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Homogenization of a thermo-poro-elastic medium

with two-components and interfacial

hydraulic/thermal exchange barrier

Abdelhamid Ainouz

Dept. of Maths, Univ. Sci. Tech. H. Boumedienne

Po Box 32, El-Alia, Bab-Ezzouar, Algiers, Algeria

Abstract

The paper addresses the homogenization of a micro-model of poroelas-ticity coupled with thermal effects for two-constituent media and with im-perfect interfacial contact.The homogenized model is obtained by means ofthe two-scale convergence technique. It is shown that the macro-thermo-poro-elasticity model with double porosity/diffusivity contains in particu-lar some extra terms accounting for the micro-heterogeneities and imper-fect contact at the local scale. Finally, a corrector result is given undermore regularity assumptions on the data.

1 Introduction

Materials like sedimentary rocks and living tissues are generally considered asporous, compressible and elastic. Generally speaking, these porous materialsallows for transport mass of some substance through their open channels suchas liquid or gas. The presence of a fluid in such diffusive porous materials affectsits mechanical responses. Their elasticity properties are then clearly highlightedby the compression resulting from the fluid pressure since any release of fluidstorage causes shrinkage of the pore volume. This approach, called poroelas-ticity, accounts for the coupling of the pore pressure field with the stresses inthe skeleton. It is first derived by K. Von Terzaghi [49] in the one-dimensionalsetting and later generalized by M. Biot [14] to the three-dimensional case. Itcombines the Hookean classic elasticity theory for the mechanical response ofthe solid with the Darcy flow diffusion model for the fluid transport within thepores. However, coupling thermal effects with poromechanical processes is ofgreat importance in real-world applications such as geomechanics, civil engi-neering, biophysics. For instance, cold water injection into a hot hydrocarbonreservoir causes changes in porosity and permeability. These heat transfer pro-cesses yield soil deformations of the geothermal well, see M. C. Suarez-Arriaga[46]. An other example is given by the effects of the temperature on concrete

1

(buildings, bridges,...) which highly influence its strength development and itsdurability. It is well known that high temperature causes cracking and strengthloss within the concrete mass, see for instance, W. Khaliq and V. Kodur [32],J. Shi & al. [42], K. Y. Shin & al. [43], G. Weidenfeld & al. [51], W. Wang& al [52]. In bio-engineering, there are many important applications of thermaldiffusion processes in biological tissues such as thermoregulation, thermother-apy, radiotherapy, ... In some situations tissues are often assumed poroelasticmaterials such as brain tissues [19], bone [24], skin [29, 54], living organs [34, 39]and tumors [8, 47].

The thermoporoelasticity theory aims to describe the thermal/fluid flows andelastic behavior of heat conductive, porous and elastic media, see for instanceR.W. Zimmerman [55], O. Coussy [23], A. H. Cheng [18]. A mathematical modelfor such conductive and porous materials is a set of governing equations whichconsists of the momentum balance, the conservation of mass and the energybalance equations. The purpose is to predict deformations, fluid pressure andthe temperature under different internal, external forces and thermal sources.It is the classical Biot’s theory of consolidation processes coupled to thermalstresses.

Generally, mathematical models of flows in porous elastic media assume thatthe domain consists of a system of single network where the pores have the samesize. However, in many natural situations such as aggregated soils or fissuredrocks, materials exhibit two (or more) dominant pore scales. Mathematicalmodeling of multiple porosity media is the subject of great research activity sincethe pioneering work of Barenblatt et al. [11]. This concept has been appliedto many areas of engineering, see for instance Bai & al. [9, 10], Berryman andWang [13], Cowin [24], Khalil and Selvadurai [31], Straughan [45] and T. D. TranNgoc & al. [48], Wilson and Aifantis [53]. In this context, it is assumed thatthere exist two porous structures: one is related to macro-porosity connected tothe pores of the material and the other to micro-porosity connected to fissuresof the skeleton. This causes different pressure fields in the micropores andmacropores. Furthermore, the main temperature effect on any kind of media(solid, liquid or gas) is to induce the phenomenon of thermal expansion thatis an increase or a shrinkage in volume. Furthermore, the temperature is alsoconsidered different in each phase. The concept of thermoelasticity with at leasttwo (or multiple) temperatures was first initiated by Chen, Gurtin and Williamssee [17] and further developed by many researchers see for instance Masters &al. [35], D. Iesan [30], H.M. Youssef [50].

The goal of this paper is to derive rigorously, by means of the two-scaleconvergence technique, a new fully coupled model of thermoporoelasticity forbiphasic media. In particular, the work contains some original and essential ad-vances in the study of homogenization problems applied to poroelasticity. Noticethat there are many works devoted to the homogenization in poroelasticity andin thermoporoelasticity. We refer the reader for instance to [2, 3, 4, 21, 26, 36].The outline of the paper is divided into 4 main sections: Firstly, a micro-modelof poroelasticity coupled with thermal effects for two-constituent media is givenin section 2. It is taken into account that contact between these constituents is

2

imperfect so that fluid and heat flows through the interface is proportional tothe jump of the pressure and temperature field, respectively. Then in section 3,the weak formulation of the microscale problem and the main results are given.In section 4, the homogenization of the micro-model is given with the help ofthe two-scale convergence technique. The obtained macro-model for thermo-poroelasticity with double porosity/diffusivity is, at my knowledge, new in theliterature. It is mainly shown that micro-heterogeneities and imperfect contactat the local scale lead to a Biot/Thermal matrices and zeroth order term atthe macroscale, giving rise to absorption/diffusion term. Finally, in section 5, acorrector result is given under more regularity assumptions on the data.

2 Derivation of the micro-model

In this section we derive the set of thermo-hydro-mechanical equations for poroe-lastic materials, for more details see for e.g. O. Coussy [23].

2.1 Linear thermoporoelasticity equations

Let Ω be a bounded and smooth domain in R3 occupying during the time

interval [0, T ], T > 0 a saturated and poroelastic body which is subjected toa given body force per unit volume f0 [N=Kg.m-2.s-2], to a source/sink termg0 [Kg.m-3.s-1] and to a heat source h0 per unit time [Kg.m-1.s-3]. Let u [m]denote its displacement, p [Pa=Kg.m-1.s-2] its pressure, θ [K] its temperatureand σ [Pa] its Cauchy stress tensor. Throughout this paper the volumetricdensity ρ is taken for simplicity to be a positive constant [Kg.m-3]. Now, weintroduce the Helmholtz free energy:

A := W − Sθ (2.1)

where W [J.m-3, J=Kg.m2.s-2] is the internal energy per unit of volume relatedto the strain work density/porosity and S [J.m-3.K-1] is the entropy per unit ofvolume. The free energy A is the basic quantity to define the material. In thetheory of poromechanics and under the infinitesimal transformation, we assumethat the internal energy is a function of the state quantities e and φ:

W = W (e, φ)

where

eij (u) =1

2

(∂uj∂xi

+∂ui∂xj

), u = (ui) , 1 ≤ i, j ≤ 3

is the (linearized) strain tensor [dimensionless] and φ [dimensionless] is theporosity. Assuming an isentropic process, that is dS = 0, we get from (2.1)that

dA = dW − Sdθ = σde + pdφ− Sdθ (2.2)

where

σij :=∂W∂eij

, p :=∂W∂φ

.

3

Let us introduce the Legendre transform of the free energy A:

η := A− φp. (2.3)

It follows then from (2.2) and (2.3) that

dη = σde− φdp− Sdθ. (2.4)

Expanding the free energy η in a Taylor series around a reference state η0which corresponds to a free strain e0 = 0, a reference pressure p0, a referencetemperature θ0 and neglecting all terms up to the second order we obtain

η = η (e, p, θ) = η0 +∂η∂eij

∣∣∣0eij +

∂η∂p

∣∣∣0(p− p0) +

∂η∂θ

∣∣∣0(θ − θ0)+

∂2η∂eij∂p

∣∣∣0(p− p0) eij +

∂2η∂eij∂θ

∣∣∣0(θ − θ0) eij +

∂2η∂p∂θ

∣∣∣0(p− p0) (θ − θ0)

+ 12

∂2η∂eij∂ekh

∣∣∣0ekheij +

12

∂2η∂p2

∣∣∣0(p− p0)

2+ 1

2∂2η∂θ2

∣∣∣0(θ − θ0)

2

(2.5)

where (and in the sequel) summation over repeated indices is used. Assumingwithout no loss of generality that the energy function η presents an equilibriumpoint at η0 = 0, that is

∂η

∂eij

∣∣∣∣0

=∂η

∂p

∣∣∣∣0

=∂η

∂θ

∣∣∣∣0

= 0,

equation (2.5) reduces then to

η = 12eij

∂2η∂eij∂ekh

∣∣∣0ekh + eij

∂2η∂eij∂p

∣∣∣0(p− p0)+

eij∂2η

∂eij∂θ

∣∣∣0(θ − θ0) +

∂2η∂p∂θ

∣∣∣0(p− p0) (θ − θ0)+

12

∂2η∂p2

∣∣∣0(p− p0)

2 + 12

∂2η∂θ2

∣∣∣0(θ − θ0)

2

which can be rewritten in a more simplified form:

η = 12 (e : Ae)− (B : e) (p− p0)− (D : e) (θ − θ0)

+α (p− p0) (θ − θ0)− 12N (p− p0)

2 − υ2θ0

(θ − θ0)2

(2.6)

where

aijkh =∂2η

∂eij∂ekh

∣∣∣∣0

, bij = − ∂2η

∂eij∂p

∣∣∣∣0

, dij = − ∂2η

∂eij∂θ

∣∣∣∣0

(2.7)

α = − ∂2η

∂p∂θ

∣∣∣∣0

, N = −∂2η

∂p2

∣∣∣∣−1

0

, υ = −θ0∂2η

∂θ2

∣∣∣∣0

(2.8)

4

and

E : F = eijfij =

3∑

i,j=1

eijfij , E = (eij)1≤i,j≤3 , F = (fij)1≤i,j≤3 .

