a model of uid ows through poroelastic media · this draft was prepared using the latex style le...

This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1

A model of fluid flowsthrough poroelastic media

Giuseppe A. Zampogna1†, Ugis Lacis2, Shervin Bagheri2 andAlessandro Bottaro1

1Dipartimento di Ingegneria Civile, Chimica e Ambientale, Universita di Genova, ViaMontallegro 1, Genova, 16145, Italy

2Linne Flow Center, Department of Mechanics, KTH, SE-100 44, Stockholm, Sweden

(Received xx; revised xx; accepted xx)

Multiscale homogenization can prove a very powerful theory to treat certain fluid-structure interaction problems involving poroelastic media. This is shown here for the caseof the interaction between a newtonian fluid and a poroelastic, microstructured material.Microscopic problems are set up to determine effective tensorial properties (elasticity,permeability, porosity, bulk compliance of the solid skeleton) of the homogenized medium,both in the interior and at its boundary with the fluid domain. The macroscopicequations which are derived by homogenization theory employ such effective propertiesthus permitting the computation of velocities and displacements within the poroelasticmixture for two representative configurations of standing and travelling waves.

Key words:

1. Introduction

Solid surfaces in nature and technological applications are never perfectly smooth;most often, when a fluid flows above a surface, it encounters a series of (more or lessrigid) irregularities, around which it has to move and with which it might interact,whose typical dimensions are often much smaller than those of the macroscopic flowstructures present in the fluid domain. Examples of the above include the surfaces ofhydrophobic and hydrophilic materials, for which the interfacial energy determines theability of the material to bond with water (i.e. its wettability), and super-hydrophobicor super-hydrophilic materials, for which the roughness or surface topography, producedby micro-fabrication and coating processes, yields enhanced properties of water adhesionor repellence. Biological surfaces represent another interesting example: they are coveredby hair, filaments, scales, feathers, leaves, needles, cilia, etc., which perform a varietyof functions. Even if such features are microscopic, their collective effect is often macro-scopic, and it is thus justified to try and apply homogenization theory to such porous andcompliant media to properly represent their large-scale effects when they interact with afluid. The simplest (and earliest) homogenized condition which has been used to considersuch interactions is Navier’s slip condition (Navier 1823). If a macroscopic interface ΓIseparating a fluid from a thin poroelastic layer can be defined, for example through a

† Present address: Institut de Mecanique des Fluides de Toulouse, 2, Allee duProfesseur Camille Soula, Toulouse, 31400, France. Email address for correspondence:[email protected]

2 G. A. Zampogna, U. Lacis, S. Bagheri & A. Bottaro

homogenization process, and if n is the unit normal vector directed into the fluid, in theframe of reference of the layer the macroscopic interface condition reads

u · n = 0, u‖ = 2λE · n − [(E · n) · n] n at ΓI ,

with u the fluid velocity, u‖ the vector of velocity components tangent to the interface,

and E = 12 (∇u +t ∇u) the rate of strain tensor. The parameter λ is a slip length,

i.e. the fictitious distance below the reference interface where the velocity vanishes: theno-slip condition is recovered for λ = 0 and perfect slip is found for λ → ∞. TheNavier’s slip condition for u‖ can be interpreted as an asymptotic approximation to treatconfigurations for which either the continuum equations break down, or the interfacegeometry presents a microstructure. The challenge is clearly that of finding the sliplength, or the slip tensor in the case of anisotropic interfaces, and to link it to thegeometrical and physical properties of the poroelastic medium.

If the layer is thick, homogenized equations for the medium itself must be establishedand solved, before the coupling with the fluid can be attempted. Situations in whichporoelastic media of large dimensions interact with a fluid abound, ranging from thegrowth of tumor cells (Penta & Ambrosi 2013) to the waving of canopies (Nepf 2012).Another interesting poroelastic medium is that formed by the plumage of birds; Sarradjet al. (2011) studied how the owl hides its acoustic signature thanks to its feathers,composed by barbs, barbules and barbicels of varying dimensions (Bachmann et al. 2007).The hierarchical structure of feathers represents the main feature of homogenization: thetypical pores (or, equivalently, the size of the solid inclusions) of the medium must bemuch smaller than a characteristic dimension of the flow, for multiple scale theory to hold.Thus, microscopic equations can be set up and used to solve for the flow and the structureat small scale in the poroelastic layer (PEL, in the following), yielding tensorial quantitiessuch as effective elasticity or permeability, and companion macroscopic simulations mustbe conducted in both the fluid and the PEL.

The history of poroelasticity, with a porous, deformable medium saturated by a liquidtreated as a unique continuum, goes back to the work by Terzaghi (1936) who introducedan effective stress tensor to quantify consolidation and fracture of soils. Later, Biot (1955)characterized a poroelastic medium via an effective elasticity tensor. Further progress inthis direction was made, among others, by Rice & Cleary (1976) and Burridge & Keller(1981), before the development of a more rigorous mathematical approach to deriveeffective equations, the homogenization theory (e.g. Lee & Mei 1997).

One of the limitations of homogenization in its early expression (see, e.g. Mei &Vernescu 2010) is that it treats only unbounded homogeneous regions. The effectiveequations in rigid, porous media have historically been coupled to the fluid equationsthrough empirical laws (often similar to the Navier’s slip law written above, but withouta theoretical basis). A review of some macroscopic interface conditions is given byZampogna & Bottaro (2016). Supported by theoretical studies (Jager & Mikelic 1996;Mikelic & Jager 2000), Lacis & Bagheri (2016) and Lacis et al. (2017) have recentlyproposed – and validated in the Stokes flow limit – consistent interface conditions basedon homogenization which are similar to those developed by Carraro et al. (2015) withdifferent arguments. In the present paper, these same conditions are used to handle theinterface between a poroelastic medium and a free fluid domain when inertia is notnegligible.

The main objective of the present work is to develop rigorous microscopic and macro-scopic models to treat fluid-structure interaction problems, when a “pure” fluid flowis in contact with a fluid-saturated poroelastic media. The model equations rely on a

A model for fluid flows through poroelastic media 3

o x1x2

x3

Vs

Vf

V ΓVTOT

lL

Figure 1. Macroscopic configuration of a poroelastic matrix and microscopic zoom over one

microstructure inside a cubic representative elementary volume V (defined by the upscaling

technique). L and l are the macroscopic and microscopic length scales; Vs and Vf are the

regions, inside V , occupied by the solid and the fluid, respectively, and Γ is the fluid-structure

interface within V .

multiple-scale expansion, both for the interior of the PEL and for its interface withthe fluid domain, ΓI . The microscopic problems arising from homogenization, once thescaling parameters are properly set, are described in detail and solved for a range ofporosities, when the medium is composed by both fiber-like and sphere-like structures.The significance of the effective properties of the PEL is highlighted. In this respect,the present work can be seen as the numerical counterpart of the work of Gopinath &Mahadevan (2011), where effective properties of poroelastic bristles, carpets and brusheswere obtained from physical modeling. Once they are known, such effective propertiescan be used in the solution of macroscopic problems.

2. A multiscale mathematical model for poroelastic lattices

A continuum model for the fluid-solid mixture denoted in figure 1 as VTOT is described.In such a macroscopic domain, an elastic porous skeleton of density ρs is saturated by anincompressible newtonian fluid of density ρf and viscosity µ; VTOT can be decomposed,at least far from its boundaries, in elementary cells over which the structure can beapproximated as periodic, as sketched in figure 1. Although the elementary cell is fixedin time, t, and centered in x, its fluid and solid portions depend on time because of thedeformation so that, in general, it follows that

V (x) = Vs(x, t ) ∪ Vf (x, t ), (2.1)

where the dependence on the centroid x of each cell is due to the conditions enforcedon the boundaries of the macroscopic poroelastic region (the · denotes a dimensionalvariable); for ease of notation we will omit both of these dependencies. The equation forthe solid motion in the unit cell must be coupled with the equations for the fluid. Thus,we consider the dimensional equation for the structure, i.e.

