Homogenization of a Conductive, Convective and [610090]

Homogenization of a Conductive, Convective and
Radiative Heat Transfer Problem in a Heterogeneous
Domain
Gr egoire Allaire, Zakaria Habibi
To cite this version:
Gr egoire Allaire, Zakaria Habibi. Homogenization of a Conductive, Convective and Radiative
Heat Transfer Problem in a Heterogeneous Domain. SIAM Journal on Mathematical Analysis,
Society for Industrial and Applied Mathematics, 2013, 45 (3), pp.1136-1178. <hal-00932950 >
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Homogenization of a Conductive, Convective and Radiative H eat
Transfer Problem in a Heterogeneous Domain∗
Gr´ egoire Allaire†and Zakaria Habibi‡
January 21, 2013
Abstract
We are interested in the homogenization of heat transfer in periodic porous media where the
fluid part is made of long thin parallel cylinders, the diameter of which is of the same order than the
period. The heat is transported by conduction in the solid part of th e domain and by conduction,
convection and radiative transfer in the fluid part (the cylinders). A non-local boundary condition
modelstheradiativeheattransferonthecylinderwalls. Toobtainth ehomogenizedproblemwefirst
use a formal two-scale asymptotic expansion method. The resultin g effective model is a convection-
diffusion equation posed in a homogeneous domain with homogenized co efficients evaluated by
solving so-called cell problems where radiative transfer is taken into account. In a second step we
rigorously justify the homogenization process by using the notion o f two-scale convergence. One
feature of this work is that it combines homogenization with a 3D to 2D asymptotic analysis since
the radiative transfer in the limit cell problem is purely two-dimensiona l. Eventually, we provide
some 3D numerical results in order to show the convergence and th e computational advantages of
our homogenization method.
Key words : Periodic homogenization, two-scale convergence, heat transfer.
1 Introduction
We study heat transfer in a very heterogeneous periodic poro us medium. Since the ratio of the
heterogeneities period with the characteristic length-sc ale of the domain, denoted by ǫ, is very small in
practice, adirectnumericalsimulationofthisphenomenon iseitheroutofreachorverytimeconsuming
on any computer (especially in 3D). Therefore, the original heterogeneous problem should be replaced
by an homogeneous averaged (or effective, or homogenized) one . The goal of homogenization theory
[8], [10], [15], [23], [25], [31], [32] is to provide a system atic way of finding such effective problems,
of reconstructing an accurate solution by introducing so-c alled correctors and of rigorously justifying
such an approximation by establishing convergence theorem s and error estimates. The purpose of this
paper is to carry on this program for a model of conductive, co nvective and radiative heat transfer
in a 3D solid domain, periodically perforated by thin parall el cylinders in which a gas is flowing (see
Figure 1 for a sketch of the geometry). Convection and radiat ive transfer are taking place only in the
gas which is assumed to be transparent for radiation and with a very small bulk diffusivity. Therefore,
the radiative transfer is modelled by a non-local boundary c ondition on the cylinder walls.
Although there are many possible physical motivations for t his study, we focus on its application
to the nuclear reactor industry and especially to the so-cal led gas-cooled reactors [19] which are a
promising concept for the 4th generation reactors. The peri odic porous medium in our study is the
∗This work has been supported by the French Atomic Energy and A lternative Energy Commission, DEN/DM2S at
CEA Saclay.
†CMAP, Ecole Polytechnique, 91128 Palaiseau, & DM2S, CEA Sac lay, 91191 Gif sur Yvette, gre-
goire.allaire@polytechnique.fr
‡DM2S/SFME/LTMF, CEA Saclay, 91191 Gif sur Yvette, & CMAP, Ec ole Polytechnique, 91128 Palaiseau, za-
karia.habibi@polytechnique.edu
1

Homogenization of a Heat Transfer Problem 2
core of such a gas-cooled reactor. It is typically made of man y prismatic blocks of graphite in which
are inserted the nuclear fuel compact. Each block is periodi cally perforated by many small channels
where the coolant (Helium) flows (the number of these gas cyli nders is of the order of 104at least).
Although the solid matrix of the porous medium is itself hete rogeneous (a mixture of graphite and
of nuclear materials), we simplify the exposition by assumi ng it is already homogenized and thus
homogeneous. The analysis would not be much more complicate d otherwise but certainly less clear
for the reader. In other words, we assume that the only source of heterogeneities is coming from the
geometry of the porous medium which is a fine mixture of solid a nd fluid parts. Since the total number
ofcylindersisverylargeandtheirdiameter isvery smallco mparedtothesizeofthecore, thenumerical
simulation of this problem requires a very fine mesh and thus a very expensive computational cost for
a real geometry of a reactor core (all the more since the radia tive transfer is modelled by an integral
operator yielding dense discretization matrices). Theref ore, our goal is to find a simpler homogenized
model in an equivalent continuous domain and, specifically, to give a clear definition of the resulting
effective parameters as well as a detailed reconstruction of a n approximate solution (involving local
correctors that take into account the geometry variation).
A similar study, in a simplified 2D setting, has previously ap peared in [3]. In this reference,
the 2D domain was a cross section of the reactor core (perpend icular to the cylinders) so that the
fluid part was a periodic collection of isolated disks. Furth ermore, convection and diffusion were
neglected in the gas. Therefore, the main novelties of the pr esent paper is, first, to take into account
convection and diffusion in the fluid, second and most importan tly, to consider a porous medium
perforated by cylinders instead of disks. This last general ization is not at all a simple extension of
the previous results of [3]. It turns out that [3] can easily b e extended to a periodic distribution of
spherical holes in 3D. On the contrary, in the case of cylinde rs, since periodicity takes place only in the
transverse directions and the holes are not isolated, but ra ther connected, in the axial direction, a new
phenomenon takes place which corresponds to a dimension red uction for the radiative operator from
3D to 2D. In other words, our asymptotic analysis is not only a problem of homogenization but also
of singular perturbation. The issue of dimension reduction is well-known in solid mechanics, where
it is a basic ingredient to deduce plate or shell models from 3 D elasticity when the thickness of the
structure is going to zero (see e.g. [14]). Here, the reason f or this dimension reduction is that, in the
homogenization process, the cylinders become infinitely lo ng compared to their diameter (which goes
to zero): thus, at a microscopic scale the 3D radiative opera tor is asymptotically invariant along the
axis of these cylinders and, in the limit, degenerates to a 2D radiative operator. Furthermore, some
radiations are escaping from the cylinders by their extremi ties: asymptotically it yields an additional
vertical homogenized diffusivity which was, of course, not se en in the 2D setting of [3]. Overall our
homogenized model is new, quite surprising and not intuitiv e, even in light of [3].
There are a number of other previous contributions on the hom ogenization of radiative trans-
fer which all correspond to different geometries or scalings o f various parameters [6], [7], [9], [13].
Let us also mention that there is a huge literature on the homo genization in perforated domains or
porous media (see [23] and references therein, [16], [18] fo r the case of non-linear Neumann boundary
conditions).
The paper is organized as follows. In Section 2, we give a prec ise definition of the geometry and
of the heat transfer model, see (8). In particular we discuss our scaling assumptions in terms of the
small parameter ǫ. Furthermore, various properties of the radiative operato r are recalled. It is an
integral operator, the kernel of which is called the view fac tor (it amounts to quantify how a point
on the cylinder wall is illuminated by the other points on thi s surface). A key ingredient for the
sequel is proved in Lemma 2.1: an asymptotic expansion of the 3D view factor, integrated along
the cylinder axis, is established in terms of the 2D view fact or. Section 3 is devoted to the formal
method of two-scale asymptotic expansions applied to our pr oblem. Its main result is Proposition 3.1
which gives the precise form of the homogenized problem. Fur thermore, it also furnishes the so-called
cell problems which define the corrector term for the homogen ized solution. It is at the basis of a
reconstruction process for an accurate and detailed approx imate solution. We emphasize that the

Homogenization of a Heat Transfer Problem 3
application of the formal method of two-scale asymptotic ex pansions is not standard for two reasons.
First, to minimize the number of required terms in the result ing cascade of equations, we rely on a
variant of the method, suggested by J.-L. Lions [26], which a mounts to introduce an ansatz in the
variational formulation rather than in the strong form of th e equations. Second, we must combine
this ansatz with the dimension reduction argument for the ra diative operator as given by the technical
Lemma 2.1. Section 4 provides a rigorous mathematical justi fication of the homogenization process by
using the method of two-scale convergence [1], [30]. Our mai n result is Theorem 4.2 which confirms
the statement of Proposition 3.1. A formal generalization t o the non-linear case is briefly sketched
in Section 5. Indeed, our mathematical rigorous justificati on holds true only for a linear model so
we choose to expose this setting. However, the true physical model of radiative transfer is non-linear
since the emitted radiations are following Stefan-Boltzma nn law of proportionality to the 4th power of
temperature. Taking into account this non-linearity is not difficult for the formal method of two-scale
asymptotic expansions. Thus we give the homogenized and cel l problems in this case too, all the
more since all our numerical computations are performed in t his non-linear setting. In Section 6, we
present some numerical results for data corresponding to ga s-cooled reactors. In particular we show
that the error between the exact and reconstructed solution s, as a function of the small parameter ǫ,
is as expected of order 1 or 1 /2, depending on the choice of norm.
2 Setting of the problem
The goal of this section is to define the geometry of the period ic porous medium and to introduce the
model of conductive, convective and radiative heat transfe r.
2.1 Geometry
For simplicity we consider a rectangular open set Ω =/producttext3
j=1(0,Lj) whereLj>0 are positive lengths.
It is however essential that the domain Ω be a cylinder with ax is in the third direction, namely that
its geometry is invariant by translation along x3. The rectangular basis/producttext2
j=1(0,Lj) is periodically
divided inN(ǫ) small cells (Λ ǫ,i)i=1…N(ǫ), each of them being equal, up to a translation and rescaling
by a factor ǫ, to the same unit periodicity cell Λ =/producttext2
j=1(0,lj) withlj>0. By construction, the
domain Ω is periodic in the two first directions and is invaria nt by translation in the third one. To
avoid unnecessary complications with boundary layers (and because this is the case in the physical
problem which motivates this study) we assume that the seque nce of small positive parameters ǫ,
going to zero, is such that the basis of Ω is made up of entire ce lls only, namely Lj/(ǫlj) is an integer
for anyj= 1,2. The cell Λ is decomposed in two parts: the holes ΛFoccupied by a fluid (see Figures
1 and 2) and the solid matrix ΛS. We denote by γthe boundary between ΛSand ΛF. Then, we define
the fluid domain ΩF
ǫas the cylindrical domain with basis composed by the collect ion of ΛF
ǫ,iand the
solid domain ΩS
ǫas the cylindrical domain with basis composed by the collect ion of ΛS
ǫ,i, where ΛF,S
ǫ,i
are the translated and rescaled version of ΛF,Sfori= 1…N(ǫ) (similar to the correspondence between
Λǫ,iand Λ). In summary we have
ΩF
ǫ=N(ǫ)/uniondisplay
i=1ΛF
ǫ,i×(0,L3),ΩS
ǫ= Ω\ΩF
ǫ=N(ǫ)/uniondisplay
i=1ΛS
ǫ,i×(0,L3), γǫ=N(ǫ)/uniondisplay
i=1γǫ,i,Γǫ=γǫ×(0,L3).
For each plane cell Λ ǫ,i, the center of mass x′
0,iof the boundary γǫ,iis defined by
/integraldisplay
γǫ,i(s′−x′
0,i)ds′= 0. (1)
For any point x= (x1,x2,x3)∈R3, we denote by x′its two first components in R2such that
x= (x′,x3). We introduce the linear projection operator PfromR3toR2and its adjoint, the

Homogenization of a Heat Transfer Problem 4
extension operator EfromR2toR3, defined by
P
v1
v2
v3
=/parenleftbiggv1
v2/parenrightbigg
andE/parenleftbiggv1
v2/parenrightbigg
=
v1
v2
0
. (2)
Eventually, we denote by ∇x′the 2D gradient operator which we shall often identify to its extension
E∇x′. Similarly, for a 3D vector field F(x′,x3) we shall use the notation div x′Ffor div x(PF).
Figure 1: Periodic domain for a gas cooled reactor core
Figure 2: 2D reference cell for a gas cooled reactor core
2.2 Governing equations
There is a vast literature on heat transfer and we refer the in terested reader to [12], [27], [33] for an
introduction to the modelling of radiative transfer. We den ote byTǫthe temperature in the domain
Ω which can be decomposed as
Tǫ=/braceleftbiggTS
ǫin ΩS
ǫ,
TF
ǫin ΩF
ǫ,
whereTǫis continuous through the interface Γ ǫ.
Convection takes place only in the thin vertical cylinders ΩF
ǫoccupied by the fluid. We thus
introduce a given fluid velocity
Vǫ(x) =V(x,x′
ǫ) in ΩF
ǫ,
where the continuous vector field V(x,y′), defined in Ω ×ΛF, is periodic with respect to y′and satisfies
the two incompressibility constraints
divxV= 0 and div y′V= 0 in Ω ×ΛF,andV·n= 0 on Ω ×γ

Homogenization of a Heat Transfer Problem 5
wherenis the unit outward normal (from ΛSto ΛF) onγ. A typical example of such a velocity field
isV= (V′(x3,y′),V3(x′,y′)) withV′= (V1,V2), divy′V′= 0 andV′·n= 0 onγ.
The thermal diffusion is assumed to be much smaller in the fluid t han in the solid. More precisely
we assume that it is of order 1 in ΩS
ǫand of order ǫin ΩF
ǫ. The conductivity tensor is thus defined by
Kǫ(x) =/braceleftbiggKS
ǫ(x) =KS(x,x′
ǫ) in ΩS
ǫ,
ǫKF
ǫ(x) =ǫKF(x,x′
ǫ) in ΩF
ǫ,(3)
whereKS(x,y′),KF(x,y′) are periodic symmetric positive definite tensors defined in the unit cell Y,
satisfying
∀v∈R3,∀y′∈Λ,∀x∈Ω, α|v|2≤3/summationdisplay
i,j=1KF,S
i,j(x,y′)vivj≤β|v|2,
for some constants 0 <α≤β. The choice of the ǫscaling in (3) is made in order to have a dominant
convection in the fluid part at the macroscopic scale. Howeve r, at the microscopic scale the convection
and the diffusion are balanced as will be clear later.
The fluid is assumed to be almost transparent, so that heat can also be transported by radiative
transfer in ΩF
ǫ. This radiative effect is modelled by a non local boundary cond ition on the interface Γ ǫ
between ΩF
ǫand ΩS
ǫ. More precisely, in addition to the continuity of temperatu re we write a balance
of heat fluxes on the interface
TS
ǫ=TF
ǫand−KS
ǫ∇TS
ǫ·n=−ǫKF
ǫ∇TF
ǫ·n+σ
ǫGǫ(TF
ǫ) on Γ ǫ, (4)
whereσ>0 is a given positive constant and Gǫis the radiative operator defined by
Gǫ(Tǫ)(s) =Tǫ(s)−/integraldisplay
Γǫ,iTǫ(x)F(s,x)dx= (Id−ζǫ)Tǫ(s)∀s∈Γǫ,i, (5)
with
ζǫ(f)(s) =/integraldisplay
Γǫ,iF(s,x)f(x)dx. (6)
The scaling ǫ−1in front of the radiative operator Gǫin (4) is chosen because it yields a perfect balance,
in the limit as ǫgoes to zero, between the bulk heat conduction and the surfac e radiative transfer (this
scaling was first proposed in [3] and is due to the fact that the operator (Id−ζǫ) has a non-trivial
kernel, see Lemma 2.1). In (6) Fis the so-called view factor (see [27], [24], [22]). The view factor
F(s,x) is a geometrical quantity between two different points sandxof the same cylinder Γ ǫ,i. Its
explicit formula for surfaces enclosing convex domains is i n 3D
F(s,x) :=F3D(s,x) =nx·(s−x)ns·(x−s)
π|x−s|4,
wherenzdenotes the unit normal at the point z. In 2D the view factor is
F(s,x) :=F2D(s′,x′) =n′
x·(s′−x′)n′
s·(x′−s′)
2|x′−s′|3
and the operator in (6) is denoted by ζ2D
ǫ. Some useful properties of the view factor are given below
in Lemma 2.1.
For simplicity we assume that the only heat source is a bulk de nsity of thermal sources in the
solid part, f∈L2(Ω),f≥0 and the external boundary condition is a simple Dirichlet c ondition.
Eventually, the governing equations of our model are


