FLUID 0<r<1~ FLUID

Homogénéisation des systèmes

couplés: fluide/structure + transport réactif

Andro Mikeli ´c

Institut Camille Jordan, UMR 5208, UFR Math ´ematiques, Universit ´e Claude Bernard Lyon 1, Lyon, FRANCE

Introduction 1

L’argile est une roche sédimentaire, composée pour une large part de minéraux spécifiques, silicates en général d’aluminium plus ou moins hydratés, qui présentent une structure feuilletée (phyllosilicates) qui explique leur

plasticité, ou bien une structure fibreuse (sépiolite et palygorskite) qui explique leurs qualités d’absorption.

On les classe selon l’épaisseur des feuillets (0,7 ou 1 ou 1,4 nm).

Questions:

i) Peut-on modéliser la filtration à travers une telle

géométrie. Perméabilité (mesure de capacité d’un matériau à laisser passer un fluide)?

ii) Peut-on modéliser un écoulement ráctif, couplé avec la filtration/déformation, à travers une telle géométrie?

iii) Peut-on ajouter des effets électrocinétiques?

Asymptotic solution for an elastic channel 1

Figure 1: Schematic of a section of length L a long

elastic channel.

Asymptotic solution for an elastic channel 2

We consider the stationary laminar flow of incompressible Newtonian fluid through a 2-D channel with linearly elastic walls. The fluid is driven by the pressure gradient.

L is the length of the channel

ℓ is the half of the channel width in the undeformed state δ is the thickness of the walls in undeformed state

ℓ ∼ δ

ε = ℓ

L << 1. ⁽¹⁾

The outer walls of the channel are fixed. Due to symmetry, only the upper half of the fluid geometry will be considered, that is, the fluid and solid occupy

Asymptotic solution for an elastic channel 3

Ω^f₀ = {(x, y), 0 < x < L, 0 < y < ℓ} , ⁽²⁾ Ω^s₀ = {(x, y), 0 < x < L, l < y < ℓ + δ} , ⁽³⁾ respectively. Further, the height of the unknown boundary of the fluid-solid interface is denoted by γ(x). As a result, the fluid in the deformed configuration will occupy:

Ω^f = {(x, y), 0 < x < L, 0 < y < γ(x)} . ⁽⁴⁾ To derive an asymptotic solution to the FSI problem first we need the model:

Find Γ^I, ^v, p and ^u such that:

Γ^I = n

y + u(y)|∀y ∈ Γ^I₀o

(the interface), ⁽⁵⁾

Asymptotic solution for an elastic channel 4

In Ω^f we have the Stokes system for the flow:

−µ_f∆v + ∇p = ~b in Ω^f, ⁽⁶⁾

∇ · v = 0 in Ω^f, ⁽⁷⁾ In Ω^s₀ we have the equilibrium equations of the linear

elasticity

−∇ · (C : E(u)) = ~b₀ in Ω^s₀, ⁽⁸⁾ S = C : E(u) = λ_str (E(u)) I + 2µ_sE(u); 2E_ij = ∂u_i

∂x_j + ∂u_j

∂x_i .

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Asymptotic solution for an elastic channel 5

On the interface we have continuity of the contact forces, written in Lagrange’s configuration:

det(∇u + I)(−pI + 2µ_fD(v(y + u(y)))) (∇u + I)^−T n₀

= (C : E) n₀ on Γ^I₀. ⁽¹⁰⁾ v satisfies the kinematic interface condition

v = 0 on Γ^I. ⁽¹¹⁾ and in addition ^v, p and ^u should also satisfy any boundary conditions that might be specified on ∂Ω\Γ^I.

Observe that the position of the interface is part of the

boundary value problem, and the solid-fluid coupling term (10) makes it a nonlinear one.

Asymptotic solution for an elastic channel 6

STRATEGY:

The system (5)-(10) will be normalized for this particular geometry.

Then a formal expansion of the field variables

(pressure, velocity and displacements) with respect to the small parameter ε will be used to obtain an

asymptotic solution of (5)-(10).

This asymptotic solution will further be interpreted as a one-dimensional Darcy flow with a nonlinear

permeability.

Dimensionless form 1

x = Lx, y˜ = ℓy, p(x, y) =˜ IPp(˜˜ x,y˜), ⁽¹²⁾ v₁(x, y) = ¯V₁v˜₁(˜x, y), v˜ ₂(x, y) = ¯V₂v˜₂(˜x,y˜), ⁽¹³⁾ u₁(x, y) = ¯U₁u˜₁(˜x, y), u˜ ₂(x, y) = ¯U₂u˜₂(˜x,y˜). ⁽¹⁴⁾ In these notations the fluid (4) and solid (3) domain are

given by

Ω_f = {(˜x,y˜) : 0 < x <˜ 1, 0 < y <˜ γ(˜˜ x)}, Ω_s =

(˜x, y) : 0˜ < x <˜ 1, γ(˜˜ x) < y <˜ 1 + δ ℓ

,

Dimensionless form 2

˜

γ(˜x) = γ(x)