In (2.7), A = (aijkh) [Kg.m-1.s-2] is the (fourth-rank) elasticity stiffness tensor,B = (bij) [dimensionless] the symmetric stress-pressure tensor, expressing thechange in porosity to the strain variation when pressure and temperature arekept constant and D is the thermal dilation (symmetric) tensor related to thesolid deformation by the following expression:

D = (dij) , dij = − ∂2η

∂eij∂ekh

∂ekh∂θ

∣∣∣∣0

= aijkhγkh

where

γkh = −∂ekh∂θ

[K-1]

In (2.8), α [K-1] expresses the volumetric thermal dilation coefficient with re-spect to the pore pressure. Furthermore N [Kg.m-1.s-2] is the inverse of thecompressibility coefficient so it refers to a modulus relating the pressure p lin-early to the porosity variation φ when the volumetric dilation is kept zero .Finally υ [Kg.m-1.s-2.K-1] is the volumetric heat capacity. From (2.4) and (2.6)we deduce the constitutive equations:

σ = ∂eη = Ae− (p− p0)B − (θ − θ0)D, (2.9)

φ = −∂pη = B : e +1

N(p− p0) + α (θ − θ0) , (2.10)

S = −∂θη = D : e + α (p− p0) +υ

θ0(θ − θ0) . (2.11)

Equation (2.9) is well-known in the literature as the Duhamel–Neumann rela-tion.

In the framework of consolidation assumption, the inertia effects are ne-glected, that is ρ∂2ttu ≃ 0. In this case, the conservation of linear momentumequation reads in its differential form as

divσ + f0 = 0. (2.12)

Notice that

σ

ij (u) = σij (u)− (bij (p− p0) + dij (θ − θ0)) = aijkhekh (u)

is the effective stress tensor and equation (2.12) becomes then

− div (Ae (u)) +B∇p+D∇θ = f0. (2.13)

In the case of homogeneous and isotropic materials, the phenomenological ten-sors A, B and D take the simplified forms:

aijkh = λδijδkh + µ (δikδjh + δihδjk) , bij = βδij , (2.14)

dij = (3λ+ 2µ) γδij , γkh = γδkh (2.15)

5

where (δij) is the Kronecker symbol, β the Biot-Willis coefficient [15], γ [K-1]is well-known as the thermal expansion coefficient and λ [Pa], µ [Pa] are theLame’s constants which are related to the Young’s modulus E [Pa] and thePoisson’s ration ν (dimensionless) through the expression:

λ = Eν

(1− 2ν) (1 + ν), µ = E

ν

2 (1 + ν).

The effective stress is then related to e through the linear Hooke’s constitutivelaw:

σ

ij = 2µeij + λekkδij .

The Biot-Willis coefficient β expresses, at constant fluid pressure, the ratio ofthe volume of fluid squeezed out of a solid to total volume change for elasticdeformation. The coefficient γ refers to the fractional change in volume perdegree of temperature change. The coefficient 2λ + 3µ is the bulk moduluswhich relates the volumetric dilation eii to σ at a constant pore pressure.

If the porous solid is saturated by a compressible fluid whose mass densityρ0 assumed to be constant, then the equation of continuity for the flow fluid isgiven by

∂tφ+ divv =g0ρ0

(2.16)

where v [m. s-1] is the velocity of the total amount of the fluid content. Forlaminar flow, the fluid flux is related to the fluid velocity v through the Darcy’slaw (neglecting the gravity force g):

v =−kfµd

∇p (2.17)

where kf is the intrinsic permeability coefficient [m2] and µd [Pa.s] is the dy-namic viscosity. The negative sign in the Darcy law is needed because fluidsflow from high pressure to low one, opposite to the direction of the pressuregradient. Using (2.10), (2.16) and (2.17) we get

∂t

(1

Np+B : e (u) + αθ

)− div

(kfµd

∇p)

=g0ρ0. (2.18)

Similarly, the classical Fourier’s law relates heat flux vector q [Kg.s-3] withtemperature gradient ∇θ by the equation

q = −λ0∇θ

where λ0 [kg.m.s-3.K-1] is the thermal conductivity. Using the first and secondlaws of thermodynamics [28], the energy balance equation reads as follows:

∂tW = σ : e (∂tu) + p∂tφ− divq+ h0 (2.19)

where the left hand side represents the instantaneous change in storage energyin the body, the first term of the right hand side is heat entering the material

6

and the last term stands for energy generation in the body. From (2.4), differ-entiating η with respect to t and taking into account the fact that S = −∂θη, itfollows that

∂tη = σ : e (∂tu)− φ∂tp− S∂tθ. (2.20)

But, using (2.1) and (2.3), we also have

∂tη = ∂tW − φ∂tp− p∂tφ− S∂tθ − θ∂tS. (2.21)

Equating (2.20) and (2.21), equation (2.19) is rewritten as

θ∂tS = −divq+ h0. (2.22)

Assuming small variations of temperature so that one can replace θ by θ0, when-ever required, and inserting (2.11) into (2.22) yield to the following equation:

θ0∂t

D : e (u) + αp+

υ

θ0θ

= div (λ0∇θ) + h0. (2.23)

Equations (2.13), (2.18) and (2.23) are then the fundamental equations of linearthermoporoelasticity. Denoting

γ = (3λ+ 2µ) γ, φ0 =1

N, κ = µ−1

d kf ,

g = ρ−1g0, λ = θ−10 λ0, c = θ−1

0 υ, h = θ−10 h0

and taking into account (2.14), (2.15), Equations (2.13), (2.18) and (2.23) arereduced to the following system:

− (λ+ µ)∇ (divu)− µ∆u+ β∇p+ γ∇θ = f0,

∂t (φ0p+ βdivu+ αθ)− κ∆p = g,

∂t (cθ + γdivu+ αp)− λ∆θ = h.

(2.24)

Remark 2.1 Note that if we let γ and α to be negligible then the system (2.24)decouples to the classical Biot system [14]:

−divσ = − (λ+ µ)∇div (u)− µ∆u+ β∇p = f0,

∂t (φp+ βdivu)− κ∆pm = g,

together with the single heat equation

c∂tθ − λ∆θ = h.

On the other hand, neglecting β and α, the system (2.24) decouples to the ther-moelasticity system:

−divσ = − (λ+ µ)∇div (u)− µ∆u+ γ∇θ = f0,

∂t (cθ + γdivu)− λ∆θ = h

with the single pressure diffusion equation

φ∂tp− κ∆p = g.

7

2.2 A mathematical model for two-components materials

In this subsection we shall derive from (2.24) the mathematical model of thermo-poroelasticity for isotropic materials made of two constituents, namely matrixand inclusions and which are in imperfect interfacial contact. The geometri-cal setting is described as follows: Let Ω be a bounded and a smooth domainof R3. We assume that Ω is divided into two open sets Ω1 and Ω2 such thatΩ = Ω1 ∪ Ω2 and ∂Ω2 ∩ ∂Ω = ∅. The medium occupying Ω is composed of twoporo-elastic solids Ω1 and Ω2 separated by the interface Σ := ∂Ω2. Accordingto (2.24), the local description is given by the following system: for each phasem = 1, 2 corresponding to the material Ωm

−divσm = fm,

σm = − (λm + µm)∇div (um)− µm∆um + βm∇pm + γm∇θm,

∂t (φmpm + βmdivum + αmθm)− κm∆pm = gm,

∂t (cmθm + γmdivum + αmpm)− λm∆θm = hm.

(2.25)

For piecewise homogeneous media, the system (2.25) is complemented by inter-face, boundary and initial conditions. They read as follows: on the interface Σ

u1 = u2, (2.26a)

σ1 · n = σ2 · n, (2.26b)

κ1∇p1 · n = κ2∇p2 · n, (2.26c)

λ1∇θ1 · n = λ2∇θ2 · n, (2.26d)

κ1∇p1 · n = −ς (p1 − p2) , (2.26e)

λ1∇θ1 · n = −ω (θ1 − θ2) (2.26f)

where n is the unit normal vector on Σ pointing outwards to Ω2. In (2.26e) ς[kg-1.m2.s] is the interfacial hydraulic permeability and in (2.26f) ω [kg.s-3.K-1]is the interface thermal conductance. Conditions (2.26a)-(2.26d) are the con-tinuity of the displacements, of the normal stresses and of the normal fluxes(hydraulic and thermal). It is assumed that the hydraulic/thermal contact be-tween these two materials is imperfect, so that the fluxes are related to thejump of pressures and temperatures, see (2.26e) and (2.26f). For example thecase ς = ∞ corresponds to a perfect hydraulic contact, so that p1 = p2 acrossthe interface and the pressure is continuous. If ς = 0 there is no hydrauliccontact across the interface, yielding no motion of the fluid relative to the solid,that is, κ1∇p1 · n = κ2∇p2 · n = 0. In the literature (2.26e) is known as theDeresiewicz-Skalak condition [25] and (2.26f) as the Newton’s cooling law [16].On the exterior boundary Γ we assume the following homogeneous Dirichletboundary conditions:

u1 = 0, p1 = 0, θ1 = 0. (2.27)

8

Finally the initial conditions are given as follows:

um (0, x) = 0, pm (0, x) = 0, θm (0, x) = 0, x ∈ Ωm. (2.28)

In summary, the system (2.25)-(2.28) is the complete set of equations for ther-moporoelastic media with two-components.

In this work we shall study a model consisting of ”very” small inclusionsembedded in a matrix, so we have to introduce a small and dimensionless pa-rameter expressing the ratio between the local scale of the inclusions and themacroscopic scale of the matrix. This will be done in the next subsection.

2.3 Scaling

We consider a poroelastic composite of dimension O(L3)where L [m] is the

characteristic length of the medium at the macroscopic scale. We assume thatthe composite has a periodic structure with period Y with dimension O

(ℓ3)

where ℓ [m] is the microscopic characteristic length. The fundamental assump-tion in the periodic homogenization theory [12, 41] is that these scales are sep-arated which in this case can be read as follows:

ε :=ℓ

L≪ 1.