−div(KS
ǫ∇TS
ǫ) =f in ΩS
ǫ
−div(ǫKF
ǫ∇TF
ǫ)+Vǫ·∇TF
ǫ= 0 in ΩF
ǫ
−KS
ǫ∇TS
ǫ·n=−ǫKF
ǫ∇TF
ǫ·n+σ
ǫGǫ(TF
ǫ) on Γ ǫ
TS
ǫ=TF
ǫ on Γǫ
Tǫ= 0 on ∂Ω.(8)

Homogenization of a Heat Transfer Problem 6
Proposition 2.1. The boundary value problem (8) admits a unique solution TǫinH1
0(Ω).
ProofThis is a classical result (see [3] if necessary) by applicat ion of the Lax-Milgram lemma. The
main point is that the operator Gǫis self-adjoint and non-negative, as stated in Lemma 2.1 bel ow./squaresolid
Remark 2.1. The solution of (8) satisfies the maximum principle, namely f≥0inΩimplies that
Tǫ≥0inΩ(see [33]). However, we shall not use this property in the seq uel.
Remark 2.2. The radiation operator introduced in (5) is a linear operator : this is clearly a simplifying
assumption. Actually, the true physical radiation operato r is non-linear and defined, on each Γǫ,i,1≤
i≤N(ǫ), by
Gǫ(Tǫ) =e(Id−ζǫ)(Id−(1−e)ζǫ)−1(T4
ǫ). (9)
whereζǫis the operator defined by (6). To simplify the exposition, we focus on the case of so-called
black walls, i.e., we assume that the emissivity is e= 1(we can find in [7] a study of this kind
of problems when the emissivity depends on the radiation fre quency). However, our analysis can
be extended straightforwardly to the other cases 0< e <1(see e.g. [20]). The formal two-scale
asymptotic expansion method can also be extended to the abov e non-linear operator, at the price of
more tedious computations [20]. However, the rigorous just ification of the homogenization process is,
for the moment, available only for the linearized form of the radiation operator. Therefore we content
ourselves in exposing the homogenization process for the li near case. Nevertheless, in Section 5 we
indicate how our results can be generalized to the above non- linear setting. Furthermore, our numerical
results in Section 6 are obtained in the non-linear case whic h is more realistic from a physical point of
view.
2.3 Properties of the view factor
We recall and establish some useful properties of the view fa ctor that we will use later.
Lemma 2.1. For pointsxandsbelonging to the same cylinder Γǫ,i, the view factor F(s,x)satisfies
1.
F(s,x)≥0, F(s,x) =F(x,s), (10)
2. /integraldisplay
γǫ,iF2D(s′,x′)ds′= 1,
3. as an operator from L2into itself,
/ba∇dblζǫ/ba∇dbl ≤1, (11)
4. /integraldisplay
γǫ,i/integraldisplay
γǫ,i(x′−x′
0,i)F2D(s′,x′)dx′ds′= 0,
5.
ker(Id−ζ2D
ǫ) =R, (12)
6. the radiative operator Gǫis self-adjoint on L2(Γǫ,i)and non-negative in the sense that
/integraldisplay
Γǫ,iGǫ(f)fds≥0∀f∈L2(Γǫ,i), (13)

Homogenization of a Heat Transfer Problem 7
7. for any given s3∈(0,L),
/integraldisplayL
0F3D(s,x)dx3=F2D(s′,x′)+O(ǫ2
L3), (14)
8. for any function g∈C3(0,L)with compact support in (0,L),
/integraldisplayL
0g(x3)F3D(s,x)dx3=F2D(s′,x′)/parenleftBig
g(s3)+|x′−s′|2
2g′′(s3)+O(ǫp)/parenrightBig
, (15)
where any 0<p<3is admissible and g′′denotes the second derivative of g. Furthermore, for
any function f∈L∞(0,L), we have
/integraldisplayL
0/integraldisplayL
0f(x3)g(s3)F3D(s,x)dx3ds3=F2D(s′,x′)/parenleftBig/integraldisplayL
0f(x3)g(x3)dx3
+1
2|x′−s′|2/integraldisplayL
0f(x3)g′′(x3)dx3+O(ǫp)/parenrightBig
.(16)
Remark 2.3. The surface Γǫ,iof each cylinder is not closed (it is only the lateral boundar y and the
two end cross-sections are missing). Therefore, the second p roperty of Lemma 2.1 does not hold in
3D, namely /integraldisplay
Γǫ,iF3D(s′,x′)ds′/\egatio\slash= 1.
Remark 2.4. The asymptotic properties (14) can be physically interprete d by saying that in a thin and
long cylinder the 3D view factor are well approximated by the 2D view factor, upon vertical integration.
Since the surface Γǫ,iis open at its extremities, there is some leakage of the radia ted energy. The
asymptotic property (15) and (16) take into account the quan tification of this leakage which corresponds
to a diffusive corrector term in the x3direction (remember that |x′−s′|2is of the order of ǫ2).
ProofThe six first properties are classical and may be found in [20] . The proof of (14) follows from
a change of variables and a Taylor expansion. At this point, t he assumption that s3does not depend
onǫand is different from the two end points 0 and Lis crucial. Indeed, because the cylinder Γ ǫ,iis
vertical, we have ns3=nx3= 0 and
I=/integraldisplayL
0nx·(s−x)ns·(x−s)
π|x−s|4dx3=n′
x·(s′−x′)n′
s·(x′−s′)
π(x′−s′)4/integraldisplayL
01
/parenleftbigg
1+(x3−s3)2
|x′−s′|2/parenrightbigg2dx3.
By the change of variables
z=x3−s3
α,whereα=|x′−s′|, (17)
and integration, we obtain
I=2
πF2D(s′,x′)/parenleftbigg
h1(L−s3
|x′−s′|)−h1(−s3
|x′−s′|)/parenrightbigg
whereh1(z) is the primitive of the previous integrand given by
h1(z) =1
2/parenleftbiggz
z2+1+arctan(z)/parenrightbigg
. (18)

Homogenization of a Heat Transfer Problem 8
By Taylor expansion we get
h1(z) =

+π
2+O(z−3) whenz→+∞,
−π
2+O(z−3) whenz→ −∞.(19)
Since|x′−s′|=O(ǫ),s3=O(L) andF2D(s′,x′) =O(ǫ−1), we deduce (14).
The proof of (15) is a little more difficult although the strate gy is the same. Let us notice that the
assumption of compact support for gallows us to avoid difficulties coming from the case when s3= 0
ors3=L. By the same change of variables (17) we obtain
I=/integraldisplayL
0g(x3)F3D(s,x)dx3=2
πF2D(x′,s′)/integraldisplay
∆g(s3+αz)
(1+z2)2dz=2
πF2D(x′,s′)/hatwideI,
where the domain of integration ∆ is given by ∆ = [−s3
α,L−s3
α]. Remark that α=O(ǫ). By using
a Taylor expansion in a neighbourhood of s3, we have
g(s3+αz) =g(s3)+αzg′(s3)+1
2α2z2g′′(s3)+O(α3z3),
and/hatwideIbecomes
/hatwideI=I1+I2+I3+I4, (20)
where,h1(z) being given by (18),
I1=g(s3)/integraldisplay
∆1
(1+z2)2dz=g(s3)
2/parenleftbigg
h1(L−s3
α)−h1(−s3
α)/parenrightbigg
=g(s3)
2/parenleftbig
π+O(α3)/parenrightbig
.
On the other hand we get
I2=αg′(s3)/integraldisplay
∆z
(1+z2)2dz=αg′(s3)/parenleftbigg
h2(L−s3
α)−h2(−s3
α)/parenrightbigg
=αg′(s3)
2O(α2),
I3=α2
2g′′(s3)/integraldisplay
Dz2
(1+z2)2dz=α2
2g′′(s3)/parenleftbigg
h3(L−s3
α)−h3(−s3
α)/parenrightbigg
=α2
4g′′(s3)(π+O(α)),
where we performed a Taylor expansion of h2(z) andh3(z) which are the primitives of the previous
integrands in I2andI3, respectively, given by
h2(z) =1
2/parenleftbigg−1
z2+1/parenrightbigg
, h3(z) =1
2/parenleftbigg−z
z2+1+arctan(z)/parenrightbigg
.
The last integral in (20) is of order O(ǫp) for any 0 <p<3 because
|I4| ≤Cα3/integraldisplay
∆z3
(1+z2)2dz≤Cα3/parenleftbigg
h4(L−s3
α)−h4(−s3
α)/parenrightbigg
,
whereh4(z) is the primitive of the previous integrand given by
h4(z) =1
2/parenleftbigg
log(z2+1)+1
1+z2/parenrightbigg
. (21)
By a Taylor expansion of (21) when z→ ±∞we get
|I4| ≤Cα3|logα| ≤Cαp∀0<p<3.
Hence the result (15) since α=O(ǫ). Eventually, (16) is immediate using (15). /squaresolid

Homogenization of a Heat Transfer Problem 9
Remark 2.5. If the function fis smooth, by integration by parts (16) becomes
/integraldisplayL
0/integraldisplayL
0f(x3)g(s3)F3D(s,x)dx3ds3=F2D(s′,x′)/parenleftBig/integraldisplayL
0f(x3)g(x3)dx3
−1
2|x′−s′|2/integraldisplayL
0f′(x3)g′(x3)dx3+O(ǫp)/parenrightBig
.(22)
Actually, (22) can be proved directly with different smoothn ess assumptions: it holds true for fandg
of classC2, one of them being with compact support.
3 Two-scale asymptotic expansion
The homogenized problem can be obtained heuristically by th e method of two-scale asymptotic ex-
pansion [10], [15], [31]. This method is based on the periodi c assumption on the geometry of the
porous medium. However here, because the radiative operato r is only 2D periodic, we shall introduce
a microscopic variable y′which is merely a 2D variable (in the plane perpendicular to t he cylinders).
Of course, denoting the space variable x= (x′,x3), the fast and slow variables are related by y′=x′/ǫ.
The radiative operator is creating an additional difficulty: since the fluid part is made of thin and long
cylinders, the 3D view factors will asymptotically be repla ced by the 2D view factors (see Lemma 2.1).
Therefore, our problem is not only an homogenization proble m but it is also a singularly perturbed
one. It can be compared to the dimension reduction issue in so lid mechanics, i.e., how a plate or shell
model can be deduced from a 3D elasticity one (see e.g. [14]).
The starting point of the method of two-scale asymptotic exp ansion is to assume that the solution
Tǫof problem (8) is given by the series
Tǫ=T0(x)+ǫ T1(x,x′
ǫ)+ǫ2T2(x,x′
ǫ)+O(ǫ3) (23)
where, fori= 1,2,y′→Ti(x,y′) is Λ-periodic and
Ti(x,y′) =/braceleftBigg
TS
i(x,y′) in Ω ×ΛS,
TF
i(x,y′) in Ω ×ΛF,(24)
with the continuity condition at the interface, TS
i(x,y′) =TF
i(x,y′) onγ=∂ΛS∩∂ΛF. As in the
classical examples of homogenization, we assume that the fir st term of the asymptotic expansion T0
depends only on the macroscopic variable x. As usual this property can be established by the same
development as below if we had assumed rather that T0≡T0(x,y′).
Introducing (23) in the equations (8) of the model, we deduce the main result of this section.
Proposition 3.1. Under assumption (23), the zero-order term T0of the expansion for the solution
Tǫof (8) is the solution of the homogenized problem
/braceleftBigg
−div(K∗(x)∇T0(x))+V∗(x)·∇T0(x) =θf(x)inΩ
T0(x) = 0 on∂Ω(25)
with the porosity factor θ=|ΛS|/|Λ|, the homogenized conductivity tensor K∗given by its entries,
forj,k= 1,2,3,
K∗
j,k(x) =1
|Λ|/bracketleftbigg/integraldisplay
ΛSKS(x,y′)(ej+∇y′ωS
j(x,y′))·(ek+∇y′ωS
k(x,y′))dy′
+σ/integraldisplay
γ(Id−ζ2D)(ωS
k(x,y′)+yk)(ωS
j(x,y′)+yj)dy′
+σ
2/integraldisplay
γ/integraldisplay
γF2D(s′,y′)|s′−y′|2dy′ds′δj3δk3/bracketrightbigg
(26)

Homogenization of a Heat Transfer Problem 10
and a vertical homogenized velocity given by
V∗(x) =e3
|Λ|/integraldisplay
ΛFV(x,y′)·e3dy′,
whereζ2Dis the unit cell view factor operator defined by
ζ2D(ω)(s′) =/integraldisplay
γF2D(s′,y′)ω(y′)dy′
and, forj= 1,2,3,ωj(x,y′)(equal toωS
jinΛSand toωF
jinΛF) is the solution of the 2D cell problem