ℓ . ⁽¹⁵⁾

∂p

∂x = IP L

∂p˜

∂x˜, ∂p

∂y = IP ℓ

∂p˜

∂y˜, ⁽¹⁶⁾

∂v_i

∂x = V¯_i L

∂v˜_i

∂x˜ , ∂v_i

∂y = V¯_i ℓ

∂v˜_i

∂y˜ , ⁽¹⁷⁾

∂u_i

∂x = U¯_i L

∂u˜_i

∂x˜ , ∂u_i

∂y = U¯_i ℓ

∂u˜_i

∂y˜ . ⁽¹⁸⁾

It is clear that the scaling parameters can not be chosen independently. Below we will discuss the relations between different scaling parameters.

Dimensionless Stokes system

IP = µV¯₁

Lε² = µ4V_1,maxℓ²L

ℓ² = 4µV_1,maxL. ⁽¹⁹⁾

V¯₂ = εV¯₁ ⁽²⁰⁾

The Stokes system can now be rewritten as

−ε²∂²v˜₁

∂x˜² − ∂²v˜₁

∂y˜² + ∂p˜

∂x˜ = 0 ⁽²¹⁾

−ε² ∂²v˜₂

∂x˜² − ∂²v˜₂

∂y˜² + ε⁻² ∂p˜

∂y˜ = 0

∂v˜₁

∂x˜ + ∂v˜₂

∂y˜ = 0.

Dimensionless Navier system

S^s =





(λ_s + 2µ_s)^U¯_L^{1 ∂}_∂^u˜_x˜¹ + λ_s^U¯_l^{2 ∂}_∂^u˜_y˜² µ_{s ¯}

U1

l ∂u˜1

∂y˜ + ^U¯_L^{2 ∂}_∂^u˜_x˜² µ_{s ¯}

U1

l ∂u˜1

∂y˜ + ^U¯_L^{2 ∂}_∂^u˜_x˜²

(λ_s + 2µ_s)^U¯_l^{2 ∂}_∂^u˜_y˜² + λ_s ^U¯_L^{1 ∂}_∂^u˜_x˜¹





Further assume that

U¯₂ = δ, U¯₁ = ε⁰U¯₂ = δ.

Using this scaling for the displacements, the stress in the solid become

S^s = δ ℓ

"

(λ_s + 2µ_s)ε^∂_∂^u˜_x˜¹ + λ_s ^∂_∂^u˜_y˜² µ_s^∂_∂^u˜_y˜¹ + µ_sε^∂_∂^u˜_x˜² µ_s ^∂_∂^u˜_y˜¹ + µ_sε^∂_∂^u˜_x˜² (λ_s + 2µ_s)^∂_∂^u˜_y˜² + λ_sε^∂_∂^u˜_x˜¹

# .

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Dimensionless Navier system 2

Further, it is also necessary to write the system of elasticity equations (8) in non-dimensional form. For an isotropic

solid, and in the absence of a body force, it is easy to see, the equation (8) reduces to

ε²δ(λ_s + 2µ_s)∂²u˜₁

∂x˜² + εδ(λ_s + µ_s) ∂²u˜₂

∂x∂˜ y˜ + δµ_s∂²u˜₁

∂y˜² = 0, ⁽²³⁾ ε²δµ_s ∂²u˜₂

∂x˜² + εδ(µ_s + λ_s) ∂²u˜₁

∂x∂˜ y˜ + δ(λ_s + 2µ_s)∂²u˜₂

∂y˜² = 0. ⁽²⁴⁾

Asymptotic expansion

Consider now, an asymptotic expansions of the field variables with respect to the small parameter ε:

˜

v_i = ˜v_i⁰ + εv˜_i¹ + ε²v˜_i² + ... ⁽²⁵⁾

˜

p = ˜p⁰ + εp˜¹ + ε²p˜² + ...

˜

u_i = ˜u⁰_i + εu˜¹_i + ε²u˜²_i + ... ⁽²⁶⁾

Asymptotic expansion for Stokes system

Substituting these expansions in Stokes system (21), and collecting terms corresponding to different powers of ε, we get:

at order ε⁻² : ∂p˜⁰

∂y˜ = 0, at orderε⁻¹ : ∂p˜¹

∂y˜ = 0. ⁽²⁷⁾

Asymptotic expansion for Stokes system

Next, at order ε⁰ we obtain:

−∂²v˜₁⁰

∂y˜² + ∂p˜⁰

∂x˜ = 0 ^(28a)

−∂²v˜₂⁰

∂y˜² + ∂p˜²

∂y˜ = 0 ^(28b)

∂v˜₁⁰

∂x˜ + ∂v˜₂⁰

∂y˜ = 0. ^(28c)

⇒ p˜⁰ = ˜p⁰(x), p˜¹ = ˜p¹(x). ⁽²⁹⁾ Integration of the equation (28a) with respect to y˜ yields:

∂v˜₁⁰

∂y˜ = ˜y∂p˜⁰

∂x˜ .