ρs∂2vi

∂t2=∂σij∂xj

on Vs, (2.2)

where vi are the components of v, the solid displacement vector, and σij are thecomponents of the solid stress tensor. Under the assumption that the structure is linearlyelastic, for small strain the following equation holds

σij = Cijklεkl(v) =1

2Cijkl

(∂vk∂xl

+∂vl∂xk

), (2.3)


with Cijkl the fourth order stiffness or elasticity tensor, whose components dependlinearly on Young’s modulus of elasticity, E, and on the Poisson ratio, νP , for a solidskeleton formed by an isotropic material. In the remainder of the paper, Cijkl is the

dimensionless counterpart of Cijkl, using E as scale. Also, we will fix the value of thePoisson ratio at 0.33 so that in Voigt’s contracted notation (Voigt 1889) the isotropicstiffness tensor of the material forming the solid skeleton is

Cijkl =

1.48 0.72 0.72 0 0 00.72 1.48 0.72 0 0 00.72 0.72 1.48 0 0 0

0 0 0 0.38 0 00 0 0 0 0.38 00 0 0 0 0 0.38

. (2.4)

Equation (2.2) must be coupled to the dimensional Navier-Stokes equations (NSE) forthe velocity field u and the pressure field p, written in the form:

ρf

(∂ui

∂t+ uj

∂ui∂xj

)=∂Σij∂xj

on Vf , (2.5)

∂ui∂xi

= 0 on Vf , (2.6)

where Σij is the canonical fluid stress tensor

Σij = −pδij + 2µεij(u), (2.7)

with ε the strain tensor for the fluid, formally defined like for the structure. The boundaryconditions on the solid-fluid interface Γ are the continuity of velocities and normalstresses:

ui =∂vi

∂tand Σijnj = σijnj , (2.8)

where n is the unit normal vector pointing from the solid to the fluid; moreover, weimpose periodicity over all boundaries of the microscopic reference volume V .

At this point we invoke the separation of scales between L and l (cf. fig 1), whichallows to define the small parameter ε = l/L 1 and to introduce two non-dimensionalvariables: the microscopic fast variable, x = x/l, and the macroscopic slow variable,x′ = εx. The use of homogenization theory (Mei & Vernescu 2010) leads us to macroscopicgoverning equations via the use of spatial averaging over V :

〈·〉 =1

V

∫V

· dV . (2.9)

The macroscopic description at leading order of the anisotropic, compressible skeleton’sdisplacement field v(x′, t) and of the fluid pressure field p(x′, t), whose mathematicaldevelopment is detailed in Appendix A, is given by

(1− θ) ρsρf

Cavi =∂

∂x′j

[Cijpqε′pq(v)− α′ijCa p

]on VTOT ,

βCap− ε2ReLKij∂2p

∂x′j∂x′i

= −αlk∂vl∂x′k

on VTOT .

(2.10)

These equations contain all the dominant terms and the presence of an ε2 term is onlyrelated to the fact that they have been rendered dimensionless by the same scales used


in the free fluid region bordering the PEL, namely the characteristic velocity within thefree fluid domain, U , the macroscopic length scale, L, the dynamic pressure, ρf U

2, andthe time scale L/U . The Reynolds number is ReL = ρf UL/µ and the Cauchy numberis Ca = ρf U

2/E. In (2.10) the porosity θ = Vf/V appears, plus some tensors whichvary only over the macroscale; they are the effective elasticity tensor, Cijkl, the effectivefluid volume fractions, αlk and α′ij , the effective permeability tensor, Kij and the bulkcompliance of the solid skeleton, β. All of these tensors (aside from Kij) depend on twofurther tensors, χpqi and ηi, which are defined on the periodic elementary volume V andsatisfy the systems:

∂

∂xjCijkl [εkl(χpq) + δkpδlq] = 0 on Vs,

Cijkl [εkl(χpq) + δkpδlq]nj = 0 on Γ,

∂

∂xj[Cijkl εkl(η)] = 0 on Vs,

[Cijkl εkl(η)]nj = −ni on Γ.(2.11)

By using the dimensionless form of (2.9), the effective elasticity tensor is defined as:

Cijpq = 〈Cijkl εkl(χpq)〉+ 〈Cijpq〉, (2.12)

where Cijkl is given by equation (2.4). The first effective fluid volume fraction of theanisotropic skeleton is

αlk = θδlk −1

2〈∂χ

pqi

∂xi〉(δlpδkq + δlqδkp); (2.13)

this form of αlk generalizes the concept introduced for isotropic media by Skotheim &Mahadevan (2004). The vector ηi is used to define the second effective fluid volumefraction, i.e.

α′ij = θδij + 〈Cijkl εkl(η)〉, (2.14)

plus the bulk compliance of the solid skeleton

β = 〈∂ηi∂xi〉. (2.15)

It has been shown by Mei & Vernescu (2010) that the two volume fractions coincide, i.e.αij = α′ij and this is verified in the present paper as a check of numerical consistency. Toclose (2.10) the effective permeability tensor Kij = 〈Kij〉 needs to be defined; it arisesfrom the solution of a Stokes problem over the fluid domain Vf :

∂Aj∂xi− ∂2Kij

∂xk∂xk= δij on Vf ,

∂Kij

∂xi= 0 on Vf ,

Kij = 0 on Γ.

(2.16)

Once (2.10) is solved, the fluid velocity over VTOT can be recovered from Darcy’s law:

〈ui〉 − θvi = −ε2ReLKij∂p

∂x′j(2.17)

for the effective velocity of the homogenized medium, 〈ui〉 − θvi. For ease of notation,from this moment on, we refer to the system formed by (2.10) and (2.17) as the FPEL(Fluid-PoroELastic) model. It should be stressed again that the FPEL equations, asgiven here, are rendered dimensionless, and later solved numerically, using the free fluidscales, and this for immediate coupling at the solid-PEL interface with the dimensionlessNavier-Stokes equations.


Figure 2. A pure fluid region and a composite region (VNS and VPEL, respectively) in contactthrough the fluid-structure macroscopic interface ΓI . Information between the NS and FPELsolvers is exchanged using equations (2.18–2.19).

2.1. Focus on the macroscopic interface

When a poroelastic material is in contact with a macroscopic domain only filled byfluid, it is convenient to introduce a macroscopic interface between these two domains,which we denote as ΓI (defined in figure 2). In order to solve for the displacement ofthe poroelastic material and for the pore pressure we need essentially two boundaryconditions, one for each equation from the equation system (2.10). For the elasticityequation from the system (2.10) we use the stress continuity condition, which has beenshown to work well for Stokes flows by Lacis et al. (2017),

Tijnj |ΓI= Σijnj , (2.18)

where Σij is the stress tensor of the free fluid, and Tij is the stress tensor of the mixture inthe interior domain (cf. eq. A 42 in Appendix A). Same condition has been also proposedpreviously by Gopinath & Mahadevan (2011). For the pore pressure equation from thesystem (2.10) we use a Dirichlet condition on the pore-pressure value. There has not beenan agreement in literature on whether the pressure should be the same as in free fluid(Lacis & Bagheri 2016) or there should be a jump in pressure (Carraro et al. 2013). Inthe current work, we select the pressure continuity

p− = p, (2.19)

where p is the free fluid pressure and p− is the pore pressure directly below the inter-face. It has been shown by Lacis et al. (2017) that pressure continuity works well forgiven geometries in poroelastic set-ups. One can also expect pressure continuity due tosymmetry reasons (Carraro et al. 2013).

To close the formulation, one also needs boundary conditions for the free fluid. Forthe standard Navier-Stokes equation system it is sufficient to provide velocity boundarycondition. We use the velocity boundary condition proposed by Lacis et al. (2017) forporoelastic materials, which in dimensional form reads

〈ui〉|ΓI=∂vi∂t−[Kij l2

µ

]〈p〉,j +

[Lijk l

](〈uj〉,k + 〈uk〉,j) . (2.20)

Making the boundary condition non-dimensional using the same scales as for governingequations, gives us the final form of the velocity boundary condition

〈ui〉|ΓI= vi − ε2ReLKij

∂p

∂x′j+ εLijk

(∂uj∂x′k

+∂uk∂x′j

), (2.21)

where the bars above the unknown tensors denotes that they are obtained using volumeaverages above the interface in an elongated interface cell (Lacis et al. 2017) near theboundary between the poroelastic medium and the free fluid (see later figures 10 and 11).Equation (2.21) represents the extended version of Navier’s slip condition. The coefficients


arising in the velocity boundary conditions are then found as

Kij =1

V

∫Vi

Kij dV , (2.22)

Lijk =1

V

∫Vi

Lijk dV , (2.23)

where Vi is the averaging volume in the interface cell (Lacis & Bagheri 2016; Laciset al. 2017). For the sake of completeness, the interface cell problems are provided in theAppendix B.

