Asymptotics of a thermal flow with highly conductive and radiant suspensions

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Asymptotics of a thermal flow with highly conductive and radiant suspensions

Fadila Bentalha, Isabelle Gruais, Dan Polisevski

To cite this version:

Fadila Bentalha, Isabelle Gruais, Dan Polisevski. Asymptotics of a thermal flow with highly conductive

and radiant suspensions. 2005. �hal-00005450v3�

Asymptotics of a thermal flow with highly conductive and radiant suspensions

Fadila Bentalha ^∗ , Isabelle Gruais ^∗∗ and Dan Poliˇsevski ^∗∗∗

Abstract. Radiant spherical suspensions have an ε-periodic distribution in a tridimensional incompressible viscous fluid governed by the Stokes-Boussinesq system. We perform the homogenization procedure when the radius of the solid spheres is of order ε ³ (the critical size of perforations for the Navier-Stokes system) and when the ratio of the fluid/solid conductivities is of order ε ⁶ , the order of the total volume of suspensions. Adapting the methods used in the study of small inclusions, we prove that the macroscopic behavior is described by a Brinkman-Boussinesq type law and two coupled heat equations, where certain capacities of the suspensions and of the radiant sources appear.

Mathematical Subject Classification (2000). 35B27, 76D07, 76S05.

Keywords. Stokes-Boussinesq system, homogenization, non local effects.

1 Preliminaries

One main achievement of homogenization theory was the ability to conceptu- ally clarify the relationship between microscopic and macroscopic properties of physical systems, at least as far as the periodic approximation could be accept- able. The major restriction was the technically impossible interplay between different scales: if some quantity varies as the power ε ^α of the size ε of the mesh, then the case where α < 0 leads to blow up at the limit. This type of problems were introduced and solved for the first time by [1] and developed by [2, 3, 4, 5, 6]. One major contribution in that direction is the paper by G. Allaire [7] who clearly underlies the role of critical discriminating scales beyond which nothing can be said, but rigidification of elastic systems for instance, and that can however generate a transition state where either ’non local’ effects [2, 5] or

’coming from nowhere’ terms [1] can emerge.

In this paper, we are insterested in the former case which has been thor-

oughly explored when non local effects concentrate on rod-like one-dimensional

submanifolds of the three-dimensional space: see [2] for the Laplacian, [5] for the

Elasticity system. This geometry enables the formulation of the limit problem

as a rod-like boundary value problem solved by the density of a Radon mea-

sure. Our question then was: what happens in other geometries, especially if

non local effects are to be supported by a cloud of little particles? The physical

opportunity was the example of thermal flows (see [8, 9]) where highly heat con-

ducting spheres are immerged in a Stokes-Boussinesq fluid. It is straightforward that for some critical size of the particles (eventually ε ³ when the period of the distribution is ε) the resulting mixture will display a specific behaviour strongly discriminating between a trivial case and a classically homogenized case. Our concern was then to develop new skills to understand how the expected non local effects would be formulated. We found out that the Dirac structure of the masses make the classical formulation in terms of a jump term updated and that it rather generates an additional source coupled with a capacitary term representative of a Brinkman-Boussinesq type law.

More precisely, the physics of the problem may be described as follows. Solid spherical suspensions are ε-periodically distributed in a tridimensional bounded domain filled with an incompressible fluid governed by the Stokes-Boussinesq system. We study the homogenization of the convective movement which is generated by highly heterogeneous radiant sources, when the radius of the sus- pensions is of ε ³ -order, that is the border case for the Navier-Stokes system (see [7]). Assuming that the conductivity and the radiant source of the fluid have ε ⁰ - order, we found that the only regular case in which we have macroscopic effects from both the conductivity and the radiation of the suspensions is when they are of ε ⁶ -order. Therefore, we have treated here strictly this case. Nevertheless, the present procedure can be easily adapted to the other cases.

Let Ω ⊂ R ³ be a bounded open set and let Y :=

− 1 2 , + 1

2 ³

.

Y _ε ^k := εk + εY, k ∈ Z ³ . Z _ε := {k ∈ Z ³ , Y _ε ^k ⊂ Ω}

The reunion of the suspensions is defined by T _ε := ∪ k∈Z

ε

B(εk, r _ε ),

where 0 < r _ε << ε and B(εk, r _ε ) is the ball of radius r _ε centered at εk, k ∈ Z _ε . The fluid domain is given by

Ω ε = Ω \ T ε .

Let e ⁽³⁾ the last vector of the canonical basis of R ³ , n the normal on ∂Ω ε in the outward direction and [·] ε the jump across the interface ∂T ε .