Asymptotic expansion for Stokes system 2

The last equation, after another integration with respect to

˜

y, from y˜ to γ˜(˜x), becomes:

˜

v₁⁰(˜x, γ˜(˜x)) − v₁⁰(˜x, y˜) = 1

2 γ˜²(˜x) − y˜² ∂p˜⁰

∂x˜ . ⁽³⁰⁾

⇒ − ∂

∂x˜

1

3γ˜³(˜x)∂p˜⁰

∂x˜

= 0. ⁽³¹⁾

Observe that the last equation can be interpreted as a Darcy flow in a 1-D porous medium. To do this, fix x˜ and define, the y˜−average operator _h·iy˜:

hφ(˜˜ x, y)˜ i^y˜ := 1 2

Z _{γ(˜˜ x)}

−γ(˜˜ x)

φ(˜˜ x, y))dy˜ ⁽³²⁾

Asymptotic expansion for Stokes system 3

where φ(˜^˜ x, y)˜ is a generic function. The average is with respect to the reference width of the fluid channel. Then:

hv˜₁(˜x)i = −1

3γ˜³(˜x)∂p˜⁰

∂x˜ ⁽³³⁾

One would now recognize that equation (31) is the

conservation of mass for a flow with flux _hv˜₁(˜x)i, driven by a pressure gradient ^∂_∂^p˜_x˜⁰ . Furthermore, the quantity K^˜:

K˜ := ˜K(˜γ³(˜x), x) =˜ −µ_f hv˜₁(˜x)i

∂_x˜p˜⁰(˜x) = µ_f

3 γ˜³(˜x) ⁽³⁴⁾

Asymptotic expansion for Stokes system 4

defined as the ratio of the mass flux and the pressure

gradient, can be interpreted as permeability of the channel.

The permeability is scaled, as usual, by the fluid viscosity, so it doesn’t depend on the fluid properties. In the rigid case (γ˜(˜x) = 1) it will coincide with the standard Darcy

permeability. When the channel is deformable however, it is not constant but will depend on the 3-rd power of the

channel opening. With the definition (34) in mind, equation (31) can be rewritten as

− ∂

∂x˜

K˜ (˜γ³(˜x))∂p˜⁰

∂x˜

= 0. ⁽³⁵⁾

We need an expression for γ˜ in terms of p˜⁰ which will close the last equation.

Asymptotic expansion for elasticity system

Here we get

˜

u⁰₁(˜x, y) =˜ − 1 + ^δ_ℓ − y˜

µ_s c₁(˜x) ⁽³⁶⁾ and

˜

u⁰₂(˜x, y) =˜ − 1 + ^δ_ℓ − y˜

λ_s + 2µ_s c₂(˜x). ⁽³⁷⁾

Expansion of solid-fluid interface condition

First for the fluid stress ^Tf we get

T^f = IP







2ε² ∂v˜₁

∂x˜ − p˜ ε∂v˜₁

∂y˜ + ε³ ∂v˜₂

∂x˜ ε∂v˜₁

∂y˜ + ε³ ∂v˜₂

∂x˜ 2ε²∂v˜₂

∂y˜ − p˜





 ⁽³⁸⁾

Asymptotics for the solid-fluid interface

Next, the expansions for velocity and pressure from (25) are substituted into the last equation (38) to obtain

T^f = µ_fV¯₁L ℓ²

"

−p˜⁰(˜x) 0 0 −p˜⁰(˜x)

#

+ O (ε) . ⁽³⁹⁾ The normal component of the zeroth order term for the

stress tensor is calculated at the interface, at the reference configuration, for which the normal is simply ⁿ₀ = e₂:

S^s,0n₀ = δ

ℓ (c₁(˜x)e₁ + c₂(˜x)e₂) . ⁽⁴⁰⁾ Note that the last equation cannot be compared directly

with (39) as the first is in Lagrangian, while the former

Asymptotics for the solid-fluid interface 2

in Eulerian coordinates. However, the point (˜x, 1), at the interface in the reference configuration, corresponds to the interface point (˜x_E, γ˜(˜x_E)) in the Eulerian formulation, where

˜

x_E = ˜x + ˜u⁰₁(˜x) and γ˜(˜x_E) = 1 + ˜u⁰₂(˜x). ⁽⁴¹⁾ The fluid stress force at the interface in the reference

configuration is given by the expression at the left hand side in the formula (10). To use equation (10) one needs the

leading order of the deformation gradient:

F = ∇u+I = 1 + ^∂u_∂x^{1 ∂u}_∂y¹

∂u2

∂x 1 + ^∂u_∂y²

!

= 1 + ε^∂_∂^u˜_x˜^{1 δ}_l ^∂_∂^u˜_y˜¹ ε^∂_∂^u˜_x˜² 1 + ^δ_l ^∂_∂^u˜_y˜²

!