−divy′P/bracketleftBig
KS(x,y′)(ej+∇y′ωS
j(x,y′))/bracketrightBig
= 0 inΛS
−divy′P/bracketleftBig
KF(x,y′)(ej+∇y′ωF
j(x,y′))/bracketrightBig
+V(x,y′)·(ej+∇y′ωF
j(x,y′)) = 0 inΛF
−P/bracketleftBig
KS(x,y′)(ej+∇y′ωS
j(x,y′))/bracketrightBig
·n=σ(Id−ζ2D)(ωS
j(x,y′)+yj)onγ
ωF
j(x,y′) =ωS
j(x,y′)onγ
y′/ma√sto→ωj(x,y′)isΛ-periodic,(27)
wherePis the 3D to 2D projection operator defined by (2). Furthermor e,T1is given by
T1(x,y′) =3/summationdisplay
j=1ωj(x,y′)∂T0
∂xj(x). (28)
Remark 3.1. As usual in homogenization, the cell problem (27) is a partia l differential equation with
respect toy′wherexplays the role of a parameter. It is proved to be well-posed in L emma 3.1 below.
We emphasize that the cell problem in Λcan be decoupled as two successive sub-problems in ΛS
andΛFrespectively. First, we solve a cell problem in ΛSusing the non local boundary condition on
γ, independently of what happens in ΛF. Second, we solve a cell problem in ΛFwith the continuity
boundary condition on γyielding a Dirichlet boundary condition. In particular, the homogenized tensor
K∗depends only on ΛS.
Remark 3.2. The homogenized tensor K∗has an extra contribution (26) for its 3,3entry depending
merely on the view factor and not on the cell solutions. It aris es from the leakage of the radiative
energy at both ends of each cylinder Γǫ,i(which are not closed as explained in Remark 2.4). This
loss of radiative energy at the cylinders extremities yield s this additional axial (or vertical) thermal
diffusion. For circular cross-section cylinders (namely γis a circle), we can explicitly compute
/integraldisplay
γ/integraldisplay
γF2D(s′,y′)|s′−y′|2dy′ds′=16
3πr3whereris the radius of γ.
If the conductivity tensor KShas a block diagonal structure, namely KS
3,j(x) =KS
j,3(x) = 0forj= 1,2,
the cell problem (27) has an obvious solution for j= 3which yields an explicit formula for K∗
3,3. More
precisely, since Pe3= 0and(Id−ζ2D)y3= 0, the solution ωS
3is a constant (with respect to y′) for
any cell geometry. This implies that K∗
3,j(x) =K∗
j,3(x) = 0forj= 1,2and
K∗
3,3(x) =1
|Λ|/parenleftbigg/integraldisplay
ΛSKS
3,3(x,y′)dy′+σ
2/integraldisplay
γ/integraldisplay
γF2D(s′,y′)|s′−y′|2dy′ds′/parenrightbigg
.
Remark 3.3. As usual in homogenization, Proposition 3.1 gives a complet e characterization of the
two first terms T0(x) +ǫT1(x,x′
ǫ)of the ansatz (23). With such an approximation, not only do we
have a correct estimate of the temperature Tǫ(x)but also of its gradient (or of the heat flux) since it

Homogenization of a Heat Transfer Problem 11
implies∇Tǫ(x)≈ ∇T0(x) +∇y′T1(x,x′
ǫ)(in this last formula the corrector ∇y′T1is of order 1 and
can not be ignored).
The proof of Proposition 3.1 shall require the consideration of the second order corrector T2but
we are not interested in its precise evaluation since it is mu ch smaller and negligible in the numerical
examples (see [4] for a more complete analysis of the second o rder corrector).
Proofof Proposition 3.1
All the difficulties are concentrated on the radiation term in which simplifications must necessarily
take placebecauseitisformallydominatingas ǫgoestozero. Consequently, insteadofusingtheformal
method of two scale asymptotic expansions in the strong form of problem (8), which is complicated
because of the non-local boundary condition (the radiation term), we follow the lead of [3] (based
on an original idea of J.-L. Lions [26]) and use a two scale asy mptotic expansion in the variational
formulation of (8), taking advantage of its symmetry. This t rick allows us to truncate the ansatz at a
lower order term and considerably simplifies the computatio ns.
The variational formulation of problem (8) is: find Tǫ∈H1
0(Ωǫ) such that
aǫ(Tǫ,φǫ) =Lǫ(φǫ) for anyφǫ∈H1
0(Ωǫ), (29)
where
aǫ(Tǫ,φǫ) =/integraldisplay
ΩSǫKS
ǫ(x)∇Tǫ(x)·∇φǫ(x)dx+ǫ/integraldisplay
ΩFǫKF
ǫ(x)∇Tǫ(x)·∇φǫ(x)dx
+/integraldisplay
ΩFǫVǫ(x)·∇Tǫ(x)φǫ(x)dx+σ
ǫ/integraldisplay
ΓǫGǫ(Tǫ)(x)φǫ(x)dx
and
Lǫ(φǫ) =/integraldisplay
ΩSǫf(x)φǫ(x)dx.
We choose φǫof the same form as Tǫin (23) (but without remainder term)
φǫ(x) =φ0(x)+ǫ φ1(x,x′
ǫ)+ǫ2φ2(x,x′
ǫ) (30)
with smooth functions φ0(x) andφi(x,y′),i= 1,2, being Λ-periodic in y′and such that
φi(x,y′) =/braceleftBigg
φS
i(x,y′) in Ω ×ΛS,
φF
i(x,y′) in Ω ×ΛF.
We also assume that φ0(x) andφi(x,y′) have compact support in x∈Ω.
Inserting the ansatz (23) and (30) in the variational formul ation (29) yields
a0(T0,T1,φ0,φ1)+ǫa1(T0,T1,T2,φ0,φ1,φ2) =L0(φ0,φ1)+ǫL1(φ0,φ1,φ2)+O(ǫ2).(31)
Thenon-conventional strategy oftheproofisthefollowing : notonlyweidentifythezero-orderterm
a0=L0but we also use the first-order identity a1=L1. The zero-order identity, a0(T0,T1,φ0,φ1) =
L0(φ0,φ1), allows us to find the homogenized problem for T0in Ω and the cell problem for TS
1in
Ω×ΛS. The first-order identity a1(T0,T1,T2,φ0,φ1,φ2) =L1(φ0,φ1,φ2) yields the cell problem for
TF
1in Ω×ΛF. We emphasize that it is crucial, for the identification of th e first-order term, that
the test functions ( φi)i=0,1,2have compact supports. Indeed, in view of Lemma 2.1, the 3D to 2D
asymptotic of the view factor has a sufficiently small remaind er term only for compactly supported
test functions.
For the sake of clarity we divide the proof in three steps. The first step is devoted to the ansatz for
the convection and diffusion terms. The second one focuses on t he radiation term, while the third one

Homogenization of a Heat Transfer Problem 12
combines these various terms to deduce the cell and homogeni zed problems by identifying equations
of the same order in powers of ǫ.
We now give the details of the proof. We rewrite the variation al formulation (29) as
aǫ(Tǫ,φǫ) =aC
ǫ(Tǫ,φǫ)+aRad
ǫ(Tǫ,φǫ) =Lǫ(φǫ)
where
aC
ǫ(Tǫ,φǫ) =/integraldisplay
ΩSǫKS
ǫ(x)∇Tǫ(x)·∇φǫ(x)dx+ǫ/integraldisplay
ΩFǫKF
ǫ(x)∇Tǫ(x)·∇φǫ(x)dx
+/integraldisplay
ΩFǫVǫ(x)·∇Tǫ(x)φǫ(x)dx
aRad
ǫ(Tǫ,φǫ) =σ
ǫN(ǫ)/summationdisplay
i=1/integraldisplay
Γǫ,iGǫ(Tǫ)(x)φǫ(x)dx.(32)
Step 1 : Expansion of aC
ǫ−Lǫ
This is a standard calculation. Plugging the ansatz (23) and (30) we obtain
aC
ǫ−Lǫ=
/integraldisplay
ΩSǫKS
ǫ[(∇xT0+∇y′T1)·(∇xφ0+∇y′φ1)]+/integraldisplay
ΩFǫVǫ·(∇xT0+∇y′T1)φ0
+ǫ/bracketleftBigg/integraldisplay
ΩSǫKS
ǫ/bracketleftbig
(∇xT1+∇y′T2)·(∇xφ0+∇y′φ1)+(∇xφ1+∇y′φ2)·(∇xT0+∇y′T1)/bracketrightbig
+
/integraldisplay
ΩFǫKF
ǫ(∇xT0+∇y′T1)·(∇xφ0+∇y′φ1)+/integraldisplay
ΩFǫVǫ·/bracketleftbig
(∇xT1+∇y′T2)φ0+(∇xT0+∇y′T1)φ1/bracketrightbig/bracketrightBigg
−/integraldisplay
ΩSǫf(φ0+ǫφ1)+O(ǫ2)(33)
where all functions are evaluated at ( x,x′/ǫ). Using Lemma 3.2 below, we deduce
|Λ|(aC
ǫ−Lǫ) =
/integraldisplay
Ω/integraldisplay
ΛSKS(x,y′)(∇xT0(x)+∇y′T1(x,y′)·(∇xφ0(x)+∇y′φ1(x,y′))dy′dx
+/integraldisplay
Ω/integraldisplay
ΛFV(x,y′)·∇xT0(x)φ0(x)dy′dx−/integraldisplay
Ω/integraldisplay
ΛSf(x)φ0(x)dy′dx
+ǫ/bracketleftbigg/integraldisplay
Ω/integraldisplay
ΛSKS(x,y′)/bracketleftbig
(∇xT1(x,y′)+∇y′T2(x,y′))·(∇xφ0(x)+∇y′φ1(x,y′))
+ (∇xφ1(x,y′)+∇y′φ2(x,y′))·(∇xT0(x)+∇y′T1(x,y′))/bracketrightbig
dy′dx
+/integraldisplay
Ω/integraldisplay
ΛFKF(x,y′)(∇xT0(x)+∇y′T1(x,y′))·(∇xφ0(x)+∇y′φ1(x,y′))dy′dx
+/integraldisplay
Ω/integraldisplay
ΛFV(x,y′)·[∇xT1(x,y′)φ0(x)+∇xT0(x)φ1(x,y′)+∇y′T1(x,y′)φ1(x,y′)]dy′dx
−/integraldisplay
Ω/integraldisplay
ΛSf(x)φ1(x,y′)dy′dx/bracketrightbigg
+O(ǫ2).(34)

Homogenization of a Heat Transfer Problem 13
Step 2 : Expansion of aRad
ǫ
A similar expansion in the 2D setting was carried out in [3]. H owever, the present 3D configuration
is different since, the fluid holes being thin long cylinder, th ere is also a 3D to 2D transition (which
did not occur in [3]) taking place. The purpose of this second step is to write a Taylor expansion of
the radiation operator, up to second order,
aRad
ǫ=arad
0+ǫarad
1+O(ǫ2) (35)
Fortunately, as we shall see later, the term arad
1does play any role in the definition of the corrector T1
in ΛF. Therefore, we don’t need to evaluate arad
1which, of course, significantly reduces the amount
of tedious calculations. The radiation term is given by
aRad
ǫ=σ
ǫN(ǫ)/summationdisplay
i=1/bracketleftBigg/integraldisplay
Γǫ,iTǫ(x)φǫ(x)dx−/integraldisplay
Γǫ,i/integraldisplay
Γǫ,iF(x,s)Tǫ(x)φǫ(s)dxds/bracketrightBigg
. (36)
In the ansatz (23) and (30) we make a Taylor expansion around e ach center of mass x′
0,iof each
boundaryγǫ,i. To simplify the notations, we drop the label iand denote by x′
0eachx′
0,i. We also
denote (x′−x′
0) byǫh′and (s′−x′
0) byǫl′. Thus we get
Tǫ(x) =T0(x′
0,x3)+ǫ/parenleftbigg
∇x′T0(x′
0,x3)·h′+T1(x′
0,x′
ǫ,x3)/parenrightbigg
+ǫ2/hatwideT2,ǫ(x)+O(ǫ3) (37)
φǫ(s) =φ0(x′
0,s3)+ǫ/parenleftbigg
∇x′φ0(x′
0,s3)·l′+φ1(x′
0,s′
ǫ,s3)/parenrightbigg
+ǫ2/hatwideφ2,ǫ(s)+O(ǫ3) (38)
where
/hatwideT2,ǫ(x) =1
2∇x′∇x′T0(x′
0,x3)h′·h′+∇x′T1(x′
0,x′
ǫ,x3)·h′+T2(x′
0,x′
ǫ,x3)
/hatwideφ2,ǫ(s) =1
2∇x′∇x′φ0(x′
0,s3)l′·l′+∇x′φ1(x′
0,s′
ǫ,s3)·l′+φ2(x′
0,s′
ǫ,s3)
The precise form of the terms /hatwideT2,ǫand/hatwideφ2,ǫis not important since the O(ǫ2)-order terms will disappear
by simplification as we shall see later. Using (37) and (38), w e obtain
Tǫ(x)φǫ(s) = (Tφ)0(x3,s3)+ǫ(Tφ)1(x,s)+ǫ2(Tφ)2(x,s)+O(ǫ3).
where
(Tφ)0(x3,s3) =φ0(x′
0,s3)T0(x′
0,x3)
(Tφ)1(x,s) =φ0(x′
0,s3)∇x′T0(x′
0,x3)·h′+T0(x′
0,x3)∇x′φ0(x′
0,s3)·l′
+φ0(x′
0,s3)T1(x′
0,x′
ǫ,x3)+φ1(x′
0,s′
ǫ,s3)T0(x′
0,x3)
(Tφ)2(x,s) =φ1(x′
0,s′
ǫ,s3)T1(x′
0,x′
ǫ,x3)+/hatwideT2,ǫφ0(x′
0,s3)+/hatwideφ2,ǫT0(x′
0,x3)
+∇x′φ0(x′
0,s3)·l′∇x′T0(x′
0,s3)·h′+φ1(x′
0,s′
ǫ,s3)∇x′T0(x′
0,s3)·h′
+T1(x′
0,x′
ǫ,x3)∇x′φ0(x′
0,s3)·l′