For a > 0 (the so-called Rayleigh number), b > 0 ( b

ε r

_ε

3

denoting the ratio of the solid/fluid conductivities), f ∈ C _c (Ω), g ∈ C _c (Ω), where

C c (Ω) := {g ∈ C(Ω); suppg is compact },

we consider the problem corresponding to the non-dimensional Stokes-Boussinesq

system governing the thermal flow of an ε-periodic distribution suspension of

solid spheres:

To find (u ^ε , p ^ε ), θ ^ε , ζ ^ε solution of

divu ^ε = 0, in Ω ε , (1)

−∆u ^ε + ∇p ^ε = aθ ^ε e ⁽³⁾ , in Ω _ε , (2)

−∆θ ^ε + u ^ε ∇θ ^ε = f, in Ω ε , (3)

−∆ζ ^ε = g, in T _ε , (4)

ζ ^ε = θ ^ε , on ∂T ε (5)

∂θ ^ε

∂n = b ε

r _ε ³ ∂ζ ^ε

∂n , on ∂T ε (6)

u ^ε = 0, on ∂Ω _ε , (7)

θ ^ε = 0, on ∂Ω. (8)

Set

V ε := {v ∈ H ₀ ¹ (Ω ε ; R ³ ), div v = 0}.

Thanks to (5), we extend θ ^ε on T ε by setting θ ^ε = ζ ^ε on T _ε . Then, the variational formulation reads:

∀(v, q) ∈ V × L ² (Ω _ε ), Z

Ω

_ε

∇u ^ε · ∇v dx = a Z

Ω

_ε

θ ^ε v ₃ dx Z

Ω

ε

q divu ^ε dx = 0

(9)

∀ϕ ∈ H ₀ ¹ (Ω _ε ), Z

Ω

_ε

∇θ ^ε ∇ϕ dx + b ε

r ε

³ Z

T

_ε

∇θ ^ε ∇ϕ dx +

Z

Ω

ε

u ^ε ϕ∇θ ^ε dx = Z

Ω

ε

f ϕdx + b ε

r _ε ³ Z

T

ε

gϕdx.

(10)

We define F _ε ∈ H ⁻¹ (Ω) by

∀ϕ ∈ H ₀ ¹ (Ω), F ε (ϕ) :=

Z

Ω

_ε

f ϕ dx + b ε

r ε

3 Z

T

_ε

gϕdx. (11) Then, for α > 0 (we shall choose a suitable value for this parameter later), we can present the variational formulation of the problem (1)–(8):

To find (u ^ε , θ ^ε ) ∈ V _ε × H ₀ ¹ (Ω) such that

∀(v, ϕ) ∈ V _ε × H ₀ ¹ (Ω), hG(u ^ε , θ ^ε ), (v, ϕ)i = F ε (ϕ) (12) where the mapping G : V ε × H ₀ ¹ (Ω) → V _ε ⁰ × H ⁻¹ (Ω) is defined by

hG(u, θ), (v, ϕ)i = α Z

Ω

_ε

∇u∇v dx − αa Z

Ω

_ε

θv ₃ dx

+ Z

Ω

ε

∇θ∇ϕ dx + Z

Ω

ε

uϕ∇θ dx + b ε

r _ε ³ Z

T

ε

∇θ∇ϕ dx.

In order to prove the existence theorem for problem (12), we make use of the following result of Gossez.

Theorem 1.1 Let X be a reflexive Banach space and G : X → X ⁰ a continuous mapping between the corresponding weak topologies. If

hGϕ, ϕi

|ϕ| X

→ ∞ as |ϕ| _X → ∞

then G is a surjection.

Acting as in the proof of Theorem 5.2.2 [8] Ch 1, Sec. 5, we find that the ex- istence of the weak solutions of problem (12) is assured if α is chosen sufficiently small.

Moreover, if (u ^ε , θ ^ε ) is a solution of problem (12), then, by using the weak maximum principle, we obtain that θ ^ε ∈ L ^∞ (Ω), (see Theorem 3.4 [8] Ch 2, Sec. 3).

Remark 1.2 For any a > 0, we have proved the existence of a solution of (12), but we do not have a uniqueness result, except if we assume that a > 0 is small enough.

In the sequel, C will denote a suitable positive constant independent of ε and which may differ from line to line.

2 Basic inequalities

Lemma 2.1 and Lemma 2.2 below are set without proof since it is an adaptation of the case p = 2 of Lemma A.3 [2] and Lemma A.4 [2] respectively but with integrals set on spheres.

Lemma 2.1 For every 0 < r 1 < r 2 , consider:

C(r 1 , r 2 ) := {x ∈ R ³ , r 1 < |x| < r 2 }.

