiting shear-thinning and low- and high-shear plateauxis obtained. For high volume fractions and high shear rates the out-of-plane forces lead to shearthickening. The results on steady fiber-scale permeability are presented in the form of phase diagramswhich show that in the typical range of parameters for textiles, the effective permeability of thedeformable case can be up 60% lower than that of the rigid case due to the formation of fiber clusters.

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1. Introduction

Accurate permeability prediction of textile preforms is remarkablyimportant for the manufacturing of fiber-reinforced thermosettingcomposites (Potter, 1997). The final quality of the component mostlydepends on process variables, whose optimization require the tex-tile permeability as an input parameter. Textile preforms for these

applications generally present a hierarchical structure and thereforedifferent length scales to be taken into account, typically rangingbetween one and three orders of magnitude. As a consequence, thenumerical solution of the fluid flow in the real geometry is compu-tationally expensive or even not affordable with standard techniqueswhen length scales diverge. A common practice consists in separatingscales and/or reducing the dimensionality of the problem. This allowsfor the so-called “constitutive” relations (Hunt et al., 2014), that is,analytical solutions or experimental correlations which serve asauxiliary means for the numerical simulations. A review of the severalexperimental and analytical techniques developed in this framework

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Chemical Engineering Science

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n Corresponding author.E-mail address: sizquierdo@itainnova.es (S. Izquierdo).

Chemical Engineering Science 126 (2015) 471–482

can be found in Åström et al. (1992) and Yazdchi and Luding (2012).One of the best established correlations was proposed by Gebart(1992), who derived analytically a permeability law for orderedarrangements of cylinders:

K

R2 ¼ C

ffiffiffiffiffiffiffiffiffiffiffiffi1�εc1�ε

r�1

" #5=2; ð1Þ

where K is the permeability, R the characteristic radius, C a geometricalfactor depending on the arrangement, ε the porosity and εc its criticalvalue (or percolation threshold). The author calculated C ¼ 16=9π

ffiffiffi2

p,

εc ¼ 1�π=4 for square arrangements and C ¼ 16=9πffiffiffi6

p,

εc ¼ 1�π=2ffiffiffi3

pfor hexagonal ones. Papathanasiou (1997) addressed

the multi-scale nature of the problem by solving numerically a squarearray layout of permeable multi-filament yarns with circular fibers andshowed that the effective permeability depends strongly on themicroscopic porosity only at low values of the macroscopic one. Heproposed a dimensionless correlation for the multi-scale permeabilityin the form (Papathanasiou, 2001):

Keff ¼ KM 1þa1KM

Km

� �n�3=2" #

; ð2Þ

where KM and Km correspond to the yarn (macro) and fiber (micro)permeabilities, respectively. The constants a1 and n are geometricalparameters that the author best fitted with numerical simulation data,obtaining: a1 ¼ 2:3 and n¼0.59 for square arrangements, a1 ¼ 3:0 andn¼0.625 for hexagonal. Analogous studies were conducted for differ-ent yarn cross-sections and arrangements (Markicevic and Papathan-asiou, 2002; Papathanasiou et al., 2002). Due to the geometricaldependence on the percolation threshold, the validity of the above(or similar) correlations is limited to strictly regular layouts, both at themacro- and microscopic scales. Consequently, their use for thenumerical simulation of textile geometries often results in an unac-ceptable loss of accuracy due to: (i) the false assumption of regulartopologies and (ii) the deformation of the structures induced by thefluid flow.

In order to overcome the first issue (i), random or realisticallyreconstructed fiber configurations have been extensively studied(Endruweit et al., 2013; Soltani et al., 2014) and statisticaldescriptors have been proposed to relate the permeability tonon-regular fiber arrangements (Chen and Papathanasiou, 2008;Yazdchi et al., 2012). The effect of several micro-structural para-meters on the effective permeability has been also investigatedusing up-scaling techniques (Yazdchi et al., 2011; Yazdchi andLuding, 2013). However, despite the intense work on configura-tions and up-scaling, the fluid–structure interaction problem(ii) has not been addressed in this framework, as far as the presentauthors know.

The flow-induced deformation of fibers, however, affects theinterconnectivity of the porous matrix and thus the percolatingpaths, which in turn affect the permeability (Hunt et al., 2014).Indeed, relevant recent work on the permeability of deformingporous matrices relies on the idea that the flow resistance ofparticle clusters (in two dimensions) is larger than that justifiableby single particle contributions. This is basically due to theentrapment of fluid within the cluster, which increases theapparent volume fraction reducing the hydraulic (or wet) area.Scholz et al. (2012) recently proposed a generalized empiricalexpression for permeability based on this concept:

K ¼ cl2c1�χo

N

� �β

; ð3Þ

where c is a constant that depends on the local pore geometry, lc is thelimiting hydrodynamic length, χo is the open-space Euler character-istic (of the conducting phase), N is the number of particles and β isthe conductivity exponent. The Euler characteristic χ is a Minkowski

functional that in this framework is defined as the difference betweenthe number of connected components of each phase (Mecke and Arns,2005); thus χo is the difference between the liquid phase and thenumber of solid components (neglecting the fluid entrapped in closedcavities). The authors best fitted β¼ 1:27 against experimental andnumerical data for quasi-two-dimensional porous structures (close tothe critical value βc ¼ 1:3 for two-dimensional structures Scholz et al.,2012). The quantity 1�χo is generally referred to as genus andrepresents the total number of clusters of single or touching particles;thus 1�χo

� �=N is the number of clusters per particle or cluster density,

which in the following will be called Ω for compactness.Based on this latter idea, in this work we propose a multi-scale

framework for the analysis of the local fiber topology induced by thefluid flow, through the cluster density. A two-dimensional meso-scopic model for the deformation of fibers subject to out-of-planemovement restrictions is derived for creeping flow conditions byanalogy with non-Brownian systems with confining potentials.Typical non-Brownian examples are the suspensions of particles,such as rods or short fibers in a polymer (for composite manufactur-ing) and also the pulp in paper manufacturing (Larson, 1998). Thedifference of the proposed model and these non-Brownian examplesis that we use out-of-plane forces, instead of long-range forces/potentials. The mesoscopic model is a homogeneous Fokker–Planckequation in a phase space of fiber configurations, for the probabilitydensity function of the fiber displacements. A fiber clusteringcriterion is then defined via autoconvolution functions of theprobability densities, which yields the local topology of the fibersand the related change in permeability through the cluster densityΩ. The resulting multi-scale hydrodynamic system is solved numeri-cally by a coupled finite-volume/lattice-Boltzmann method (thelatter accelerated on graphic processing units). Due to the lack ofexperimental or analytical means for its validation, the behavior ofthe proposed model is assessed in terms of a non-Newtonianreduced viscosity related to the Brinkman closure for porous media.The resulting rheology shows qualitative agreement with that of wetgranular media.

The work is organized as follows. The theoretical model isexplained in Section 2: firstly, the modeling framework is intro-duced, then the macroscopic equations are derived from themicroscopic scale via volume-averaging technique and the modelsfor the fiber dynamics and clustering are detailed. A very briefdescription of the numerical methods follows in Section 3. Section4 is dedicated to the introduction and discussion of the results,which comprehend: a parametric analysis of the proposed model;the assessment of its behavior in terms of the reduced viscosityand the results obtained for the permeability of multi-scale fibrousmedia. The conclusions and an outlook on further work concludethe paper (Section 5).

2. Theoretical model

Let us consider the modeling framework shown in Fig. 1. Weconsider two scales: a macro-scale (the yarn scale in Fig. 1(a)) anda micro-scale (the fiber scale in Fig. 1(b)). The relative two-dimensional representative elementary volumes (REV) are shownon the right, respectively REVM and REVm. A macroscopic porosityεM is defined as the void fraction in REVM and a microscopic oneεm as the void fraction in REVm.

The fibers are assumed to be clamped at both ends, thus theycan bend under the effect of the perpendicular flow field. Thecross-section of the yarn results in a domain of two-dimensionalinteracting particles suspended in the fluid (Fig. 1(a)), whosemovement is restricted by the out-of-plane constraint.