3. Microscopic problems

In this section the solutions of the microscopic problems (2.11) and (2.16) for theinternal region of the poroelastic medium, and of problems (B 1–B 4) for the interfacialregion are presented. For each problem we have considered two kinds of microscopicstructures, shown in figure 3: both are cubic symmetric structures, the former is builtstarting from an isotropic geometry, composed by spheres linked by transversal cylindricalbars (denoted as “spheres” in the following) and the latter is built starting from atransversely isotropic geometry, composed by a main vertical cylinder aligned along x3and traversed by two slender cylinders with axes parallel to x1 and x2, respectively(denoted as “cylinders”). The ratio between the radius rc of the connecting cylinders andthat of the sphere or of the vertical cylinder, r, is here fixed for each porosity (rc/r = 0.4).The limit value of porosity is constrained by the fact that the connected network needsto be permeable in all directions, i.e. the fluid in Vf must be able to flow throughoutthe grid; this is realized by porosities in the interval (0.04, 1) for the case of “spheres”and (0.3, 1) for “cylinders”. The need of considering fully permeable porous media (i.e.connected fluid regions) has its solid counterpart in the need of using connected skeletons:if this were not the case, there would be no elastic response to a deformation along thedirection in which the structure is disconnected (cf. Hoffmann et al. 2004).

The general form of the (symmetric) effective stiffness tensor in Voigt’s notation forthe microscopic structures examined, is

Cijkl =

C1111 C1122 C1133 0 0 0C1122 C1111 C1133 0 0 0C1133 C1133 C3333 0 0 0

0 0 0 C2323 0 00 0 0 0 C1313 00 0 0 0 0 C1212

,

with C2323 = C1313 and C1212 6= 12 (C1111−C1122) as a consequence of the lateral connecting

cylindrical rods. There are thus, at the most, six independent elements. In the “spheres”case the independent terms reduce to three, since C3333 = C1111, C1122 = C1133, andC1212 = C1313 = C2323. This is the case of the simplest possible anisotropic structure,that with cubic symmetry. In the “cylinders” case all six coefficients must be solved for.The transverse isotropy in this case is broken by the presence of the C1212 term, whichis independent of C1111 and C1122; the reason is that not all planes through the axis ofsymmetry x3 of the main cylinder are planes of symmetry of the whole structure.


Figure 3. The two different microscopic structures analyzed in the present work, shown herein the limit case of very small and very large porosities.

Figure 4. Magnitude of the independent components of χ in the case of “spheres” for θ = 0.8.

3.1. Solution of the interior problem

A detailed parametric study is presented here for the interior microscopic problem†for both the cases of “spheres” and “cylinders”, when rc/r = 0.4. Results for other radiiratios are presented by Zampogna & Bottaro (2017).

Attention is initially focussed on problems (2.11). The magnitude of the independententries of the microscopic field χ are represented in figure 4 for a set of parameters usedlater. Since only the components of the average over Vs of the jacobian of χ and η,J(χij)kl, J(η)kl, i, j, k, l = 1, 2, 3, are present in the macroscopic equations, we need tolook at these elements carefully. In the case of “spheres” the non-zero components ofJ(χij)kl and J(η)kl are listed in the following equations, highlighting their symmetriesbecause of the isotropy of the material and the (cubic) symmetry of the microscopicstructure:

J(χ11)11 = J(χ22)22 = J(χ33)33, (3.1)

J(χll)kk = J(χ11)22, l, k = 1, 2, 3, l 6= k,

J(χlk)lk = J(χlk)kl = J(χ12)12, l, k = 1, 2, 3, l 6= k,

J(η)11 = J(η)22 = J(η)33.

The values of the four independent coefficients are represented in figure 5 for porosities

† The solution of the interior problem has been carried out using OpenFoam, with an adhoc solver already described in Zampogna & Bottaro (2017). The numerical convergence of theresults has been verified using three different mesh sizes with up to 8×106 cells per unit volume.Some results have been independently verified by a solver based on FreeFem++ (Hecht 2012).


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

3

102

101

100

1 θ=V /V

|J(χ11)11|

|J(χ11)22|

|J(χ12)12|

|J(η)11|

Figure 5. Absolute values of the four independent parameters identified by equations (3.1) asfunction of the porosity for the “spherical” microscopic structure.

0 0.2 0.4 0.6 0.8 110

5

104

103

102

101

100

101

1 θ=VS/V

C1111

C1122

C1212

0 0.2 0.4 0.6 0.8 1

1-θ=VS/V

0

0.2

0.4

0.6

0.8

1

αii,α' ii

αii

α 'ii

Figure 6. Components of the effective elasticity tensor Cijkl (left) and effective fluid volumefractions αij and α′ij (right) in the case of “spheres”.

ranging in the interval (0.04, 0.97). Combining equation (3.1) with (2.12) we obtain thatCijkl is defined by three independent elements.

The sudden slope discontinuity which appears in all the curves in figures 5 and 6 closeto 1 − θ = 0.6 is due to a change in the topology of the main central sphere in theunit cell: for θ > 0.6 the presence of the cylinders to connect the microscopic structureis not needed anymore and Vs can be simply realized via the intersection between aunitary cube and a sphere centered in it. Figure 6 demonstrates how the skeleton behavesfollowing a deformation: the elastic response (i.e. the intensity of the bulk and shearmoduli) becomes smaller when the porosity increases, i.e. when the central sphere issmaller (or when the main cylinders are slender). Moreover, it is important to note thateven if we are evaluating a cubic symmetric structure on a cubic elementary cell, theinformation about the isotropy of the initial geometry, the sphere, is maintained: therelation C1212 = C1313 = C2323 holds, which, in general, is not true for cubic symmetricstructures. Even if αij and α′ij are defined in a completely different way, they assume thesame numerical value, since for generic microscopic structures it is

〈εii(χpq)〉 = −〈Cpqlkεlk(η)〉 p = q = 1, 2, 3. (3.2)

It should also be noted that the quantities shown in eq. (3.2) are of the same order of


Figure 7. Magnitude of the independent components of χ in the case of “cylinders” forθ = 0.8.

the porosity itself, especially for intermediate values of θ, so that the effective fluidvolume fraction αii = α′ii displays changes with respect to the porosity because ofcompressibility effects. If the compressibility of the material decreases (i.e. νP approaches0.5), η and χ decrease and the difference between θ and α becomes smaller. Focusing onthe definitions of α and α′ in eqs. (2.13, 2.14), we observe that α varies with the porositybecause of internal deformations of the structures; such internal deformations are dueto external forcing on Γ , whose dependence on θ is measured by α′ (cf. eq. (A 37)).This interpretation of the effective fluid volume fractions confirms why the intensity ofvariations with respect to θ are the same for α and α′, as stated in (3.2).

In the “cylinders” case, the magnitude of the independent components of the micro-scopic field χ are represented in figure 7 for a given set of parameters. In this case,since the initial microscopic structure is transversely isotropic, there are ten independentcoefficients of the jacobians of χ and η, listed below and represented in figure 8 forvarying values of θ:

J(χ11)11 = J(χ22)22, (3.3)

J(χ11)22 = J(χ22)11,

J(χ11)33 = J(χ22)33,

J(χ33)11 = J(χ33)22,

J(χ33)33,

J(χlk)kl = J(χlk)lk = J(χ12)12, l, k = 1, 2, l 6= k,

J(χl3)l3 = J(χ3l)l3 = J(χ13)13, l = 1, 2,

J(χ3l)3l = J(χl3)3l = J(χ13)31, l = 1, 2,

J(η)11 = J(η)22,

J(η)33.