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Asymptotics for the solid-fluid interface 3

therefore, ^F = F⁰ + O (ε), where F⁰ = 1 ^δ_l ^∂_∂^u˜_y˜⁰¹

0 1 + ^δ_l ^∂_∂^u˜_y˜⁰²

!

and ^F0−T =

1 + δ l

∂u˜⁰₂

∂y˜

₋¹ 1 + ^δ_l ^∂_∂^u˜_y˜⁰²

−^δ_l ^∂_∂^u˜_y˜⁰¹

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The insertion of (39), (40) and (43) into (10) now results in:

−IPp˜⁰(˜x)e₂ = δ

ℓ (c₁(˜x)e₁ + c₂(˜x)e₂) + O(ε). ⁽⁴⁴⁾

⇒ u˜⁰₁ = c₁(˜x) = 0,

˜

γ(˜x) = 1 + ˜u⁰₂(˜x, 1) = 1 + IP

λ_s + 2µ_s p˜⁰(˜x). ⁽⁴⁵⁾

Permeability of a long elastic channel

Equivalently, in dimensional variables, γ(x) = ℓ + ℓ

λ_s + 2µ_s p⁰(x). ⁽⁴⁶⁾

Permeability of a long elastic channel

K = K(p⁰(x), x) = 1 3ℓ

l + l

λ_s + 2µ_s p⁰(x) 3

= ℓ² 3

1 + p⁰(x) λ_s + 2µ_s

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Long elastic channel: A typical solution

Comments:

For more details see: O. Iliev, A. Mikeli´c, P. Popov ^{: On}

upscaling certain flows in deformable porous media, Multiscale Model. Simul., Vol. 7 (2008), no. 1, p. 93-123.

For application to the blood flow see: A. Mikeli´c, G. Guidoboni, S.

Cani´c : Fluid-Structure Interaction in a Pre-Stressed Tube with Thickˇ Elastic Walls I: The Stationary Stokes Problem, Networks and

Heterogeneous Media, Vol. 2 (2007), p. 397 - 423.

In the above reference the following law was derived:

∆p(z) =

K ₍₁₋^Eh_σ2)R² − P_eR

u⁰_s(R, z). ⁽⁴⁸⁾

where

K = 1 2

2 + h/R (1 + h/R)²,

Open questions:

u⁰_s(R, z) is the radial displacement at the interface, and

∆p(z) is the transmural pressure. Notice that for K = 1 this is exactly the Law of Laplace.

Open questions:

Nonlinear elastic solid structure?

Electrokinetic effects?

Coupling with reactive flows?

Dynamic Permeability 1

Fluid-saturated porous media arise in contexts as

sedimentary rocks and soils and also in study of metallic foam structures, polymer gels and catalytic beds. One of basic properties of porous media is the fluid permeability .

This key macroscopic property of porous media is described by the well-known Darcy law

U~ (x) = −k_dc

µ ∇p₀(x), ⁽⁴⁹⁾

U~ is the average fluid velocity (= the filtration velocity)

∇p₀ is the pressure gradient µ is the dynamic fluid viscosity k_dc is the static fluid permeability.

Dynamic Permeability 2

Darcy’s law is valid for the slow flow of viscous fluids. It corresponds to the homogenization of the complex fluid motions taking place at the microscopic scale in the pore structure. Hence the permeability tensor k_dc depends

nonlocally on the morphology of the pore space.

We note the considerable progress in understanding the Darcy law (49) by homogenizing the Stokes and the

Navier-Stokes fluid equations from the microscopic pore level to the macroscopic level, see e.g.

G. Allaire, in : One-phase newtonian flow, Homogenization and porous media, ed. U. Hornung, Springer 1997, pp. 45 - 68.

A. Mikeli´c, in : Homogenization theory and applications to filtration through porous media, Filtration in Porous Media and Industrial

Applications, Lecture Notes in Mathematics Vol. 1734, Springer, 2000, pp. 127-214.

Dynamic Permeability 3

Another important transport phenomena involve electrical conduction. Electrical formation factor F is given by

F = σ

σ^∗ , ⁽⁵⁰⁾

where σ^∗ is the effective electrical conductivity of a porous medium containing fluid of conductivity σ and insulating solid phase.

Relationship between the permeability and conductivity:

k_dc = c₁ ℓ²_c

F . ⁽⁵¹⁾

(see Katz and Thompson, Phys. Rev. B 1986. ) ℓ_c is a

characteristic length related to the threshold pressure in a mercury injection experiment.

Dynamic Permeability 4

c₁ is a constant, estimated to be 4.4_×10⁻³. ℓ_c is well defined only when the porous medium can be modeled as a

distribution of cylindrical pores on a lattice.

The formal analogy between Darcy’s law (49) and Ohm’s law for bulk electrical transport in porous media suggests a relationship between electricity and hydrodynamics at the macroscopic level. In this direction Johnson, Koplik and

Schwartz introduced a geometrical pore-size parameter Λ, which is always well defined for any porous medium and

which describes the effects of an internal boundary layer on a variety of processes such as electrical surface conduction etc. Let E be the local electrical field, then the parameter Λ is a weighted pore volume-to-surface area ratio

Λ = 2 R

|E(r)| dV

, ⁽⁵²⁾

Dynamic Permeability 5

where dV denotes an integration over the pore volume and dS denotes an integration over the pore-solid surface.