Each fiber can bend up to ξmax, which is a function of thedistance z from REVm to the clamped end (Fig. 1(b)). The length of

L. Bergamasco et al. / Chemical Engineering Science 126 (2015) 471–482472

the fiber Lz is imposed by the structure of the textile, for example,by knit points. The two-dimensional model is intended to beapplied to real 3D fabrics.

The bending direction is imposed by the flow field, so ingeneral the section of each fiber on REVm can move in a circlewhose radius is ξmax. The bending of each fiber affects that of theneighboring ones by hydrodynamic interaction, so the relativemovement can lead to an increase or a reduction in the distancebetween them (Fig. 1(c)). Contact is approached as the relativedistance ξc tends to zero. The resulting topology affects the fluidpercolating paths, as long as clustered structures or preferentialchannels can form, which reduce or increase the hydraulic con-ductivity of the yarn.

In order to derive the theoretical model, we start by consider-ing the transport equations at the micro-scale and proceed byupscaling them to the macro-scale via volume-averaging techni-que. The fluctuating stress tensor resulting from the fluid–struc-ture interaction problem is closed with the proposed models forthe fiber dynamics and clustering (which are detailed in dedicatedsections).

2.1. Volume-averaged equations

Let us consider the representative elementary volume at themicro-scale (REVm) in Fig. 1(b). The section of the fiber isrepresented by the solid σ-phase, suspended in the fluid ν-phase.The σ-region is therefore a mono-disperse solid phase withoutinterconnectivity. The flow in the fluid ν-region is assumed to beNewtonian, isothermal and incompressible. The inertial effects areneglected in creeping flow conditions. The continuum transportequations for the fluid ν-region are given by the steady-state

Stokes system:

∇x � v¼ 0; ð4Þ

0¼ �∇xpþμ∇2xv; ð5Þ

where v is the velocity vector, p is the pressure and μ is theviscosity. The subscript x on operators indicates that they act inphysical space. For solving this system of equations the no-slipboundary condition is set at the ν–σ interface, vjνσ ¼ 0. In order toaccount for the porous media description, the superficial averageof a quantity φ is defined as

φ� �¼ 1

VZVν

φ dV ; ð6Þ

and the intrinsic volume average as

φ� �ν ¼ 1

Vν

ZVν

φ dV ; ð7Þ

where Vν is the volume of the fluid ν-phase in the total averagingvolume V (extending over the length Lz). The relationship betweensuperficial and intrinsic volume-averaged quantities is φ

� �¼εm φ� �ν, with εm the porosity for the ν-phase defined as

εm ¼ Vν=V. The superficial average of the continuity equation (4)is expanded by applying the spatial averaging theorem (Howesand Whitaker, 1985):

∇x � vh iþ1VZSνσ

nνσ � v dS¼ 0; ð8Þ

where Sνσ is the interface area between the fluid ν-phase and thesolid σ-phase and nνσ its unit normal. Eq. (8) can be simplified byimposing the boundary condition vjνσ ¼ 0 at the ν–σ-interface(Whitaker, 1986), which yields

∇x � vh i ¼ 0: ð9ÞThe superficial average is applied to the momentum equation (5),which yields

0¼ � ⟨∇p⟩þμ⟨∇2xv⟩: ð10Þ

Applying twice the spatial averaging theorem (Howes andWhitaker, 1985) and arranging terms in such a way that thesuperficial averaged velocity and the intrinsic averaged pressureare the main variables in the equation, the averaged momentumequation is obtained (Whitaker, 1986; Ochoa-Tapia and Whitaker,1995) as

0¼ �∇x p� �νþ μ

εm∇2

x vh i� μεm

∇xεm �∇x vh iνð Þ

� 1Vν

ZSνσ

nνσ � ½�Iðp� p� �νÞ

þμð∇xv�∇x vh iνÞ� dS; ð11ÞI being the unit tensor. The last term is the position-dependentfluctuating stress tensor (Valdés-Parada et al., 2007), whichrepresents the drag force exerted by the solid phase onto the fluidphase (Fdνσ-ν) within the averaging volume Vν. This term needs tobe closed in order to solve the equations. Brinkman's approxima-tion for this term is (Breugem, 2007):

Fd;Brνσ-ν � �μK�1m ðx; εmÞ � vh iþðμeff �μÞ

εm∇2

x vh i; ð12Þ

where the first term is a Darcy drag, while the second is acorrection that accounts for randomness in disordered mediathrough an effective viscosity μeff . In this work we adopt thefollowing closure:

Fd;Ωνσ-ν � �μK�1m ðx; εm;ΩÞ � vh i; ð13Þ

where the permeability tensor is a function of the porosity and ofthe local topology of the solid phase through the cluster densityΩ.

Fig. 1. Schematics of the scales considered and relative representative elementaryvolumes (REV): yarn macro-scale (a) and fiber micro-scale (b). Fiber configurations(c) and the corresponding cluster density Ω, which is defined as the number ofsingle or connected components per fiber.

L. Bergamasco et al. / Chemical Engineering Science 126 (2015) 471–482 473

This latter parameter accounts for the degree of aggregation of thesolid phase, thus intrinsically for apparent variations in porositythrough the open-space Euler characteristic χo (Scholz et al.,2012). Finally, for constant porosity εm, the equations to be solvedin the yarn domain at the macro-scale (Fig. 1(a)) read:

∇x � vh i ¼ 0; ð14Þ

0¼ �∇x p� �νþ μ

εm∇2

x vh i�μK�1m ðx; εm;ΩÞ � vh i; ð15Þ

where the permeability tensor is recovered by solving the modelsfor fiber dynamics and clustering, as explained in the nextsections.

2.2. Fiber dynamics

The equations to reproduce the movement of the solid σ-phasein Fig. 1(b) are derived from Newton's Second Law in a statisticalway. As shown in the figure, the bending of the fibers is modeledby a mass-spring system, whose spring represents the resistanceto bending and is thus related to the out-of-plane forces. Thedynamic equilibrium on the mass is regulated by: (i) a drag force(FdðtÞ), due to the movement through the viscous solvent; (ii) theconnector force (FcðtÞ), which represents the resistance to bend-ing; (iii) a stochastic diffusion term (FhðtÞ), which accounts forhydrodynamic dispersion. Inertial terms are neglected in the limitof small Stokes numbers, which in this case we define as(Marchisio and Fox, 2013)

St ¼ 2ρorLzvin18μR

51; ð16Þ

ρo being the density of the material of the fibers and vin thevelocity at the inlet of REVM (Fig. 1(a)). The velocity vin results fromimposing creeping flow conditions through a Reynolds numberbased on the yarn radius R (see Section 4.3 for details). Because theflow is at a low Stokes number and the fibers have a constrainedmovement around a reference point, the friction forces associatedto the relative motion of the fibers can be neglected. As a result, fornegligible Stokes numbers fibers tend to move together ratherthan relative to each other and the inertial term can be canceled.

Indicating with rc the position of the fixed end of the spring(i.e. the undeformed state) and rðtÞ the position vector of the mass(see Fig. 1(b)), the drag force is

FdðtÞ ¼ ζLz vðrðtÞ; tÞ�drdtðtÞ

� �; ð17Þ

with ζ being an Oseen drag coefficient (Chwang and Wu, 1976), Lzthe length of the fiber and the term inside the brackets thevelocity of the mass relative to the viscous solvent. The readershould notice that, here (and in the following) we omit thevolume-averaged notation for readability; however, the solventvelocity is a volume-averaged one, obtained by solving Eqs. (14)and (15).

The connector force is given as

FcðtÞ ¼Hðrc�rðtÞÞ; ð18Þwhere the connector force law H will be discussed later. The lastforce results from hydrodynamic interactions:

FhðtÞ ¼ σdWðtÞ; ð19Þσbeing the standard deviation of the Wiener process WðtÞ. In thiswork we use this term to model the diffusive random fluctuationsof the fibers due to hydrodynamic dispersion; for this purpose weadopt a diffusion coefficient D and write σ ¼

ffiffiffiffiffiffiffi2D

pfor conven-

ience. In the limit of small Stokes number, lubrication forces(Israelachvili, 2011) are neglected, assuming that the relativevelocity between the fibers is small.