Homogenization of a Heat Transfer Problem 14
Since the test functions φihave compact support in Ω, we can use formula (16) of Lemma 2.1 (or
formula (22) of Remark 2.5) for the 3D to 2D asymptotic behavi or of the view factor. Thus we deduce
/integraldisplay
Γǫ,i/integraldisplay
Γǫ,i(Tφ)0(x3,s3)F(s,x)dsdx=/integraldisplay
γǫ,i/integraldisplay
γǫ,iF2D(s′,x′)/integraldisplayL3
0(Tφ)0(x3,x3)dx3
−1
2/integraldisplay
γǫ,i/integraldisplay
γǫ,iF2D(s′,x′)|x′−s′|2/integraldisplayL3
0∂φ0
∂x3(x′
0,x3)∂T0
∂x3(x′
0,x3)dx3+|γǫ,i|2O(ǫp−1),
with 0<p<3. Then, since |γǫ,i|=ǫ|γ|,
1
ǫ/parenleftBigg/integraldisplay
Γǫ,i(Tφ)0(x3,x3)dx−/integraldisplay
Γǫ,i/integraldisplay
Γǫ,i(Tφ)0(x3,s3)F(s,x)dsdx/parenrightBigg
=1
2ǫ/integraldisplay
γǫ,i/integraldisplay
γǫ,iF2D(s′,x′)|x′−s′|2/integraldisplayL3
0∂φ0
∂x3(x′
0,x3)∂T0
∂x3(x′
0,x3)dx3+|γ|2O(ǫp).(39)
A similar computation, taking into account the various symm etry properties of the view factor, yields
1
ǫ/parenleftBigg/integraldisplay
Γǫ,iǫ(Tφ)1(x,x)dx−/integraldisplay
Γǫ,i/integraldisplay
Γǫ,iǫ(Tφ)1(x,s)F(s,x)dsdx/parenrightBigg
=O(ǫ3), (40)
and
1
ǫ/parenleftBigg/integraldisplay
Γǫ,iǫ2(Tφ)2(x,x)dx−/integraldisplay
Γǫ,i/integraldisplay
Γǫ,iǫ2(Tφ)2(x,s)F(s,x)dsdx/parenrightBigg
=ǫ/parenleftBigg/integraldisplayL3
0/integraldisplay
γǫ,iφ1(x′
0,s′
ǫ,x3)/bracketleftBigg
T1(x′
0,s′
ǫ,x3)−/integraldisplay
γǫ,iT1(x′
0,x′
ǫ,x3)F2D(s′,x′)dx′/bracketrightBigg
ds′dx3
+/integraldisplayL3
0/integraldisplay
γǫ,i(∇x′φ0(x′
0,x3)·l′)∇x′T0(x′
0,x3)·/bracketleftBigg
l′−/integraldisplay
γǫ,ih′F2D(s′,x′)dx′/bracketrightBigg
ds′dx3
+/integraldisplayL3
0/integraldisplay
γǫ,i∇x′φ0(x′
0,x3)·l′/bracketleftBigg
T1(x′
0,s′
ǫ,x3)−/integraldisplay
γǫ,iT1(x′
0,x′
ǫ,x3)F2D(s′,x′)dx′/bracketrightBigg
ds′dx3
+/integraldisplayL3
0/integraldisplay
γǫ,iφ1(x′
0,s′
ǫ,x3)∇x′T0(x′
0,x3)·/bracketleftBigg
l′−/integraldisplay
γǫ,ih′F2D(s′,x′)dx′/bracketrightBigg
ds′dx3/parenrightBigg
+O(ǫ3).(41)
In (40) and (41), we do not give the explicit form of the remain der terms (including the diffusive term
coming from the 3D to 2D limit in the view factor) which are neg ligible after rescaling and summation
over all cells as soon as they are of order O(ǫq) withq>2.
Thus Lemma 3.2, the changes of variables y′=x′/ǫandz′=s′/ǫin (39), (40), (41) and summing
over all cells, yield
σ
ǫ
N(ǫ)/summationdisplay
i=1/integraldisplay
Γǫ,iTǫ(x)φǫ(x)dx−/integraldisplay
Γǫ,i/integraldisplay
Γǫ,iTǫ(x)φǫ(s)F(s,x)dsdx
=arad
0+O(ǫp−2) (42)

Homogenization of a Heat Transfer Problem 15
with
arad
0=σ
|Λ|/parenleftbigg1
2/integraldisplay
Ω∂φ0
∂x3(x)∂T0
∂x3(x)dx/integraldisplay
γ/integraldisplay
γF2D(z′,y′)|z′−y′|2dy′dz′
+/integraldisplay
Ω∇x′φ0(x)·/integraldisplay
γ/parenleftBig
y′⊗y′−/integraldisplay
γy′⊗z′F2D(z′,y′)dz′/parenrightBig
dy′∇x′T0(x)dx
+/integraldisplay
Ω∇x′T0(x)·/integraldisplay
γφ1(x,y′)/parenleftBig
y′−/integraldisplay
γz′F2D(z′,y′)dz′/parenrightBig
dy′dx
+/integraldisplay
Ω∇x′φ0(x)·/integraldisplay
γT1(x,y′)/parenleftBig
y′−/integraldisplay
γz′F2D(z′,y′)dz′/parenrightBig
dy′dx
+/integraldisplay
Ω/integraldisplay
γ/integraldisplay
γ(δ(y′−z′)−F2D(y′,z′))T1(x,z′)φ1(x,y′)dz′dy′dx/parenrightbigg(43)
whereδis the Dirac mass. Remark that the last term in (43) can also be written
/integraldisplay
γ/integraldisplay
γ(δ(y′−z′)−F2D(y′,z′))T1(x,z′)φ1(x,y′)dz′dy′=/integraldisplay
γφ1(x,y′)/parenleftBig
(Id−ζ2D)T1/parenrightBig
(x,y′)dy′.
Remark 3.4. As already said, in the spirit of our proof we should also comp ute the next order term
arad
1in the asymptotic expansion aRad
ǫ=arad
0+ǫarad
1+O(ǫ2). The computation of arad
1is tedious
and require to carry the expansions of Tǫandφǫto one more order in ǫ, a formidable task that is
not pursued here (similar computations can be found in [4] fo r a 2D-configuration). Fortunately, the
radiation term contributes merely to the boundary conditio n for the cell problem in the solid part ΛS
and does not play any role for the cell problem in the fluid part ΛF. Since the first-order terms a1,L1
are used to deduce the fluid cell problem, it is perfectly legi timate not to compute arad
1.
Step 3 : Identification of the limit variational formulations
The zero-th order ǫ0-term of (31) is
a0(T0,T1,φ0,φ1) =L0(φ0,φ1)
which is equivalent to
/integraldisplay
Ω/integraldisplay
ΛSKS(x,y′)(∇xT0(x)+∇y′T1(x,y′)·(∇xφ0(x)+∇y′φ1(x,y′))dy′dx
+/integraldisplay
Ω/integraldisplay
ΛFV(x,y′)·∇xT0(x)φ0(x)dy′dx
+σ
2/integraldisplay
Ω∂φ0
∂x3(x)∂T0
∂x3(x)dx/integraldisplay
γ/integraldisplay
γF2D(z′,y′)|z′−y′|2dy′dz′
+σ/integraldisplay
Ω∇x′T0(x)·/integraldisplay
γφ1(x,y′)/parenleftBig
y′−/integraldisplay
γz′F2D(z′,y′)dz′/parenrightBig
dy′dx
+σ/integraldisplay
Ω∇x′φ0(x)·/integraldisplay
γT1(x,y′)/parenleftBig
y′−/integraldisplay
γz′F2D(z′,y′)dz′/parenrightBig
dy′dx
+σ/integraldisplay
Ω∇x′φ0(x)·/integraldisplay
γ/parenleftBig
y′⊗y′−/integraldisplay
γy′⊗z′F2D(z′,y′)dz′/parenrightBig
∇x′T0(x)dy′dx
+σ/integraldisplay
Ω/integraldisplay
γ/integraldisplay
γ(δ(y′−z′)−F2D(y′,z′))T1(x,z′)φ1(x,y′)dz′dy′dx
=|ΛS|/integraldisplay
Ωf(x)φ0(x)dx(44)

Homogenization of a Heat Transfer Problem 16
We recognize in (44) the variational formulation of the so-c alled two-scale limit problem which is a
combination of the homogenized and cell problems (in ΛSonly).
We recover the cell problem in ΛSby takingφ0= 0 in the limit of the variational formulation (44)
/integraldisplay
Ω/integraldisplay
ΛSKS(x,y′)(∇xT0(x)+∇y′T1(x,y′))·∇y′φ1(x,y′)dy′dx
+σ/integraldisplay
Ω∇x′T0(x)·/integraldisplay
γφ1(x,y′)(y′−/integraldisplay
γz′F2D(z′,y′)dz′)dy′dx
+σ/integraldisplay
Ω/integraldisplay
γ/integraldisplay
γ(δ(y′−z′)−F2D(y′,z′))T1(x,z′)φ1(x,y′)dz′dy′dx= 0 (45)
The solution T1of the above variational formulation is given by (28) in ΛSwhereωj≡ωS
j(x,y′), for
1≤j≤3, are the solutions of the cell problems in the 2D solid media ΛS


−divy′P/bracketleftbig
KS(x,y′)(ej+∇y′ωS
j(y′))/bracketrightbig
= 0 in ΛS
−P/bracketleftbig
KS(x,y′)(ej+∇y′ωS
j(y′))/bracketrightbig
·n=σ(Id−ζ2D)(ωS
j(y′)+yj) onγ
y′/ma√sto→ωS
j(y′) is Λ- periodic.(46)
Remark 3.5. As already said, the macroscopic variable xplays the role of a parameter in (46).
Therefore, for the sake of notational simplicity we shall oft en forget the dependence on xfor the
solutionsωjof the cell problems.
To recover the homogenized problem we now substitute φ1by 0 in (44). We obtain
/integraldisplay
Ω/integraldisplay
ΛS3/summationdisplay
k,j=1KS(x,y′)(∇y′ωk(y′)+ek)·(∇y′ωj(y′)+ej)∂T0
∂xk(x)∂φ0
∂xj(x)dy′dx
+/integraldisplay
Ω/integraldisplay
ΛF3/summationdisplay
k=1VK(x,y′)∂T0
∂xk(x)φ0(x)dy′dx
+σ
2/integraldisplay
Ω∂φ0
∂x3(x)∂T0
∂x3(x)dx/integraldisplay
γ/integraldisplay
γF2D(z′,y′)|z′−y′|2dy′dz′
+σ/integraldisplay
Ω/integraldisplay
γ3/summationdisplay
k,j=1(Id−ζ2D)(ωk(y′)+yk)(ωj(y′)+yj)∂T0
∂xk(x)∂φ0
∂xj(x)dy′dx
=|ΛS|/integraldisplay
Ωf(x)φ0(x)dx (47)
which is the variational formulation of the homogenized pro blem (25) where K∗andV∗are given by
the formulas of Proposition 3.1.
We now turn to the first order ǫ1-term of (31) which yields the cell problem in ΛF. Indeed, up to
this point, the zero-th order term of (31) has given the cell p roblem in ΛS, as well as the homogenized
problem for T0in the domain Ω. Nonetheless, as we already said in Remark 3.3 , we want to compute
everywhere the corrector T1of the solution Tǫ, not merely in the solid part. Therefore, we look at the
next,ǫ1-order term of (31)
a1(T0,T1,T2,φ0,φ1,φ2) =L1(φ0,φ1,φ2)

Homogenization of a Heat Transfer Problem 17
where we shall keep only the terms coming from the fluid part (t hose coming from the solid part will
contribute to the determination of T2which we do not pursue here). It is equivalent to
/integraldisplay
Ω/integraldisplay
ΛSKS(x,y′)/bracketleftBig/parenleftbig
∇xT1(x,y′)+∇y′T2(x,y′)/parenrightbig
·/parenleftbig
∇xφ0(x)+∇y′φ1(x,y′)/parenrightbig
+/parenleftbig
∇xφ1(x,y′)+∇y′φ2(x,y′)/parenrightbig
·/parenleftbig
∇xT0(x)+∇y′T1(x,y′)/parenrightbig/bracketrightBig
dy′dx
+/integraldisplay
Ω/integraldisplay
ΛFKF(x,y′)/parenleftbig
∇xT0(x)+∇y′T1(x,y′)/parenrightbig
·/parenleftbig
∇xφ0(x)+∇y′φ1(x,y′)/parenrightbig
dy′dx
+/integraldisplay
Ω/integraldisplay
ΛFV(x,y′)·/parenleftbig
∇xT1(x,y′)φ0(x)+∇xT0(x)φ1(x,y′)+∇y′T1(x,y′)φ1(x,y′)/parenrightbig
dy′dx
+arad
1
=/integraldisplay
Ω/integraldisplay
ΛSf(x)φ1(x,y′)dy′dx(48)
Note that, by virtue of Lemma 3.2, the approximation of an int egral on ΩS,F
ǫby a double integral on
Ω×ΛS,Fis of orderǫ2and thus does not interact with the first order ǫ1-term of (31).
In (48) we take φ0≡0 andφ2≡0 everywhere, and φ1= 0 in ΛSonly. It thus becomes the
variational formulation of
/braceleftBigg
−divy′P/bracketleftbig
KF(x,y′)(∇T0+∇y′T1)/bracketrightbig
+V(x,y′)·(∇xT0+∇y′T1) = 0 in ΛF,
T1is continuous through γ.(49)
Therefore, the solution T1of (49) is given by (28) in ΛFwhereωj≡ωF
j(x,y′), for 1≤j≤3, are the
solutions of the cell problems in ΛF


−divy′P/bracketleftbig
KF(x,y′)(ej+∇y′ωF
j(y′))/bracketrightbig
+V(x,y′)·(ej+∇y′ωF
j(y′)) = 0 in ΛF
ωF
j=ωS
jonγ
y′/ma√sto→ωF
j(y′) is Λ-periodic.(50)
Combining (46) and (50), we get (27). /squaresolid
Lemma 3.1. Each of the cell problems (27) admits a unique solution, up to a constant, in H1
#(Λ).
ProofFirst, we recall that each cell problem in Λ is decoupled into two independents cell problems,
(46) in ΛSwith a radiative boundary condition and (50) in ΛFwith a Dirichlet boundary condition.
Forφ∈H1
#(ΛS), the variational formulation of (46) is given by
/integraldisplay
ΛSKS(x,y′)/parenleftbig
∇y′ωS
i(y′)+ei/parenrightbig
·∇y′φ(y′)+σ/integraldisplay
γ(Id−ζ2D)/parenleftbig
ωS
i(y′)+y′
i/parenrightbig
φ(y′) = 0.(51)
Using (11), we deduce that the bilinear form of (51) is coerci ve onH1
#(ΛS)/R
a(φ,φ) =/integraldisplay
ΛSKS∇y′φ·∇y′φ+σ/integraldisplay
γ(Id−ζ2D)φφ≥C/ba∇dbl∇yφ/ba∇dblL2
#(ΛS)≡C/ba∇dblφ/ba∇dblH1
#(ΛS)/R.(52)
Furthermore, since (12) implies that (51) holds true when th e test function φis a constant, the
Fredholm alternative (see [31]) yields existence and uniqu eness inH1
#(ΛS)/R(i.e., up to a constant)
of the cell problem (46) solution.
Theexistenceofauniquesolutionin H1
#(ΛF)ofthefluidcellproblems(50), withanon-homogeneous
Dirichlet boundary condition, is completely standard for t his simple convection-diffusion equation
(note that, for our geometry in Figure 2, the periodic bounda ry condition does not appear in the fluid
cell ΛF). Of course, since ωS
jis defined up to a constant, so is ωF
j, but with the same constant. /squaresolid