Based on plausible physical arguments, and backed by detailed computer simulations on random materials,

Johnson, Koplik and Schwartz found the formula k_dc = c₂ Λ²

8F , ⁽⁵³⁾

where c₂ is a constant of order 1. We note that Λ/2 6= V_P /S, where V_p is the pore volume and S is the surface area of the pore-grain interface. Λ is a length that is directly related to transport,; it is a measure of the dynamically connected part of the pore space.

Random Porous Media

Let us define a random porous medium after the article M. Avellaneda, S. Torquato : Flow in random porous media :

mathematical formulation, variational principles, and rigorous bounds , J. Fluid Mech. , Vol. ²⁰⁶ (1989) , p. 25-46.

The random porous medium is a domain of space _V(ω) ∈ IR³ (where the realization ω is taken from probability space Ω) which consists of two regions : the pore space _V1(ω),

through which fluid flows, of the porosity ϕ and a

solid-phase region _V2(ω). Let ∂V(ω) be the surface between V¹(ω) and _V2(ω). The characteristic function of the pore

region is given by

I(x, ω) =

(1 x ∈ V¹(ω),

0 x ∈ V²(ω). ⁽⁵⁴⁾

Random Porous Media 2

We assume that our porous medium is a realization of a

statistically homogeneous ergodic random porous medium, scaled by a small parameter a . This means that by means of Birkhoff’s ergodic theorem we equate ensemble

averages with infinite volume averages, i.e. we have

< f(x₀) >= lim

|C|→∞

1

|C|

Z

C

f(x₀ + x) dx ⁽⁵⁵⁾ where < · · · > denotes ensemble averaging. For periodic porous media, this corresponds to the averaging over a unit cell.

We note that

ϕ =< I >= lim

|V¹|,|V|→∞

|V¹|

|V| ; S =< M >= lim

|∂V|,|V|→∞

|∂V|

|V|

Expos ´e `a l’Atelier ”ANALYSE MULTIECHELLE EN ELECTROCINETIQUE ET APPLICATIONS AUX MILIEUX POREUX”, CMAP, Ecole Polytechnique, Palaiseau, le 22 mars 2010 – p. 33/72(56)

Random Porous Media 3

Let us recall that the fluid permeability arising in Darcy’s law (49) can be expressed in terms of a certain scaled auxiliary velocity field for periodic media (see e.g. the minicours by G. Allaire) and random media (see e.g. papers by S.

Kozlov). Following Avellaneda and Torquato, we write it as (k_dc)_ij =< w_jⁱ >, ⁽⁵⁷⁾ where the auxiliary scaled velocity wⁱ satisfies

∆wⁱ = ∇πⁱ − eⁱ in _V1, ⁽⁵⁸⁾

div wⁱ = 0 in _V1, ⁽⁵⁹⁾

wⁱ = 0 on ∂V ⁽⁶⁰⁾

{wⁱ, πⁱ} are bounded functions . ⁽⁶¹⁾

Random Porous Media 4

Now we consider a fluid-filled porous medium, with

connected pore space, under the excitation of an external harmonic source with frequency ω.

The pore structure at microlevel is characterized by a

typical length of the pore scale a. In order to be as close to the reality as possible, we suppose that the fluid has

dynamic viscosity µ, bulk modulus K and density ρ_f.

Intrinistic viscous relaxation time is t₀ = ρ_fa²/µ the fluid sound velocity c₀ = p

K/ρ_f.

We note that t₀ is the time scale at which the inertial force density inside a pore, ρ_fa/t²₀ equals the viscous force

density µ/(t₀a). The product of these two quantities defines the large scale L = t c .

Dynamic Permeability 6

The displacements and the Reynolds number are small and the linearized fluid/structure interaction is considered. A

dimensionless small parameter is ε = a/L. After making the change of variables y = x/a and τ = t/t₀ and looking for

time harmonic motions with angular frequency ω, we find the following dimensionless form of the problem

−iωε~v = Div σ in _V1 ⁽⁶²⁾ σ = −pI + 2εD(~v) in _V1 ⁽⁶³⁾

iωεp = div ~v in _V1 ⁽⁶⁴⁾

εΘ = CD(~u) in _V2 ⁽⁶⁵⁾

− ρ_s

ρ_f ω²ε~u = Div Θ in _V2 ⁽⁶⁶⁾

~v = −iω~u and σ~n = Θ~n on ∂V. ⁽⁶⁷⁾

Dynamic Permeability 7

Here C is the fourth-order elastic tensor in the units of K and p, σ and Θ are in units of Lµ²/(ρ_fa³).

Using the modified expansion of Burridge and Keller, Jour.