We assume that the stiffness tensors A1 and A2, permeabilities κ1 and κ2,thermal conductivities λ1 and λ2 are of the same order of magnitude. Pressuresp1, p2 and temperatures θ1, θ2 are also considered of the same order. Moreprecisely, we assume that

|A1| = |A2| = O(ε0), |κ1| = |κ2| = O

(ε0),∣∣∣λ1∣∣∣ =

∣∣∣λ2∣∣∣ = O

(ε0),

p1 = p2 = O(ε0), θ1 = θ2 = O

(ε0).

Equations (2.26e) and (2.26f) give at the microscale two dimensionless (Biot)numbers:

Bi1 =|ς (p1 − p2)||κ1∇p1|

=ςl

κ1, Bi2 =

|ω (θ1 − θ2)|∣∣∣λ1∇θ1∣∣∣

=ωl

λ1.

Since the area of Σε is of order ε−1, a convenient scaling of those Biot numbersis Bi1 = Bi2 = O (ε), see [6, 37].

Remark 2.2 Note that other assumptions could be studied as well. One also

could consider the following case: |A1| = |A2| = |κ1| =∣∣∣λ1∣∣∣ = O

(ε0), p1 =

p2 = θ1 = θ2 = O(ε0)and |κ2| =

∣∣∣λ2∣∣∣ = O

(ε2). See for instance [3, 4] and

for more general situations we refer the reader to [40] .

9

2.4 Problem statement

In this subsection a micro-model for a thermoporoelastic medium with two-components and with interfacial hydraulic/thermal exchange barrier is pre-sented. As before, we consider Ω a bounded domain in R

3 with a smoothboundary Γ. The region Ω represents a part of a medium made of two con-stituents: the matrix and the inclusions, separated by a thin and periodic layerso that the hydraulic/thermal flux are proportional to the jump of the pres-sure/temperature field. To describe the periodicity of the medium, we considerY :=]0, 1[3 as the generic cell of periodicity divided as Y := Y1 ∪ Y2 ∪ Σ whereY1, Y2 are two connected, open and disjoint subsets of Y and Σ := ∂Y1 ∩ ∂Y2is a smooth surface that separates them. We assume that Y2 ⊂ Y . Let χm

denote the Y -periodic characteristic function of Ym (m = 1, 2). Let ε > 0 be asufficiently small parameter and set

Ωεm := x ∈ Ω : χm(

x

ε) = 1, Σε := Ωε

1 ∩Ωε2.

We assume that Ωε2 ⊂ Ω. The space-time regions are denoted by

Q := (0, T )× Ω, ΓT := (0, T )× Γ, ΣT := (0, T )× Σ,

Qεm := (0, T )× Ωε

m, ΣεT := (0, T )× Σε.

In view of (2.25), the thermoporoelasticity system is given in each phase Qεm by

− divσεm = fm, σε

m = (σm,ij)ij , (2.29)

∂t (φmpεm + βmdivuε

m + αmθεm)− κm∆pεm = gm, (2.30)

∂t (cmθεm + γmdivuε

m + αmpεm)− λm∆θεm = hm (2.31)

whereσεm,ij (u

ε) = am,ijklekl (uεm)− (βmp

εm + γmθ

εm) δij (2.32)

with (am,ijkl)1≤i,j,k,l≤3 the elasticity tensor stiffness satisfying the Hooke’s lawfor isotropic materials:

am,ijkl = λmδijδkl + µm (δikδjl + δilδjk) , m = 1, 2, 1 ≤ i, j, k, l ≤ 3.

As in (2.26a)-(2.26f), the transmission conditions on the interface ΣεT are as

follows:

uε1 = uε

2, (2.33a)

σε1 · nε = σε

2 · nε, (2.33b)

κ1∇pε1 · nε = κ2∇pε2 · nε, (2.33c)

λ1∇θε1 · nε = λ2∇θε2 · nε, (2.33d)

κ1∇pε1 · nε = −ςε (pε1 − pε2) , (2.33e)

λ1∇θε1 · nε = −ωε (θε1 − θε2) (2.33f)

10

where nε is the unit normal of Σε pointing outwards of Ωε2. Furthermore, bound-

ary conditions (2.27) on ΓT read as:

uε1 = 0, θε1 = pε1 = 0. (2.34)

Observe that no boundary conditions on Γ are required for the phase 2 since theinclusions Ωε

2 are strictly embedded in Ω. Finally, the initial conditions (2.28)give

uεm (x, 0) = 0, θεm (x, 0) = pεm (x, 0) = 0 in Ωε

m. (2.35)

We assume that the elastic modulii am,ijkl, the Biot-Willis parameter βm, thethermal dilation coefficients γm and αm, the compressibility φm, the diffusivitycoefficients κm, λm and the heat capacity cm are positive constants. We alsoassume that the body force fm is in L2 (Ω)3, the sink source gm and the heatsource hm are in L2 (Ω). Furthermore, the interface hydraulic permeability andthe interface thermal conductance are such that ςε (x) = ες

(xε

)and ωε (x) =

εω(xε

)(see Sec. 2.3) where ς and ω are continuous on R

3, Y−periodic andbounded from below: ∃C > 0 such that for all y ∈ R

3

ς (y) ≥ C, ω (y) ≥ C.

In what follows, C will denote a positive constant independent of ε.

3 Statement of the main results

We first introduce some notations: if E is a Banach space then for p = 2,∞,LpT (E) denotes the Bochner space Lp(0, T ;E) of (class of ) functions u : t 7−→

u (t) defined a.e. on (0, T ) with values inE such that ‖u (t)‖pL

p

T(E) :=

∫ T

0 ‖u (t)‖pE dt

is finite (dt denotes the Lebesgue measure on the interval (0, T )). Let L2#(Y )

(resp. L2#(Ym)) be the space of (class of) functions belonging to L2

loc(R3) (resp.

L2loc(Zm)) which are Y -periodic, where Zm = ∪~k∈Z3

(Ym + ~k

). Let H1

#(Y )

(resp. H1#(Ym)) to be the space of those functions together with their deriva-

tives belonging to L2#(Y ) (resp. L2

#(Ym)) having the same trace on the oppositefaces of ∂Y (resp. ∂Ym ∩ ∂Y ). Let

V: = H10 (Ω)

3, Hε := L2 (Ωε

1)× L2 (Ωε2) ,

H1Γ (Ω

ε1) :=

q ∈ H1 (Ωε

1) : q|Γ = 0,

V ε := V ε1 × V ε

2 = H1Γ (Ω

ε1)×H1 (Ωε

2) .

The space V ε is equipped with the inner product:

(q, ψ)V ε :=∫Ωε

1

∇q1∇ψ1dx+∫Ωε

2

∇q2∇ψ2dx+

ε∫Σε (q1 − q2) (ψ1 − ψ2) ds

ε (x) , q = (q1, q2) , ψ = (ψ1, ψ2) ∈ V ε

11

where dx and dsε (x) stands for the Lebesgue measure in R3 and the surfacic

measure on Σε, respectively. Let us denote for a.e. t ∈ (0, T )

uε(t, x) =

uε1(t, x), x ∈ Ωε

1,

uε2(t, x), x ∈ Ωε

2,

pε(t, x) = (pε1(t, x), pε2(t, x)) , θ

ε(t, x) = (θε1(t, x), θε2(t, x))

and let us define

Aε(x) := χ1(xε)A1 + χ2(

xε)A2,

fε(x) := χ1(xε)f1 (x) + χ2(

xε)f2 (x) ,

gε(x) := χ1(xε)g1 (x) + χ2(

xε)g2 (x) ,

hε(x) := χ1(xε)h1 (x) + χ2(

xε)h2 (x)

(3.1)

where Am := (am,ijkl)1≤i,j,k,l≤3. Now we are in position to give the weakformulation.

Definition 3.1 A weak solution of the micro-model (2.29)-(2.35) is a triple(uε, pε, θε) ∈ L∞

T (V) × L2T (V

ε)2 such that pε, θε ∈ L∞T (Hε) and for m = 1, 2

∂t (φmpεm + βmdivuε

m + αmθεm) ∈ L2

T (Vεm

∗),

∂t (cmθεm + γmdivuε

m + αmpεm) ∈ L2

T (Vεm

∗)

and for all v ∈ V, (q1, q2) ∈ V ε, we have the three following coupled systems:for a.e. t ∈ (0, T ) ,

∫ΩAεe(uε)e(v)dx +

∫Ωε

1

(β1∇pε1 + γ1∇θε1)vdx

+∫Ωε

2

(β2∇pε2 + γ2∇θε2)vdx =∫Ωfεvdx,

(3.2)

〈∂t(φ1pε1 + β1divuε + α1θ

ε1), q1〉V ε

1∗,V ε

1

+〈∂t(φ2pε2 + β2divuε + α2θ

ε2), q2〉V ε

2∗,V ε

2

+∫Ωε

1

κ1∇pε1∇q1dx+∫Ωε

2

κ2∇pε2∇q2dx

+∫Σε ς

ε(pε1 − pε2)(q1 − q2)dsε(x) =

∫Ωgεqεdx

(3.3)

and

〈∂t(c1θε1 + γ1divuε + α1p

ε1), q1〉V ε

1∗,V ε

1

〈∂t(c2θε2 + γ2divuε + α2p

ε2), q2〉V ε

2∗,V ε

2

+∫Ωε

1

λ1∇θε1∇q1dx+∫Ωε

2

λ2∇θε2∇q2dx

+∫Σε ω

ε(θε1 − θε2)(q1 − q2)dsε(x) =

∫Ωε h

εqεdx

(3.4)

12

with the initial conditions:

uε(0, ·) = 0,pε1(0, x) = θε1(0, x) = 0, x ∈ Ωε

1,pε2(0, x) = θε2(0, x) = 0, x ∈ Ωε

2

(3.5)

where we have denoted

qε (x) =

q1 (x) , x ∈ Ωε

1,q2 (x) , x ∈ Ωε

2.