Homogenization of a Heat Transfer Problem 18
We recall a classical lemma used in the proof of Proposition 3 .1.
Lemma 3.2. For a smooth function fand any integer p≥0we have
i./integraldisplay
γǫ,if(x′
ǫ)(x′
k−x′
0,k)pdx′=ǫ1+p/integraldisplay
γf(y′)(y′
k−y′
0,k)pdy′
ii. ǫ2N(ǫ)/summationdisplay
i=1/integraldisplayL3
0f(x′
0,i,x3)dx3=1
|Λ|/integraldisplay
Ωf(x)dx+O(ǫ2)
iii./integraldisplay
ΩS,F
ǫf(x,x′
ǫ)dx=1
|Λ|/integraldisplay
Ω/integraldisplay
ΛS,Ff(x,y′)dy′dx+O(ǫ2)
ProofThefirstformulais immediate by asimplechange of variables . For thesecond one, we perform
a Taylor expansion of f(x′) (which is assumed to be C2) aroundx′
0,ithe center of mass of each cell
Λǫ,i
f(x′,x3) =f(x′
0,i,x3)+(x′−x′
0,i)∇x′f(x′
0,i,x3)+O(ǫ2) (53)
which becomes by integration in Λ ǫ,i
/integraldisplay
Λǫ,if(x′,x3)dx′=ǫ2|Λ|f(x′
0,i,x3)+O(ǫ4)
because|Λǫ,i|=ǫ2|Λ|and/integraltext
Λǫ,i(x′−x′
0,i)dx′= 0. Aftersummation, andintegration between0and L3in
x3, we obtain the desired result. The third formula is a consequ ence of the Riemann-Lebesgue lemma
which states that the integral of a periodically oscillatin g smooth function converges to its average
with a convergence speed smaller than any power of ǫ(this is obtained by repeated integration by
parts since any zero-average periodic function of the fast yvariable is the divergence of a zero-average
periodic vector field in y). /squaresolid
4 Convergence
The results of the previous section are only formal. They are based on the assumption that the
temperature Tǫadmits the asymptotic expansion (23). Therefore, to comple te our study, we need a
rigorous mathematical justification of Proposition 3.1. He re, we prove a convergence result using the
two-scale convergence method [1], [30].
4.1 A priori estimates
To use the two-scale convergence method, we first need to esta blish some a priori estimates on the
unknownTǫ.
Proposition 4.1. LetTǫbe the solution of problem (8). There exists a constant C, not de pending on
ǫ, such that
/ba∇dblTǫ/ba∇dblL2(Ω)+/ba∇dbl∇Tǫ/ba∇dblL2(ΩSǫ)+√ǫ/ba∇dbl∇Tǫ/ba∇dblL2(ΩFǫ)+√ǫ/ba∇dblTǫ/ba∇dblL2(Γǫ)≤C (54)
ProofTakingφǫ=Tǫin the variational formulation (29) of (8) we obtain
/integraldisplay
ΩSǫKS
ǫ|∇Tǫ|2dx+ǫ/integraldisplay
ΩFǫKF
ǫ|∇Tǫ|2dx+/integraldisplay
ΩFǫVǫ·∇TǫTǫdx+σ
ǫ/integraldisplay
ΓǫGǫ(Tǫ)Tǫds=/integraldisplay
ΩSǫfTǫdx.(55)

Homogenization of a Heat Transfer Problem 19
Since divVǫ= 0 in ΩF
ǫ,Vǫ·n= 0 on Γ ǫandTǫ= 0 on∂Ω, we have
/integraldisplay
ΩFǫVǫ·∇TǫTǫdx= 0.
Furthermore, since Gǫis a positive operator (see Lemma 2.1)
/integraldisplay
ΓǫGǫ(Tǫ)Tǫds≥0.
Consequently, by the coercivity of Kǫ, we obtain
/ba∇dbl∇Tǫ/ba∇dbl2
L2(ΩSǫ)+ǫ/ba∇dbl∇Tǫ/ba∇dbl2
L2(ΩFǫ)≤C/ba∇dblTǫ/ba∇dblL2(ΩSǫ). (56)
Using Lemma 4.1 we deduce
/ba∇dbl∇Tǫ/ba∇dblL2(ΩSǫ)≤C. (57)
On the other hand, using Lemma 4.3 and formula (56) yields
/ba∇dblTǫ/ba∇dbl2
L2(Ω)≤C/bracketleftBig
/ba∇dblTǫ/ba∇dbl2
L2(ΩSǫ)+ǫ2/ba∇dbl∇Tǫ/ba∇dbl2
L2(Ω)/bracketrightBig
≤C/bracketleftBig
1+ǫ/ba∇dbl∇Tǫ/ba∇dbl2
L2(ΩFǫ)/bracketrightBig
≤C/bracketleftbig
1+/ba∇dblTǫ/ba∇dblL2(Ω)/bracketrightbig
(sinceǫ<1) from which we deduce
/ba∇dblTǫ/ba∇dblL2(Ω)≤C. (58)
By (58), and using (56) again, we get
√ǫ/ba∇dbl∇Tǫ/ba∇dblL2(ΩFǫ)≤C (59)
Using (58) and (57) and Lemma 4.2 we deduce
√ǫ/ba∇dblTǫ/ba∇dblL2(Γǫ)≤C. (60)
Combining (57), (58), (59) and (60) we obtain the desired a pr iori estimate (54). /squaresolid
Lemma 4.1. (see Lemma A.4 in [5]) There exists a constant C >0, not depending on ǫ, such that
for any function u∈H1(ΩS
ǫ)satisfyingu= 0on∂Ω∩∂ΩS
ǫ
/ba∇dblu/ba∇dblL2(ΩSǫ)≤C/ba∇dbl∇u/ba∇dblL2(ΩSǫ).
Lemma 4.2. (see Lemma 4.2.4 in [3]) There exists a constant C >0, not depending on ǫ, such that
√ǫ/ba∇dblu/ba∇dblL2(Γǫ)≤C/ba∇dblu/ba∇dblH1(ΩSǫ)∀u∈H1(ΩS
ǫ). (61)
Lemma 4.3. There exists a constant C >0, not depending on ǫ, such that
/ba∇dblu/ba∇dblL2(ΩFǫ)≤C/bracketleftBig
/ba∇dblu/ba∇dblL2(ΩSǫ)+ǫ/ba∇dbl∇u/ba∇dblL2(Ωǫ)/bracketrightBig
∀u∈H1(Ω). (62)

Homogenization of a Heat Transfer Problem 20
ProofThe proof of Lemma 4.3 is similar to those of the previous lemm as so we content ourselves in
briefly sketching it. We denote by Y= Λ×(0,1) a 3D unit cell and similarly YF,S= ΛF,S×(0,1).
By an obvious rescaling and summation argument, it is enough to prove that there exists a constant
C, not depending on ǫ, such that
/ba∇dblu/ba∇dbl2
L2(YF)≤C/bracketleftBig
/ba∇dblu/ba∇dbl2
L2(YS)+/ba∇dbl∇u/ba∇dbl2
L2(Y)/bracketrightBig
∀u∈H1(Y). (63)
We prove (63) by contradiction. Indeed, we suppose that it do es not hold true, namely there exists a
sequenceφn∈H1(Y), forn≥1, such that
/ba∇dblφn/ba∇dblL2(YF)= 1 and /ba∇dblφn/ba∇dbl2
L2(YS)+/ba∇dbl∇φn/ba∇dbl2
L2(Y)<1
n. (64)
Up to a subsequence, φnconverges weakly in H1(Y) to a limit φ, and by Rellich theorem this conver-
gence is strong in L2(Y). However, (64) tells us that ∇φnconverges strongly to 0 in L2(Y). Therefore,
∇φ= 0 andφis constant in Y. Once again, (64) implies that this constant is zero in YSbut this is
a contradiction with the fact that /ba∇dblφ/ba∇dblL2(YF)= limn/ba∇dblφn/ba∇dblL2(YF)= 1. /squaresolid
4.2 Two scale convergence
In this section we first recall the notion of two-scale conver gence [1], [30]. Here, since there is no
periodicityinthethirdspacedirection, weslightlymodif ythedefinitionoftwo-scale convergence(these
changes do not affect the proofs in any essential way). Second, we prove a rigorous homogenization
result, using the two-scale convergence method, to confirm t he result obtained in the previous section.
Definition 4.1. A bounded sequence uǫinL2(Ω)is said to two-scale converge to a function u0(x,y′)∈
L2(Ω×Λ)if there exists a subsequence still denoted by uǫsuch that
lim
ǫ→0/integraldisplay
Ωuǫ(x)ψ(x,x′
ǫ)dx=1
|Λ|/integraldisplay
Ω/integraldisplay
Λu0(x,y′)ψ(x,y′)dxdy′(65)
for anyΛ-periodic test function ψ(x,y′)∈L2(Ω;C#(Λ)).
This notion of ”two-scale convergence” makes sense because of the next compactness theorem [1],
[30].
Theorem 4.1. From each bounded sequence uǫinL2(Ω), we can extract a subsequence and there
exists a limit u0(x,y′)∈L2(Ω×Λ)such that this subsequence two-scale converges to u0.
The extension of Theorem 4.1 to bounded sequences in H1(Ω) is given next.
Proposition 4.2. From each bounded sequence uǫinH1(Ω), we can extract a subsequence and there
exist two limits u0∈H1(Ω)andu1(x,y′)∈L2(Ω;H1
#(Λ))such that, for this subsequence, uǫconverges
weakly tou0inH1(Ω)and∇uǫtwo-scale converges to ∇xu0(x)+∇y′u1(x,y′).
Two-scale convergence can be extended to sequences defined o n periodic surfaces [2], [29].
Proposition 4.3. For any sequence uǫinL2(Γǫ)such that
ǫ/integraldisplay
Γǫ|uǫ|2dx≤C, (66)
there exist a subsequence, still denoted uǫ, and a limit function u0(x,y′)∈L2(Ω;L2
#(γ))such thatuǫ
two-scale converges to u0in the sense
lim
ǫ→0ǫ/integraldisplay
Γǫuǫ(x)ψ(x,x′
ǫ)dx=1
|Λ|/integraldisplay
Ω/integraldisplay
γu0(x,y′)ψ(x,y′)dxdy′(67)
for anyΛ-periodic test function ψ(x,y′)∈L2(Ω;C#(γ)).

Homogenization of a Heat Transfer Problem 21
Remark 4.1. Ifuǫis a bounded sequence in H1(Ωǫ), then the uniform bound (66) holds true. It is
then easy to check that the two different two-scale limits u0given by Propositions 4.2 and 4.3 coincide
[2].
Our main results in this section is the following.
Theorem 4.2. LetTǫbe the sequence of solutions of (8). Let T0(x)be the solution of the homogenized
problem (25) and T1(x,y′)be the first corrector defined by (28). Then Tǫtwo-scale converges to T0
andχS
ǫ∇Tǫtwo-scale converges to χS(y′)(∇xT0(x) +∇y′T1(x,y′))whereχS
ǫ(x) =χS(x′/ǫ)is the
characteristic function of ΩS
ǫandχS(y′)that ofΛS.
ProofThe a priori estimate (54) implies that, up to a subsequence, Tǫtwo-scale converges to a
functionT0∈H1
0(Ω) andχS
ǫ∇Tǫtwo-scale converges to χS(y′)(∇xT0(x) +∇yT1(x,y′)) whereT1∈
L2(Ω;H1
#(Λ)). Furthermore, Tǫtwo-scale converges to T0on the periodic surface Γ ǫ, in the sense of
Proposition 4.3.
Although we use the same notations, we still have to show that T0is a solution of the homogenized
problem (25) and that T1is the first corrector defined by (28). Convergence of the enti re sequence
(and not merely of an extracted subsequence) will follow fro m the uniqueness of the solution of (25).
In a first step, we compute the corrector T1in terms of ∇xT0by choosing the test function
φǫ(x) =ǫφ1(x,x′
ǫ), whereφ1(x,y′) is any smooth function, compactly supported in xand Λ-periodic
iny′, in the variational formulation (29) which becomes (using t he self-adjoint character of Gǫ)
/integraldisplay
ΩSǫKS
ǫ∇Tǫ·∇y′φ1+σ/integraldisplay
ΓǫTǫGǫ(φ1) =o(1) (68)
where, thanks to the a priori estimate (54), o(1) is a small remainder term going to 0 with ǫ. By virtue
of a lower order truncation of formula (15) in Lemma 2.1, the r adiative operator can be approximated
as
Gǫ(φ1) = (Id−ζǫ)(φ1) = (Id−ζ2D
ǫ)(φ1)+O(ǫ2).
Then, to pass to the two-scale limit in the radiative term, we rely on Lemma 4.4 below which gives
us a smooth periodic vector-valued function θ(x,y′) such that


−divy′θ(x,y′) = 0 in ΛS,
θ(x,y′)·n= (Id−ζ2D)φ1(x,y′) onγ,
y′→θ(x,y′) is Λ-periodic.
Furthermore, θ(x,y′) has the same compact support than φ1(x,y′) with respect to x∈Ω. However,
sinceζ2D
ǫis an integral operator, we usually have a difference between t he two terms below
ζ2D
ǫ/parenleftbigg
φ1(x,x′
ǫ)/parenrightbigg
/\egatio\slash=/parenleftBig
ζ2D(φ1(x,y′))/parenrightBig
(y′=x′
ǫ).
Therefore, we need to use a Taylor expansion of φ1
φ1(x,x′
ǫ) =φ1(x0,i,x′
ǫ)+(x′−x′
0,i)·∇x′φ1(x0,i,x′
ǫ)+O(ǫ2),
wherex0,i= (x′
0,i,x3) andx′
0,iis the center of mass of each boundary γǫ,i, defined by (1). Then, the
following equality holds true
ζ2D
ǫ/parenleftbigg
φ1(x,x′
ǫ)/parenrightbigg
=/parenleftBig
ζ2D(φ1(x0,i,y′))/parenrightBig
(y′=x′
ǫ)+ǫ/parenleftBig
ζ2D(y′·∇x′φ1(x0,i,y′))/parenrightBig
(y′=x′
ǫ)+O(ǫ2).
Then, we can rewrite the radiative term in (68) as
σ/integraldisplay
ΓǫTǫGǫ(φ1) =σ/integraldisplay
ΓǫTǫθ(x0,i,x′
ǫ)·n+σǫ/integraldisplay
ΓǫTǫ/parenleftBig
(Id−ζ2D)(y′·∇x′φ1(x0,i,y′))/parenrightBig
(y′=x′
ǫ)+O(ǫ).(69)