Then, if u ∈ H ¹ (C(r 1 , r 2 )), the following estimate holds true:

|∇u| ² _C(r

1

,r

₂

) ≥ 4πr 1 r 2

r ₂ − r ₁ Z

−

S

r2

u dσ − Z

−

S

r1

u dσ

2

, (13)

where

Z

−

S

_r

· dσ := 1 4πr ²

Z

S

_r

· dσ.

Lemma 2.2 There exists a positive constant C > 0 such that: ∀(R, α) ∈ R ⁺ × (0, 1), ∀u ∈ H ¹ (B(0, R)),

Z

B(0,R)

|u − Z

−

S

_αR

u dσ| ² dx ≤ C R ²

α |∇u| ² _B(0,R) .

From now on, we denote by R ε a radius with the property r ε << R ε << ε, that is :

ε→0 lim r _ε R ε

= lim

ε→0

R _ε

ε = 0. (14)

Obviously, its existence is insured by the assumption 0 < r _ε << ε.

We introduce the measure dm ^ε := 3

4π ε

r ε

³

1 T

_ε

(x) dx and denote the norm in L ² _m

_ε

by:

|ϕ| ² _m

ε

:=

Z

|ϕ| ² dm _ε .

We denote the domain confined between the spheres of radius a and b by C(a, b) := {x ∈ R ³ , a < |x| < b}

and correspondingly

C ^k (a, b) := εk + C(a, b), We also use the following notations:

C ε := ∪ k∈Z

ε

C ^k (r _ε , R _ε ).

S _r ^k

_ε

= ∂B(εk, r ε ), S r

_ε

:= ∪ k∈Z

_ε

S _r ^k

_ε

, S _R ^k

ε

= ∂B(εk, R _ε ), S _R

_ε

:= ∪ k∈Z

ε

S _R ^k

ε

,

Consider the piecewise constant functions defined after some θ ∈ H ₀ ¹ (Ω) by

˜

τ ^ε (x) = X

k∈Z

ε

Z

−

S

^k_rε

θ dσ

! 1 _Y

k

ε

(x), (15)

θ ˜ ^ε (x) = X

k∈Z

ε

Z

−

S

^k

Rε

θ dσ

! 1 _Y

k

ε

(x). (16)

Lemma 2.3 For every θ ∈ H ₀ ¹ (Ω), we have Z

Ω

|θ − θ ˜ ^ε | ² dx ≤ C ε ³ R ε

Z

Ω

|∇θ| ² dx, (17) Z

T

_ε

|θ − τ ˜ ^ε | ² dx ≤ Cr ² _ε Z

T

_ε

|∇θ| ² dx (18)

Z

Ω

| θ ˜ ^ε − τ ˜ ^ε | ² dx ≤ C ε ³ r ε

Z

Cε

|∇θ| ² dx. (19) where θ ˜ ^ε and τ ˜ ^ε are defined by (15) and (16).

Moreover:

Z

Ω

| θ ˜ ^ε | ² dx = Z

| θ ˜ ^ε | ² dm ε , Z

Ω

|˜ τ ^ε | ² dx = Z

|˜ τ ^ε | ² dm ε . (20)

Proof. Notice that by definition:

Z

Ω

|θ− θ ˜ ^ε | ² dx = X

k∈Z

_ε

Z

Y

_ε^k

|θ−

Z

−

S

^k

Rε

θ dσ| ² dx ≤ X

k∈Z

_ε

Z

B(εk,

^ε

√ 3 2

)

|θ−

Z

−

S

^k

Rε

θ dσ| ² dx

where we have used that

Y _ε ^k ⊂ B(εk, ε √ 3 2 ) for every k ∈ Z ε . We use Lemma 2.2 with

R = ε √ 3

2 , α = 2R ε

ε √ 3 to deduce that

Z

Ω

|θ − θ ˜ ^ε | ² dx ≤ C ε √ 3 2

! ² ε √

3 2R ε

X

k∈Z

ε

Z

B(εk,

^ε

√ 3 2

)

|∇θ| ² dx

≤ C ε ³ R ε

X

k∈Z

ε

Z

B(εk,

^ε

√3 2

)

|∇θ| ² dx ≤ C ε ³ R ε

Z

Ω

|∇θ| ² dx

which shows (17).

To establish (18), we recall the definition:

Z

T

_ε

|θ − τ ˜ ^ε | ² dx = X

k∈Z

ε

Z

B(εk,r

_ε

)

|θ − Z

−

S

^k_rε

θ dσ| ² dx

Applying Lemma 2.2 with R = r ε and α = 1, we get the result Z

T

ε

|θ − τ ˜ ^ε | ² dx ≤ Cr ² _ε X

k∈Z

ε

Z

B(εk,r

ε

)

|∇θ| ² dx ≤ Cr ² _ε Z

T

ε

|∇θ| ² dx.