Existence and uniqueness results for the system (3.2)-(3.5) can be performedby using the Galerkin technique or the semi-group method. For more details,we refer the reader for instance to Showalter and Momken [44]. Hence we givewithout proof the following result.

Theorem 3.1 There exists a unique (uε, pε, θε) ∈ L∞T (V) × L2

T (Vε)2 with

(pε, θε) ∈ L∞T (Hε)2 solution of the weak system (2.29)-(2.35) such that

‖uε‖L∞

T(V) ≤ C,

‖pε‖L2

T(V ε) + ‖pε‖L∞

T(Hε) ≤ C,

‖θε‖L2

T(V ε) + ‖θε‖L∞

T(Hε) ≤ C.

(3.6)

The key ingredient of our convergence results are the uniform estimates (3.6)and the use of the two-scale convergence technique, see Sect. 4.1 below. In orderto give our main result, we introduce some notations related to the local scalemodels: let dkl = (ykδil)1≤i≤3 and wij ∈ (H1

#(Y )/R)3 denote the solution tothe microscopic system:

−divy(A(ey(wij + dij

))) = 0 a.e. in Y,[

wij]= 0 a.e. on Σ,

y 7−→ wij(x, y) is Y − periodic(3.7)

where

A (y) :=

A1, y ∈ Y1,A2, y ∈ Y2.

and [·] denotes the jump on Σ. Let us define the ”micro-pressure” πim ∈

H1#(Ym)/R (i = 1, 2, 3) to be the solution of

−div(κm(ei +∇yπ

im (y)

))= 0 in Ω× Ym,[

κm(ei +∇yπ

im (y)

)]· ν (y) = 0 on Ω× Σ,

y 7−→ πim (y) is Y − periodic.

(3.8)

Here ei is the ith vector of the standard basis of R3. Likewise, let the ”micro-temperature” ϑim ∈ H1(Ym)/R to be the solution of

−div(λm (y)

(ei +∇yϑ

im

))= 0 in Ω× Ym,[

λm (y)(ei +∇yϑ

im

)]· ν = 0 on Ω× Σ,

y 7−→ ϑim is Y − periodic.

(3.9)

13

Remark 3.1 It is worthwhile noticing that these three boundary value problems(3.7), (3.8) and (3.9) are well-posed in the sense that each problem admits aunique weak solution (see for instance A. Bensoussan & al. [12]).

Let us denote

Aijkl :=

∫

Y

Aey(wij + dij

)ey(wkl + dkl

)dx, σ0

ij (u) := Aijklekh(u). (3.10)

In other words

Aijkh =

3∑

r,s=1

∫

Y

aijrs(y)(δirδjs + ers,y(wkh)(y))dy. (3.11)

Put

Bm = (Bm,ij)1≤i,j≤3 , Dm = (Dm,ij)1≤i,j≤3 , Cm = (Cm,ij)1≤i,j≤3 ,

Km = (Km,ij)1≤i,j≤3 , Lm = (Lm,ij)1≤i,j≤3

where

Bm,ij : = βm

∫

Y

χm (y)

(δij +

∂πim

∂yj(y)

)dy, (3.12)

Dm,ij : = γm

∫

Y

χm (y)

(δij +

∂ϑim∂yj

(y)

)dy, (3.13)

Cm,ij : =

∫

Ym

(δij + div

(wij))

dy (3.14)

and

Km,ij :=

∫

Ym

κm

(δji +

∂πmj

∂yi

)dy, Lm,ij :=

∫

Ym

λm

(δji +

∂ϑmj

∂yi

)dy. (3.15)

Set

f∗ := |Y1| f1+ |Y2| f2, g∗m := |Ym| gm, h∗m := |Ym|hm,

c∗m := |Ym| cm, ϕ∗m := |Ym|φm, γ∗m := |Ym| γm,

ω∗ :=∫Σω (y) ds (y) , ζ∗ :=

∫Σς (y) ds (y) . α∗

m := |Ym|αm

(3.16)where |Ym| denotes the volume of Ym and ds (y) the surfacic measure of Σ.

Now we state the main result of the paper.

Theorem 3.2 Let (uε, pε, θε) ∈ L∞T (V) × L2

T (Vε)2 be the weak solution of

(2.29)-(2.35). Then, up to a subsequence, there exist u ∈ L∞T (V), p1, θ1 ∈

14

L∞T (H1

0 (Ω)) and p2, θ2 ∈ L∞T (H1(Ω)) such that the following weak limits holds:

for a.e. t ∈ (0, T )

uε(t, ·) u(t, ·), weakly in L∞T (V),

χεmp

εm(t, ·) χmpm (t, ·), θεm(t, ·) 2−s

χmθm (t, ·), weakly in L∞T (H1(Ω)).

Furthermore, the weak limit (u, pm, θm) is solution of the following homogenizedsystem:

−div(σ0 (u)

)+B1∇p1 +B2∇p2 +D1∇θ1 +D2∇θ2 = f , a.e in Q,

∂t (φ∗1p1 + β1C1 : e (u) + α∗

1θ1)− div (K1∇p1) + ζ∗ (p1 − p2) = g∗1 , a.e. in Q,

∂t (φ∗2p2 + β2C2 : e (u) + α∗

2θ2)− div (K2∇p2) + ζ∗ (p2 − p1) = g∗2 , a.e.in Q,

∂t c∗1θ1 + γ1C1 : e (u) + α∗1p1 − div (L1∇θ1) + ω∗ (θ1 − θ2) = h∗1, a.e.in Q,

∂t c∗2θ2 + γ2C2 : e (u) + α∗2p2 − div (L2∇θ2) + ω∗ (θ2 − θ1) = h∗2, a.e.in Q,

u = 0, θ1 = p1 = 0 a.e. on ΓT ,

L2∇θ2 · ν = K2∇p2 · ν = 0, a.e. on ΓT ,

u(0, x) = 0, θm (0, x) = pm(0, x) = 0 a.e. in Ω.(3.17)

The next section is devoted to the proof of Theorems 3.2.

4 Derivation of the homogenized model

In this section, we shall first recall the two scale convergence technique. Thenwe shall use the uniform estimates (3.6) to show that the solution to the mi-croscopic model (2.29)-(2.35) converges in the two-scale sense to the solution ofthe homogenized problem (3.17).

4.1 Two scale convergence

The two-scale convergence method was first introduced by G. Nguetseng [38]and later developed by G. Allaire [5]. This technique is intended to handlehomogenization problems involving periodic microstructures. Hereafter, we re-call its definition and its main results. For more details, we refer the reader to[5, 33, 38].

We denote C#(Y ) to be the space of all continuous functions in R3 which are

Y -periodic. Let C∞# (Y ) denote the subspace of C#(Y ) of infinitely differentiable

functions.

15

Lemma 4.1 Let q ∈ L2(Ω;C#(Y )). Then q (x, x/ε) ∈ L2 (Ω) and

∫

Ω

∣∣∣q(x,x

ε)∣∣∣2

dx ≤∫

Ω

supy∈Y

|q(x, y)|2 dx,

limε→0

∫

Ω

∣∣∣q(x,x

ε)∣∣∣2

dx = limε→0

∫

Ω×Y

q(x, y)2dxdy.

Such a function will be called in the sequel an admissible test function.

Definition 4.1 A sequence (vε) in L2(Ω) two-scale converges to v ∈ L2(Ω×Y )

and we write vε2−s v if, for any q ∈ L2(Ω;C#(Y )),

limε→0

∫

Ω

vε(x)q(x,x

ε)dx =

∫

Ω×Y

v(x, y)q(x, y)dxdy.

Theorem 4.1 Let (vε) be a sequence of functions in L2(Ω) which is uniformlybounded. Then, there exist v ∈ L2(Ω × Y ) and a subsequence of (vε) whichtwo-scale converges to v.

Remark 4.1 Thanks to Theorem 4.1, it is easy to see that for all q ∈ L2(Ω;C#(Y ))

limε→0

∫

Ωεm

fm(x)q(x,x

ε)dx =

∫

Ω×Ym

fm(x)q(x, y)dxdy

since ∫

Ωεm

fm(x)q(x,x

ε)dx =

∫

Ω

fm(x)χm

(xε

)q(x,

x

ε)dx

and x 7−→ χm

(xε

)q(x, x

ε) is an admissible test function.

Theorem 4.2 Let (vε) be a uniformly bounded sequence in H1(Ω) (resp. H10 (Ω)).

Then there exist v ∈ H1(Ω) (resp. H10 (Ω)) and v ∈ L2(Ω;H1

#(Y )/R) such that,up to a subsequence,

vε2−s v; ∇vε 2−s

∇v +∇y v. (4.1)

In (4.1) and in the sequel the subscript y on a differential operator as in ∇y

indicates that the differentiation acts only on y.

Theorem 4.3 operators act only on those variables

We now extend the notion of two-scale convergence to periodic surfaces[6, 37]:

Definition 4.2 A sequence (wε) in L2(Σε) two-scale converges to w0(x, y) ∈L2 (Ω× Σ) if for any q ∈ D

(Ω; C∞

# (Σ))

limε→0

ε

∫

Σε

wε(x)q(x,x

ε)dsε (x) =

∫

Ω×Σ

w0(x, y)q(x, y)dxds (y) .

16

We state the following compactness result:

Theorem 4.4 Let (wε) be a sequence in L2(Σε) such that

√ε

∫

Σε

|wε(x)|2 dsε (x) ≤ C.

Then, up to a subsequence, there exists w0(x, y) ∈ L2 (Ω× Σ) such that (wε)two-scale converges in the sense of Definition 4.2 to w0(x, y) ∈ L2 (Ω× Σ).

Corollary 4.1 Let v (y) ∈ L2# (Σ). Then for any q ∈ H1 (Ω)

limε→0

∫

Σε

εv(x

ε)q(x)dsε (x) =

∫

Ω×Σ

v(y)q(x)dxds (y) .

Theorem 4.5 Let (wε) be a sequence of functions in H1(Ω) such that

‖wε‖L2(Ω) + ε ‖∇wε‖L2(Ω)N ≤ C.