Finally, the dynamic equilibrium of forces on the mass yieldsthe stochastic differential equation (Langevin equation):

drdtðtÞ ¼ vðrðtÞ; tÞþ H

ζLzðrc�rðtÞÞþ

ffiffiffiffiffiffiffi2D

p dWdt

ðtÞ: ð20Þ

Applying the forward Kolmogorov equation (Öttinger, 1996) andletting ξ¼ rc�rðtÞ yields the diffusion (Fokker–Planck) equationfor the probability density ψ ðξ; tÞ of the local fiber displacementwith respect to the undeformed state (see Appendix A)

∂ψ∂t

þ∇ξ � ∇xv � ξ�HξζLz

� �ψ

¼D∇2

ξψ : ð21Þ

In this work we assume the connector force law to follow thefinitely extensible non-linear elastic model (Öttinger, 1996),therefore:

H¼ h

1� JξJ2=ξ2max

; ð22Þ

with h being the spring constant and ξmax the maximum exten-sibility (see Fig. 1(b)).

2.3. Model parameters

Let us now focus on the calculation of the maximum extensi-bility ξmax and the contact distance ξc (see Fig. 1(b)). Consideringthe fiber as a high-aspect-ratio hyperstatic beam subject to adistributed load, the bending (and thus the maximum extensi-bility) is given as (Young et al., 2011)

ξmax ¼ζz2

24EJLz�zð Þ2; ð23Þ

E being the Young modulus of the material and J ¼ πr4=4 thesecond moment. The drag force per unit length ζ can be recoveredby the Oseen formula (Chwang and Wu, 1976) as

ζ ¼ 4πμvinlog ð4=RemÞ�γþ0:5

; ð24Þ

where Rem is the Reynolds number based on the fiber radius and γthe Euler constant. The maximum bending ξmax depends on thegeometry of the fibers, on their material and on the flow, whichmakes this parameter representative of the out-of-plane forces.

On the other hand, the contact distance ξc is a geometricalfunction of the porosity. In this work we consider the fibers to bein a square arrangement when undeformed (see Fig. 1(c)); thus thecontact distance can be written as

ξc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiπr2

1�εm

s�2r: ð25Þ

An overview of the material properties and dimensions used inthis work are shown in Table 1 (the symbol C indicates the rangeof values explored). The yarn radius, for example, ranges between100 and 500 μm, depending on the macroscopic porosity εM (seeFig. 1(a)).

2.4. Dimensional analysis

Considering a characteristic time-scale θ¼ ζLz=2h of the fiberrelaxation and a characteristic length-scale of the mass–spring systemℓ¼

ffiffiffiffiffiffiffiffiffi2θD

p(see Fig. 1(b)), the maximum extensibility and contact

distance are made dimensionless respectively as ξmax ¼ ξ2max=2θD andξc ¼ ξc=

ffiffiffiffiffiffiffiffiffi2θD

p. The dimensionless form of Eq. (21) is

∂ψ∂t

þ∇ξ � θ ∇xvð Þ � ξ�12H ξ

� �ψ

¼ D∇2

ξψ ; ð26Þ

where D is a dimensionless diffusion coefficient which is equal to 1/2due to scaling and θ ∇xvð Þ is the dimensionless shear-rate. The

L. Bergamasco et al. / Chemical Engineering Science 126 (2015) 471–482474

dimensionless connector force in Eq. (22) reads

H ¼ 1� J ξ J2=ξmax

h i�1; ð27Þ

thus the support of the probability density function is a disc of radius

ξ�1=2

max . The initial distribution for Eq. (26) is given as the analyticalsolution for θ ∇xvð Þ ¼ 0, which, considering unitary normalization,reads

ψ0ðξÞ ¼ H� ξmax=2

ZH

� ξmax=2 dξ �1

: ð28Þ

An overview of the model parameters is shown in Table 2. Therelaxation time θ is estimated on the basis of a viscous numberdefined as (Jop et al., 2006; Boyer et al., 2011)

IM ¼ θvinR; ð29Þ

which represents the ratio between the time scale of the fiberrearrangement and the time scale of the flow. In this work, we setthe maximum viscous number IM¼3, which is the maximum value forwhich we found stability of the numerical methods (see Section 3).The dimensionless shear rate at the fiber scale (i.e. in the porousregion), is then defined in scalar form as

Im ¼ θII _γ ; ð30ÞII _γ being the second invariant of the rate of strain. The dimensionlessshear rate Im can be seen as a microscopic viscous number (whosetypical value is IM=εm).

The proposed model does not explicitly account for the contactamong the fibers, therefore the maximum extensibility isrestricted to ξmaxrξcþr (Fig. 1(b)). This physically means thatwhen the contact is approached, the fiber cannot extend anyfurther. The condition in dimensionless form reads (from Eq. (25))

ξmaxrffiffiffiffiffiffiffiffiffiffiffiffiffiffiπr2

1�εm

s�r

0@

1A 2θD� ��1

: ð31Þ

Considering that the porosity εm is bounded between the percola-tion threshold εcm ¼ 1�π=4 (for a square arrangement) and 1,and assuming for the diffusion coefficient D values in the range

10�12C10�9 m2=s, Eq. (31) yields for the maximum extensibilitythe values given in Table 2.

2.5. Fiber clustering

The clustering criterion is defined on the basis of topologicalarguments, that is on the basis of the configuration of the fibersobtained from Eq. (26) through the probability of their displace-ment from the initial position. The distribution functions areassociated with the section of the fibers in Fig. 1(b) and (c), thusa contact zone is defined by the overlap of the probability of onefiber with its neighboring ones. The probability that a fiber be incontact range with the neighbors is given by

Ω¼ 1� 1�mnf

� �∑nf

i ¼ 1ψ ðξÞnψ ðξþ ξcniÞ; ð32Þ

where nf is the number of neighboring fibers, m is the reciprocal ofthe maximum number of fibers in a cluster, ni is the unit vector tothe i-th neighboring fiber and n is the convolution operator. In thiswork we assume a square (undeformed) configuration, and thuseach fiber has 8 neighbors (nf¼8) on a square reference topolo-gical unit (Fig. 1(c) left). Notice that the topological unit isrepresentative of the configuration of a number of fibers N,through the probability density, and thus the parameter m variesbetween 1/2 (for a cluster of 2 fibers) and 0 (for infinitely largecluster). In this work we analyze the worst condition for perme-ability, that is m¼0. WhenΩ¼ 1 no clustering occurs (rigid fibers)while when Ω¼ 0 the fibers form a single cluster (impermeablestructure).

According to the chosen definition for overlap, Ω representsthe quantity 1�χo=N

� �in Eq. (3), with N in this case being the

number of fibers represented by the topological unit. Therefore,considering that in our case the limiting hydrodynamic length lc inEq. (3) is the contact distance ξc in Eq. (25), the permeability canbe rewritten as

Km ¼ c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiπr2

1�εm

s�2r

24

352

Ωβ ; ð33Þ

where for the local pore geometry constant we adopt the samevalue (c¼1/12) as for Eq. (3) (Scholz et al., 2012), which corre-sponds to a confinement between parallel plates (Bruus, 2007).

Finally, Eq. (33) yields the permeability as a function of theporosity εm and of the cluster density Ω, which is used for theclosure of the hydrodynamic system (Eq. (15)).

3. Numerical methods

The numerical approach for solving the theoretical modelconsists in a coupled finite-volume/lattice-Boltzmann solution:finite volume method for the fluid flow equations and lattice

Table 1Geometrical parameters and physical properties of the glass fibers considered. Thesymbol C indicates the range of values explored.

Parameter / Property Symbol Units Value

GeometryYarn radius R μm 100C500Fiber radius r μm 10Fiber length Lz μm 800MaterialDensity ρo g/cm3 2.4Young modulus E MPa 85

Table 2Summary table of the model parameters. The symbol C indicates the range of values explored.