We come to (19). Indeed, applying Lemma 2.1 and (14):

Z

Ω

| θ ˜ ^ε − τ ˜ ^ε | ² dx = X

k∈Z

ε

Z

Y

_ε^k

| Z

−

S

^k

Rε

θ dσ − Z

−

S

^k_rε

θ dσ| ² dy

≤ X

k∈Z

ε

Z

Y

_ε^k

(R ε − r ε ) 4πR ε r ε

dy Z

C

^k

rε,Rε

|∇θ| ² dx = (R ε − r ε ) 4πr ε R ε

X

k∈Z

ε

ε ³ Z

C

^k

rε,Rε

|∇θ| ² dx

= Cε ³ (R ε − r ε ) 4πr _ε R _ε

Z

Cε

|∇θ| ² dx ≤ C ε ³ r _ε

Z

Cε

|∇θ| ² dx.

Finally, a direct computation yields (20).

Proposition 2.4 For any θ ∈ H ₀ ¹ (Ω), there holds true:

Z

|θ| ² dm ε ≤ C max (1, ε ³ r ε

) Z

Ω

|∇θ| ² dx.

Proof. We have:

Z

|θ| ² dm ε ≤ 2 Z

|θ − τ ˜ ^ε | ² dm ε + 2 Z

|˜ τ ^ε | ² dm ε

= 2 Z

|θ − τ ˜ ^ε | ² dm _ε + 2 Z

Ω

|˜ τ ^ε | ² dx

≤ Cr _ε ² Z

|∇θ| ² dm _ε + 4 Z

Ω

|˜ τ ^ε − θ ˜ ^ε | ² dx + 8 Z

Ω

| θ ˜ ^ε − θ| ² dx + 8 Z

Ω

|θ| ² dx

≤ Cr ² _ε ε

r ε

3 Z

T

_ε

|∇θ| ² dx+C ε ³ r ε

Z

C

ε

|∇θ| ² dx+C ε ³ R ε

Z

Ω

|∇θ| ² dx+C Z

Ω

|∇θ| ² dx

≤ C ε ³

r _ε + ε ³ R _ε + 1

Z

Ω

|∇θ| ² dx ≤ C max (1, ε ³ r _ε )

Z

Ω

|∇θ| ² dx

Lemma 2.5 For ϕ ∈ C c (Ω) consider the piecewise constant function:

ϕ ^ε (x) := X

k∈Z

ε

Z

−

Y

_ε^k

ϕ dx

!

1 B(εk,r

_ε

) (x).

Then:

ε→0 lim |ϕ − ϕ ^ε | m

ε

= 0.

Proof. Notice that

|ϕ − ϕ ^ε | ² _m

_ε

= 3 4π

ε r _ε

³ X

k∈Z

ε

Z

B(εk,r

ε

)

|ϕ − Z

−

Y

_ε^k

ϕ dy| ² dx.

As we have also

|B(εk, r _ε )| = 4π

3 r _ε ³ , card(Z _ε ) ' |Ω|

ε ³ then, by the uniform continuity of ϕ on Ω, the result follows.

3 A priori estimates

In the sequel, we denote

γ ε := r _ε

ε ³ (21)

and we assume that

ε→0 lim γ _ε = γ ∈]0, +∞[. (22)

We denote F ∈ H ⁻¹ (Ω) by F (ϕ) :=

Z

Ω

f ϕ dx + 4πb 3

Z

Ω

gϕ dx (23)

Proposition 3.1 We have

F ε * F weakly in H ⁻¹ (Ω) Proof. For ϕ ∈ H ₀ ¹ (Ω) it follows

|F ε (ϕ)| ≤ |f | Ω

_ε

|ϕ| Ω

_ε

+ C ε

r _ε ³

|g| _∞ Z

T

ε

ϕ dx

≤ C|ϕ| Ω + C Z

ϕ dm ε

(24) with

Z

ϕ dm ε

≤ ( Z

dm ε ) ^1/2 Z

|θ ^ε | ² dm ε

1/2

= p

|Ω|

Z

|ϕ| ² dm ε

1/2

. (25) Notice that due to (22), Proposition 2.4 also reads

Z

|ϕ| ² dm ε ≤ C|∇ϕ| ² _Ω . (26) Substituting (25) and (26) into the right-hand side of (24), we get, using Poincar´ e’s inequality,

|F ε (ϕ)| ≤ C|∇ϕ| Ω . (27)

Now, let ϕ ∈ D(Ω). By the Mean Theorem, there exist ξ _ε ^k ∈ B(εk, r ε ) such that F _ε (ϕ) =

Z

Ω

ε

f ϕ dx + b ε

r _ε ³

X

k∈Z

^ε

Z

B(εk,r

ε

)

g(x)ϕ(x) dx

= Z

Ω

ε

f ϕ dx + b ε

r _ε ³

X

k∈Z

^ε

4π

3 r ³ _ε g(ξ _ε ^k )ϕ(ξ _ε ^k )

= Z

Ω

ε

f ϕ dx + 4πb 3

X

k∈Z

^ε

|Y _ε ^k |g(ξ _ε ^k )ϕ(ξ _ε ^k ).