Homogenization of a Heat Transfer Problem 22
We can pass to the two-scale limit in the second term in the rig ht hand side of (69) by applying
Proposition 4.3 (replacing y′·∇x′φ1(x0,i,y′) by the suitable two-scale test function y′·∇x′φ1(x,y′) up
to another O(ǫ) error). For the first term, we use a similar Taylor expansion forθ
σ/integraldisplay
ΓǫTǫθ(x0,i,x′
ǫ)·n=σ/integraldisplay
ΓǫTǫθ(x,x′
ǫ)·n−ǫσ/integraldisplay
ΓǫTǫ/parenleftBig
y′·∇x′θ(x,y′)/parenrightBig
(y′=x′
ǫ)·n+O(ǫ)
=σ/integraldisplay
ΩSǫdiv/parenleftbigg
Tǫθ(x,x′
ǫ/parenrightbigg
−ǫσ/integraldisplay
ΓǫTǫ/parenleftBig
y′·∇x′θ(x,y′)/parenrightBig
(y′=x′
ǫ)·n+O(ǫ).(70)
For the second integral in (70) we can pass to the two-scale li mit by another application of Proposition
4.3. Concerning the first integral, we develop
div/parenleftbigg
Tǫ(x)θ(x,x′
ǫ)/parenrightbigg
=∇Tǫ(x)·θ(x,x′
ǫ)+Tǫ(x)(divxθ)(x,x′
ǫ),
and we can pass to the two-scale limit, thanks to Proposition 4.2. All in all, after some integration by
parts, and recalling that ker(Id−ζ2D
ǫ) =R, we get
lim
ǫ→0σ/integraldisplay
ΓǫTǫGǫ(φ1) =σ
|Λ|/integraldisplay
Ω/integraldisplay
ΛS/parenleftBig
θ(x,y′)·(∇T0(x)+∇y′T1(x,y′))+T0(x)divxθ(x,y′)/parenrightBig
dy′dx
+σ
|Λ|/integraldisplay
Ω/integraldisplay
γT0(x)/parenleftBig
(Id−ζ2D)(y′·∇x′φ1(x,y′))−y′·∇x′θ(x,y′)/parenrightBig
dy′dx
=σ
|Λ|/integraldisplay
Ω/integraldisplay
γθ(x,y′)·n/parenleftBig
T1(x,y′)+y′·∇x′T0(x)/parenrightBig
dy′dx
=σ
|Λ|/integraldisplay
Ω/integraldisplay
γ(Id−ζ2D)(φ1(x,y′))/parenleftBig
T1(x,y′)+y′·∇x′T0(x)/parenrightBig
dy′dx.
Therefore, the two-scale limit of (68) is
/integraldisplay
Ω/integraldisplay
ΛSKS(x,y′)(∇xT0(x)+∇y′T1(x,y′))·∇y′φ1(x,y′)dy′dx
+σ/integraldisplay
Ω/integraldisplay
γ(Id−ζ2D)φ1(x,y′)/parenleftBig
T1(x,y′)+y′·∇x′T0(x)/parenrightBig
dy′dx= 0
which is precisely the variational formulation (45) for T1. Therefore, we have proved that
T1(x,y′) =3/summationdisplay
k=1∂T0
∂xk(x)ωk(y′) in Ω ×ΛS.
Remark that our convergence proof does not justify formula ( 28) forT1(x,y′) in the fluid part Ω ×ΛF.
Remark 4.2. The first step of our proof (which gives formula (28) for T1(x,y′)) was missing in the
proof of Theorem 4.6 in [3]. Our above argument works also in th e simpler 2D setting of [3] and is
thus filling this gap.
In a second step, we recover the homogenized problem for T0by choosing another test function
φǫ(x) in the variational formulation (29) given by
φǫ(x) =φ0(x)+ǫφ1(x,x′
ǫ) withφ1(x,y′) =3/summationdisplay
k=1∂φ0
∂xk(x)ωk(y′)

Homogenization of a Heat Transfer Problem 23
whereφ0∈C∞
c(Ω) andωjare solutions of the cell problems (27). The variational for mulation (29)
becomes/integraldisplay
ΩSǫKS
ǫ∇Tǫ·(∇xφ0+∇y′φ1)+ǫ/integraldisplay
ΩFǫKF
ǫ∇Tǫ·(∇φ0+∇y′φ1)
+/integraldisplay
ΩFǫVǫ·∇Tǫ(φ0+ǫφ1)+σ
ǫ/integraldisplay
ΓǫGǫ(Tǫ)(φ0+ǫφ1) =/integraldisplay
ΩSǫfφ0+o(1) (71)
whereo(1) is a small remainder term going to 0 with ǫ. Passing to the two-scale limit in all terms,
except the radiative one, is standard (see [1] if necessary) . Therefore, we focus only on the radiative
termσ
ǫ/integraldisplay
ΓǫGǫ(Tǫ)(φ0+ǫφ1) =σ
ǫ/integraldisplay
ΓǫTǫGǫ(φ0+ǫφ1) (72)
for which we generalize an argument of [3]. We write a Taylor e xpansion of φǫ, with respect to
the macroscopic variable only, around the center of mass x0,i= (x′
0,i,x3) of each boundary γǫ,i(for
conciseness we drop the index iin the sequel)
φǫ(x) =φ0(x0)+∇x′φ0(x0)·(x′−x′
0)+ǫφ1(x0,x′
ǫ)+ǫ∇x′φ1(x0,x′
ǫ)·(x′−x′
0)
+1
2∇x′∇x′φ0(x0)(x′−x′
0)·(x′−x′
0)+O(ǫ3).
We go up to second order in this Taylor expansion since, upon d ividing byǫas in (72) and summing
over all boundaries γǫ,i, they will have a non-zero limit according to Proposition 4. 3. Recall that the
solution of the cell problem ωS
3in thex3direction is a constant in ΛS(see Remark 3.2): we can choose
this constant to be zero so that ωS
3(y′) = 0 in ΛSand thus on the boundary γtoo. Therefore, in the
boundary integral (72) we can write that the test function φ1is just
φ1(x,y′) =2/summationdisplay
k=1∂φ0
∂xk(x)ωk(y′) onγ,
without any contribution in the x3direction. Thus, the radiation term is given by
1
ǫGǫ(φ0+ǫφ1)(x) =1
ǫ(Id−ζǫ)(φ0+ǫφ1)(x) =ǫ/parenleftBig
ψ0,ǫ(x)+ψ1,ǫ(x)+ψ2,ǫ(x)+O(ǫ)/parenrightBig
where
ψ0,ǫ(x) =1
ǫ2(Id−ζǫ)/bracketleftbig
φ0(x0)/bracketrightbig
ψ1,ǫ(x) =1
ǫ2/summationdisplay
k=1(Id−ζǫ)/bracketleftbigg/parenleftbigg
ωk(x′
ǫ)+xk−x0,k
ǫ/parenrightbigg∂φ0
∂xk(x0)/bracketrightbigg
(73)
ψ2,ǫ(x) = (Id−ζǫ)/bracketleftbigg1
2∇x′∇x′φ0(x0)·(x′−x′
0)
ǫ⊗(x′−x′
0)
ǫ
+2/summationdisplay
k=1∇x′∂φ0
∂xk(x0)·/parenleftbigg(x′−x′
0)
ǫωk(x′
ǫ)/parenrightbigg/bracketrightBigg
.
In (73), when the integral operator ζǫis applied to a function depending on x0= (x′
0,x3), it is meant
thatx′
0is constant (for all points on Γ ǫ,iwhilex3is varying. In other words, for a given function g,
we have
ζǫ/bracketleftbig
g(x0)/bracketrightbig
(s) =/integraldisplay
Γǫ,iF(s,x)g(x′
0,x3)dx.

Homogenization of a Heat Transfer Problem 24
Remark 4.3. At this point, for simplicity we assume that the periodic diff usion coefficients KS,Fand
the velocity Vdo not depend on x. Otherwise, this would add further terms in (73) correspond ing to
thexderivatives of the cell solutions ωk. Our arguments would still work but we prefer to simplify the
exposition.
The termψ0,ǫis new compared to the 2D setting in [3] (where it was vanishin g). Furthermore, the
main additional difficulty with respect to [3] is that we need t o approximate the 3D view factor in ζǫ
by the 2D view factor which is appearing in the homogenized li mit. For this goal we rely on Lemma
2.1. First, by virtue of (15), for any 0 <p<3 we have
ψ0,ǫ(x) =−1
2ǫ2∂2φ0
∂x2
3(x0)/integraldisplay
γǫ,iF2D(s′,x′)|x′−s′|2ds′+O(ǫp−2).
Second, by a lower order truncation of (15), and since ωkdoes not depend on x3,
ψ1,ǫ(x) =1
ǫ2/summationdisplay
k=1∂φ0
∂xk(x0)(Id−ζ2D
ǫ)/parenleftbigg
ωk(x′
ǫ)+xk−x0,k
ǫ/parenrightbigg
+O(ǫ)
and
ψ2,ǫ(x) =1
2∇x′∇x′φ0(x0)·(Id−ζ2D
ǫ)/parenleftbigg(x′−x′
0)
ǫ⊗(x′−x′
0)
ǫ/parenrightbigg
+2/summationdisplay
k=1∇x′∂φ0
∂xk(x0)·(Id−ζ2D
ǫ)/parenleftbigg(x′−x′
0)
ǫωk(x′
ǫ)/parenrightbigg
+O(ǫ2).
In order to recover continuous functions, we use the followi ng Taylor expansions
∂φ0
∂xk(x0) =∂φ0
∂xk(x)−∇x′∂φ0
∂xk(x)·(x′−x′
0)+O(ǫ2),
∂2φ0
∂x2
3(x0) =∂2φ0
∂x2
3(x)+O(ǫ).
We get
ψ0,ǫ(x) =−1
2∂2φ0
∂x2
3(x)/integraldisplay
γǫ,iF2D(s′,x′)|x′−s′|2
ǫ2ds′+O(ǫp−2),
ψ1,ǫ(x) =1
ǫ2/summationdisplay
k=1/parenleftbigg∂φ0
∂xk(x)−∇x′∂φ0
∂xk(x)·(x′−x′
0)/parenrightbigg
(Id−ζ2D
ǫ)/parenleftbigg
ωk(x′
ǫ)+xk−x0,k
ǫ/parenrightbigg
+O(ǫ),
ψ2,ǫ(x) =1
2∇x′∇x′φ0(x)·(Id−ζ2D
ǫ)/parenleftbigg(x′−x′
0)
ǫ⊗(x′−x′
0)
ǫ/parenrightbigg
+2/summationdisplay
k=1∇x′∂φ0
∂xk(x)·(Id−ζ2D
ǫ)/parenleftbigg(x′−x′
0)
ǫωk(x′
ǫ)/parenrightbigg
+O(ǫ).
The leading term of ψ0,ǫ(x) is precisely an oscillating test function for two-scale co nvergence
ψ0,ǫ(x) =ψ0/parenleftbigg
x,x′
ǫ/parenrightbigg
+O(ǫp−2) withψ0(x,y′) =−1
2∂2φ0
∂x2
3(x)/integraldisplay
γF2D(z′,y′)|y′−z′|2dz′.(74)

Homogenization of a Heat Transfer Problem 25
The same is true for ψ2,ǫ(x) which is also an oscillating test function for two-scale co nvergence
ψ2,ǫ(x) =ψ2/parenleftbigg
x,x′
ǫ/parenrightbigg
+O(ǫ)
with
ψ2(x,y′) =1
2∇x′∇x′φ0(x)·(Id−ζ2D)/parenleftbig
y′⊗y′/parenrightbig
+2/summationdisplay
k=1∇x′∂φ0
∂xk(x)·(Id−ζ2D)/parenleftbig
y′ωk(y′)/parenrightbig
.
Rewriting the radiative term (72) as
σ
ǫ/integraldisplay
ΓǫTǫGǫ(φ0+ǫφ1) =σǫ/integraldisplay
ΓǫTǫ/parenleftBig
ψ0,ǫ(x)+ψ1,ǫ(x)+ψ2,ǫ(x)+O(ǫ)/parenrightBig
, (75)
wecanpasstothetwo-scale limitinthefirstandthirdtermin therighthandsideof(75) byapplication
of Proposition 4.3. We obtain
lim
ǫ→0σǫN(ǫ)/summationdisplay
i=1/integraldisplay
Γǫ,iψ0,ǫTǫ=−σ
2|Λ|/integraldisplay
ΩT0(x)∂2φ0
∂x2
3(x)/integraldisplay
γ/integraldisplay
γF2D(y′,z′)|y′−z′|2dy′dz′dx. (76)
and
lim
ǫ→0σǫN(ǫ)/summationdisplay
i=1/integraldisplay
Γǫ,iψ2,ǫTǫ=σ
|Λ|/integraldisplay
ΩT0(x)/integraldisplay
γψ2(x,y′)dy′dx= 0 (77)
because, by the second property of Lemma 2.1, we have/integraltext
γψ2(x,y′)dy′= 0.
It remains to pass to the limit in the second term of (75) invol vingψ1,ǫ. Following [3] we use
the classical trick of H-convergence [28] which amounts to make a comparison with th e variational
formulationofthecellproblems(46) withthetestfunction Tǫ∂φ0
∂xk(recallthat φ0hascompactsupport).
From (46), after rescaling and integration with respect to x3, we obtain for k= 1,2
σ/integraldisplay
Γǫ(Id−ζ2D
ǫ)/parenleftbigg
ωk(x′
ǫ)+xk−x0,k
ǫ/parenrightbigg/parenleftbigg
Tǫ∂φ0
∂xk/parenrightbigg
=−/integraldisplay
ΩSǫKS
ǫ/parenleftbigg
∇y′ωk(x′
ǫ)+ek/parenrightbigg
·∇x′/parenleftbigg
Tǫ∂φ0
∂xk/parenrightbigg
,
which implies
ǫσ/integraldisplay
Γǫψ1,ǫ(x)Tǫ(x) =−2/summationdisplay
k=1/integraldisplay
ΩSǫKS
ǫ/parenleftbigg
∇y′ωk(x′
ǫ)+ek/parenrightbigg
·∇x′/parenleftbigg
Tǫ∂φ0
∂xk/parenrightbigg
(78)
−σǫ2/summationdisplay
k=1/integraldisplay
Γǫ(Id−ζ2D
ǫ)/parenleftbigg
ωk(x′
ǫ)+xk−x0,k
ǫ/parenrightbigg/parenleftbigg
∇x′∂φ
∂xk(x)·x′−x′
0
ǫ/parenrightbigg
Tǫ.
It is now possible to pass to the two-scale limit in the right h and side of (78) and, summing up those