Parameter Dimensional Dimensionless

Symbol Eq. Units Value Symbol Scaling Value

Macro, micro porosities – – – – εM ; εm – 1 – π=4C0:99Maximum extensibility ξmax (23) μm 0C36 ξmax ξmax ¼ ξ2max=2θD 0C6:5� 104

Contact distance ξc (25) μm 0C157 ξc ξc ¼ ξc=ffiffiffiffiffiffiffiffiffi2θD

p0C1:1� 103

Macro viscous number – (29) – – IM – 0C3Fiber relaxation time θ – s 0.01 C 15 – – –

Micro viscous number – (30) – – Im – 0C14Diffusion coefficient D – m2/s 10�12C10�9 D D ¼D=2D 1/2

L. Bergamasco et al. / Chemical Engineering Science 126 (2015) 471–482 475

Boltzmann method for the Fokker–Planck equation. This hybridmethodology has already been applied and validated for thenumerical simulation of viscoelastic suspensions (Bergamascoet al., 2013); therefore here we briefly recall the approach andrefer the reader to the prior article for further details.

3.1. Fluid flow equations

The volume-averaged transport equations (14) and (15) aresolved by the finite volume method (FVM) using the commercialsolver ANSYS Fluents. In this paper we adopt a third orderquadratic upwind scheme for momentum and a second orderscheme for pressure interpolation. The semi-implicit method forpressure linked equations is used for the pressure–velocitycoupling.

3.2. Fiber dynamics model

The Fokker–Planck equation in the configuration space (Eq. (26))is solved by a lattice Boltzmann method (LBM) using a lattice-BGKequation. In this work we adopt a 16,384 DoF lattice (D2Q5) and arelaxation time τ¼ 0:55 (Bergamasco et al., 2013). For computa-tional efficiency, the lattice Boltzmann solution is accelerated ona Graphic Processing Unit (GPU) by a Compute Unified DeviceArchitecture (CUDA) implementation (Nvidia Developer Zone,2012). The salient details of the implementation are given inAppendix B.

3.3. Multi-scale system

The multi-scale hydrodynamic system resulting from Eqs. (14),(15) and (26) is solved by coupling the two previous methods. Thecompiled CUDA code of the lattice Boltzmann solution is dynami-cally called at cell centers from the FVM solver through a compiledUser Defined Function (UDF). The sub-grid simulation is driven by

passage and retrieval of the required variables between the twocompiled codes through a stream process. The details of thecoupled algorithm are given in Appendix C.

4. Results and discussion

4.1. Micro-scale model

Let us first focus on the analysis of the proposed model for thefiber dynamics (Eq. (26)) and clustering (Eq. (32)). A parametricstudy has been carried out using a set of 2000 random realizationsof the velocity gradient tensor ∇v, which accounts for localinhomogeneity of the flow field inside the porous medium. Foreach realization within the selected range of II _γ three independentcomponents of the velocity gradient tensor (11, 12, 21) aregenerated and component 22 is set to ensure incompressibility.The study has been performed by varying the parameters θ (fiberrelaxation time), ξmax (maximum bending) and the ratio ξc=ξmax

(contact/bending ratio). The cluster density Ω in Eq. (32) iscomputed for each realization, which allows the analysis of itsdependence on each parameter.

The data obtained for varying θ as a function of the invariant of∇v is shown in the inset of Fig. 2(a): the data collapse into a singlecurve for θII _γ , which is the dimensionless rate of strain ormicroscopic viscous number Im. The spread of the data is due tothe definition of Im based on the invariant, for which, the samevalue can be yield by different ∇v.

The effect on Ω of varying ξc=ξmax and ξmax is shown inFig. 2(b) and relative inset, while Fig. 2(c)–(e) shows an exampleconfiguration of the probability densities at the correspondingpoints in the figure.

In order to interpret physically the results, let us firstly recallthat Ω is bounded between 0 and 1, which respectively corre-sponds to the maximum and minimum fiber clustering, that is, to

0 2 4 6 8 10 12 140.4

0.5

0.6

0.7

0.8

0.9

1

Im Im

= 0.1 = 0.5 = 1.0 = 3.0

0 3.5 7 10.5 140.4

0.6

0.8

1

0 2 4 6 8 10 12 140.4

0.5

0.6

0.7

0.8

0.9

1

0 3.5 7 10.5 140.2

0.4

0.6

0.8

1

Im

(e)

(d)

(c)

Fig. 2. (a) Numerical data obtained for Ω as a function of the viscous number Im. The main figure shows the collapse of the data in the inset with the relaxation time θ

(ξmax ¼ 10, ξc=ξmax ¼ 0:5). (b) Cluster density Ω as a function of the contact/bending ratio ξc=ξmax for ξmax ¼ 10 and of the maximum extensibility ξmax for ξc=ξmax ¼ 0:3(inset). Figures (c)–(e) show an example configuration of the probability densities at the corresponding points in (b). (For interpretation of the references to color in thisfigure caption, the reader is referred to the web version of this paper.)

L. Bergamasco et al. / Chemical Engineering Science 126 (2015) 471–482476

the impermeable and rigid conditions of the fibrous medium. IfIm-0, the strain rate (or the relaxation time) tends to zero,therefore Ω is given by the overlap of the initial probabilitydensities of Eq. (28) (see Fig. 2(c)). For increasing Im, the prob-ability densities present more localized features, thus the overlapreduces and Ω increases (see Fig. 2(d) and (e)). This physicallyrecovers the increasing anisotropy and stiffness of the fibers due tothe higher strain rate, which leads the fibers towards the rigidcondition (Ω¼1).

With regards to the dependence of Ω on the maximumbending ξmax and contact/bending ratio ξc=ξmax, Fig. 2(b) showsthat for constant ξmax and increasing ξc, the contact probability isreduced and Ω increases. This is justified by the fact that, forincreasing ξc , the porosity increases (see Eq. (31)), lowering thecontact probability. Similarly, the inset in the figure shows theeffect of varying ξmax with constant ξc=ξmax.

In order to present the results in a compact manner, amicroscopic phase diagram is built (Fig. 3(a)). The numerical datais best fitted and a reduced number of points is plotted for eachdimension to allow a proper visualization. The diagram showsthe contours of the cluster density Ω for D¼ 10�12 m2=s as afunction of the microscopic viscous number Im, the maximumextensibility parameter ξmax and a normalized microscopic poros-ity ~εm defined as

~εm ¼ εm�εcmεmaxm �εcm

; ð34Þ

where the porosity εm as a function of the model parameters isgiven by Eq. (31), and ξc=ξmax ranges between 0 and 1. Themaximum porosity εmax

m is given by ξc ¼ ξmax.The plane corresponding to ξmax ¼ 10 in the phase diagram is

shown in Fig. 3(b). The contours show the iso-Ω values as afunction of the viscous number Im and normalized porosity ~εm.For Ωo0:1 the fibrous medium tends to the impermeable stateand for Ω40:9 to the rigid one. In the intermediate range, thecluster density Ω for a constant porosity increases with theviscous number, and decreases with the maximum extensibility(Fig. 3(a)). The first effect is related to the increasing stiffnessof the fibers and thus to the lower contact probability, whilethe second one is due to the increasing maximum bending ofthe fibers. Finally, the increase inΩ observed for very high Im canbe associated with the jamming of the fibers for very lowporosities.

4.2. Model assessment

We will now focus on the assessment of the behavior of theproposed model. We should first indicate that a framework for thevalidation is not available, neither experimentally nor analytically.Therefore we compare the results of our model with the rheologyof wet granular media in terms of an equivalent reduced viscosity.Boyer et al. (2011) have recently proposed a phenomenological lawfor the rheology of dense suspensions of spherical particles interms of volume fraction ϕm and a dimensionless viscous numberIv (which is formally equivalent to our definition for Im). Theauthors deduced the dimensionless shear viscosity in the follow-ing form:

η ¼ 1þ52ϕm 1� ϕm

ϕmaxm

!�1

þνðϕmÞϕ

ϕmaxm �ϕm

!2

; ð35Þ

where ϕmaxm ¼0.585 is the maximum volume fraction at the

jamming point and ϕm ¼ϕmaxm =ð1þ I1=2v Þ. The first two terms on

the right-hand side of Eq. (35) represent the hydrodynamiccontribution ηh to the rheology and they tend to the Einsteinviscosity ηE ¼ 1þ5ϕm=2 at OðϕmÞ. The third term represents solidcontact contributions ηc, where

νðϕmÞ ¼ c1þc2�c1

1þ I0ϕ2mðϕmax

m �ϕmÞ�2; ð36Þ

with c1 ¼ 0:32, c2 ¼ 0:7 and I0 ¼ 0:005 fitting rheological para-meters (Boyer et al., 2011). The expression for the contactcontribution has been chosen to be similar to an analogous onefor dry granular media (Jop et al., 2006).