There follows

∀ϕ ∈ D(Ω), lim

ε→0 F ε (ϕ) = Z

Ω

f ϕ dx + 4πb 3

Z

Ω

gϕ dx = F(ϕ). (28) The proof is completed by (27) and the density of D(Ω) in H ₀ ¹ (Ω).

Proposition 3.2 If (u ^ε , θ ^ε ) ∈ V _ε × H ₀ ¹ (Ω) is a solution of the problem (12), and if u ˆ ^ε stands for u ^ε continued with zero to Ω, then we have

ˆ

u ^ε and θ ^ε are bounded in H ₀ ¹ (Ω). (29) Moreover,

|∇θ ^ε | ² _Ω

ε

+ b ε

r ε

³

|∇θ ^ε | ² _T

ε

≤ C. (30)

Proof. Substituting v = u ^ε in (9) and noticing that Z

Ω

_ε

u ^ε θ ^ε ∇θ ^ε dx = Z

Ω

_ε

u ^ε ∇ |θ ^ε | ²

2

dx = − Z

Ω

_ε

div(u ^ε ) |θ ^ε | ²

2

dx = 0, we get:

|∇u ^ε | Ω

ε

≤ a|θ ^ε | Ω

ε

, (31) Seting ϕ = θ ^ε in (10) and taking into account Proposition 3.1, we find

|∇θ ^ε | ² _Ω

ε

+ b ε

r _ε ³

|∇θ ^ε | ² _T

ε

= F _ε (θ ^ε ) ≤ C|∇θ ^ε | _Ω (32) Noticing that b

ε r

ε

³

>> 1, we deduce from (32):

|∇θ ^ε | ² _Ω ≤ |∇θ ^ε | ² _Ω

_ε

+ b ε

r _ε ³

|∇θ ^ε | ² _T

_ε

≤ C|∇θ ^ε | Ω . Therefore

|∇θ ^ε | Ω ≤ C (33)

and thus

|θ ^ε | Ω ≤ C. (34)

Then, (30) follows from (32). Finally, (29) is completed by the estimates (31) and (34).

Proposition 3.3 There exist u ∈ H ₀ ¹ (Ω; R ³ ), θ ∈ H ₀ ¹ (Ω) and τ ∈ L ² (Ω) such that, on some subsequence,

ˆ

u ^ε * u in H ₀ ¹ (Ω; R ³ ), θ ^ε * θ in H ₀ ¹ (Ω),

˜

τ ^ε * τ in L ² (Ω), θ ^ε dm _ε * ^? τ dx in M _b (Ω),

where M b (Ω) is the set of bounded Radon measures on Ω and where * ^? denotes the weak-star convergence in the measures.

Proof. From (29), we get, on some subsequence, the following convergences:

θ ^ε * θ in H ₀ ¹ (Ω) (35)

θ ^ε → θ in L ² (Ω). (36)

ˆ

u ^ε * u in H ₀ ¹ (Ω; R ³ ). (37) Moreover, (17) yields

|θ ^ε − θ ˜ ^ε | ² _Ω ≤ C r ε

ε ³ r ε

R ε

|∇θ ^ε | ² _Ω which obviously yields

ε→0 lim |θ ^ε − θ ˜ ^ε | ² _Ω = 0.

Combining with (36), we infer that

θ ˜ ^ε → θ in L ² (Ω). (38)

We set

τ ^ε := 3 4π

ε r ε

3

θ ^ε 1 T

_ε

(x), (39)

and hence

θ ^ε dm ε = τ ^ε dx.

Taking (29) and (26) into account, we obtain Z

|θ ^ε | ² dm _ε ≤ C.

We also remark that for any ϕ ∈ C c (Ω), we have Z

ϕ dm ε → Z

Ω

ϕdx.