The components of the effective stiffness tensor are represented in figure 9. Theconsiderations valid for the case of spheres, about the behavior of the elastic responseof the structure, hold also in the cylinder case. In particular, the information about thetransverse isotropy of the starting geometry, the main vertical cylinder, are conservedin the shear modulus, i.e. C1313 = C2323 6= C1212. Also in the case of “cylinders” αii andα′ii are equal to within numerical accuracy, as it should be, and this represents a furthercheck on the reliability of the numerics.

Comparing the effective elasticity tensor, for both the structures analyzed, in thelimit for Vf which approaches zero, the components of the microscopic tensor Cijklare recovered; furthermore, αij and α′ij tend to θδij . This fact can be easily proven


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1- =VS/V

10-4

10-3

10-2

10-1

100

|J(11

)11

|

|J(11

)22

|

|J(11

)33

|

|J(33

)11

|

|J(33

)33

|

|J(12

)12

|

|J(13

)13

|

|J(13

)31

|

|J( )11

|

|J( )33

|

Figure 8. “Cylindrical” microscopic structure: absolute values of ten independent parametersidentified by equations (3.3) as function of the porosity.

0 0.2 0.4 0.6 0.8

1- =VS/V

10-5

100

C1111

C1122

C1133

C3333

C1212

C1313

0 0.2 0.4 0.6 0.8

1- =VS/V

0.65

0.7

0.75

0.8

0.85

0.9

0.95

ii,

' ii

11

33

'11

'33

Figure 9. Non-zero entries of Cijkl (left), αij and α′ij (right) for different porosities in the caseof “cylinders”.

by observing that, for Vf = 0, Γ disappears and problems (2.11) become homogeneous,so that χ and η are equal to zero.

3.2. Solution of the interface problem

As for the interior problem, we consider two different geometries for the interfaceproblem†, “spheres” and “cylinders”, but we focus on a single value of the porosity, θ =0.8, to be later used in all macroscopic simulations. All non-zero components determinedfrom the simulation as volume averages above the interface and volume averages in theperiodic unit cell below the interface are summarized in table 1.

In the current work, since we focus on a two-dimensional macroscopic problem, onlythe components K11, K33 and L113 are required. The underlying fields for K11 are shownin figure 10 for both geometries. There we can see that, in both cases, the solutionrapidly approaches the periodic interior solution. The interface permeability is obtainedas a volume average above the interface location (green plane). For comparison purposes,

† The solution of the interface problem has been computed using an ad hoc solver based onFreeFem++. The numerical details of this solver are described in Lacis et al. (2017).


Figure 10. Solutions of interface problems for the coefficients of the Darcy term. The interfaceis located exactly at the tip of the last solid structure ys = 0.0 (green, transparent plane).Porosity θ = 0.8 and radius ratio rc/r = 0.4. Sphere based (left) and cylinder based (right)microscale geometries. The plots are cut after the second microstructure below the interfacealthough the full solution domain consists of five microstructures.

Figure 11. Solutions of interface problems for the coefficients of the Navier-slip term. Theinterface is located exactly at the tip of the last solid structure ys = 0.0 (green, transparentplane). Sphere based (left) and cylinder based (right) microscale geometries.

K11 K33 L113

“Spheres”Interface 1.01 · 10−2 1.71 · 10−2 1.08 · 10−1

Interior 1.79 · 10−2 1.72 · 10−2 –

“Cylinders”Interface 1.31 · 10−2 2.51 · 10−2 8.79 · 10−2

Interior 1.33 · 10−2 2.50 · 10−2 –

Table 1. Effective tensor components arising from the interface problem. Both geometries areconsidered. Only non-zero components, later used in the macroscopic simulations, are reported.

we give in the table also the interior macroscopic values of the coefficients, evaluated atthe second microstructure below the interface as volume average over the representativevolume element. Such internal values are identical to those obtained with the proceduredescribed in section 3.1. Similarly, the underlying fields for L113 are shown in figure 11for both geometries. There we see that, for both geometries, the field rapidly dissipateswithin the interior, practically reaching a zero value already at the level of the secondmicrostructure. The interface coefficient is obtained again as a volume average above theinterface location.

4. Macroscopic applications

In this section we solve for two different configurations of the flow to show theapplicability of the FPEL model to real physical cases. In the first example the relationsbetween the characteristic scales of the fluid and of the structure’s motion, which are atthe basis of our homogenization approach, are strictly taken into account (cf. AppendixA); in the latter, these scaling relations are weakened to allow us to compare qualitatively


the numerical solution of the PEL model with experiments on canopy flows taken fromthe literature.

4.1. Standing waves

Macroscopic solutions are obtained with a domain-decomposition-based solver, de-scribed in Zampogna & Bottaro (2017), whose fundamental idea is to solve iterativelythe Navier Stokes equations within VNS and the FPEL equations within the VPELregion, defined in figure 2. A two-way coupling between the two regions is performedthanks to the interface conditions, imposed at ΓI , which is the intersection between thetwo domains: (2.18) and (2.19) are used to go from the NS to the PEL domain and(2.21) is used in the opposite sense. A validation of the FPEL model for some steadyconfigurations is provided in Zampogna & Bottaro (2017). Here, we aim to understandthe time-dependent behavior of the model, and solve an unsteady configuration of theflow, without any further assumption, except for the fact that the unknowns do notdepend on the transverse direction, x2.

The configuration chosen allows to evaluate the horizontal and vertical componentsof the velocity in VNS and VPEL. In particular, within VNS , a vertical velocity whichin the absence of the poroelastic medium would be zero, develops. A channel, of non-dimensional size 2 × 2, is filled in the upper half by a fluid, and in the lower half by asaturated poroelastic medium with spherical inclusions and porosity equal to 0.8. Themotion of the fluid is due to an oscillating pressure gradient which has the dimensionlessform: ∂1p(t) = α cos(ωt). In dimensional terms the box dimensions are taken to be1m× 1m. The fluid contained within it is an aqueous solution of glycerol at about 98%(ρf = 1250 kg/m3, ν = 8 × 10−4 m2/s, cf. Segur & Obestar 1951) and, for ReL =ρf UL/µ = 100, the characteristic speed U in the pure fluid domain is 0.8 m/s. Thisyields a pressure scale which is of order P ∼ ρf U

2 = 8 Pa both in the fluid andin the poroelastic region, with a consequent displacement scale of the pores of orderPL/E = 27 µm for materials with very low Young modulus, E = 3 × 105 Pa, such assylicon-based polymers (ρs = 16 Kg/m3). There is thus a factor of about 105 between thescale of the displacement of the pores and the wavelength of the standing wave which iscreated in the channel. The velocity within the poroelastic medium scales as U ∼ ε2PL/µand, if ε = 2.14× 10−2, it has a value of order 0.0037 m/s which is about twenty timessmaller than the velocity in the pure fluid region. This is clearly related to the Reynoldsnumber and to ε = l/L, since it is U ∼ ε2ReLU (cf. Appendix A). The effect of ε onthe amplitude of the displacement will be examined later on. The Cauchy number whichensues from the parameters chosen is Ca = ρf U

2/E = 2.7× 10−5.The central image of figure 12 shows the horizontal velocity averaged throughout the

PEL, as function of time. The dimensionless mean velocity attains values of order 10−2,consistent with the scalings just discussed. In the same figure, eight instants during theperiodic cycle are provided for the profile of the horizontal velocity in the free fluidregion, sampled at x1 = 1. At the horizontal interface ΓI , positioned at x3 = 0, a weakslip flow arises. The vertical velocity which ensues from the presence of the PEL is drawnin figure 13, in both the PEL region and in the free fluid domain; such a componentis smaller than its horizontal counterpart and is negligible in the PEL. This is borneout by equation (2.17) since the imposed pressure gradient acts along x1. Figure 13 alsodisplays the standing oscillation of the interface, with the vertical displacement dividedby Ca/2 for the oscillation to be visible in the scale of the figure. In physical dimensionsthe maximum displacement is very small (consistent with the scalings of the previousparagraph), but the waveform is clear and numerically well resolved.