In order to assess the behavior of our model, we compare thenumerical results obtained with the above mentioned contribu-tions, namely ηE , ηh and ηhþ c , according to the following rheolo-gical description:

η ¼ μeff

μ� 1þεmA Krig

m

� ��1� Kdef

m

� ��1

ð37Þ

obtained by comparison of Brinkman's closure (Eq. (12)) and ours(Eq. (13)). The comparison is based on a simplified 1D approxima-tion of the two closures for unidirectional flow within the porousmedium, which yields A¼ �1:47r2 (see Appendix D for the fullderivation of this value).

The comparison is shown in Figs. 4 and 5, where the perme-abilities in the deformable and rigid case are given by Eq. (33).

05

10100

101102

1030

0.25

0.5

0.75

1

Im

0 0.2 0.4 0.6 0.8 1

0 2 4 6 8 10 12 140

0.2

0.4

0.6

0.8

1

Im

0.1

0.20.3

0.40.5

0.60.7

0.8

0.9

Fig. 3. Micro-scale phase diagram for deformable fibrous media (a). The diagram shows the iso-colors of the cluster density Ω as a function of the microscopic viscousnumber Im, the normalized microscopic porosity ~εm and the maximum extensibility ξmax (ξc=ξmax ¼ 0C1 and D¼ 10�12 m2=s). The numerical data are best fitted and areduced number of points per dimension are shown to allow a proper visualization. Figure (b) shows the iso-Ω contours on the plane ξmax ¼ 10 in (a). (For interpretation ofthe references to color in this figure caption, the reader is referred to the web version of this paper.)

L. Bergamasco et al. / Chemical Engineering Science 126 (2015) 471–482 477

In the rigid case Ω¼ 1 (constant). The volume fraction has beennormalized as it is done for the porosity in Eq. (34). The normal-ization allows a better comparison of the results, as long as themaximum volume fraction at jamming for wet granular media isϕmax

m ¼ 0:585, while in our case it can be assumed to beϕmax

m ¼ 1�εcm (due to the ordered packing).Let us first focus on the assessment of Eq. (37) for varying Ω in

Kdefm , shown in Fig. 4(a). For Ω¼1, no increase in viscosity is

observed (rigid case). ForΩ¼0.99, the increase in viscosity shownby the deformable fibrous medium lies within that of granularsuspensions. ForΩ¼0.9, the deformable fibrous medium presentsa rheological behavior which is qualitatively comparable withthat of wet granular suspensions. For Ω40:9, the out-of-planeforces dominate the behavior yielding higher reduced viscositiesthat of wet granular suspensions. The results are shown up toΩ¼0.01.

The numerical data is shown in Fig. 4(b), where the analyticalbounding values Ω¼1 and Ω¼0.01 are reported. In order to plot

these data, the Ω values are extracted from the micro-scale phasediagram in Fig. 3(b), for constant viscous number Im or constantporosity ~εm. The extracted Ω values are then used for Kdef

m inEq. (37). This procedure allows us to get the equivalent rheologicalbehavior and to relate Eq. (37) to Im. The data show qualitativeagreement with the rheology of wet granular suspensions; theincrease in viscosity is larger than that of granular media, with abehavior which lies within the bounding values. For a constantviscous number Im and increasing volume fractions, the reducedviscosity increases, due to the increasing clustering of the fibers(i.e. reduced mobility).

The effect of considering different values of the diffusioncoefficient D for Im¼0 is shown in Fig. 5(a). The results arereported for the range explored in this paper (see Table 2). Thelarger the diffusion, the larger the increase in viscosity for a givenvolume fraction. In the range explored, the increase in viscositylies between the analytical bounds (exception made only for thecase D¼1e�12 m2/s and ~ϕm ¼1, which presents a higher value).

0 0.2 0.4 0.6 0.8 1100

101

102

103

104

105

106

107

Wet granular (Eq. (35))

Def. Fib. Med. (Eq. (37))

constant (analytical)

0.5

0.9

0.1

= 0.01

10.99

0 0.2 0.4 0.6 0.8 1100

101

102

103

104

105

106

107

Wet granular (Eq. (35))

Def. Fib. Med. (Eq. (37))

Im

= 0 (numerical)

Im

= 5 (numerical)

Im

= 10 (numerical)

Fig. 4. Comparison of the reduced viscosity given by the proposed model for deformable fibrous media (Eq. (37)) with that of wet granular media (Eq. (35)). (a) Viscosity as afunction of the volume fraction ~ϕm for constant Ω in Eq. (37) (analytical). (b) Viscosity as a function of the volume fraction ~ϕm given by the numerical model for differentviscous numbers Im. The numerical data lie within the analytical bounds for Ω¼0.01 and Ω¼1 (compare Fig. 4(a)). The maximum extensibility is ξmax ¼ 10, the diffusioncoefficient is D¼ 10�10 m2=s and A¼ �1:47r2 in Eq. (37)) (see Appendix D for the derivation of this value). (For interpretation of the references to color in this figure caption,the reader is referred to the web version of this paper.)

0 0.2 0.4 0.6 0.8 1100

101

102

103

104

105

106

107 Wet granular (Eq. (35)) Def. Fib. Med. (Eq. (37))

D = 1e−09 m2/s (numerical)

D = 1e−10 m2/s (numerical)

D = 1e−11 m2/s (numerical)

D = 1e−12 m2/s (numerical)

Im

= 0

0.1 1 10100

101

102

103

104

105

Im

Wet granular (Eq. (35)) Def. Fib. Med. (Eq. (37))

Fig. 5. Comparison of the reduced viscosity given by the proposed model for deformable fibrous media (Eq. (37)) with that of wet granular media (Eq. (35)). (a) Viscosity as afunction of the volume fraction ~ϕm given by the numerical model for Im¼0 and different values of the diffusion coefficient. (b) Viscosity as a function of the viscous numberIm given by the numerical model for different values of volume fraction. The maximum extensibility is ξmax ¼ 10, A¼ �1:47r2 in Eq. (37)) (see Appendix D for the derivationof this value) and in (b) the diffusion coefficient is D¼ 10�9 m2=s. (For interpretation of the references to color in this figure caption, the reader is referred to the web versionof this paper.)

L. Bergamasco et al. / Chemical Engineering Science 126 (2015) 471–482478

Finally, and to conclude the discussion, we show the behavior ofthe proposed model for constant values of the volume fraction andvarying viscous number (Fig. 5(b)). We show the results for threerepresentative values of volume fractions: low range ( ~ϕm ¼ 0:130),intermediate range ( ~ϕm ¼ 0:525) and upper-intermediate range( ~ϕm ¼ 0:635). The numerical data is always extracted from the phasediagram, Fig. 3, and used for Kdef

m in Eq. (37), as previously explained.For constant volume fraction and increasing viscous number, shear-thinning is always observed. This physically recovers the increasingstiffness of the fibers and thus the lower clustering probability.Fig. 5(b) shows that, for very low volume fractions ( ~ϕm ¼ 0:130), anear Newtonian-like behavior is obtained. For an intermediatevolume fraction ( ~ϕm ¼ 0:525), the non-Newtonian reduced viscosityobtained exhibits shear-thinning behavior with low- and high-shearplateaux. The reduced viscosity is of the order of that of the rheologyof wet granular media. For high volume fractions ( ~ϕm ¼ 0:635) andhigh shear rates, the behavior changes abruptly from shear thinningto shear thickening (compare Fig. 3(b)). This phenomenon is relatedto the restriction caused by the out-of-plane forces, which leads thefibers close to jamming.