Then, using Lemma A-2 of [2], we find that there exists some τ ∈ L ² (Ω) such that, on some subsequence, the following convergence holds:

θ ^ε dm ε

* τ dx, ? M b (Ω). (40)

Moreover, recall that from (18) we have, taking into account (30):

Z

|θ ^ε − τ ˜ ^ε | ² dm ε ≤ Cr _ε ² Z

|∇θ ^ε | ² dm ε ≤ Cr _ε ² . (41) This implies:

(θ ^ε − τ ˜ ^ε ) dm ε

* h dx, ? M b (Ω) for some h ∈ L ² (Ω) and

|h| ² _Ω ≤ lim inf

ε→0

Z

|θ ^ε − τ ˜ ^ε | ² dm ε = 0, that is:

(θ ^ε − τ ˜ ^ε ) dm ε

* ? 0, M b (Ω). (42)

Notice that from (19):

|˜ τ ^ε | ² _Ω ≤ 2|˜ τ ^ε − θ ˜ ^ε | ² _Ω + 2| θ ˜ ^ε | ² _Ω ≤ C ε ³ r ε

|∇θ ^ε | ² _C

_ε

+ C ≤ C, (43) and hence, for some ˜ τ ∈ L ² (Ω),

˜

τ ^ε * τ ˜ in L ² (Ω). (44)

Combining (40) and (42), we arrive at

˜ τ ^ε dm ε

* τ dx, ? M b (Ω).

It remains to show that

˜

τ = τ. (45)

To that aim, let ϕ ∈ C c (Ω) and let ϕ ^ε (x) := X

k∈Z

ε

Z

−

Y

_ε^k

ϕ dy

!

1 B(εk,r

_ε

) (x).

We have

| Z

Ω

(τ ^ε − τ ˜ ^ε ) ϕ dx| =

3 4π

ε r _ε

³ Z

T

ε

θ ^ε ϕ dx−

Z

Ω

X

k∈Z

ε

Z

−

S

^k_rε

θ ^ε dσ

! 1 _Y

k

ε

ϕdx

= Z

θ ^ε ϕ dm _ε − ε ³ X

k∈Z

ε

Z

−

S

^k_rε

θ ^ε dσ

! Z

−

Y

_ε^k

ϕ dx

= Z

θ ^ε ϕ dm _ε − Z

˜

τ ^ε ϕ ^ε dm _ε

≤ Z

Ω

(θ ^ε − ˜ τ ^ε )ϕ dm ε

+ Z

Ω

˜

τ ^ε (ϕ − ϕ ^ε ) dm ε

≤ |θ ^ε − ˜ τ ^ε | m

_ε

|ϕ| m

_ε

+ |˜ τ ^ε | m

_ε

|ϕ − ϕ ^ε | m

_ε

. (46) From (20) and (43), we deduce that

|˜ τ ^ε | m

ε

= |˜ τ ^ε | Ω ≤ C.

Moreover, ϕ ∈ C c (Ω) yields

|ϕ| m

_ε

≤ C.

Then, (46) becomes

| Z

Ω

(τ ^ε − τ ˜ ^ε ) ϕ dx| ≤ C|θ ^ε − ˜ τ ^ε | m

_ε

+ C|ϕ − ϕ ^ε | m

_ε

. From (41), we infer that

| Z

Ω

(τ ^ε − τ ˜ ^ε ) ϕ dx| ≤ Cr ε + C|ϕ − ϕ ^ε | m

ε

. (47) Thus (47) and Lemma 2.5 yield

ε→0 lim Z

Ω

(τ ^ε − τ ˜ ^ε ) ϕ dx = 0.

As this holds for every ϕ ∈ C c (Ω), the density of C c (Ω) in L ² (Ω) together with (44) and (40) imply that τ ^ε * τ ˜ = τ in L ² (Ω).

4 The two macroscopic heat equations

The aim of this section is to pass to the limit as ε → 0 in the variational formulation

∀Φ ∈ H ₀ ¹ (Ω), Z

Ω

_ε

∇θ ^ε ∇Φ dx + b ε

r _ε ³ Z

T

_ε

∇θ ^ε ∇Φ dx+

+ Z

Ω

_ε

u ^ε ∇θ ^ε Φ dx = F ε (Φ).

(48)

Let ϕ, ψ ∈ D(Ω) and set

ϕ ^ε (x) = X

k∈Z

ε

Z

−

S

^k_rε

ϕ dσ

! 1 _Y

k

ε

(x), (49)

ψ ^ε (x) = X

k∈Z

ε

Z

−

S

^k_rε

ψ dσ

! 1 _Y

k

ε

(x). (50)

Let W ^ε denote the fundamental solution of the Laplacian, namely

∆W ^ε = 0 in C(r ε , R ε ), (51)

W ^ε = 1 in r = r _ε , (52)

W ^ε = 0 in r = R ε . (53)

The same arguments as in the proof of Lemma A.3 [2] yield W ^ε (r) = r _ε

(R ε − r ε ) R _ε

r − 1

if y ∈ C(r ε , R _ε ) and |y| = r. (54) Then, we set

w ^ε (x) :=







0 in Ω _ε \ C _ε ,

W ^ε (x − εk) in C _ε ^k , ∀k ∈ Z _ε , 1 in T _ε .