Then, there exist a subsequence of (wε), still denoted by (wε), and w0(x, y) ∈L2(Ω;H1

#(Y ))

such that wε 2−s w0 and ε∇wε 2−s

∇yw0 and for every q ∈D(Ω; C∞

# (Y ))we have

limε→0

ε

∫

Σε

wε (x) qε (x) dsε (x) =

∫

Ω×Σ

w0 (x, y) q (x, y) dxds (y) .

Remark 4.2 Notice that two-scale convergence can also handle problems in-volving a parameter without affecting the results stated above. Therefore, weshall use this technique to study the homogenization of our model which involvesthe time parameter t. For more details, see [20].

4.2 Homogenization process

In this subsection, we shall prove Theorem 3.2. The proof is divided into Lem-mata 4.2-4.8.

Lemma 4.2 There exists a subsequence of (uε, θε, pε), still denoted (uε, θε, pε),and there exist

u ∈ L∞T (H1

0 (Ω)3), u ∈ L∞

T

(L2(Ω;H1

#(Y )/R)3), pm, θm ∈ L∞

T (H10 (Ω))

andpm, θm ∈ L2(Q;H1

#(Ym)/R)

17

such that, for a.e. t ∈ (0, T )

uε(t, x)2−s u(t, x), (4.2)

χεm (x) pεm(t, x)

2−s χm (y) pm (t, x), (4.3)

χεm (x) θεm(t, x)

2−s χm (y) θm (t, x), (4.4)

∇uε(t, x)2−s ∇u(t, x) +∇yu(t, x, y), (4.5)

χεm (x)∇pεm(t, x)

2−s χm (y) (∇pm(t, x) +∇y pm(t, x, y)), (4.6)

χεm (x)∇θεm(t, x)

2−s χm (y) (∇θm(t, x) +∇y θm(t, x, y)). (4.7)

Moreover, for any ψ ∈ D(Q; C#(Y )) we have

limε→0

∫Σε

T

ζε (x) (pε1 (t, x)− pε2 (t, x))ψε (t, x) dsε (x) dt

=∫Q×Σ ς (y) (p1 (t, x)− p2 (t, x))ψ (t, x, y) dtdxds (y) ,

(4.8)

limε→0

∫Σε

T

ωε (x) (θε1 (t, x)− θε2 (t, x))ψε (t, x) dsε (x) dt

=∫Q×Σ

ω (y) (θ1 (t, x) − θ2 (t, x))ψ (t, x, y) dtdxds (y)(4.9)

where we have denoted ψε(t, x) = ψ(x, t, x/ε).

Proof. The two-scale limits (4.2)-(4.9) are a straightforward application of thea priori estimates (3.6) and the compactness Theorems 4.1, 4.2, 4.4 and 4.5.

Lemma 4.3 The corrector displacement u can be written as:

u(t, x, y) = wij(y)eij(u)(t, x) for a.e. (t, x, y) ∈ Q × Y (4.10)

where wij ∈ (H1#(Y )/R)3 is the solution to the microscopic system (3.7).

Proof. We choose adequate test functions in (3.2): Let

v(x) := vε(x) = εv(x,x

ε)

where v ∈ D(Ω; C∞# (Y ))3. Then, we have for a.e. t ∈ (0, T )

∫ΩA(xε

)e(uε) (t, x)

εex (v) (x,

x

ε) + ey (v) (x,

x

ε)dx

+ε∫Ωε

1

(β1∇pε1 (t, x) + γ1∇θε1 (t, x)) v(x,x

ε)dx

+ε∫Ωε

2

(β2∇pε2 (t, x) + γ2∇θε2 (t, x)) v(x,x

ε)dx

= ε∫Ωε

1

f1 (x) v(x,x

ε)dx+ ε

∫Ωε

2

f2 (x) v(x,x

ε)dx.

(4.11)

18

In view of (4.5), we pass to the limit in the first term of the l.h.s. of (4.11) toget for a.e. t ∈ (0, T )

limε→0

∫

Ω

Aε (x) e(uε) (t, x)εex (v) (x,

x

ε) + ey (v) (x,

x

ε)dx

=

∫

Ω×Y

A (y) (e(u) (t, x) + ey (u) (t, x, y)) ey (v) (x, y)dxdy. (4.12)

Next, since∣∣∣∣∣ε∫

Ωεm

βm∇pεm (t, x) v(x,x

ε)dx

∣∣∣∣∣ ≤ εβm

∥∥∥v(x,x

ε

∥∥∥L2(Ω)

‖∇pεm‖L2(Ωεm) ,

∣∣∣∣∣ε∫

Ωεm

γm∇θεm (t, x) v(x,x

ε)dx

∣∣∣∣∣ ≤ εγm

∥∥∥v(x,x

ε

∥∥∥L2(Ω)

‖∇θεm‖L2(Ωεm)

and taking into account (3.6) together with Lemma 4.1, we see that for a.e.t ∈ (0, T )

limε→0

ε

∫

Ωεm

(βm∇pεm (t, x) + γm∇θεm (t, x)) v(x,x

ε)dx = 0.

In the same way, letting ε→ 0 in the r.h.s. of (4.11) we obtain:

limε→0

ε

(∫

Ωε1

f1 (x) v(x,x

ε)dx+

∫

Ωε2

f2 (x) v(x,x

ε)dx

)= 0. (4.13)

Collecting (4.12), (4.13) and passing to the limit in (4.11) yield for a.e. t ∈ (0, T )∫

Ω×Y

A (y) (e(u) (t, x) + ey (u) (t, x, y)) ey (v) (x, y)dx = 0

which gives after an integration by parts the following boundary value problem:

−divy A (y) (e(u) (t, x) + ey (u) (t, x, y)) = 0 a.e. in Q× Y ,

y 7−→ u(t, x, y) is Y − periodic.(4.14)

According to the linearity of the system (4.14), we see that u can be written interms of u through the following scale separation expression:

u(t, x, y) = eij(u)(t, x)wij (y), a.e. (t, x, y) ∈ Q× Y

where wij ∈ (H1#(Y )/R)3 is the solution to the microscopic system defined by

(3.7), see for instance [12, page 15]. Hence (4.10) is proved.

Lemma 4.4 The corrector pressure pm satisfies

pm(t, x, y) = πim(y)

∂pm∂xi

(t, x), a.e. (t, x, y) ∈ Q× Ym (4.15)

where πim(y) i = 1, 2, 3, m = 1, 2 are defined by (3.8).

19

Proof. Let qm ∈ D(Q; C∞# (Y )). Taking

qm (t, x) := qεm(t, x) = εϕm(t, x,x

ε)

in (3.3) and integrating with respect to t ∈ (0, T ), we get

〈∂t(φ1pε1 + β1divuε + α1θ

ε1), q

ε1〉V ε

1∗,V ε

1

+〈∂t(φ2pε2 + β2divuε + α2θ

ε2), q

ε2〉V ε

2∗,V ε

2

+∫Qε

1

κ1∇pε1 (t, x)(ε∇xq1(t, x,

xε) +∇y q1(t, x,

xε))dtdx

+∫Qε

2

κ2∇pε2 (t, x)(ε∇xq2(t, x,

xε) +∇y q2(t, x,

xε))dtdx

+εRε = 0

(4.16)

whereRε = −

∫Qε

1

(φ1pε1 + β1divu

ε + α1θε1)∂tq

ε1dtdx

−∫Qε

2

(φ2pε2 + β2divu

ε + α2θε2)∂tq

ε2dtdx

+∫Σε

T

ες(xε

)(pε1 − pε2)(q

ε1 − qε2)dtds

ε(x)

−∫Qε

1

g1qε1dtdx−

∫Qε

2

g2qε2dtdx

(4.17)

and qεm (t, x) := qm (x, t, x/ε). From (4.8) it is easy to see that∣∣∣∣∣

∫

ΣεT

ες(xε

)(pε1 − pε2)(q

ε1 − qε2)dtds

ε(x)

∣∣∣∣∣ ≤ C. (4.18)

Furthermore, thanks to Lemma 4.1 we have

limε→0

(∫

Qε1

g1qε1dtdx+

∫

Qε2

g2qε2dtdx

)=

∫

Q×Y1

g1q1dtdxdy +

∫

Q×Y2

g2q2dtdxdy. (4.19)

By virtue of the uniform estimates (3.6), the sequences pεmε , divuεε andθεmε are uniformly bounded in L2 (Qε

m) so that

∣∣∣∣∣

∫

Qεm

(φmpεm + βmdivuε + αmθ

εm)∂tq

εmdtdx

∣∣∣∣∣ ≤ C. (4.20)

Taking into account (4.18)-(4.20) we get from (4.17) that

limε→0

εRε = 0.

20

On the other hand passing to the limit in (4.16) and taking into account (4.6)we are led to

∫

Q×Y

χ1 (y)κ1 (∇p1 (t, x) +∇y p1 (t, x))∇y q1(t, x, y)dtdxdy

+

∫

Q×Y

χ2 (y)κ2 (∇p2 (t, x) +∇y p2 (t, x, y))∇y q2(t, x, y)dtdx = 0,

so that an integration by parts yields:

−div (κ1 (∇p1 (t, x) +∇y p1 (t, x, y))) = 0 a.e. in Q× Y1,

−div (κ2 (∇p2 (t, x) +∇y p2 (t, x, y))) = 0 a.e. in Q× Y2,

[κm (∇pm (t, x) +∇y pm (t, x, y))] · ν (y) a.e. on Q× Σ,

y 7−→ p1, p2 are Y − periodic.

(4.21)

As in Lemma 4.3 and thanks to the linearity of the system (4.21) we can writethat

pm(t, x, y) = πim(y)

∂pm∂xi

(t, x) a.e. (t, x, y) ∈ Q× Ym

where πim(y) ∈ πm i ∈ H1

#(Ym)/R is the solution of (3.8). The Lemma is thenproved.