Acoustic Soc. Amer. 1981 , the field quantities are expanded as a two-scale perturbation series in ε :

~v = v₀(x, y) + εv₁(x, y) + . . . , with x = r/L and y = r/a. Then the derivatives are expressed with respect to x and y

variables.

By equating terms with equal powers of ε, a hierarchy of equations for the quantities v₀, v₁, u₀, u₁, σ₀, σ₁, p₀, p₁ etc is obtained. The relevant lowest-order (ε⁻¹) equations are :

Div_yσ₀ = 0 and σ₀ = −p₀I ⁽⁶⁸⁾ div_yv₀ = 0 and D_y(u₀) = 0. ⁽⁶⁹⁾

Dynamic Permeability 8

Now we conclude that u₀ = u₀(x) , σ₀ = σ₀(x) and, as far as the fluid is concerned, to the lowest order it may be ^regarded

as incompressible on the y scale.

Next following Biot’s articles v₀ is expressed in terms of u₀ and a relative fluid permeability w

v₀ = −iω

u₀(x) + w(x, y)~

. ⁽⁷⁰⁾

Now we introduce the corresponding auxiliary problem :

−iω∆_yW ^↔ = ∇^yP^↔ − ω²W^↔ − I^↔ in _V1, ⁽⁷¹⁾

div W ^↔ = 0 in _V1, ⁽⁷²⁾

W^↔ = 0 on ∂V ⁽⁷³⁾

{W^↔, P^↔} are bounded functions . ⁽⁷⁴⁾

Dynamic Permeability 9

The average flow rate U^~ is obtained from 70 by averaging w over the pore scale y :

U~ = −iω < ~w >_y= −iω < W^↔(x, y) >_y

−∇^xp₀+ω²u₀

(75)

Equation (75) may be viewed as a generalization of the

Darcy’s law to deformable elastic porous media. The elastic solid displacement actes as an additional source term of

the form ε²u₀. Comparing (75) with the Darcy’s law implies the following definition of the dynamic permeability

κ^↔(ω) = −iω < W^↔(x, y) >_y ⁽⁷⁶⁾

Dynamic Permeability 10

The real and imaginary parts of κ^↔(ω) behave very differently. The real component is dominant at low

frequencies, where viscous stresses are important. The imaginary component dominates at high frequencies, because in this regime the fluid behaves as if it were

inviscid in almost all of the pore space and viscous effects are confined to a boundary layer of width ^p2µ/(ρ_fω) near the pore surface. More precisely,

ℜκ^↔(ω) ∼ k_dc and Im κ^↔(ω) ∼ ωρ_f

µ < |W^↔(y, 0)|² >_y, for ω ≪

(77)

where W^↔(y, 0) is the solution for (71)-(73) for ω = 0.

The high frequencies limit was studied in number of papers.

Dynamic Permeability 11

The main result is the following : ℜκ^↔(ω) ∼

√2 ΛF

ωρ_f µ

₋3/2

and Im κ^↔(ω) ∼ µ

ωF ρ_f , for ω ≫ 1,

(78)

where Λ is is a weighted pore volume-to-surface area ratio given by (52) and electrical formation factor F is given by (50). The limit is calculated using the observation

ω²W ^↔ ∼ −I^↔ + ∇^yP^↔ ∼ E, ω ≫ 1, ⁽⁷⁹⁾ where E the electrical field satisfying

curl E = 0, div E = 0 ⁽⁸⁰⁾

in the pore geometry and the boundary condition E · n = 0

Dynamic Permeability 12

We note that the boundary value problem for the field E can be interpreted as the stationary potential flow of an ideal

fluid through the microstructure arising from an imposed pressure drop I^↔.

Next, from dimensional analysis, the dynamic permeability can be written in the scaled form

κ^↔(ω) = R²

8F f(ωρ_fR²

µ ), ⁽⁸¹⁾

where R is the length scale defining the effective hydrodynamic pore radius of the microstructure.

As remarked by Avellaneda et al, the determination of

κ^↔(ω) requires ideally an infinite set of measurements. At the other hand, empiric theory requires only the

measurements of Λ and F.

Dynamic Permeability 13

The approach is based on a hypothesis of universality of the dynamic response , supposed to be valid for a wide class of porous media.

Explicit formula of Johnson, Koplik and Dashen 1986:

κ(ω) = Λ²

F g(ωρ_fΛ²

µ ) with g(˜ω) = 1 8p

1 − iω/16˜ − iω˜ . ⁽⁸²⁾ The permeability calculation for periodic porous media:

κ(ω) =

(κ₀ + iC₁ωρ_f/µ as ω → 0,

√2

ΛF (ωµ/ρ_f)^−3/2 + iµ/(F ρ_fω) as ω → ∞. ⁽⁸³⁾

Geometry

Periodic 2-scale conv for porous media pb 1

Description of a periodic porous medium

We consider a periodic porous medium Ω =]0, L[³ in IR³ with a periodic arrangement of the pores. The formal description goes along the following lines: First we define the

geometrical structure inside the unit cell _Y =]0, 1[³. Let _Ys (the solid part) be a closed subset of _Y^¯ and _Yf = Y\Y^s (the fluid part). Now we make the periodic repetition of _Ys all

over IR³ and set _Ys^k = Y^s + k, k ∈ ZZ³. Obviously the set E_S = S

k∈ZZ³ Ys^k is a closed subset of IR³ and E_F = IR³\E_S is an open set in IR³. We make the following assumptions on Y^f and E_F :

(i) _Yf is an open connected set of strictly positive measure, with a Lipschitz boundary and _Ys has strictly positive

measure in _Y^¯ as well.