(55)

Proposition 4.1 We have

|∇w ^ε | Ω ≤ C (56)

Proof. Indeed, direct computation shows

|∇w ^ε | ² _Ω = X

k∈Z

ε

Z

C

^k

rε,Rε

|∇w ^ε | ² dx

= X

k∈Z

ε

Z 2π 0

dΦ Z π

0

sin Θ dΘ Z R

ε

r

ε

dr r ²

r ε R ε

R _ε − r _ε ²

≤ C |Ω|

ε ³ 1

r ε

− 1 R ε

r ε R ε

R ε − r ε

²

≤ C γ ε

(1 − _R ^r

^ε

ε

) . The proof is completed by (14) and (22).

For ϕ, ψ ∈ D(Ω), let us define

Φ ^ε = (1 − w ^ε )ϕ + w ^ε ψ ^ε . (57) Lemma 4.2 We have

ε→0 lim |Φ ^ε − ϕ| Ω = 0.

Proof. First notice that w ^ε → 0 in L ² (Ω). Indeed:

|w ^ε | Ω = |w ^ε | _C

_ε

_∪T

_ε

≤ |C ε ∪ T ε | = |Ω|

ε ³ 4π

3 R ³ _ε

and lim _ε→0 ^R _ε

^ε

= 0 by assumption (14). As an immediate consequence:

(1 − w ^ε )ϕ → ϕ in L ² (Ω).

Moreover, the uniform continuity of ψ over Ω implies that

ε→0 lim |ψ ^ε − ψ| _∞ = 0 so that

w ^ε ψ ^ε = w ^ε (ψ ^ε − ψ) + w ^ε ψ → 0 in L ² (Ω).

This achieves the proof.

Proposition 4.3 If θ ^ε is solution of (12) and Φ ^ε is given by (57) for any ψ, ϕ ∈ D(Ω), then we have

ε→0 lim Z

Ω

ε

∇θ ^ε · (∇Φ ^ε + Φ ^ε u ^ε ) dx

= Z

Ω

∇θ · (∇ϕ + ϕu) dx + 4πγ Z

Ω

(θ − τ )(ψ − ϕ) dx.

Proof. First consider Z

Ω

_ε

\C

ε

∇θ ^ε · (∇Φ ^ε + Φ ^ε u ^ε ) dx which reduces to

Z

Ω

_ε

\C

ε

∇θ ^ε · (∇ϕ + ϕu ^ε ) dx = Z

Ω

∇θ ^ε · ∇ϕ1 Ω

_ε

\C

ε

+ ϕ1 Ω

_ε

\C

ε

u ^ε dx.

Lebesgue’s dominated convergence theorem yields ∇ϕ1 _Ω

_ε

_\C

_ε

→ ∇ϕ in L ² (Ω).

Thus, taking (35) into account:

Z

Ω

∇θ ^ε · ∇ϕ1 _Ω

_ε

_\C

_ε

dx → Z

Ω

∇θ · ∇ϕ dx.

Moreover,

|1 Ω

_ε

\C

ε

u ^ε − u| Ω ≤ |u ^ε − u| Ω + |u| C

ε

∪T

ε

and the right-hand side converges to zero because (37) yields

u ^ε → u in L ² (Ω) (58)

and we apply Lebesgue’s dominated convergence theorem to conclude with the second term. Thus

1 _Ω

_ε

_\C

_ε

u ^ε → u in L ² (Ω). (59) Now, as ϕ ∈ C c (Ω), ϕ1 _Ω

_ε

_\C

_ε

u ^ε → ϕu in L ² (Ω). Thus, using (35) again,

Z

Ω

∇θ ^ε · ϕ1 Ω

_ε

\C

ε

u ^ε dx → Z

Ω

∇θ · ϕu dx.

As a result:

ε→0 lim Z

Ω

ε

\C

ε

∇θ ^ε · (∇Φ ^ε + Φ ^ε u ^ε ) dx = Z

Ω

∇θ · (∇ϕ + ϕu) dx. (60) Now, we come to the remaining part, namely

Z

Cε

∇θ ^ε · (∇Φ ^ε + Φ ^ε u ^ε ) dx = Z

Cε

∇θ ^ε · (∇ϕ + ϕu ^ε ) dx +

Z

Cε

∇θ ^ε ·(∇w ^ε (ψ ^ε − ϕ) + w ^ε (−∇ϕ) + w ^ε u ^ε (ψ ^ε − ϕ)) dx := I 1 + I 2

(61)

We have I 1 =

Z

Cε

∇θ ^ε · (∇ϕ + ϕu ^ε ) dx = Z

Cε

∇θ ^ε · ∇ϕ dx + Z

Cε

∇θ ^ε · ϕu ^ε dx. (62) In the first term, 1 _C

_ε

∇ϕ → 0 in L ² (Ω) and ∇θ ^ε * ∇θ in L ² (Ω) imply

Z

Cε

∇θ ^ε · ∇ϕ dx → 0. (63)

The second term in (62) is handled by using the estimate:

|u ^ε | _C

_ε

= |1 _C

_ε

u ^ε | Ω ≤ |u ^ε − u| Ω + |u| _C

_ε

,

where the right hand side tends to zero due to (58). Using ∇θ ^ε * ∇θ in L ² (Ω) again, we deduce that

Z

Cε

∇θ ^ε · ϕu ^ε dx → 0, (64) and hence I 1 tends to zero.