Lemma 4.5 The corrector temperature θm is related to the homogenized tem-perature θm via the linear relation:

θm (t, x, y) = ϑim(y)∂θm∂xi

(t, x) + Cte, a.e. (t, x, y) ∈ Q× Ym (4.22)

where, for i = 1, 2, 3, the ”micro-temperature” ϑim ∈ H1(Ym)/R is the solutionof (3.9).

The proof of this Lemma follows the same lines as that of Lemma 4.4 andtherefore will not be given.

Lemma 4.6 The macroscopic balance equation reads as follows:

−div(σ0 (u)

)+B1∇p1 +B2∇p2 +D1∇θ1 +D2∇θ2 = f in Ω,

u = 0 on ∂Ω(4.23)

where A, Bm, Dm and f are defined by (3.11), (3.12), (3.13) and (3.16) re-spectively.

Proof. The convergence results obtained in Lemma 4.2 allow us to derive themacroscopic equations (3.17). To do so, we first determine the limiting equations

21

of (3.2)-(3.5). Let v ∈D(Ω)3. Multiplying (3.2) by v and passing to the limit,we find by virtue of (4.5)-(4.7) and Remark 4.1 that for a.e. t ∈ (0, T )

∫

Ω×Y

A[e(u) + ey(u)]e(v)dxdy + β1

∫

Ω×Y1

(∇p1 +∇y p1)vdxdy

+β2

∫

Ω×Y2

(∇p2 +∇y p2)vdxdy + γ1

∫

Ω×Y1

(∇θ1 +∇y θ1

)vdxdy

+γ2

∫

Ω×Y2

(∇θ2 +∇y θ2

)vdxdy

=

∫

Ω×Y1

f1vdxdy +

∫

Ω×Y2

f2vdxdy. (4.24)

Let us rewrite the first integral term in the l.h.s. of (4.24) with the help of(3.10), (3.11) and (4.10). We have

∫

Ω×Y

Aijkh [e(u) + ey(u)]e(v)dxdy =

∫

Ω

σhomij (u) eij(v)dx. (4.25)

Likewise, by using (4.15) and (4.22), there holds

β1

∫

Ω×Y1

(∇p1 +∇y p1)vdx + β2

∫

Ω×Y2

(∇p2 +∇y p2)vdxdy

= β1

∫

Ω×Y

χ1

(δik +

∂πi1

∂yk

)∂p1∂xi

vkdx

+β2

∫

Ω×Y

χ2

(δik +

∂πi2

∂yk

)∂p2∂xi

vkdx (4.26)

where vk is the kth component of v. After simple algebraic calculations, (4.26)becomes then

β1

∫

Ω×Y1

(∇p1 +∇y p1)vdxdy + β2

∫

Ω×Y2

(∇p2 +∇y p2)vdxdy

=

∫

Ω

B1∇p1 (x)vdx +

∫

Ω

B2∇p2vdx. (4.27)

In the same way, one can show that∫

Ω×Y1

γ1

(∇θ1 +∇y θ1

)vdxdy + γ2

∫

Ω×Y2

(∇θ2 +∇y θ2

)vdxdy

=

∫

Ω

D1∇θ1vdx +

∫

Ω

D2∇θ2vdx. (4.28)

Finally, inserting (3.16), (4.25), (4.27) and (4.28) into (4.24) and using the fact

thatD(Ω)3 is dense inH10 (Ω)

3, we deduce the homogenized balance formulation:∫ΩAijkhekh(u)eij(v)dx +

∫ΩB1∇p1vdx +

∫ΩB2∇p2vdx

+∫ΩD1∇θ1vdx +

∫ΩD2∇θ2vdx =

∫Ω fvdx

(4.29)

which by an integration by parts yields (4.23). The Lemma is then proved.

22

Lemma 4.7 The macroscopic mass conservation equation is given by:

∂t (φ∗1p1 + β1C1 : e (u) + α∗

1θ1)− div (K1∇p1) + ζ∗ (p1 − p2) = g∗1 , in Q,

∂t (φ∗2p2 + β2C2 : e (u) + α∗

2θ2)− div (K2∇p2) + ζ∗ (p2 − p1) = g∗2 , in Q,

p1 = 0, on ΓT ,

K2∇p2 · ν = 0 on ΓT ,

p1 (0, .) = p2 (0, .) = 0 in Ω(4.30)

where Cm, Km and (φ∗m, α∗m, ζ

∗, g∗m) are given in (3.14), (3.15) and (3.16) re-spectively.

Proof. Let qm(t, x) ∈ D((0, T )×Ω). Integration by parts in (3.3) with respectto the time variable t ∈ (0, T ) yields:

∫Qε

1

(φ1pε1 (t, x) + β1divu

ε (t, x) + α1θε1 (t, x))∂tq1 (t, x) dtdx

+∫Qε

1

(φ2pε2 (t, x) + β2divu

ε (t, x) + α2θε2 (t, x))∂tq2 (t, x) dtdx

+∫Qε

1

κ1∇pε1 (t, x)∇q1 (t, x) dtdx+∫Qε

2

κ2∇pε2 (t, x)∇q2 (t, x) dtdx

+∫Σε

T

ςε (x) (pε1 (t, x)− pε2 (t, x))(q1 (t, x)− q2 (t, x))dsε(x)dt

=∫Qε

1

g1 (x) q1 (t, x) dtdx+∫Qε

2

g2 (x) q2 (t, x) dtdx.

(4.31)

Using (4.3) and (4.4) we obtain:

limε→0

∫

Qεm

(φmpεm (t, x) + αmθ

εm (t, x))∂tqm (t, x) dtdx

=

∫

Q×Ym

(φmpm (t, x) + αmθm (t, x))∂tqm (t, x) dtdxdy. (4.32)

Furthermore, thanks to (4.5) and (4.6) we see that

limε→0

∫Qε

m

βmdivuε (t, x) ∂tqm (t, x) dtdx

=∫Q×Ym

βm (divu (t, x) + divyu (t, x, y)) ∂tqm (t, x) dtdxdy

(4.33)

and

limε→0

∫Qε

mκm∇pεm (t, x)∇qm (t, x) dxdt

=∫Q×Ym

(κm ∇pm (t, x) +∇y pm (t, x, y))∇qm (t, x) dtdxdy.

(4.34)

23

Now, using (4.10) we observe that∫Q×Ym

βm (divu (t, x) + divyu (t, x, y)) ∂tqm (t, x) dtdxdy

=∫QβmCm : e (u) (t, x) ∂tqm (t, x) dtdx.

(4.35)

In the same way, using (4.15) we find that∫Q×Ym

(κm ∇pm (t, x) +∇y pm (t, x, y))∇qm (t, x) dxdydt

=∫QKm∇pm (x, t)∇qm (x, t) dtdx.

(4.36)

Now, taking into account (4.8) we have

limε→0

∫Σε

T

ες(xε

)(pε1 (x, t)− pε2 (x))(q1 (x, t)− q2 (x, t))ds

ε(x)dt

=∫Q×Σ

ς (y) (p1 (x, t)− p2 (x))(q1 (t, x)− q2 (t, x))dtdxds(y).

(4.37)

Using Remark 4.1, we infer that

limε→0

(∫Qε

1

g1 (x) q1 (t, x) dtdx+∫Qε

2

g2 (x) q2 (t, x) dtdx)

=∫Qg1q1 (t, x) dtdx+

∫Qg2q2 (t, x) dtdx.

(4.38)

Finally, having in mind (4.32)-(4.38), we can now pass to the limit in (4.31) toget

∫Q(φ∗1p1 (t, x) + β1C1 : e (u) (t, x) + α∗

1θ1 (t, x) ∂tq1 (t, x) dtdx+∫Q(φ∗2p2 (t, x) + β2C2 : e (u) (t, x) + α∗

2θ2 (t, x) ∂tq2 (t, x) dtdx+∫QK1∇p1 (t, x)∇q1 (t, x) dtdx+

∫QK2∇p2 (t, x)∇q2 (t, x) dtdx

+∫Qζ∗(p1 (t, x)− p2 (t, x))(q1 (t, x)− q2 (t, x))dtdx

=∫Qg∗1 (x) q1 (t, x) dtdx+

∫Qg∗2 (x) q2 (t, x) dtdx

which by integration by parts yields (4.30). This gives the desired result.Likewise, as in the proof of Lemma 4.7, one can analogously show the fol-

lowing result:

Lemma 4.8 The macroscopic heat equation is given by:

∂t c∗1θ1 + γ1C1 : e (u) + α∗1p1 − div (L1∇θ1) + ω∗ (θ1 − θ2) = h∗1, in Q,

∂t c∗2θ2 + γ2C2 : e (u) + α∗2p2 − div (L2∇θ2) + ω∗ (θ2 − θ1) = h∗2, in Q,

θ1 = 0, on ΓT ,

L2∇θ2 · ν = 0 on ΓT ,

θ1 (0, .) = θ2 (0, .) = 0 in Ω

where Lm and (c∗m, ω∗, h∗m) are given in (3.15) and (3.16) respectively.

24

Proof of Theorem 3.2. Collecting all the results given in Lemmata 4.6, 4.7,4.8, we arrive at the homogenized system (3.17).

5 A corrector result

Let u, pm and θm to be the corrector terms of uε, pεm and θεm respectively. Theyare given by (4.5), (4.6) and (4.7) respectively. Now we first give the followingcorrector result.

Theorem 5.1 Assume that A ∈ L∞ (Y ). Then ey(u) is an admissible testfunction and the sequence [e(uε) (x)− e(u) (x)− ey(u)

(x, x

ε

)] converges strongly

to 0 in L2 (Ω)3×3

. We also have

limε→0

∥∥χm

(xε

) ∇pεm (t, x)−∇pm (t, x)−∇y pm

(t, x, x

ε

)∥∥0,QT

= 0,

limε→0

∥∥∥χm

(xε

)∇θεm (t, x)−∇θm (t, x)−∇y θm

(t, x, x

ε

)∥∥∥0,QT

= 0

and

limε

(∫ΩAεe(uε)e(uε)dx +

∑m

∫Ωε

m(αm∇pεm + γm∇θεm)uεdx

)

=∫ΩAe(u)e(u)dx +

∑m

∫Ω (ξm∇θm + ηm∇pm)u0dx.