4.3. Multi-scale model

In order to assess the accuracy of the FVM numerical solution,we firstly analyze the permeability of a dual-scale rigid porousmedium (Fig. 1(a)). The domain is periodic, and the macroscopicReynolds number ReM ¼ ρvinR=μ is kept constant at 10�3 (creepingflow). The microscopic permeability is computed according toEq. (1) considering a square fiber arrangement. The effectivepermeability is then recovered by Darcy's law as

Keff ¼ � μΔp

vin; ð38Þ

where Δp is the stream-wise pressure drop per unit length. Thenumerical results (Fig. 6) show very good agreement with thecorrelation for multi-scale fibrous media of Eq. (2). As expected,the effect of the microscopic porosity εm on the effective perme-ability is important for low macroscopic porosities, in this case forεMo0:6. In this range, the effective permeability can vary up totwo orders of magnitude, depending on the microscopic porosityεm.

At the microscopic scale, the permeability can vary with respectto the rigid case according to the topology of the fibers induced bythe fluid flow. The entity of this variability is shown in the inset ofFig. 6, which shows that the permeability of the rigid case given byEq. (33) forΩ¼ 1 slightly underestimates that given by Eq. (1). Thepermeability decreases with the clustering of the fibers, that is,with decreasing Ω. If Ω tends to zero, the permeability also tendsto zero, which means that the yarn is impermeable (we show onlyΩ¼ 0:1 for illustration). Therefore, better conditions for infiltra-tion are achieved for high Ω, that is, high (dimensionless) strainrates (compare Fig. 3(b)).

In order to analyze this effect on the effective permeability, amulti-scale phase diagram (Fig. 7) is built with the effectivevolume fraction ϕeff , which is a commonly used quantity fortextiles since it is experimentally measurable through the weightand volume of the textile. For a given ϕeff , the relationshipbetween micro and macroscopic porosities cannot be easilycomputed; however, the effective volume fraction can be writtenin terms of the porosities in the following form:

ϕeff ¼ 1�εmð Þ 1�εMð Þ; ð39Þwhich describes all the possible configurations of the textile. Thediagram in Fig. 7 shows the whole range of micro- and macro-scopic porosities, down to the percolation thresholds (εcm and εcM).The blue area corresponds to εM40:6, where the microscopic

porosity has a negligible effect on permeability (Fig. 6). The greenarea identifies the range of microscopic porosity from the percola-tion threshold to 0.5, where the effects of fiber deformations arenegligible due to the high packing of fibers and thus to theirreduced mobility. The red area represents the range of micro- andmacroscopic porosities where the effect of fiber deformationand thus of the clustering significantly modifies the effectivepermeability.

The inset shows the pathlines of the flow field on top of the Ωcontours obtained with the multi-scale model in the deformablecase at the corresponding point (p) in the diagram. The modelparameters are: ξmax ¼ 10, εM ¼ 0:36, εm ¼ 0:8, IM¼0.6 andξc=ξmax ¼ 0:36 with D¼ 10�11 m2=s. For this case, the effectivepermeability is 10% lower than in the rigid case due to theclustering of fibers in the yarn.

0.2 0.4 0.6 0.8 110−6

10−4

10−2

100

102

M

Kef

frig

/ R

2

m = 0.3

m = 0.5

m = 0.8

m = 0.9

0.2 0.4 0.6 0.8 110−6

10−4

10−2

100

102

m

K/r

2

Eq. (1)Eq. (33)

= 1

= 0.1

Fig. 6. Comparison of the numerically predicted effective permeability on therepresentative elementary volume REVM in Fig. 1(a) with the analytical solution ofEq. (2) for rigid porous media. In the inset, comparison of the microscopicpermeability given by Eqs. (1) and (33) is shown. The permeability range varieswith Ω (red zone) according to Eq. (33). Higher permeabilities correspond to Ω-1;lower permeabilities for Ω-0. (For interpretation of the references to color in thisfigure caption, the reader is referred to the web version of this paper.)

Fig. 7. Multi-scale phase diagram for fibrous media. In the blue area, the micro-scopic porosity has a negligible effect on permeability; in the green area, the effectsof fiber deformations are negligible; in the red area, the effect of fiber deformationsignificantly modifies the effective permeability. In the inset, flow field configura-tion at point (p) is shown. The model parameters for the simulation are: ξmax ¼ 10,εM ¼ 0:36, εm ¼ 0:8, IM¼0.6, ξc=ξmax ¼ 0:36, D¼ 10�11 m2=s and z¼Lz=2. The color-bar shows the Ω scale for the contours and the velocity magnitude (withinparentheses) for the pathlines. (For interpretation of the references to color in thisfigure caption, the reader is referred to the web version of this paper.)

L. Bergamasco et al. / Chemical Engineering Science 126 (2015) 471–482 479

In order to present the results in a compact manner, we show theeffect of the clustering of fibers in the yarn on the effective perme-ability using the phase diagrams in Fig. 8(b) and (a). The effectivepermeability is given by Eq. (2), where the macro-permeability KM isgiven by Eq. (1) and the microscopic Km by Eq. (33). The qualitativelimiting values shown in Fig. 4a are well recovered, that is, fiberdeformation is important in the red region. Fig. 8(a) shows that fortypical textile parameters, namely intermediate cluster density(Ω¼ 0:5), low to intermediate effective volume fractions (ϕeff o0:4)and intermediate to high micro-to-macro porosities (εm=εM ¼1:5C2:5), the effective permeability of the deformable case can beup to 60% lower than that of the rigid case.

5. Conclusions

A two-dimensional mesoscopic model for fiber deformation andclustering formation in hierarchical fibrous media has been presented,which allows us to compute the micro-scale steady permeability as afunction of the topology of the porous media. The model has beenderived for creeping flow conditions by analogy with non-Browniansuspensions of particles with confining potentials. The resulting multi-scale hydrodynamic system is numerically solved by a coupled finite-volume/lattice-Boltzmann method. The behavior of the proposedmodel has been compared with the rheology of wet granular suspen-sions through a non-Newtonian reduced viscosity and qualitativeagreement has been found.

The microscopic permeability of the fibrous medium has beencharacterized in terms of porosity, dimensionless shear rate anddimensionless out-of-plane forces. The best conditions for infiltra-tion have been found for high shear rates and high out-of-planeforces, that is for rigid fibers and thus reduced clustering.

The effective permeability has been shown to be sensitivelyaffected by clustering over the whole range of volume fractions; inparticular for typical values for textiles, the effective permeabilityof the deformable case can be up to 60% lower than that in therigid case. The results obtained suggest that a better insight on thephysics of the fibers can be helpful in identifying best operatingconditions for infiltration in hierarchical fibrous media.

The present work could be improved in several ways. A non-exhaustive list of issues that deserve further research are: (i) thedrag coefficient in Eq. (23) is a non-confined one, and correctionfactors (typically in a range between 1 and 2) to take into accountconfinement should be included (Clift et al., 2005); (ii) further

physical or chemical potentials within the fibers could be consi-dered, including the effect of capillary forces appearing in unsa-turated conditions; and (iii) in order to reduce the computationalcost, model order reduction techniques could be applied whenusing the mesoscale model in the macroscale framework.