When the horizontal velocity is averaged over only the interface, a temporal oscillation


-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1t =1

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1t =1.29

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1t =1.58

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1t =1.968

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1t =0.064

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1t =0.258

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1t =0.548

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1t =0.838

3.5 4 4.5 5 5.5 6

Number of periodic cycles

-0.02

-0.01

0

0.01

0.02

Figure 12. Central image: mean horizontal component of the effective velocity field in thepooelastic domain as function of time. The insets represent the horizontal velocity profile in thefluid domain at x1 = 1 (midline through the domain), sampled at the instants of the periodiccycle indicated in the central image. Except for a thin layer immediately above the interface, tothis scale the present solution is almost indistinguishable from Womersley’s exact solution.

such as that shown in figure 14 appears; after one period the solution has already attaineda limit-cycle behavior. Here, the effect of the homogenized boundary condition can beappreciated, since the solid line (current solution) displays the solution obtained byemploying eq. (2.21), while the dashed line is obtained by treating the interface withthe simplified condition (L113 = 0) used by Zampogna & Bottaro (2017). With the newcondition the interface velocity is about 50% larger than with the simplified condition,stressing the importance of an appropriate treatment of the fluid-PEL boundary, withaccount of both an interfacial permeability, Kij , and a tensorial Navier slip length, Lijk.

From the point of view of the solid motion, an analysis of the behavior of the skeleton isprovided in figure 15. The value of the horizontal displacement averaged over the interfaceis plotted versus time in the top frame. Since the vertical displacement has zero mean,we have chosen to represent the time variation of its maximum and minimum valuesinstead, in the bottom frame of the figure. The dimensionless amplitudes are very small(cf. vertical axes of the figure), conforming to the scales and the normalization chosen.Since the interface ΓI moves with the pressure gradient which forces the macroscopicflow, its displacement can be written as

v1 = A(x1)cos(ωt+ φ1) and v3 = B(x1)cos(ωt+ φ2) at x3 = 0 (4.1)

where A and B are amplitudes and φ1, φ2 are a priori unknown phases.It is clear from Appendix A that ε is strictly related to ReL and Ca in order for the

homogenization hypotheses to be satisfied. Going beyond the scaling expressed by eq.(A 6), we have varied ε and Ca to examine qualitatively the response of the structure tovariations of these two parameters. Figure 16 shows that the amplitude becomes smalleras ε increases. This fact means that the smaller is the characteristic size of the microscopicstructure, for given macroscopic scales, the larger is its elastic deformation to an externalforcing. Alternatively, an increase of Ca is equivalent to a decrease of Young’s modulus E


Figure 13. Contours of the vertical component of the effective velocity field in the whole domain.The periodic cycle is sampled at 8 instants within a period of oscillations. The fluid-structureinterface ΓI is positioned at rest at x3 = 0. The colors refers to the pure fluid region, VNS , whilethe grayscale indicates the contours of 〈u3〉−θv3 in the VPEL The thick solid lines represent thevertical displacement of the macroscopic interface ΓI , (Ca/2)−1v3, for the respective snapshots.


0 5 10 15


-0.04

-0.02

0

0.02

0.04

Figure 14. Mean horizontal fluid velocity at the interface as function of time: comparisonbetween the result of the present work (solid line) and that computed with the simplifiedcondition by Zampogna & Bottaro (2017) (dashed line).

0 5 10 15


-4

-2

0

2

10-4

0 5 10 15


-1

0

110

-7

Figure 15. Top frame: mean horizontal displacement of the interface in time. Bottom frame:maximum and minimum values of the vertical displacement of the interface in time for the sameconfiguration.

and this provokes a sizeable deformation of the elastic skeleton. This intuitive behaviourcan be shown by analyzing the interface conditions expressed by (2.18), imposed at ΓI ,which are used to transfer information from the fluid region to the PEL. In dimensionlessterms they read

C1313 ε13(v) =Ca

εReLε13(u|F )

and

C3311 ε11(v) + C3333 ε33(v)− α33 Ca p|H = −Ca p|F +Ca

εReε33(u|F ), (4.2)

with the solid strain at the interface ΓI which is thus inversely proportional to ε anddirectly proportional to Ca. In conclusion, the deformation of the structure with ε andCa displays a trend which agrees with what one would expect on the basis of physicalarguments.

4.2. Travelling waves

Alongside the low-Re configuration just shown, a different case, characterized bycomplex phenomena associated to turbulence, is tested. This case, which stretches thelimits of applicability of the model equations, concerns vegetated aquatic flows, commonin nature and studied extensively in the past (e.g. Raupach & Shaw 1982; De Langre 2008;Ghisalberti & Nepf 2009; Nepf 2012). A key point of the cited works consists in modellingthe presence of the vegetation and its effect on the fluid flows. A very common choice is


10-4

10-3

10-2

10-1

10-5

10-4

10-3

10-2

A

Ca=9.15 10-8

Ca=2.7 10-5

10-4

10-3

10-2

10-7

10-6

B

Ca=9.15 10-8

Ca=2.7 10-5

Figure 16. Maximum, along x1 and for x3 = 0, of the displacement amplitudes, defined in eq.(4.1). For each ε two different values of Ca are shown.

to model the drag exerted by the canopy on the fluid in the context of the mixing layeranalogy (Raupach et al. 1996), introducing a fictitious drag coefficient in the vegetatedregion. In recent papers, Zampogna et al. (2016) and Luminari et al. (2016) have shownthat a sufficiently dense canopy can be approximated as a rigid homogeneous porousmedium and modelled by a generalized Darcy’s law, proposing a new strategy to analyzethese kinds of environmental flows. In this section we try to make a further step forwardby considering the canopy as a porous and deformable (as opposed to rigid) medium,using the continuum model developed. This is the very first step in this direction and, asmentioned above, it requires using the PEL model for a range of parameters beyond thosewhich have led to the scalings presented in Appendix A. In particular, vegetated flowshave typically large porosities and display significant fluid inertial effect. We proceednonetheless, with the limitations well in mind but strong of the positive feedback whicha similar approach (in the simpler, rigid limit) has received †.

The experimental setup developed by Ghisalberti & Nepf (2002), meant to repro-duce the behavior of oceanic seagrass meadows, is considered. Artificial seagrass, withcharacteristics similar to the real one (blades made from a polyethylene film withρs = 920 kg/m3 and Young’s modulus E = 3.0 × 108 Pa), is placed at the bottom ofa flume and a unidirectional water flow is generated by gravity over and through theblades. The dimensions of each blade are: thickness t = 10−4m, width w = 3 × 10−3mand height h = 0.127 m; the mean spacing between successive blades is 0.023 m, yieldinga porosity which exceeds 0.99. The hydraulic diameter of the blades can be computedto be equal to 1.94 × 10−4; for the geometrical arrangement of cylindrical inclusionsconsidered here a porosity equal to that of the experimental set up is realized whenl = 8.6× 10−4 m.

Several experimental configurations have been tested by Ghisalberti & Nepf and, forcertain parameters of the flow, the onset of monami waves has been observed. In suchcases the amplitude of the wave and the vortex mechanics have been evaluated. We focuson case B in their paper for which ReL = ρf UL/µ ∼ 3.6×104, with U = 5.66×10−2 m/sthe water velocity above the vegetated layer. The measured dimensional period of themonami is 224 s for a wavelength of 2L = 1.31m (giving a phase speed slightly smallerthan U). With these data we have ε = l/L = 1.32 × 10−3 and the velocity through the

† See “Editor Highlight” on the paper by Zampogna et al. (2016),http://agupubs.onlinelibrary.wiley.com/hub/article/10.1002/2016WR018915/editor-highlight/


0 1 2 3 4 5 6 7 8 9 10 11

x1

0

0.05

0.1

0.15

0.2

x3

Figure 17. Computational domain together with the boundary conditions employed. Theinterface between the PEL domain and the free fluid region is positioned at x3 = 0.194. Thearrows represents the displacement field at a certain temporal instant. Image not to scale.

canopy is U = 6.53 × 10−3m/s, yielding a microscopic Reynolds number of 5.6. Thus,Re is not of order ε and the theory - as developed here - is formally not applicable.