Periodic 2-scale conv for porous media pb 2

(ii) E_F and the interior of E_S are open sets with the

boundary of class C^1,1, which are locally located on one side of their boundary. Moreover E_F is connected and the solid part, E_S, is supposed connected in IR³

Now we see that Ω =]0, L[³ is covered with a regular mesh of size ε, each cell being a cube _Yi^ε = ε(Y + i), with

1 ≤ i ≤ N(ε) = |Ω|ε⁻³[1 + 0(1)]. We define _Ys^ε_i = (Π^ε_i)⁻¹(Y^s) and _Yf^ε

i = (Π^ε_i)⁻¹(Yf). For sufficiently small ε > 0 we

suppose L/ε ∈ IN and consider T_ε = {k ∈ ZZ³|Y_s^ε_k ⊂ Ω} and define

Ω^ε_s = [

k∈Tε

Ys^εk, Γ^ε = ∂Ω^ε_s, Ω^ε_f = Ω \ Ω^ε_s.

Periodic 2-scale conv for porous media pb

Obviously, ∂Ω^ε_f = ∂Ω ∪ Γ^ε. The domains Ω^ε_s and Ω^ε_f

represent, respectively, the solid and fluid parts of a porous medium Ω.

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00000000 00000000 00000000 11111111 11111111 11111111

00000000 00000000 00000000 11111111 11111111 11111111 00

11 00 11 00 11 00 11 00 11

000 111 000 111 000 111 000 111 000 111

00000000 11111111 00

00 00 00

11 11 11 11

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0000 00 1111 11

0000 00 1111 11

0000 00 1111 11

0000 00 1111 11 000000

000 111111 111

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0000 00 1111 11

0000 00 1111 11

0000 00 1111 11

0000 00 1111 11 000000

000 111111 111

000000 000 111111 111

000000 000 111111 111

0000 00 1111 11

0000 00 1111 11

0000 00 1111 11

0000 00 1111 11 000000

000 111111 111

000000 000 111111 111

000000 000 111111 111

0000 00 1111 11

0000 00 1111 11

0000 00 1111 11

0000 00 1111 11 000000

000 111111 111

000000 000 111111 111

000000 000 111111 111

0000 00 1111 11

0000 00 1111 11

0000 00 1111 11

0000 00 1111 11 000000

000 111111 111 000000 000 111111 111 000000 000 111111 111 000000 000 111111 111 000000 000 111111 111

000000 000 111111 000111 000000 111111 111

000000 000 111111 111

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000000 000 111111 111

0000 00 1111 11

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000 111 000 111 000 111 000 111

00 11

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0000 00 1111 11

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0000 00 1111 000 11

111

z r

fluid

FLUID

0 < y < 1

0<r<1~ FLUID

FLUID

2ε

PERIODIC DOMAIN IN TUBE FLOW PERIODIC DOMAIN IN POROUS MEDIA FLOW

Biot’s law without dissipation 1

Obviously, ∂Ω^ε_f = ∂Ω ∪ Γ^ε. The domains Ω^ε_s and Ω^ε_f

represent, respectively, the solid and fluid parts of a porous medium Ω.We suppose small deformations and the

displacements are described by linearized equations in both media. More precisely, in Ω^ε_f we have the linearized momentum equation for the time derivative of the fluid displacement u^ε, in Eulerian variables.

Ω^ε_s is the reference configuration of a deformable elastic

body and the equation of linear elasticity for the deformation w^ε are in Lagrangean variables. In both domains all

quadratic terms are neglected. The interface between two media changes with the deformation of the structure.

Biot’s law without dissipation 2

If the pores contain an inviscid fluid then the kinematic interface condition is continuity of the normal velocities. It reads

∀t, ∀x ∈ Γ^ε, ∂u^ε

∂t (x + w^ε(x, t), t) · ν = ∂w^ε

∂t (x, t) · ν.