Homogenization of a Heat Transfer Problem 26
limits, we deduce
lim
ǫ→0σ
ǫ/integraldisplay
ΓǫGǫ(Tǫ)(φ0+ǫφ1)
=−σ
2|Λ|/integraldisplay
ΩT0(x)∂2φ0
∂x2
3(x)/integraldisplay
γ/integraldisplay
γF2D(y′,z′)|y′−z′|2dy′dz′dx
−1
|Λ|2/summationdisplay
k=1/integraldisplay
Ω/integraldisplay
ΛSKS(ek+∇y′ωk(y′))·/parenleftbigg
∇x′(T0∂φ0
∂xk)(x)+∂φ0
∂xk(x)∇y′T1(x,y′)/parenrightbigg
dy′dx
−σ
|Λ|2/summationdisplay
k=1/integraldisplay
Ω/integraldisplay
γ(Id−ζ2D)(ωk+yk)y′·∇x′∂φ0
∂xkT0dy′dx
=−σ
2|Λ|/integraldisplay
ΩT0(x)∂2φ0
∂x2
3(x)/integraldisplay
γ/integraldisplay
γF2D(y′,z′)|y′−z′|2dy′dz′dx
−1
|Λ|/integraldisplay
Ω/integraldisplay
ΛSKS(∇x′T0+∇y′T1)·(∇x′φ0+∇y′φ1)dy′dx−/integraldisplay
ΩK∗T0∇x′∇x′φ0dx.
=/integraldisplay
ΩK∗(x)∇xT0(x)·∇xφ0(x)dx−1
|Λ|/integraldisplay
Ω/integraldisplay
ΛSKS(∇xT0+∇y′T1)·(∇xφ0+∇y′φ1)dy′dx(79)
Thetwo last equalities in(79) holdtruethanksto thefollow ing equivalent formulafor thehomogenized
conductivity
K∗
j,k(x) =1
|Λ|/bracketleftbigg/integraldisplay
ΛSKS(x,y′)(ej+∇y′ωS
j(x,y′))·ekdy′+σ/integraldisplay
γ(Id−ζ2D)(ωS
j(x,y′)+yj)ykdy′/bracketrightbigg
which is obtained by a combination of (26) and of the variatio nal formulation of the cell problems.
The two-scale limits of the other terms in the variational fo rmulation (71) are easily obtained
lim
ǫ→0/integraldisplay
ΩSǫKS
ǫ∇Tǫ·(∇xφ0+∇y′φ1)dx=1
|Λ|/integraldisplay
Ω/integraldisplay
ΛSKS(∇xT0+∇y′T1)·(∇xφ0+∇y′φ1)dy′dx,
lim
ǫ→0ǫ/integraldisplay
ΩFǫKF
ǫ∇Tǫ·(∇xφ0+∇y′φ1)dx= 0,
lim
ǫ→0/integraldisplay
ΩFǫVǫ·∇Tǫ(φ0+ǫφ1)dx =−lim
ǫ→0/integraldisplay
ΩFǫTǫVǫ·(∇xφ0+∇y′φ1)dx
=−1
|Λ|/integraldisplay
Ω/integraldisplay
ΛFT0V(y′,x)·(∇xφ0+∇y′φ1)dy′dx
=1
|Λ|/integraldisplay
Ω/integraldisplay
ΛFV(y′,x)·∇xT0φ0dy′dx,
by integration by parts and our assumptions on the velocity V. Summingup all those terms we deduce
that the limit of the variational formulation (71) is, up to s ome integration by parts, the variational
formulation (47) of the homogenized problem. /squaresolid
Remark 4.4. In the course of the proof of Theorem 4.2 we use in an essential wa y the fact that the
boundary condition on ∂Ωis of Dirichlet type. For example, it was crucial that the tes t function had
a compact support (at least in x3) to apply Lemma 2.1 on the 3D to 2D reduction of the view factor .
We do not know if the convergence proof can be extended to the c ase of Neumann boundary conditions.
We now state and prove a technical result which was required i n the previous proof.

Homogenization of a Heat Transfer Problem 27
Lemma 4.4. Letφ1(x,y′)be a smooth function, compactly supported in x∈ΩandΛ-periodic in y′.
There exists at least one smooth vector-valued function θ(x,y′)(with values in R2) such that


−divy′θ(x,y′) = 0 inΛS,
θ(x,y′)·n= (Id−ζ2D)φ1(x,y′) onγ,
y′→θ(x,y′)isΛ-periodic.(80)
ProofIt is enough to look for a solution under the form θ(x,y′) =∇y′η(x,y′). To solve the 2D
elliptic equation for η(inH1
#(ΛS)/R), corresponding to (80), we just have to check the compatibi lity
condition of the data (or Fredholm alternative). By virtue o f the second property of Lemma 2.1 we
can check that, indeed,
/integraldisplay
γ(Id−ζ2D)φ1(x,y′)dy′=/integraldisplay
γφ1(x,y′)dy′−/integraldisplay
γ/integraldisplay
γF2D(y′,s′)φ1(x,y′)dy′ds′= 0.
There is no uniqueness of the solution θ(x,y′) to which we can add any solenoidal field with zero
normal trace. /squaresolid
4.3 Strong convergence
Our main result, Theorem 4.2, gives only a weak convergence ( or two-scale convergence) of the se-
quencesTǫand∇Tǫ. The goal of our next result is to improve this weak convergen ce into a strong
one. As usual in homogenization theory it requires some addi tional smoothness assumptions. More
precisely, we need T1(x,x′/ǫ) to belong to the space H1(Ω) (but not to be uniformly bounded). This
is true, of course, if T1(x,y′) is a smooth function of ( x,y′). In view of formula (28) for T1, it is enough
that either the homogenized solution T0(x) or the cell solutions ωk(y′) be smooth. To establish our
strong convergence result we rely on the usual energy conver gence trick (as described in [1] in the
context of two-scale convergence) which is inspired from th e notion of Γ-convergence [17].
Theorem 4.3. Assuming that T1(x,y′)is smooth enough and denoting by χS
ǫthe characteristic func-
tion of the solid part ΩS
ǫ, the sequence/parenleftbigg
∇Tǫ(x)−∇T0(x)−∇y′T1(x,x′
ǫ)/parenrightbigg
χS
ǫconverges strongly to
zero inL2(Ω)dand the sequence (Tǫ(x)−T0(x))converges strongly to zero in L2(Ω).
ProofWe develop the ”energy” of the difference Tǫ(x)−T0(x)−ǫT1(x,x′
ǫ) and we get, using the
energy equality (55)
/integraldisplay
ΩSǫKS
ǫ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇Tǫ(x)−∇T0(x)−∇y′T1(x,x′
ǫ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
dx+ǫ/integraldisplay
ΩFǫKF
ǫ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇Tǫ(x)−∇T0(x)−∇y′T1(x,x′
ǫ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
dx
+σ
ǫ/integraldisplay
ΓǫGǫ/parenleftbigg
Tǫ(x)−T0(x)−ǫT1(x,x′
ǫ)/parenrightbigg/parenleftbigg
Tǫ(x)−T0(x)−ǫT1(x,x′
ǫ)/parenrightbigg
dx=/integraldisplay
ΩSǫfǫ(x)Tǫ(x)dx
+/integraldisplay
ΩSǫKS
ǫ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇T0(x)+∇y′T1(x,x′
ǫ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
dx+ǫ/integraldisplay
ΩFǫKF
ǫ/vextendsingle/vextendsingle/vextendsingle/vextendsingle∇T0(x)+∇y′T1(x,x′
ǫ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2
dx
+σ
ǫ/integraldisplay
ΓǫGǫ/parenleftbigg
T0(x)+ǫT1(x,x′
ǫ)/parenrightbigg/parenleftbigg
T0(x)+ǫT1(x,x′
ǫ)/parenrightbigg
dx−2σ
ǫ/integraldisplay
ΓǫGǫ/parenleftbigg
T0(x)+ǫT1(x,x′
ǫ)/parenrightbigg
Tǫ(x)dx
−2/integraldisplay
ΩSǫKS
ǫ∇Tǫ(x)·/parenleftBig
∇T0(x)+∇y′T1(x′,x
ǫ)/parenrightBig
dx−2ǫ/integraldisplay
ΩFǫKF
ǫ∇Tǫ(x)·/parenleftbigg
∇T0(x)+∇y′T1(x,x′
ǫ)/parenrightbigg
dx(81)
Using the coercivity condition of KS,F
ǫon the left hand side, as well as the positivity of the operato r
Gǫ, and passing to the two scale limit in the right hand side of (8 1) we obtain an upper bound for
αlim
ǫ→0/vextenddouble/vextenddouble(∇Tǫ−∇T0−∇y′T1)χS
ǫ/vextenddouble/vextenddouble2
L2(Ω)+αlim
ǫ→0ǫ/vextenddouble/vextenddouble(∇Tǫ−∇T0−∇y′T1)χF
ǫ/vextenddouble/vextenddouble2
L2(Ω)

Homogenization of a Heat Transfer Problem 28
Then, a combination of Lemmas 4.1 and 4.3 yields a bound for
lim
ǫ→0/ba∇dblTǫ−T0−ǫT1/ba∇dbl2
L2(Ω).
We now have to prove that these upper bounds are all zero, i.e. , that the two-scale limit of the right
hand side of (81) vanishes.
Indeed, by virtue of (42) and (79) we have
lim
ǫ→0σ
ǫ/integraldisplay
ΓǫGǫ/parenleftbigg
T0(x)+ǫT1(x,x′
ǫ)/parenrightbigg/parenleftbigg
T0(x)+ǫT1(x,x′
ǫ)/parenrightbigg
dx
=σ
|Λ|/parenleftbigg1
2/integraldisplay
Ω∂T0
∂x3(x)∂T0
∂x3(x)dx/integraldisplay
γ/integraldisplay
γF2D(z′,y′)|z′−y′|2dy′dz′
+/integraldisplay
Ω/integraldisplay
γ/parenleftbig
∇x′T0(x)·y+T1(x,y′)/parenrightbig
(Id−ζ2D)(T1(x,y′)+∇x′T0(x)·y)dy′dx
=/integraldisplay
ΩK∗(x)|∇xT0(x)|2dx−1
|Λ|/integraldisplay
Ω/integraldisplay
ΛSKS(x,y′)/vextendsingle/vextendsingle∇xT0(x)+∇y′T1(x,y′)/vextendsingle/vextendsingle2dy′dx
and
lim
ǫ→0σ
ǫ/integraldisplay
ΓǫGǫ/parenleftbigg
T0(x)+ǫT1(x,x′
ǫ)/parenrightbigg
Tǫ(x)dx=/integraldisplay
ΩK∗(x)|∇xT0(x)|2dx
−1
|Λ|/integraldisplay
Ω/integraldisplay
ΛSKS(x,y′)/vextendsingle/vextendsingle∇xT0(x)+∇y′T1(x,y′)/vextendsingle/vextendsingle2dy′dx
Passing to the two scales limit in the right hand side of (81) y ields
−/integraldisplay
ΩK∗(x)∇xT0(x)·∇xT0(x)dx+θ/integraldisplay
Ωf(x)T0(x)dx
which is equal to zero thanks to the variational formulation of the homogenized problem (25). Hence
the result. /squaresolid
5 Non-linear case
As already discussedinRemark 2.2, thetruephysical proble minvolves anon-linear radiation operator,
defined by formula (9) instead of (5). The study of the linear c ase was a simplifying assumption in
order to rigorously prove the convergence of the homogeniza tion process. However, the formal method
of two-scale asymptotic expansion is perfectly valid in the non-linear case too (see [3] and [20] if
necessary). In this section we give, without proofs, the hom ogenization result in the non-linear case.
When the radiation operator, defined by formula is given by (9 ) instead of (5), the non-linear
equivalent of Proposition 3.1 is the following.
Proposition 5.1. Under assumption (23) the leading term T0is the solution of the homogenized
problem
/braceleftBigg
−div(K∗(x,T3
0)∇T0(x))+V∗(x)·∇T0(x) =θf(x)inΩ
T0(x) = 0 on∂Ω(82)

Homogenization of a Heat Transfer Problem 29
with the porosity factor θ=|ΛS|/|Λ|, the homogenized conductivity given by its entries, for j,k=
1,2,3,
K∗
j,k(x,T3
0) =1
|Λ|/bracketleftbigg/integraldisplay
ΛSKS(x,y′)(ej+∇yωj(y′))·(ek+∇yωk(y′))dy′
+4σT3
0(x)/integraldisplay
γG(ωk(y′)+yk)(ωj(y′)+yj)
+ 2σT3
0(x)/integraldisplay
γ/integraldisplay
γF2D(s′,y′)|s′−y′|2dy′ds′δj3δk3/bracketrightbigg
and an homogenized velocity given by
V∗
k=1
|Λ|/integraldisplay
ΛFV(x,y′)·ekdy′
where/parenleftbig
ωj(x,T3
0(x),y′)/parenrightbig
1≤j≤3are the solutions of the cell problems


−divyKS(x,y′)(ej+∇yωS
j(y′)) = 0 inΛS
−divyKF(x,y′)(ej+∇yωF
j(y′))+V(x,y′)·(ej+∇yωF
j(y′)) = 0 inΛF
−KS(y′,x3)(ej+∇yωS
j(y′))·n= 4σT3
0(x)G(ωS
j(y′)+yj)onγ
ωF
j(y′) =ωS
j(y′)onγ
y′/ma√sto→ωj(y′)isΛ-periodic,(83)
andT1is given by
T1(x,y′) =3/summationdisplay
j=1ωj(x,T3
0(x),y′)∂T0
∂xj(x). (84)
The homogenized problem (82) is a non-linear convection-di ffusion model where the non-linearity
appears only in the conductivity tensor K∗which depends on the third power of the temperature. As
usual in homogenization, the cell problems are linearized, depending on the value of the macroscopic
temperature at each macroscopic point x. The linearization of the Stefan-Boltzmann law (giving the
irradiance as the fourth power of the temperature) yields th e coefficient 4 σT3
0in the radiation operator
of the cell problem (83). Therefore, the cell solutions, as w ell as the homogenized tensor K∗, depend
on the third power of the temperature T3
0.
6 Numerical results
In this section we describe some numerical experiments to st udy the asymptotic behaviour of the heat
transfer model (8) in the non-linear case, i.e., when the rad iation operator is defined as in Remark
2.2. Our goal is to show the efficiency of our proposed homogeni zation procedure, to validate it by
comparing the reconstructed solution of the homogenized mo del with the numerical solution of the
exact model (8) for smaller and smaller values of ǫand to exhibit a numerical rate of convergence
in terms of ǫ. While the computations in [3] were restricted to the 2D sett ing, here we perform 3D
numerical simulations of (8). All computations have been do ne with the finite element code CAST3M
[11] developed at the French Atomic and Alternative Energy C ommission (CEA).
6.1 Changing variables for the numerical simulation
Usually, in homogenization theory, we solve a problem in a fix ed domain Ω with cells of size ǫ, which
tends to 0 (see Figure 3). However, in practice for our nuclea r reactor problem, the sizes of the

Homogenization of a Heat Transfer Problem 30
gas cylinders and cell assemblies are fixed by manufacturing constraints. Therefore, following [3], we
proceed differently: we fix the size of the periodical cell (ind ependent of ǫ) and we increase the total
number of cells, i.e., the size of the global domain which is o f orderǫ−1. In other words, instead of
using the macroscopic space variable x∈Ω, we use the microscopic space variable y=x/ǫ. In this
new frame of reference, all periodicity cells are of unit siz e and the computational domain is ǫ−1Ω
which is increasing as ǫgoes to 0 (see Figure 4).
Figure 3: Standard homogenization in a fixed domain Ω
Figure 4: Rescaled process of homogenization with constant periodicity cell and increasing domain
/hatwideΩ =ǫ−1Ω
If the fixed domain is denoted by Ω =/producttext3
j=1(0,Lj), our rescaled computational domain is /hatwideΩ =
ǫ−1Ω =/producttext3
j=1(0,Lj/ǫ), where there exist integers Njsuch thatLj/ǫ=Njℓj, forj= 1,2 (so that only
entire cells belong to /hatwideΩ). For any function u(x) defined on Ω, we introduce the rescaled function /hatwideu(y),
defined on/hatwideΩ by
/hatwideu(y) =u(ǫy) =u(x), (85)
which satisfies ∇y/hatwideu(y) =ǫ(∇xu)(ǫy) =ǫ∇xu(x). All quantities defined in /hatwideΩ are denoted with a hat /hatwide
and, for simplicity, we drop the dependence on ǫ. For example, we define the conductivity tensor /hatwideK
as
/hatwideK(y) =/braceleftBigg/hatwidestKS(y) =KS(ǫy,y′) in/hatwideΩS,
/hatwidestKF(y) =ǫKF(ǫy,y′) in/hatwideΩF,(86)
and the fluid velocity
/hatwideV(y) =ǫV(ǫy,y′) in/hatwideΩF. (87)
We also define /hatwideΩS,/hatwideΩF,/hatwideΓ and∂/hatwideΩ by the same change of variables relating Ω and /hatwideΩ. In this new frame