It is however interesting to see whether a complex motion, such as a monami wave,shows up in the displacement field of the poroelastic medium and how the disturbancethrough the canopy can propagate. For our purposes it is not necessary to consider themacroscopic fluid-structure interaction problem, described previously by the introductionof conditions (2.18, 2.21 and 2.19) and we thus simulate only the poroelastic domain. ThePEL model is solved over a rectangular elongated computational box of 7.2m× 0.127m,corresponding to the whole vegetated region in the recirculating flume of Ghisalberti &Nepf (2002). The flow field is assumed to undergo the observed monami effect, and thisis borne by the conditions imposed at the upper boundary of the computational box.The complete set of boundary conditions used in the computational configuration aresummarized in figure 17 together with a snapshot in time of the displacement field; theyare:• a Neumann inlet-outlet condition for the displacement of the homogeneous structure

at the vertical boundaries of the domain

∂v1∂x1

=∂v3∂x1

= 0; (4.3)

• a Neumann inlet-outlet condition for the horizontal fluid velocity at the verticalboundaries of the domain; in terms of pressure, this means

∂2p

∂x21= 0; (4.4)

• a Dirichlet condition to force to zero the displacement at the bottom of the flume,

v1 = v3 = 0; (4.5)

• a Neumann condition for the normal stresses is imposed at ΓI

∂2v1∂x23

=∂2v3∂x23

= 0; (4.6)

• the horizontal and vertical velocities are travelling waves and this can be achieved


Figure 18. Horizontal and vertical components of the displacement vector (first two frames)and vector representation of the effective velocity (third frame) at the same temporal instantof figure 17. Clockwise and counterclockwise vortices, centered at the interface between canopyand free fluid region, are sketched in red and blue to guide the eye. Figure not to scale.

by imposing the following conditions for the pressure at ΓI :

p = −αcos(kx1 − ωt) (4.7)

and

∂2p

∂x23= −K11

K33αkcos(kx1 − ωt) +

θ

ε2ReLK33(∂v1∂x1

+∂v3∂x3

). (4.8)

Equation (4.8) is necessary to satisfy continuity at ΓI and can be deduced by substitutingeq. (4.7) into eq. (2.17) and taking the divergence. The parameters k and ω are knownfrom the experiments and, in dimensionless form, are k = π and ω = 2.02. The amplitudeα = 10 is chosen in order for the velocity to be of the order of that measured within thecanopy. The remaining important parameter which has to be set is the Cauchy number,which is Ca = ρf U

2/E = 1.07× 10−8.We already know from Zampogna & Bottaro (2017) and Lacis et al. (2017) that the

microscopic inclusions must be connected along all directions for a collective movementof the PEL to arise. We thus consider linked structures, as done in the previous section,with rc/r = 0.4, despite the absence in the real physical problem of such connections.To use a structure as close as possible to that of the experiments, we take the values forthe effective tensors valid for the case of linked cylinders, with porosity θ = 0.99. Thecorresponding values of Cijkl and αij are available from figure 9, while K11 = 0.15 andK33 = 0.18 (Zampogna & Bottaro 2016).

Travelling vortical structures arise in both velocity and displacement fields. Figure 17shows that the periodicity of the vortices 2π/k = 2, imposed as a pressure condition at


ΓI , is maintained by the displacement field, which is itself perfectly synchronized withthe motion of the fluid (compare figure 17 and the third frame of figure 18). The flowvortices, as observed by Ghisalberti & Nepf (2002), are approximately elliptical in cross-section and expand in the whole VPEL. A snapshot of the displacement field within thewhole PEL is displayed in the top two frames of figure 18.

In the experiments, the maximum deflection of the canopy is about 10% of its height,while our poroelastic medium shows a maximum horizontal displacement which is abouthalf, for the same structural stiffness. This is probably related to the parameters of theexperiments, to the different solid skeletons present in the two configurations (bladesversus connected cylinders) and to the use of a theory based on linear elastic equations.Future work will address the issue of large deformations by the use of nonlinear poroe-lasticity since, as recently shown by MacMinn et al. (2016), the errors committed usinga theory based on linear poroelasticity in phenomena where the kinematics of the solidskeleton is affected by nonlinearities can be very significant. For the time being we mustthus satisfy ourselves with the qualitative agreement observed.

5. Conclusions

The present work describes a model to study the interactions between a fluid and aporoelastic layer, in the linear elasticity limit and for conditions of non-negligible fluidinertia. This paper thus completes the work by Lacis et al. (2017), where the Stokes limitwas treated and validated. The equations posed here are based on two nested points ofview: a microscopic and a macroscopic one. From the coupling of these two points of viewsignifical physical quantities arise: the effective elasticity tensor Cijkl and the permeabilitytensor Kij . Others quantities found, the effective fluid volume fraction, αij and the bulkcompliance of the solid skeleton, β, describe mainly the variation of porosity due to thedeformation of the microscopic skeleton and its compressibility.

The detailed parametric study performed for the microscale problems points to abehavior coherent with fundamentals principles of solid mechanics. The symmetries ofthe effective elasticity tensor are respected on the basis of the classification of the typeof material chosen and on the shape of the skeleton (cf. Cowin 2013). The values of Cijklfor non-porous (θ = 0) materials are recovered. Since the only restrictions required forthe type of material and for its shape are weak compressibility and connectivity of theskeleton in each direction, the model could be used, in the limit of small deformations, fora reasonably wide range of materials. The microscopic counterpart of the fluid motion,instead, is represented by the permeability tensor Kij (Zampogna & Bottaro 2016). Find-ing microscopic tensors with a clear physical significance confirms that homogenizationis a viable theoretical tool to analyze complex multiscale phenomena.

One further coupling issue treated here is relative to the macroscopic interface betweenthe two regions, VNS and VPEL. While in the case of rigid porous media (cf. Zampogna& Bottaro 2016; Lacis & Bagheri 2016) the interface conditions have been validatedby direct numerical simulations with full account of the fluid motion through individualpores, here the computational complexity renders extremely difficult the same procedure,and only a validation in the case of Stokes flow has been possible (Lacis et al. 2017).

This work is but a first step in the development of a complete theory capable toaddress the macroscopic interactions between a fluid region and a poroelastic mediumsaturated by a fluid. The main achievement has been that of providing a completecharacterization of the microscopic tensors needed for the successive macroscopic study.Two large-scale configurations have been considered and have provided a first test of thetheory. Such applications – though interesting in their own right – permit to test the


equations without representing yet a validation of the model. Hopefully, this work willstimulate future experimental activities with poroelastic materials under conditions forwhich the hypotheses of the present theory are satisfied.

Appendix A. Development of the interior model

We show here the mathematical foundations of the continuum model formed byequations (2.10) and (2.17) supplied with the microscopic problems (2.11) and (2.16).The homogenization technique is applied to (2.2), (2.5) and (2.6), with equation (2.8)as microscopic interface condition between the two phases. The approach follows thatoutlined by Mei & Vernescu (2010).

A.1. Scaling relations

First of all, we need to understand the order (in ε) of each term in the dimensionalgoverning equations. If U , V and TS are the fluid velocity, solid displacement and solidtime scales, respectively, from equation (2.8) it is:

U ∼ VTS. (A 1)

We use Young’s modulus of elasticity, E, to scale the elastic tensor and denote by P thepressure scale. If we assume that macroscopic solid stresses in the poroelastic mediumare balanced by fluid pressure, we have

EPl2

µL2TS ∼ P, (A 2)

provided that macroscopic pressure forces are equilibrated by microscopic viscous dissi-pation i.e.