Appendix A. Derivation of the Fokker–Planck equation (Eq. (21))

Let us consider an Ito stochastic differential equation (SDE) inthe form:

dy¼ A dtþB dW ; ðA:1Þwhere A and B represent respectively the drift and diffusioncoefficients. Applying Ito's formula and integrating by parts, theFokker–Planck (FP) equation for the probability density ψ satisfiesthe form Öttinger (1996)

∂ψ∂t

¼ � ∂∂y

Aψ� �þ1

2∂2

∂y2B2ψ� �

: ðA:2Þ

Let us now consider our Ito SDE (Eq. (20)):

drdtðtÞ ¼ vðrðtÞ; tÞþ H

ζLzðrc�rðtÞÞþ

ffiffiffiffiffiffiffi2D

p dWdt

ðtÞ ðA:3Þ

according to the general form of Eq. (A.2), the FP equation in ourcase (in r-coordinates) reads

∂ψ∂t

þ∇r � vðrðtÞ; tÞþ HζLz

ðrc�rðtÞÞ� �

ψ

¼D∇2rψ : ðA:4Þ

For simplicity, let us consider only the r1 coordinate:

∂ψ∂t

þ ∂∂r1

vðr1; tÞþHðrc1 �r1Þ

ζLz

� �ψ

¼D

∂2

∂r21ψ ; ðA:5Þ

and let us call x1 ¼ xc1 and ξ1 ¼ r1�rc1 . We now apply the chainrule on derivatives:

∂v∂r1

¼ ∂v∂x1

∂x1∂r1

þ ∂v∂ξ1

∂ξ1∂r1

¼ ∂v∂ξ1

; ðA:6Þ

and obtain

∂ψ∂t

þ ∂∂ξ1

vðx1þξ1; tÞþHð�ξ1ÞζLz

� �ψ

¼D

∂2

∂r21ψ : ðA:7Þ

We now adopt the local homogeneity assumption, that is, thevelocity is linear in space. This is a plausible assumption, since thelength scale of the dumbbell (fiber bending) is typically orders of

00.10.20.30.40.50.60

1

2

3

4

5

eff

m /

M0 0.2 0.4 0.6 0.8 1

Mc

mc

Keffdef/Keff

rig

00.10.20.30.40.50.60

1

2

3

4

5

eff

m /

M

0 0.2 0.4 0.6 0.8 1

Mc

mc

Keffdef/Keff

rig

Fig. 8. Multi-scale permeability maps of deformable fibrous media: Ω¼ 0:5 (a) and Ω¼ 0:1 (b). The colorbars show the ratio between the effective permeabilities in thedeformable and rigid cases for the representative elementary volume REVM in Fig. 1(a). The permeability in the deformable case is lower than that in the rigid case due to theclustering of fiber. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

L. Bergamasco et al. / Chemical Engineering Science 126 (2015) 471–482480

magnitude smaller than the macroscopic scale. Let us then linearlyexpand the velocity in space:

vðx1þξ1; tÞ � vðx1Þ x1 ¼ xc1

þξ1∂v∂x1

ðx1Þ x1 ¼ xc1

; ðA:8Þ

where vðx1Þjx1 ¼ xc1¼ 0. Passing to tensor notation ðv�∇xv � ξÞ and

substituting we obtain Eq. (21)

∂ψ∂t

þ∇ξ � ∇xv � ξ�HξζLz

� �ψ

¼D∇2

ξψ : ðA:9Þ

Appendix B. CUDA shared memory implementation of thelattice Boltzmann method for the GPU server

Our implementation of the lattice Boltzmann method for themicro-scale equation uses shared memory (CUDA, 2012a), anextremely fast on-chip memory. The main issue to take intoaccount for the implementation of the lattice Boltzmann methodis that it is of limited size and that data is shared only betweenthreads belonging to the same block. For our card (NVIDIAQuadro& 600) the shared memory is 48 kB. The code relies on asingle kernel for collision and propagation (Kuznik et al., 2010) andthe mesh is processed by rows; therefore each block correspondsto a mesh row and the number of threads to the mesh width. Inthis way the size of the data loaded on shared memory per block islimited to that required to process one row. Collision and propaga-tion are then performed on shared memory, where the horizontalpropagation of the distributions is straightforward within theblock, while the vertical one is achieved with a correct alignmentbetween global and shared memory. The coalescence of globalmemory accesses and the lack of bank conflicts on shared memoryis checked using the CUDA (2012b) visual profiler (we do not gointo the details of these issues; the interested reader can refer tothe CUDA, 2012a Programming Guide). With this implementationthe speed-up reaches nearly 60x with respect to the CPU(Bergamasco et al., 2013). We remark that this implementation istailored for small meshes and may not be applicable in other cases.Furthermore, fixing the block size equal to the mesh width sets aconstraint on the choice of the number of threads (which plays akey role in performance).

Appendix C. Coupled finite-volume/lattice-Boltzmann method

The numerical approach for solving the theoretical modelconsists in a coupled finite-volume/lattice-Boltzmann solution.The algorithm has already been applied and validated for non-homogeneous Fokker–Planck equations in dilute viscoelastic sus-pensions (Bergamasco et al., 2013). In this work we deal with ahomogeneous Fokker–Planck equation, therefore the algorithmreduces to the following:

1. solution of Eqs. (14) and (15) using the finite volume method(on the CPU);

2. solution of Eq. (26) using the lattice Boltzmann method (onthe GPU);

3. solution of Eq. (32) for the clustering model (on the GPU);4. correction of Eq. (33) for the local permeability (on the CPU).

The initial guess for the loop is given by the corresponding caseof a rigid porous medium (i.e. without the sub-grid model). Thenumerical algorithm is repeated until global convergence of Eqs.(14) and (15), which in this case requires 4–6 complete loops. Eachloop requires around 2 h of GPU time for a 20–25,000 cellFVM mesh.

Appendix D. 1D model derivation of Eq. (37)

Let us consider a unidirectional flow in the fibrous medium. Letus consider the limit of a single fiber and adopt the representativevolume in Fig. D1 (on the left). We write the equations for thecorresponding volume-averaged description on the right. To thisend, we consider the 1D approximation of the two closures:Brinkman's closure of Eq. (12) and the proposed closure ofEq. (13). We proceed by solving analytically the ordinary differ-ential equations, with the boundary conditions in Fig. D1. Let usthen first integrate twice the ordinary differential equation withour closure (from Eq. (15)):

0¼ �dpdx

þ μεm

d2u

dy2�μK �1

Ω u; ðD:1Þ

0¼ �Z

dpdx

dyþZ

μεm

d2u

dy2dy�

ZμK �1

Ω u dy; ðD:2Þ

0¼ �dpdx

yþ μεm

dudy

�μK �1Ω uyþC1; ðD:3Þ

0¼ �Z

dpdx

y dyþZ

μεm

dudy

dy

�ZμK �1

Ω uy dyþZ

C1 dy; ðD:4Þ

0¼ �dpdx

y2

2þ μεm

u�μK �1Ω u

y2

2þC1yþC2: ðD:5Þ

Let us now apply the boundary conditions to get an explicit formfor the integration constants:

ujy ¼ 0 ¼ 0-C2 ¼ 0; ðD:6Þ

dudy

y ¼ H

¼ 0-C1 ¼dpdx

HþμK �1Ω uH: ðD:7Þ

Substituting the integration constants, we obtain

0¼ �dpdx

y2

2þ μεm

u�μK �1Ω u

y2

2

þdpdx

HyþμK �1Ω uHy; ðD:8Þ

0¼ �dpdx

y2

2�Hy

� �þ μεm

u�μK �1Ω u

y2

2�Hy

� �: ðD:9Þ

Let us now consider Brinkman's closure:

0¼ �dpdx

þ μεm

d2u

dy2�μK �1

D uþ μeff �μ� �

εmd2u

dy2; ðD:10Þ

and proceed again by integrating twice and applying boundaryconditions as above:

0¼ �Z

dpdx

dyþZ

μεm

d2u

dy2dy

�ZμK �1

D u dyþZ μeff �μ� �

εmd2u

dy2dy; ðD:11Þ

Fig. D.9. Scheme for the 1D analytical solution. X-axis is in the flow direction andy-axis is perpendicular to x-axis.