To prove Theorem 5.1, we begin by establishing an integral identity.

Proposition 5.1 We have

−∫

Ω

φ∗1(p1)2 (T, x) dx−

∫

Q×Y1

[∂t(β1 (divu+ divyu) + α1θ1)] p1dtdxdy

−∫

Ω

φ∗2(p2)2 (T, x) dx−

∫

Q×Y2

[∂t(β2 (divu+ divyu) + α2θ2)] p2dtdxdy

+

∫

Q×Y1

K1∇p1∇p1dtdxdy +∫

Q×Y2

K2∇p2∇p2dtdxdy

+

∫

Q

ς∗ (p1 − p2)2dtdx =

∫

Ω

g∗1p1dx+

∫

Ω

g∗2p2dx (5.1)

and

−∫

Ω

c∗1(θ1)2 (T, x) dxdy −

∫

Q×Ym

[∂t(γ1 (divu+ divyu) + α1p1)] θ1dtdxdy

−∫

Ω×Y2

c∗2(θ2)2 (T, x) dxdy −

∫

Q×Ym

[∂t(γ2 (divu+ divyu) + α2p2)] θ2dtdxdy

+

∫

Q×Y1

L1∇θ1∇θ1dtdxdy +∫

Q×Y2

L2∇θ2∇θ2dtdxdy +∫

Q

ω∗ (θ1 − θ2)2dtdx =

∫

Ω

h∗1θ1dx+

∫

Ω

h∗2θ2dx. (5.2)

25

Proof. Take qm ∈ L2(0, T ;H1 (Ωε

m)), m = 1, 2 in (3.3) and integrate by parts

to yield:

−∫Qε

1

(φ1pε1 (t, x) + β1divu

ε (T, x) + α1θε1 (t, x))∂tq1 (t, x) dtdx

−∫Qε

2

(φ2pε2 (t, x) + β2divu

ε (t, x) + α2θε2 (t, x))∂tq2 (t, x) dtdx

+∫Ωε

1

(φ1pε1 (T, x) + β1divu

ε (T, x) + α1θε1 (T, x))q1 (T, x) dx

+∫Ωε

2

(φ2pε2 (T, x) + β2divu

ε (T, x) + α2θε2 (T, x))q2 (T, x) dtdx

−∫Ωε

1

(φ1pε1 (0, x) + β1divu

ε (0, x) + α1θε1 (0, x))q1 (0, x) dx

−∫Ωε

2

(φ2pε2 (0, x) + β2divu

ε (0, x) + α2θε2 (0, x))q2 (0, x) dx

∫Qε

1

κ1∇pε1 (t, x)∇q1 (t, x) dtdx +∫Qε

2

κ2∇pε2 (t, x)∇q2 (t, x) dtdx

+∫Σε

T

ςε (x) (pε1 (t, x)− pε2 (t, x))(q1 (t, x)− q2 (t, x))dsε(x)dt

=∫Qε

1

g1 (x) q1 (t, x) dtdx+∫Qε

2

g2 (x) q2 (t, x) dtdx.

Passing to the limit in the last identity we get

−∫Q1

(φ1p1 (t, x) + β1 (divu (t, x) + divyu1 (t, x, y)) + α1θ1 (t, x))∂tq1 (t, x) dtdxdy

−∫Q2

(φ2p2 (t, x) + β2 (divu (t, x) + divyu1 (t, x, y)) + α2θ2 (t, x))∂tq2 (t, x) dtdxdy

+∫Ω×Y1

(φ1p1 (T, x) + β1 (divu (T, x) + divyu1 (T, x, y)) + α1θ1 (T, x))q1 (T, x) dtdxdy

+∫Ω×Y2

(φ2p2 (T, x) + β2 (divu (t, x) + divyu1 (t, x, y)) + α2θ2 (T, x))q2 (T, x) dxdy

−∫Ω×Y1

(φ1pε1 (0, x) + β1 (divu (0, x) + divyu1 (0, x, y)) + α1θ1 (0, x))q1 (0, x) dxdy

−∫Ω×Y2

(φ2p2 (0, x) + β2 (divu (0, x) + divyu1 (0, x, y)) + α2θ2 (0, x))q2 (0, x) dxdy

+∫Q1

κ1 (∇p1 (t, x) +∇y p1 (t, x, y))∇q1 (t, x) dtdxdy

+∫Q2

κ2 (∇p2 (t, x) +∇y p2 (t, x, y))∇q2 (t, x) dtdxdy

+∫ΣT

ς (y) (p1 (t, x)− p2 (t, x))(q1 (t, x)− q2 (t, x))dtdxds(y)

=∫Q1×Y

g1 (x) q1 (t, x) dxdydt+∫Q2×Y

g2 (x) q2 (t, x) dtdxdy.

Now, taking any sequence (qnm)n converging to pm in the last identity and

26

integrating by parts with respect to t yield (5.1). The identity (5.2) goes alongthe same lines as that of (5.1).

Proposition 5.2 ey(u) is an admissible test function and the sequence [e(uε) (x)−e(u) (x)− ey(u)

(x, x

ε

)] converges strongly to 0 in L2 (Ω)

3×3.

Proof. We argue as in [5]. From (3.7) -(4.10) and standard results on regularityof elliptic equations [27], ey(u) is an admissible test function. Applying (3.2)yields for a.e. t ∈ (0, T )

∫

Ω

A(xε

)[e(uε) (x)− e(u) (x)− ey(u)

(x,x

ε

)]×

[e(uε) (x)− e(u) (x)− ey(u)(x,x

ε

)]dx =

−∫

Ω

(A+tA

)e(uε)[e(u) + ey(u)

(x,x

ε

)]dx

+

∫

Ω

A[e(u) + ey(u)(x,x

ε

)][e(u) + ey(u)

(x,x

ε

)]dx

∫

Ω

F1uεdx−

∫

Ω

(β1pεm + γ1θ

ε1) divu

ε)dx

∫

Ω

F2uεdx−

∫

Ω

(β2pε2 + γ2θ

ε2) divu

ε)dx. (5.3)

Using the strong convergences in L2 (Ω) of uε, pεm and θεm to u, pm and θmrespectively and the weak convergence of χε

mdiv(uε) to

∫Ym

(divu+divyu) and

taking the limit in the last term of the r.h.s. of (5.3) we obtain

limε→0

∫

Ω

Fmuεdx−∫

Ω

(βmpεm + γmθ

εm) divuε)dx

=

∫

Ω

Fmu−∫

Ω

(βmpm + γmθm)

∫

Ym

(divu+divyu) . (5.4)

Now, as (x, y) 7−→ ey(u) (x, y) is an admissible test function, the first two termsof the right hand side of (5.3) converges to

−∫

Ω×Y

(a+ta

)[e(u) + ey(u) (x, y)] [e(u) + ey(u) (x, y)]dxdy

+

∫

Ω×Y

a[e(u) + ey(u) (x, y)][e(u) + ey(u) (x, y)]dxdy

= −∫

Ω×Y

a[e(u) + ey(u) (x, y)][e(u) + ey(u) (x, y)]dxdy. (5.5)

Finally, thanks to the coercivity of A, (4.24) (with v = u) and (5.4)-(5.5) we

27

find that

α0

∥∥e(uε) (x) − e(u) (x)− ey(u)(x, x

ε

)∥∥L2(Ω)3×3 ≤

∫Ω A

(xε

)[e(uε) (x) − e(u) (x)− ey(u)

(x, x

ε

)][e(uε) (x)− e(u) (x)− ey(u)

(x, x

ε

)] →

∑m

∫Ω Fmu−

∫Ω (βmpm + γmθm)

∫Ym

(divu+divyu)

−∫ΩA[e(u) + ey(u) (x, y)][e(u) + ey(u) (x, y)] = 0

where α0 = miny∈Y A (y). Hence the proposition is proved.We now establish some corrector results on the mass conservation equation.

Let us first give some technical results.

Lemma 5.1 (G. Allaire and F. Murat [7, Lemma A.4]) There exists a con-stant C > 0 such that for all q1 ∈ H1

0 (Ω) ∩H1 (Ωε1) we have

‖q1‖0,Ωǫ1

≤ C ‖∇q1‖0,Ωǫ1

. (5.6)

Lemma 5.2 There exists a constant C > 0 such that for all q ∈ H1 (Ωε1) we

haveε ‖q‖20,Σε ≤ C

(ε2 ‖∇q‖20,Ωε

1

+ ‖q‖20,Ωε1

), (5.7)

andε ‖q‖20,Σε ≤ C

(‖∇q‖20,Ωε

1

). (5.8)

Proof. We argue as in [22]. Using the trace theorem on Y1 (see for e.g. R. A.Adams and J. F. Fournier [1]), we know that there exists a constant C (Y1) > 0such that for every ψ ∈ H1 (Y1)

∫

Σ

|ψ|2 dσ ≤ C

(∫

Y1

|∇ψ|2 dy +∫

Y1

|ψ|2 dy).

Then, using the change of variables y := x/ε we have for every q ∈ H1(Y ǫk1

)

ε

∫

Σǫk

|q|2 dσε ≤ C

(ε2∫

Y εk1

|∇q|2 dx+

∫

Y ǫk1

|q|2 dx)

(5.9)

where Y ǫk1 = ε (k +Ωε

1) and Σǫk = ε (k +Σ) k ∈ Z3. We mention that C

appearing in the inequality (5.9) is independent of k ∈ Z3. Summing up these

inequalities (5.9) over all Y ǫk1 contained in Ω, we get (5.3). To obtain (5.8),

it suffices to write (5.7) for sufficiently small ε, ε ≪ 1 and use the Friedrichinequality (5.6).