P

L∼ µU

l2. (A 3)

Thus, the solid time scale can be defined by

TS = ε−2µ

E. (A 4)

Furthermore, from (2.2) we define TS so that

ρsT 2S

=E

L2, (A 5)

on the assumption that inertia of the solid is of the same order as the solid stress overthe macroscale. Combining equations (A 4) and (A 5) we obtain:

ε−2 =ρsEl

2

µ2. (A 6)

Relation (A 3) implies that the fluid velocity scale can be written as

U = εP l

µ. (A 7)

Using this last equation together with equations (A 1) and (A 4) we can define the soliddisplacement scale:

V = εPL2

El=PL

E. (A 8)


We are now ready to introduce the relations between the dimensional and dimensionlessvariables (the latter without hat); in principle, two adimensional times can be introduced,one for the fluid, tf = Ut/l, and one for the solid, with ts = ε2Et/µ. Furthermore:

x = lx, p = Pp, u = εP l

µu, v =

PL

Ev. (A 9)

Substituting these definitions in the momentum equation for the fluid phase we have:

εRe

(∂ui∂tf

+ uj∂ui∂xj

)= − ∂p

∂xi+ 2ε

∂εij(u)

∂xj, (A 10)

where Re = (ρfUl)/µ. Applying the same procedure to the equation for the solid weobtain:

ε2∂2vi∂t2s

=∂σij∂xj

on Vs. (A 11)

The boundary condition on the normal stresses becomes:

−p ni + 2ε εij(u)nj =

[1

εCijklεkl(v)

]nj on Γ. (A 12)

The continuity equation for the fluid (2.6), the boundary condition (2.8a) and V -periodicity remain unchanged.

A.2. Multiple scales analysis

At this point we can perform a multiple scale expansion recalling the fast and slowvariables, x and x′ = εx, respectively, and the expansions

f = f (0) + εf (1) + ε2f (2) + . . . where f = ui, vi, p,Σij , σij. (A 13)

Moreover we note that the strain tensor (for either the solid or the fluid) becomes

εij + εε′ij (A 14)

where

εij(w) =1

2

(∂wi∂xj

+∂wj∂xi

)(A 15)

and

ε′ij(w) =1

2

(∂wi∂x′j

+∂wj∂x′i

). (A 16)

A.3. Small Reynolds number

Using equations (A 13) and (A 14), and assuming that Re = O(ε), we obtain at orders0 and 1 in ε the following system for the fluid:

∂u(0)i

∂xi= 0, (A 17)

∂u(1)i

∂xi+∂u

(0)i

∂x′i= 0, (A 18)

0 = −∂p(0)

∂xi, (A 19)


0 =∂Σ

(0)ij

∂x′j+∂Σ

(1)ij

∂xj= −∂p

(1)

∂xi− ∂p(0)

∂x′i+

∂2u(0)i

∂xj∂xj, (A 20)

on Vf , and for solid at order 0, 1 and 2:

∂σ(0)ij

∂xj= 0, (A 21)

0 =∂σ

(1)ij

∂xj+∂σ

(0)ij

∂x′j, (A 22)

∂2v(0)i

∂t2s=∂σ

(2)ij

∂xj+∂σ

(1)ij

∂x′j, (A 23)

on Vs, plus the boundary conditions on Γ :

u(0)i =

∂v(0)i

∂ts, (A 24)

u(1)i =

∂v(1)i

∂ts, (A 25)

σ(0)ij nj = 0, (A 26)

σ(1)ij nj = Σ

(0)ij nj = −p(0)ni, (A 27)

σ(2)ij nj = Σ

(1)ij nj . (A 28)

It is useful to specify the form of the tensors σ and Σ at the two leading powers in ε:

σ(0)ij = Cijkl(εkl(v

(0))), (A 29)

σ(1)ij = Cijkl(εkl(v

(1))) + Cijkl(ε′kl(v

(0))), (A 30)

Σ(0)ij = −p(0)δij , (A 31)

Σ(1)ij = −p(1)δij + 2εij(u

(0)). (A 32)

We now observe that, from equation (A 19), the pressure at leading order is independentof the microscale, i.e. p(0) = p(0)(x′, t); the system formed by (A 21) and (A 26) impliesthat

σ(0)ij = 0 ∀ i, j i.e. Cijkl(εkl(v

(0))) = 0 ∀ i, j, (A 33)

and from this we have

εkl(v(0)) = 0 ∀ k, l (A 34)

which implies that also the solid displacement at leading order is not a function of themicroscale, i.e. v(0) = v(0)(x′, t).

A.4. The effective elasticity tensor

Coupling the O(ε) solid momentum equation (A 22) with the boundary condition(A 27) we obtain a system which, using (A 30) and (A 33), can be rewritten in thefollowing way:

∂

∂xj

Cijkl

[εkl(v

(1)) + ε′kl(v(0))]

= 0 on Vs, (A 35)

24 G. A. Zampogna, U. Lacis, S. Bagheri & A. BottaroCijkl

[εkl(v

(1)) + ε′kl(v(0))]

nj = −p(0)ni on Γ, (A 36)

plus V -periodicity. Since the system above is a linear differential equation for v(1) forcedby v(0) and p(0), we formally express v(1) in terms of v(0) and p(0):

v(1)(x,x′, t) = χpqi (x)ε′pq(v(0))(x′, t)− ηi(x)p(0)(x′, t) on Vs, (A 37)

where χ is a third order tensor and η a vector. Substituting (A 37) into (A 35) and (A 36)we have:

∂

∂xj

Cijkl

[εkl(χ

pq)ε′pq(v(0))]− Cijkl

[εkl(η)p(0)

]+

+ Cijkl

[ε′kl(v

(0))]

= 0 on Vs, (A 38)

Cijkl

[εkl(χ

pq)ε′pq(v(0))]− Cijkl

[εkl(η)p(0)

]+

+ Cijkl

[ε′kl(v

(0))]

nj = −p(0)ni on Γ. (A 39)

A solution of this system can be found by solving the two systems in equation (2.11).The averaging operator in equation (2.9) can be written in adimensional form as

〈fp〉 =1

V

∫Vp

fp dV, (A 40)

where the function fp is defined on Vp and the subscript p means either the fluid (f) orthe solid (s) phase. From this moment on we will not distinguish between the solid orfluid phase averaging unless the distinction is ambiguous. Our target is to deduce a setof equations to determine the solution up to order ε. In order to do this we can take thesolid-phase average to express the solid stress:

〈σ(1)ij 〉 = (A 41)

= 〈Cijkl εkl(v(1))〉+ 〈Cijkl〉ε′kl(v(0)) =

= [〈Cijkl εkl(χpq)〉+ 〈Cijkl〉δkpδlq] ε′pq(v(0))− 〈Cijkl εkl(η)〉p(0) =

= Cijpq ε′pq(v(0))− 〈Cijkl εkl(η)〉p(0),

where Cijkl has already been introduced in (2.12).

A.5. Momentum balance for the composite

Equation (A 41) is the macroscale Hooke’s law in the solid phase forced by the fluidpressure. To deduce the momentum equation for the composite we can define a newtensor

Tij =

Σij on Vf

ε−1σij on Vs.(A 42)

where σ is divided by ε because we take into account the scaling relation (A 12) between

σ and Σ. It is useful at this point, to compute the average of T(0)ij , i.e. 〈T (0)

ij 〉 =∫VT

(0)ij dV/V :

〈T (0)ij 〉 = 〈σ(1)

ij 〉+ 〈Σ(0)ij 〉 = 〈σ(1)

ij 〉 − θp(0)δij = (A 43)

= [〈Cijkl εkl(χpq)〉+ 〈Cijkl〉δkpδlq] ε′pq(v(0))− (〈Cijkl εkl(η)〉+ θδij)p(0) = (A 44)


= Cijpq ε′pq(v(0))− α′ijp(0), (A 45)

where α′ij is defined by

α′ij = θδij + 〈Cijkl εkl(η)〉. (A 46)

Adding the averages of (A 20) and (A 23) we get

〈∂2v

(0)i

∂t2s〉 = 〈

∂T(0)ij

∂x′j〉+ 〈

∂Σ(1)ij

∂xj〉+ 〈

∂σ(2)ij

∂xj〉. (A 47)

Observing that we can exchange integral and derivative only if the integration domainand variables do not depend on the differentiation variable, we obtain

(1− θ)∂2v

(0)i

∂t2s=∂〈T (0)

ij 〉∂x′j

+1

V

∫Vf

∂Σ(1)ij

∂xjdV +

1

V

∫Vs

∂σ(2)ij

∂xjdV,

which, using Gauss theorem, becomes

(1− θ)∂2v

(0)i

∂t2s=∂〈T (0)

ij 〉∂x′j

+1

V

∫Γ

(σ(2)ij −Σ

(1)ij

)nj dΩ. (A 48)

The last integral in the equation above is zero by the boundary condition (A 28), so thatthe average momentum balance of the composite can be rewritten as

(1− θ)∂2v

(0)i

∂t2s=

∂

∂x′j

[Cijpqε′pq(v(0))− α′ijp(0)

]. (A 49)

Equation (A 49) can be solved after solving the equations for χ and η which yield C.