L. Bergamasco et al. / Chemical Engineering Science 126 (2015) 471–482 481

0¼ �dpdx

yþ μεm

dudy

�μK �1D uy

þ μeff �μ� �

εmdudy

þC1; ðD:12Þ

0¼ �Z

dpdx

y dyþZ

μεm

dudy

dy�Z

μK �1D uy dy

þZ μeff �μ� �

εmdudy

dyþZ

C1 dy; ðD:13Þ

0¼ �dpdx

y2

2þ μεm

u�μK �1D u

y2

2

þ μeff �μ� �

εmuþC1yþC2: ðD:14Þ

Applying boundary conditions:

ujy ¼ 0 ¼ 0-C2 ¼ 0; ðD:15Þ

dudy

y ¼ H

¼ 0-C1 ¼dpdx

HþμK �1D uH; ðD:16Þ

substituting the integration constants, we obtain

0¼ �dpdx

y2

2þ μεm

u�μK �1D u

y2

2þdpdx

Hy

þμK �1D uHyþ μeff �μ

� �εm

u; ðD:17Þ

0¼ �dpdx

y2

2�Hy

� �þ μεm

u

�μK �1D u

y2

2�Hy

� �þ μeff �μ� �

εmu: ðD:18Þ

Comparing now the two closures (Eqs. (D.9) and (D.18)), we obtain

0¼ þμK �1Ω u

y2

2�Hy

� ��μK �1

D uy2

2�Hy

� �

þ μeff �μ� �

εmu; ðD:19Þ

0¼ þK �1Ω

y2

2�Hy

� ��K �1

Dy2

2�Hy

� �

þ 1εm

μeff

μ�1

� �; ðD:20Þ

μeff

μ¼ 1þεm

y2

2�Hy

� �K �1D �K �1

Ω

� �; ðD:21Þ

where KD¼Krigm and KΩ¼Kdef

m . Let us evaluate the first term inround brackets on the right-hand side as the weighted-mean:

1H

Z H

0

y2

2�Hy

� �dy¼ �H2

3: ðD:22Þ

Using the definition of porosity on the representative volume, forsquare fiber arrangement H2 can be written as

H2 ¼ πr2

4 1�εmð Þ: ðD:23Þ

Finally, considering a weighted-mean over the porosity:

A¼ 1εmaxm �εcm

Z εmaxm

εcm�13

πr2

4 1�εmð Þ dεmffi�1:47r2; ðD:24Þ

and the reduced-viscosity assumes the following form (Eq. 37):

η ¼ μeff

μ� 1þεmA Krig

m

� ��1� Kdef

m

� ��1

: ðD:25Þ

References

Åström, B.T., Pipes, R.B., Advani, S.G., 1992. On flow through aligned fiber beds andits application to composites processing. J. Compos. Mater. 26 (9), 1351–1373.

Bergamasco, L., Izquierdo, S., Ammar, A., 2013. Direct numerical simulation ofcomplex viscoelastic flows via fast lattice-Boltzmann solution of the Fokker–Planck equation. J. Non-Newton. Fluid Mech. 201 (0), 29–38.

Boyer, F., Guazzelli, É., Pouliquen, O., 2011. Unifying suspension and granularrheology. Phys. Rev. Lett. 107, 188301.

Breugem, W.-P., 2007. The effective viscosity of a channel-type porous medium.Phys. Fluids 19 (10).

Bruus, H., 2007. Theoretical Microfluidics. Oxford University Press, USA.Chen, X., Papathanasiou, T.D., 2008. The transverse permeability of disordered fiber

arrays: a statistical correlation in terms of the mean nearest interfiber spacing.Transp. Porous Media 71 (2), 233–251.

Chwang, A.T., Wu, T.Y., 1976. Hydromechanics of low-Reynolds-number flow. Part 4.Translation of spheroids. J. Fluid Mech. 75, 677–689.

Clift, R., Grace, J.R., Weber, M.E., 2005. Bubbles, Drops, and Particles. DoverPublications, New York, NY, USA.

CUDA Toolkit Downloads, 2012a. URL: ⟨https://developer.nvidia.com/cuda-downloads⟩.CUDA Visual Profiler, 2012b. URL: ⟨https://developer.nvidia.com/nvidia-visual-profiler⟩.Endruweit, A., Gommer, F., Long, A.C., 2013. Stochastic analysis of fibre volume

fraction and permeability in fibre bundles with random filament arrangement.Composites Part A 49 (0), 109–118.

Gebart, B.R., 1992. Permeability of unidirectional reinforcements for RTM.J. Compos. Mater. 26 (8), 1100–1133.

Howes, F.A., Whitaker, S., 1985. The spatial averaging theorem revisited. Chem. Eng.Sci. 40 (8), 1387–1392.

Hunt, A., Ewing, R., Ghanbarian, B., 2014. Percolation Theory for Flow in PorousMedia, 3rd edition Springer, Cham, Switzerland, p. 2014.

Israelachvili, J.N., 2011. Intermolecular and Surface Forces, 3rd edition, AcademicPress, Burlington, MA, USA.

Jop, P., Forterre, Y., Pouliquen, O., 2006. A constitutive law for dense granular flows.Nature 441 (7094), 727–730.

Kuznik, F., Obrecht, C., Rusaouen, G., Roux, J.-J., 2010. LBM based flow simulationusing GPU computing processor. Comput. Math. Appl. 59 (7), 2380–2392.

Larson, R.G., 1998. The Structure and Rheology of Complex Fluids. Oxford UniversityPress, New York, NY, USA.

Marchisio, D.L., Fox, R.O., 2013. Computational Models for Polydisperse Particulateand Multiphase Systems. Cambridge University Press, New York, NY, USA.

Markicevic, B., Papathanasiou, T.D., 2002. On the apparent permeability of regulararrays of nonuniform fibers. Phys. Fluids 14 (9), 3347–3349.

Mecke, K., Arns, C.H., 2005. Fluids in porous media: a morphometric approach.J. Phys.: Condens. Matter. 17 (9), S503.

Nvidia Developer Zone, 2012. URL: https://developer.nvidia.com/category/zone/cuda-zone.Ochoa-Tapia, J.A., Whitaker, A., 1995. Momentum transfer at the boundary between

a porous medium and a homogeneous fluid—I. Theoretical development.Int. J. Heat Mass Transf. 38, 2635–2646.

Öttinger, H.C., 1996. Stochastic Processes in Polymeric Fluids. Springer, Heidelberg,Germany.

Papathanasiou, T.D., 1997. On the effective permeability of square arrays ofpermeable fiber tows. Int. J. Multiph. Flow 23 (1), 81–92.

Papathanasiou, T.D., 2001. Flow across structured fiber bundles: a dimensionlesscorrelation. Int. J. Multiph. Flow 27 (8), 1451–1461.

Papathanasiou, T.D., Gravel, E.M., Barwick, S.C., Dendy, E.D., 2002. Non-isotropicstructured fibrous media: the permeability of arrays of fiber bundles ofelliptical cross section. Polym. Compos. 23 (4), 520–529.

Potter, K., 1997. Resin Transfer Moulding, 1st edition Springer, London, UK, p. 1997.Scholz, C., Wirner, F., Götz, J., Rüde, U., Schröder-Turk, G.E., Mecke, K., Bechinger, C.,

2012. Permeability of porous materials determined from the Euler character-istic. Phys. Rev. Lett. 109, 264504.

Soltani, P., Johari, M.S., Zarrebini, M., 2014. Effect of 3D fiber orientation on perme-ability of realistic fibrous porous networks. Powder Technol. 254 (0), 44–56.

Valdés-Parada, F.J., Goyeau, B., Ochoa-Tapia, J.A., 2007. Jump momentum boundarycondition at a fluid-porous dividing surface: derivation of the closure problem.Chem. Eng. Sci. 62, 4025–4039.

Whitaker, S., 1986. Flow in porous media I: a theoretical derivation of Darcy's law.Transp. Porous Media 1 (1), 3–25.

Yazdchi, K., Luding, S., 2012. Towards unified drag laws for inertial flow throughfibrous materials. Chem. Eng. J. 207–208, 35–48.

Yazdchi, K., Luding, S., 2013. Upscaling and microstructural analysis of the flow–

structure relation perpendicular to random, parallel fiber arrays. Chem. Eng. Sci.98 (0), 173–185.

Yazdchi, K., Srivastava, S., Luding, S., 2011. Microstructural effects on the perme-ability of periodic fibrous porous media. Int. J. Multiph. Flow 37 (8), 956–966.

Yazdchi, K., Srivastava, S., Luding, S., 2012. Micro–macro relations for flow throughrandom arrays of cylinders. Composites Part A 43 (11), 2007–2020.

Young, W., Budynas, R., Sadegh, A., 2011. Roark's Formulas for Stress and Strain, 8thedition McGraw-Hill Professional, New York, NY, USA, p. 2011.

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