28

Lemma 5.3 We have

limε→0

∫

ΣεT

ες(pεm − pm)2dsε(x) = 0, (5.10)

limε→0

∫

ΣεT

ες (p1 − p2)2dsε(x) =

∫

Q

ζ (p1 − p2)2dx. (5.11)

limε→0

∫

ΣεT

ες(pε1 − pε2)2dsε(x) =

∫

Q

ζ (p1 − p2)2 dx. (5.12)

Proof. Using (5.9) yields

ε ‖(pεm − pm)‖20,Σε ≤ C(ε2 ‖∇(pεm − pm)‖20,Ωε

m+ ‖pεm − pm)‖20,Ωε

m

). (5.13)

We know that ‖∇(pεm − pm)‖0,Ωεm

is uniformly bounded with respect to ε and

thanks to Rellich’s Theorem, χεmp

εm converges strongly to χmpm in L2 (Ω). So,

by passing to the limit in (5.13) we easily arrive at (5.10). The convergencein (5.11) is a direct consequence of Corollary 4.1. To complete the proof itremains to show (5.12). Let us first observe that, according to Cauchy-Schwarzinequality and (5.10), we have

∣∣∣∣∣

∫

ΣεT

ες(pεm − pm) (p1 − p2) dsε(x)

∣∣∣∣∣ ≤ C√ε ‖pεm − pm‖0,Σε ‖p1 − p2‖0,Σε → 0.

(5.14)Hence, from (5.10), (5.11) and (5.14) we obtain

∫ T

0

∫Σε

T

ες(pε1 − pε2)2dsε(x)dt =

∫ T

0

∫Σε ες(p

ε1 − p1)

2dsε(x)dt

+∫Σε

T

ες(pε2 − p2)2dsε(x)dt + 2

∫Σε

T

ες(pε1 − p1) (p1 − p2) dsε(x)dt

+2∫Σε

T

ες(pε2 − p2) (p1 − p2) dsε(x)dt

+∫Σε

T

ες (p1 − p2)2dsε(x)dsε(x)dt

→∫Qζ (p1 − p2)

2 dx.

Hence (5.12).

Proposition 5.3 We have

limε→0

∥∥∥χm

(xε

)∇pεm (t, x)−∇pm (t, x)−∇y pm

(t, x,

x

ε

)∥∥∥0,QT

= 0.

29

Proof. As in the proof of Proposition 5.2, we first write that

∫

Qε1

K1

(xε

) ∣∣∣∇pε1 (t, x)−∇p1 (t, x)−∇y p1

(t, x,

x

ε

)∣∣∣2

dtdx

+

∫

Qε2

K2x

ε

∣∣∣∇pε2 (t, x)−∇p2 (t, x) −∇y p2

(t, x,

x

ε

)∣∣∣2

dtdx

=

∫

Qε1

K1

(xε

)|∇pε1 (t, x)|2 dtdx+

∫

Qε1

Km

(xε

)|∇pε1 (t, x)|2 dtdx

−2

∫

Qε1

K1

(xε

)∇pε1 (t, x) (∇p1 (t, x) +∇y p1 (t, x)) dtdx

−2

∫

Qε2

K2

(xε

)∇pε2 (t, x)

(∇p2 (t, x) +∇y p2

(t, x,

x

ε

))dtdx

∫

Qε1

K1

(xε

) ∣∣∣∇p1 (t, x) +∇y p1

(t, x,

x

ε

)∣∣∣2

dtdx

+

∫

Qε2

K2

(xε

) ∣∣∣∇p2 (t, x) +∇y p2m

(t, x,

x

ε

)∣∣∣2

dtdx. (5.15)

It is easy to show that the last two terms of the right hand side of (5.15) convergeto

−2

∫

Qεm

Km

(xε

)∇pεm (t, x)

(∇pm (t, x) +∇y pm

(t, x,

x

ε

))+

∫

Qεm

Km

(xε

) ∣∣∣∇pm (t, x) +∇y pm

(t, x,

x

ε

)∣∣∣2

−→−∫

Q×Ym

Km (y) |∇pm (t, x) +∇y pm (t, x, y)|2 , m = 1, 2. (5.16)

Now by (3.3) one can see that

∫

Qεm

K1

(xε

)∇pε1∇pε1dtdx +

∫

Qεm

K2

(xε

)∇pε2∇pε2dtdx

−∫

Qε1

∂t(φ1pε1 + β1divu

ε1 + γ1θ

ε1)p

ε1dtdx

−∫

Qεm

∂t(φ2pε2 + β2divu

ε2 + γ2θ

ε2)p

ε2dtdx

+

∫

ΣεT

ες(pε1 − pε2)2dsε(x)dt =

∫

Qεm

Gmpεmdtdx. (5.17)

Taking the limit in the r.h.s. of (5.17) gives

limε

∫

Qεm

Gmpεmdtdx = |Ym|

∫

Q

Gmpmdtdx, m = 1, 2. (5.18)

30

Using Lemma 5.3, and more precisely (5.12) and passing in the last term of thel.h.s. of (5.17) yields

limε

∫

ΣεT

ες(pε1 − pε2)2dsε(x)dt =

∫

Q×Σ

ς (y) (p1 − p2)2dxds (y) dt.

Regarding the second term of the right hand side of (5.17) we proceed as follows:Firstly, integrating par parts with respect to the time variable t we have form = 1, 2

∫

Qεm

∂t(φmpεm)pεmdtdx =

∫

Qεm

φm(pεm)2 (T, x) dx

→∫

Q×Ym

φm(pm)2 (T, x) dtdx. (5.19)

Secondly, since χεm∂t(divu

εm) converges weakly to χm ∂t(divum + divyum)

and χεmp

εm converges strongly to χmpm it follows that

limε

∫

Qεm

βm∂t(divuεm)pεmdtdx

=

∫

Q×Ym

βm ∂t(divum + divyum) pmdtdx. (5.20)

Furthermore, as χεm∂tθ

εm converges strongly to χm∂tθm, we see that

limε

∫

Qεm

γm∂tθεmp

εmdtdx =

∫

Q×Ym

γm∂tθmpmdtdx. (5.21)

Using the integral identity (5.1) and (5.18)-(5.21) we pass to the limit in (5.17)to get

limε

∫

Qε1

Km

(xε

)∇pε1∇pε1dtdx+

∫

Qε2

K2

(xε

)∇pε2∇pε2dtdx

= |Y1|∫

Q

G1p1dtdx+ |Y2|∫

Ω

G2p2dtdx

+

∫

Ω×Y1

φ1(p1)2 (T, x) dx+

∫

Ω×Y2

φ2(p2)2 (T, x) dx+

∫

Q×Y1

[∂t(β1 (divu1 + divyu1) + γ1θ1))] p1dtdx

+

∫

Q×Ym

[∂t(β2 (divu2 + divyu2) + γ2θ2))] p2dtdx

−∫

Q×Σ

ς (y) (p1 − p2)2dtdxds (y) . (5.22)

31

Next, using (4.30), the convergence in (5.22) can be reduced to

limε

∫

Qε1

K1

(xε

)∇pε1∇pε1dtdx+

∫

Qε2

K2

(xε

)∇pε2∇pε2dtdx

+

=

∫

Q×Y1

K1 (y) (∇p1 +∇y p1)(∇p1 +∇y p1))dtdxdy +

∫

Q×Y2

K2 (y) (∇p2 +∇y p2)(∇p2 +∇y p2))dtdxdy. (5.23)

Finally, collecting all the limits (5.16) and (5.23), Equation (5.15) becomes

limε

∑

m=1,2

∥∥∥χm

(xε

)∇pεm (t, x)−∇pm (t, x)−∇y pm

(t, x,

x

ε

)∥∥∥2

0,Q≤

limε

∑

m=1,2

∫

Qεm

Km

∣∣∣∇pεm (t, x)−∇pm (t, x)−∇y pm

(t, x,

x

ε

)∣∣∣2

= 0.

This gives the desired result.Now we state the corrector result for the heat equation whose proof follows

the same lines as that of Proposition 5.2 and is therefore omitted.

Proposition 5.4 We have

limε→0

∥∥∥χm

(xε

)∇θεm (t, x)−∇θm (t, x) −∇yθm

(t, x,

x

ε

)∥∥∥0,Q

= 0.

Finally we state the asymptotic behavior of the energies.

Proposition 5.5 One has for a.e. t ∈ (0, T )

limε

(∫

Ω

A(xε

)e(uε)e(uε)dx+

∑

m=1,2

∫

Ωεm

(βm∇pεm + γm∇θεm)uεdx

)

=

∫

Ω

ahome(u)e(u)dx+∑

m=1,2

∫

Ω

(Am∇θm + Bm∇pm)udx. (5.24)

Proof. Taking v =uε in (3.2) gives∫

Ω

a(xε

)e(uε) (x) e(uε)dx+

∑

m=1,2

∫

Ωεm

(βm∇pεm + βm∇θεm)uεdx

−∫

Ω

Fuε = 0

which, by taking into account (4.29), tends to∫

Ω

ahome(u)e (u) dx +∑

m=1,2

∫

Ω

(Am∇θm + Bm∇pm)udx

−∫

Ω

Fudx=0.

Hence (5.24).

32

6 Conclusion

In this paper we derived by a homogenization technique a more general modelof thermoporoelasticity with double porosity and two temperatures. More pre-cisely, we studied a micro-model of fluid and thermal flows in two componentporoelastic media consisting of matrix and inclusions with the same order ofpermeabilities and conductivities, separated by a periodic and thin layer whichforms an exchange fluid/thermal barrier. In particular, we have shown that theBiot-Willis and thermal expansion parameters are in that case matrices and nolonger scalars, see for instance [2, 3]. Let us mention that the result of the paperremains valid if one considers non homogeneous initial and/or Dirichlet con-ditions. An interesting problem is to investigate the limiting behavior of suchmedia when the flow potential in the inclusions is rescaled by ε2. This occursespecially when the flow in the inclusions presents very high frequency spatialvariations due to a relatively very low permeability, see Remark 2.2.Acknowledgment The author acknowledges the support of the Algerian min-istry of higher education and scientific research through the AMNEDP Labora-tory.

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