A.6. Continuity equation for the composite

Another equation is deduced using the continuity equation for the fluid at order ε. Letus take the fluid-phase average of (A 18):

〈∂u(0)i

∂x′i〉 = − 1

V

∫Vf

∂u(1)i

∂xidV. (A 50)

As before we can exchange derivative and integral; using Gauss theorem, we obtain thatthe equation above is equivalent to

∂〈u(0)i 〉∂x′i

=1

V

∫Γ

u(1)i ni dΩ =

1

V

∫Γ

∂v(1)i

∂tsni dΩ =

1

V

∫Vs

∂

∂xi

∂v(1)i

∂tsdV = 〈 ∂

∂xi

∂v(1)i

∂ts〉,

(A 51)by the boundary condition (A 25). Equation (A 37) tells us that

〈 ∂∂xi

∂v(1)i

∂ts〉 = 〈 ∂

∂xi

∂

∂ts

[χpqi (x)ε′pq(v

(0))(x′, ts)− ηi(x)p(0)(x′, ts)]〉 = (A 52)

= 〈 ∂∂xi

χpqi (x)ε′pq

(∂v(0)

∂ts

)(x′, ts)〉 − 〈

∂

∂xiηi(x)

∂p(0)

∂ts(x′, ts)〉 = (A 53)

= 〈∂χpqi

∂xi〉ε′pq

(∂v(0)

∂ts

)− 〈∂ηi

∂xi〉∂p

(0)

∂ts. (A 54)

The only time variable present is relative to the solid, i.e. ts = t/TS , indicated simply as tin the following. A dot above a variable denotes derivation with respect to t. Substituting


the equation above in (A 51) we obtain a relation between the solid stress and thepressure:

∂〈u(0)i 〉∂x′i

= 〈∂χpqi

∂xi〉ε′pq

(∂v(0)

∂ts

)− 〈∂ηi

∂xi〉∂p

(0)

∂ts. (A 55)

Also this equation can be solved, once we know χ and η.

A.7. Momentum equation for the effective flow

In order to obtain a momentum equation for the fluid phase we consider the problemformed by (A 17), (A 20), the boundary condition (A 24) plus periodicity over V . Thisproblem can be viewed as a Stokes problem for u(0) and p(1) forced by the gradient of

p(0) and by ∂v(0)j /∂ts, quantities defined only over the macroscale x′, so that the most

general solution takes the form:

u(0)i = −Kij

∂p(0)

∂x′j+Hij

∂v(0)j

∂tsand p(1) = −Aj

∂p(0)

∂x′j+Bj

∂v(0)j

∂ts. (A 56)

Substituting into equations (A 17), (A 20) and (A 24) we obtain

∂Hij

∂xi

∂v(0)j

∂ts=∂Kij

∂xi

∂p(0)

∂xj, (A 57)

(∂Bj∂xi− ∂2Hij

∂xk∂xk

)∂v

(0)j

∂ts=

(∂Aj∂xi− δij −

∂Kij

∂xk∂xk

)∂p(0)

∂x′j, (A 58)

and

(Hij − δij)∂v

(0)j

∂ts= Kij

∂p(0)

∂x′jon Γ. (A 59)

A solution of this problem can be found imposing that left- and the right-hand-sides ofequations (A 57) and (A 58) vanish, yielding two different problems which must be solvedin the unit cell over Vf :

∂Aj∂xi− ∂Kij

∂xk∂xk= δij ,

∂Kij

∂xi= 0, Kij = 0 on Γ, (A 60)

and∂Bj∂xi− ∂Hij

∂xk∂xk= 0,

∂Hij

∂xi= 0, Hij = δij on Γ. (A 61)

In order to guarantee unicity of the solution, since only the gradients of Aj and Bj appearin (A 60) and (A 61), we impose 〈Aj〉 = 0 and 〈Bj〉 = 0. The solution of (A 61) is simplyHij = δij ; the volume average of Kij and Hij , denoted, respectively, as Kij and Hij are:

Kij = 〈Kij〉 and Hij = 〈Hij〉 = θδij . (A 62)

Finally, the volume average of equation (A 56) for the fluid phase is

〈u(0)i 〉 − θ∂v

(0)i

∂ts= −Kij

∂p(0)

∂x′j, (A 63)

〈p(1)〉 = −Aj∂p(0)

∂x′j+ Bj

∂v(0)j

∂ts. (A 64)


Equation (A 63) is Darcy’s law for the unknown 〈u(0)i 〉− θ∂v(0)i /∂ts, which is the average

velocity of the fluid relative to the solid.

It is useful, in view of the numerical resolution of the equations, to express the solidtime derivative in (A 49), (A 55) and (A 63) in terms of the fluid time derivative. Since,by equations (A 2), (A 3) and (A 5) it follows that

TfTS

=ρfρs

(A 65)

the following relation holds

∂

∂ts=ρsρf

∂

∂tf. (A 66)

Re-normalizing length and displacement with respect L, the pressure with respect ρf U2,

with U the velocity scale in the pure fluid region, and using (A 66) in (A 49), (A 55) and(A 63), we obtain three fundamentals equations, respectively:

- the first equation of system (2.10);- the following equation

∂〈u(0)i 〉∂x′i

= 〈∂χpqi

∂xi〉ε′pq(v(0))− 〈∂ηi

∂xi〉Ca p(0); (A 67)

- equation (2.17);

a dot over a variable denotes ∂/∂tf which is the only time variable left. Substituting(2.17) into (A 67) and using the definition (2.13), we obtain the second equation ofsystem (2.10).

Appendix B. The interface conditions

In order to find the effective interface coefficients defined in (2.22) and (2.23), micro-scopic problems must be solved within the interface cell introduced in section 3.2. Theprincipal idea to deduce these problems, which is explained in Lacis & Bagheri (2016)and in Lacis et al. (2017) for both rigid and elastic porous media, is to perform a multiplescale expansion of the physical quantities above (denoted with ·+) and below (·−) themacroscopic interface ΓI and match the two solutions. The coefficients and the respectiveproblems derived using this procedure are given below. The microscopic permeabilitytensor K±ij and the vector A±j solve

∂A+j

∂xi−

∂K+ij

∂xk∂xk= 0,

∂K+ij

∂xi= 0, K+

ij = 0 on ΓI , (B 1)

∂A−j∂xi

−∂K−ij∂xk∂xk

= δij ,∂K−ij∂xi

= 0, K−ij = 0 on ΓI , (B 2)

K−ij = K+ij on ΓI(

−A−k δij +∂K−ik∂xj

+∂K−jk∂xi

)nj =

(−A+

k δij +∂K+

ik

∂xj+∂K+

jk

∂xi

)nj on ΓI ,

where, in contrast to the previous problem, instead of surface forcing on the boundariesof the solid material, now we have volume forcing in the whole fluid volume within themicroscopic cell at the interface. Finally, the interface problem for the slip tensor L±ijk


and the tensor P±ij is

∂P+jk

∂xi−

∂L+ijk

∂xn∂xn= 0,

∂L+ijk

∂xi= 0, L+

ijk = 0 on ΓI , (B 3)

∂P−jk∂xi

−∂L−ijk∂xn∂xn

= 0,∂L−ijk∂xi

= 0, L−ijk = 0 on ΓI , (B 4)

L−ijk = L+ijk on ΓI(

−P−klδij +∂L−ikl∂xj

+∂L−jkl∂xi

)nj =

(−P+

klδij +∂L+

ikl

∂xj+∂L+

jkl

∂xi

)nj + δiknl on ΓI ,

where we have again a surface forcing, which this time is located at the interface betweenthe poroelastic medium and the free fluid, expressed as a stress jump between these twodifferent domains. Once these quantities are calculated, they are rendered macroscopicvia eqs. (2.22) and (2.23) and are used in the macroscopic interface condition (2.21).

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a model of uid ows through poroelastic media · this draft was prepared using the latex style le...

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