Homogenization of a Heat Transfer Problem 31
of reference, problem (8) becomes


−div(/hatwidestKS∇/hatwideTǫ) =ǫ2/hatwidef in/hatwideΩS
−div(/hatwidestKF∇/hatwideTǫ)+/hatwideV·∇/hatwideTǫ= 0 in /hatwideΩF
−/hatwidestKS∇/hatwideTǫ·n=−/hatwidestKF∇/hatwideTǫ·n+σG(/hatwideTǫ4) on/hatwideΓ
/hatwideTǫ= 0 on ∂/hatwideΩ
/hatwideTǫis continuous through /hatwideΓ.(88)
The homogenized problem (82) becomes
/braceleftBigg
−div(/hatwideK∗(/hatwiderT03)∇/hatwiderT0)+ǫ/hatwideV∗·∇/hatwiderT0=ǫ2θ/hatwidefin/hatwideΩ,
/hatwiderT0= 0 on ∂/hatwideΩ.(89)
Furthermore, we also define
/hatwiderT1(y) =ǫT1(ǫy,y′) =ǫT1(x,x′
ǫ) =3/summationdisplay
i=1∂/hatwiderT0
∂yi(y)ωi(y′) (90)
where/hatwiderT1is purposely scaled as ǫso that the ǫ-factor disappears in the last equality of (90). Finally,
the homogenization approximation Tǫ(x)≃T0(x)+ǫT1(x,x/ǫ) becomes
/hatwideTǫ(y)≃/hatwiderT0(y)+/hatwiderT1(y). (91)
Since a factor ǫd
2appears when changing variables y=x/ǫin theL2-norms, we compute relative
errors between the exact and reconstructed solutions in the sequel. The relative errors are invariant
by our change of variables
/vextenddouble/vextenddouble/vextenddoubleTǫ(x)−(T0(x)+ǫT1(x,x
ǫ))/vextenddouble/vextenddouble/vextenddouble
L2(Ω)
/ba∇dblTǫ(x)/ba∇dblL2(Ω)=/vextenddouble/vextenddouble/vextenddouble/hatwideTǫ(y)−(/hatwiderT0(y)+/hatwiderT1(y))/vextenddouble/vextenddouble/vextenddouble
L2(/hatwideΩ)/vextenddouble/vextenddouble/vextenddouble/hatwideTǫ(y)/vextenddouble/vextenddouble/vextenddouble
L2(/hatwideΩ)(92)
and
/vextenddouble/vextenddouble/vextenddouble∇Tǫ(x)−∇(T0(x)+ǫT1(x,x
ǫ))/vextenddouble/vextenddouble/vextenddouble
L2(Ω)
/ba∇dbl∇Tǫ(x)/ba∇dblL2(Ω)=/vextenddouble/vextenddouble/vextenddouble∇/hatwideTǫ(y)−∇(/hatwiderT0(y)+/hatwiderT1(y))/vextenddouble/vextenddouble/vextenddouble
L2(/hatwideΩ)/vextenddouble/vextenddouble/vextenddouble∇/hatwideTǫ(y)/vextenddouble/vextenddouble/vextenddouble
L2(/hatwideΩ). (93)
6.2 Algorithm and computational parameters
We first give our methodology for the numerical simulations o f the homogenization process.
1. Solve the 3 cell problems (83) for a range of values of /hatwideT0(see Figure 5).
2. Compute the homogenized conductivity (as a function of te mperature) and the homogenized
velocity.
3. Solve the homogenized problem (82) by a fixed point algorit hm (see Figure 7).
4. Compute the corrector /hatwideT1(y) =3/summationdisplay
i=1∂/hatwiderT0
∂yi(y)ωi(y′).
5. Reconstruct an approximate solution: /hatwideT0(y)+/hatwideT1(y) (see Figure 7).

Homogenization of a Heat Transfer Problem 32
We now give our computational parameters for a reference com putation corresponding to ǫ=ǫ0=
1
4. The geometry corresponds to a cross-section of a typical fu el assembly for a gas-cooled nuclear
reactor (see [20] for further reference). The domain is /hatwideΩ =ǫ−1Ω =/producttext3
j=1(0,Lj/ǫ), withL3= 0.025m
and, forj= 1,2,Lj/ǫ=NjℓjwhereN1= 3,N2= 4 andℓ1= 0.04m,ℓ2= 0.07m. Each periodicity
cell contains 2 hollow cylinders (holes) (see Figure 1), the radius of which is equal to 0 .0035m. The
emissivity of the holes boundaries is equal to e= 1. The thermal source fis set to zero (we refer to
[21] for other computations, including ones with f/\egatio\slash= 0). We enforce periodic boundary conditions in
thex1direction and non-homogeneous Dirichlet boundary conditi ons in the other directions which are
given by/hatwideTǫ(y) =ǫ(3200y1+400y2)+800 on theboundariescorrespondingto y2= 0, y2=L2/ǫ, y3= 0
andy3=L3/ǫ. This boundary condition depends on ǫ, as a function of y, in such way that, as a
function of x=ǫy, it is independent of ǫ.
Thephysicalvaluesoftheisotropicconductivityare30 Wm−1K−1inthesolidpartand0 .3Wm−1K−1
in the fluid part. Since it is much smaller in the fluid than in th e solid, we decided to scale it by ǫ,
see (3) and (86). In other words, the conductivity tensor defi ned in (86) takes the values
/hatwideK=/braceleftBigg
30Wm−1K−1in/hatwideΩS,
ǫ
ǫ00.3Wm−1K−1in/hatwideΩF.
On a similar token, the physical value of the fluid velocity (a ssumed to be constant and parallel to
the cylinders axis) is 80 ms−1. By the scaling of (87), the numerical value of the velocity i s
/hatwideV=
0
0
ǫ
ǫ080
ms−1.
Remark that it is only for the reference computation ǫ0= 1/4 that/hatwideKand/hatwideVare equal to their physical
values. While the rescaled coefficients/hatwidestKF(y) and/hatwideV(y) are varying with ǫ, the original coefficients
KF(x) andV(x) are independent of ǫ. The fact that the numerical values of /hatwideKFand/hatwideVare not the
physical ones for ǫ/\egatio\slash=ǫ0= 1/4 is not a problem, since our convergence study (as ǫgoes to 0) is purely
a numerical verification of our mathematical result.
As explained in Section 6.1 we shall check numerically the co nvergence of the homogenization
process when ǫgoes to zero, or more precisely when the number of cells goes t o infinity. We thus
compare the solution /hatwideTǫof (8) (obtained by a costly numerical computation) with the homogenized
reconstructed solution /hatwideT0(y)+/hatwideT1(y) (whichismuchcheaper tocompute). Furthermore, weshall o btain
speed of convergences for the relative errors (92) and (93) p lotted in Figures 11 and 12. To avoid an
excessive computational burden, we have chosen periodicbo undarycondition in the x1direction which
implies that it is not necessary to add cells in the x1direction. Therefore, N1= 3 is fixed and we
simply add cells in the x2direction, increasing N2from 4 to 10 with a unit step. In other words, we
define
ǫ=1
N2.
Note that the vertical size of /hatwideΩ isL3/ǫ, which is thus increasing as ǫgoes to zero.
All computations are performedwithrectangular Q1finiteelements (4 nodes in 2D, 8nodes in 3D).
A boundary integral method is used for the radiative term (wh ich involves a dense matrix coupling
all nodes on the surface enclosing a fluid part). The typical n umber of nodes for the 2D cell problem
is 1 027 (from which 72 are on the radiative boundary γ); it is 6 336 for the 3D homogenized problem
(which has no radiative term); it is 96 480 for the original pr oblem (8) with ǫ=ǫ0=1
4(from which
6 912 are on the radiative boundary Γ ǫ).
6.3 Simulation results
In Figure 5 we plot the solutions of the cell problems (83) for an homogenized temperature T0= 800K.
Recall that, in the non linear case, the solutions of the cell problems depend on the macroscopic

Homogenization of a Heat Transfer Problem 33
temperature. We recognize that ωS
3is a constant in Figure 5 (right).
Figure 5: Solutions of the cell problems for T0= 800K
Thecell solutionsallow ustoevaluatethehomogenized cond uctivitywhichturnsouttonumerically
be a diagonal tensor (at least for temperatures T0≤1E+05Kwith a precision on 14 digits). However,
for larger (extreme) temperatures, /hatwideK∗is not any longer a diagonal tensor [3]. The diagonal entries of
/hatwideK∗are plotted on Figure 6 and two typical values are
/hatwideK∗(T0= 50K) =
25.907 0.0.
0.25.914 0.
0.0.30.05
,/hatwideK∗(T0= 20000K) =
49.801 0.0.
0.49.781 0.
0.0.3680.7
.
The homogenized velocity is a simple volume average, equal t o
/hatwideV∗=
0
0
15.134
ms−1.

Homogenization of a Heat Transfer Problem 34
Figure 6: Homogenized conductivities as a function of the ma croscopic temperature: K∗
11(top left),
K∗
22(top right), K∗
33(bottom).
By afixedpoint algorithm (thehomogenized conductivity /hatwideK∗is evaluated with thepreviousiterate
for the temperature), we solve the homogenized problem (it r equires of the order of 5 iterates). By a
Newton method we solve also the direct model (8) (it requires of the order of 15 iterates). In Figure
7 we plot the direct, homogenized and reconstructed solutio ns computed for a value of ǫ=ǫ0= 1/4,
as well as the error between the direct and reconstructed tem perature. The error is clearly small
and mostly concentrated on the domain boundaries. The modul i of the temperature gradients are
displayed on Figure 8. Clearly the reconstructed solution /hatwideT0+/hatwideT1is a much better approximation of
the true solution /hatwideTǫthan the mere homogenized solution /hatwideT0. The error on the temperature gradient
is larger and again concentrated on the domain boundaries (t his is consistent with the presence of
boundary layers not taken into account in our asymptotic ana lysis).

Homogenization of a Heat Transfer Problem 35
Figure 7: Solutions in /hatwideΩ
Figure 8: Modules of the solution gradients in /hatwideΩ

Homogenization of a Heat Transfer Problem 36
Notice on Figure 7 that the reconstructed temperature is sli ghtly fluctuating on the boundary
y2= 0 while the true solution is linear. This is due to the fact th at the corrector /hatwideT1does not
satisfy a Dirichlet boundary condition. This well-known effe ct in homogenization can be corrected by
introducing further terms, called boundary layers [8], [10 ]. We shall not dwell on this issue, all the
more since other boundary layers are involved in our approxi mation. Indeed, the dimension reduction
which applies to the radiative operator (which is truly 3D in the direct model (8) and only 2D in the
cell problems) certainly generates boundary layers close t o the top and bottom boundaries y3= 0 and
y3=L3/ǫ. Nevertheless, if we plot the solutions in a smaller domain /hatwide∆ (which is obtained from /hatwideΩ
by removing one row of cells close to each boundary face norma l to thex2direction and a layer of
thickness 0.025mat the top and bottom faces) we obtain a better agreement betw een/hatwideT0+/hatwideT1and/hatwideTǫ
(see Figure 9) and between ∇(/hatwideT0+/hatwideT1) and∇/hatwideTǫ(see Figure 10).
Figure 9: Solutions in the reduced domain /hatwide∆
Now, to check the convergence of our homogenization process and to obtain a numerical speed of
convergence, we display in Figures 11 and 12, as a function of ǫon a log-log plot, the relative errors
(92) and (93) related to temperature ERR(T) and temperature gradient ERR(∇T). We compare
these errors with the slopes of ǫand√ǫ. This has to be compared with the classical error estimate
for a pure diffusion problem (without radiative transfer) as g iven in [10]


ERR(T)/hatwideΩ=/vextenddouble/vextenddouble/vextenddouble/hatwideTǫ(y)−(/hatwideT0(y)+/hatwideT1(y))/vextenddouble/vextenddouble/vextenddouble
L2(/hatwideΩ)/vextenddouble/vextenddouble/vextenddouble/hatwideTǫ(y)/vextenddouble/vextenddouble/vextenddouble
L2(/hatwideΩ)≤Cǫ,
ERR(∇T)/hatwideΩ=/vextenddouble/vextenddouble/vextenddouble∇/hatwideTǫ(y)−(∇/hatwideT0(x)+∇/hatwideT1(y))/vextenddouble/vextenddouble/vextenddouble
L2(/hatwideΩ)/vextenddouble/vextenddouble/vextenddouble∇/hatwideTǫ(y)/vextenddouble/vextenddouble/vextenddouble
L2(/hatwideΩ)≤C√ǫ.
Our errors ERR(T) andERR(∇T) are in accordance with those theoretically predicted for a pure

Homogenization of a Heat Transfer Problem 37
Figure 10: Modules of the solution gradients in the reduced d omain/hatwide∆
diffusion problem, namely they behave like ǫand√ǫ, respectively. In particular, it implies that the
additional boundary layers caused by the dimension reducti on effect (due to the radiative term) have
an impact on the error compararable or smaller than that the h omogenization boundary layers.
Even for moderate-size computations, like the ones in this s ection, the gain in memory and CPU
time for our homogenization method is enormous compared to a direct simulation. This is a well-
known fact in the homogenization of diffusion problem but the g ain is all the more extreme because
of the radiative transfer involved in our model. Indeed, the direct model (8) involves a 3D radiative
transfer operator which implies that full matrices connect ing all nodes on the surface of one cylinder
have to be stored and inverted (of course they are coupled thr ough the diffusive rigidity matrix in the
solid part). Typically, one Newton iteration in our referen ce computation takes about 80 minon a
computer which has a memory of 37 .2GBand 12 processors with CPU= 2.67GHz. On the other
hand, the cell problems (27) are merely 2D, so very cheap to so lve (typically, one solution for a given
temperature T0takes 18.E−04minwith the same computer), and the homogenized problem (82)
features no radiative term (one Newton iteration in our refe rence computation takes 12 .E−02min
with the same computer). Therefore, our algorithm of Subsec tion 6.2 is very competitive and is able
to treat very large cases, like a full nuclear core computati on.

Homogenization of a Heat Transfer Problem 38
Figure 11: Relative error (92) for the temperature
Figure 12: Relative error (93) for the temperature gradient

Homogenization of a Heat Transfer Problem 39
Acknowledgement. The authors thank A. Stietel (CEA Saclay) for her useful coll aboration.
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