After linearization and neglecting the term

∇^x ∂u^ε

∂t (x, t) · w^ε(x, t), we get continuity of the normal displacements at Γ^ε.

example 3: Biot’s law without dissipation 3

Let F ∈ C^∞([0, T]; L²(Ω)³) and curl F ∈ C^∞([0, T]; L²(Ω)³) . Then we consider the system

ρ_s ∂²w^ε

∂t² − div(σ(w^ε)) = F ρ_s and σ(w^ε) = AD(w^ε) in Ω^ε_s×]0, T[,

(84)

ρ_f ∂²u^ε

∂t² + ∇p^ε = F ρ_f and div ^∂u

ε

∂t = 0 in Ω^ε_f×]0, T[, ⁽⁸⁵⁾ (u^ε − w^ε) · ν = 0 and (p^εI + σ(w^ε)) · ν = 0 on Γ_ε×]0, T[ ⁽⁸⁶⁾ {u^ε, w^ε, p^ε} are L − periodic. ⁽⁸⁷⁾ u^ε(x, 0) = ∂u^ε

∂t (x, 0) = 0 in Ω^ε_f; w^ε(x, 0) = ∂w^ε

∂t (x, 0) = 0 in Ω^ε_s

(88)

Biot’s law without dissipation 4

The above system has a unique variational solution in V = {ϕ ∈ L²(Ω)³ ; ϕ ∈ H¹(Ω^ε_s)³, div ϕ = 0 in Ω^ε_f,

div ϕ ∈ L²(Ω) and ϕ is L-periodic_}. It is arbitrary smooth with respect to t.

After taking u^ε ∈ H³(0, T; L²(Ω^ε_f)³) ∩ H²(0, T; H¹_per (Ω^ε_f)³) , w^ε ∈ H³(0, T; L²(Ω^ε_s)³) and p^ε ∈ H¹(0, T; L²(Ω^ε_f)) as the test function in (84)-(88) we obtain

∂u^ε

∂t

L^∞(0,T;L²(Ω^ε_f)³)+

∂w^ε

∂t

L^∞(0,T;L²(Ω^ε_s)³)+

∇w^ε

L^∞(0,T;L²(Ω^ε_s)⁹) ≤

(89)

Biot’s law without dissipation 5

The linearized incompressible Euler system doesn’t involve derivatives of the velocity field with respect to x and an

H¹-estimate for the velocity doesn’t follow directly. One way to proceed is to use the H(div; Ω^ε_f) estimate in the fluid part.

Nevertheless, after taking the curl of the linearized Euler system, we get

div u^ε = 0 and curl u^ε

L^∞(0,T;L²(Ω^ε_f)³) ≤ C ⁽⁹⁰⁾ and it is natural to estimate the L²-norm of the _∇u^ε by the L²-norms of div u^ε and curl u^ε. Such estimate requires a boundary condition at the interface. In our model it is a given normal component u^ε · ν and then the H¹-estimate uniform with respect to ε holds if and only if the first Betti number of Ω^ε is zero.

Biot’s law without dissipation 6

By assumptions, ∂Ω^ε_f is a connected 2D manifold. Each

such manifold is homeomorphic to a sphere with ”handles"

and the first Betti number or the genus of ∂Ω^ε_f is the number of handles. For a simply connected domain, the first Betti number is zero. Our domain is multiply connected and the estimate requires some effort. By supposing that

∂Yf ∈ C^1,1 we get

∇u^ε

L^∞(0,T;L²(Ω^ε_f)⁹) ≤ C

ε . ⁽⁹¹⁾

For the proof see [FeMi03] J. L. Ferr´ın , A. Mikeli ´c :

Homogenizing the Acoustic Properties of a Porous Matrix Containing an Incompressible Inviscid Fluid , Mathematical Methods in the Applied

Sciences, Vol. 26 (2003), p. 831-859.

Biot’s law without dissipation 7

We note that for isolated fluid parts the velocity gradient is uniformly bounded, leading to different results. For more details we refer to [FeMi] . The extension of the pressure field p^ε to Ω^ε_s×]0, T[ is given by

˜

p^ε(x, t) =











p^ε(x, t) − _|Ω¹_| R

Ω^ε_f

p^ε(x, t) dx, x ∈ Ω^ε_f,

−_|Ω¹_| R

Ω^ε_f

p^ε(x, t)dx, x ∈ Ω^ε_s. ⁽⁹²⁾

Then ^R

Ω

˜

p^εdiv ϕ = R

Ω^ε_f

p^εdiv ϕ, _∀ϕ ∈ H¹_per (Ω)³, and

p˜^ε

H¹(0,T;L²₀(Ω)) +

∇p˜^ε

H¹(0,T;H⁻per¹ (Ω)³) ≤ C. ⁽⁹³⁾

Biot’s law without dissipation 8

If we have two different estimates for gradients in the solid and in the fluid part, then the classical way to proceed is by extending the deformation from Ω^ε_f to Ω and then passing to the limit ε → 0. For precise recent results on

homogenization of Neumann problems in perforated domains with Lipschitz perforations we refer to

[Ace92] Acerbi, E., Chiadò Piat, V., Dal Maso, G., Percivale D. : An extension theorem from connected sets, and

homogenization in general periodic domains. Nonlinear Anal., TMA, 18 481–496 (1992) and to the book by Jikov, Kozlov and Oleinik [JKO].

Here we need the " triple field compactness " result from [All92], modified in [FeMi03]