It remains to study the integral I ₂ in (61). To that aim, first notice that I 2 =

Z

Cε

∇θ ^ε · ∇w ^ε (ψ ^ε − ϕ) dx =

= Z

Cε

∇θ ^ε · ∇w ^ε (ψ ^ε − ϕ ^ε ) dx + Z

Cε

∇θ ^ε · ∇w ^ε (ϕ ^ε − ϕ) dx (65) where ϕ ^ε has been defined by (49). The second term in the right-hand side of (65) may be estimated by

| Z

Cε

∇θ ^ε · ∇w ^ε (ϕ − ϕ ^ε ) dx| ≤ |∇θ ^ε | Ω |∇w ^ε | Ω |ϕ − ϕ ^ε | ∞ . (66) As (w _ε ) is bounded in H ¹ (Ω), (see Proposition 4.1), the right hand side of (66) tends to zero by the uniform continuity of ϕ over Ω.

Going back to the first term in the right hand side of (65), we may write Z

Cε

∇θ ^ε · ∇w ^ε (ψ ^ε − ϕ ^ε ) dx

= X

k∈Z

ε

Z 2π 0

dΦ Z π

0

sin Θ dΘ Z R

ε

r

ε

∂θ ^ε

∂r

C

^k

(r

_ε

,R

_ε

)

dw ε

dr r ² dr Z

−

S

^k_rε

ψ dσ − Z

−

S

^k_rε

ϕ dσ

!

= r ε R ε

(R _ε − r _ε ) X

k∈Z

_ε

Z

S

1

(θ ^ε | _|x−εk|=r

_ε

− θ ^ε | _|x−εk|=R

_ε

) Z

−

S

^k_rε

ψ dσ − Z

−

S

^k_rε

ϕ dσ

! dσ 1

= 4πr _ε R _ε ε ³ (R ε − r ε )

Z

Ω

(˜ τ ^ε − θ ˜ ^ε )(ψ ^ε − ϕ ^ε ) dx = 4πγ _ε

1 − _R ^r

^ε

ε

Z

Ω

(˜ τ ^ε − θ ˜ ^ε )(ψ ^ε − ϕ ^ε ) dx from which we infer that I 2 is converging to

4πγ Z

Ω

(τ − θ)(ψ − ϕ) dx, and the proof is completed.

We are in the position to state a part of our main result:

Corollary 4.4 The limit (u, θ, τ ) verifies the following equations:

u∇θ − ∆θ + 4πγ(θ − τ) = f in Ω, (67)

γ(τ − θ) = b

3 g in Ω. (68)

Proof. Consider the variational formulation (10) with the test function Φ = Φ ^ε defined by (57) for any ϕ, ψ ∈ D(Ω). Then, the left-hand side tends to

Z

Ω

∇θ · (∇ϕ + ϕu) dx + 4πγ Z

Ω

(τ − θ)(ψ − ϕ) dx. (69) This is a direct consequence of Proposition 4.3 together with the remark that

Z

T

ε

∇θ ^ε ∇Φ ^ε dx = 0 since Φ ^ε is constant on every B(εk, r _ε ), k ∈ Z _ε .

The convergence of the right-hand side is obtained by using the uniform conti- nuity of ψ and by Proposition 3.1. Thus we find the variational formulation of (67)-(68) and the proof is completed.

5 The homogenized problem

Proposition 3.3 yields the existence of some u ∈ H ₀ ¹ (Ω; R ³ ) with div(u) = 0 and for which the following convergence holds on some subsequence

ˆ

u ^ε * u in H ₀ ¹ (Ω; R ³ ).

From [7], we find that there exists an extension of the pressure (denoted by ˆ p ^ε ) and some p ∈ L ² (Ω) such that

ˆ

p ^ε * p in L ² (Ω)/R.

We denote by (w _ε ^k , q ^k _ε ) ∈ H ¹ (C(r ε , ^ε ₂ )) × L ² ₀ (C(r ε , ^ε ₂ )) the only solution of the following Stokes problem

divw ^k _ε = 0 in C(r ε , ε

2 ),

−∆w ^k _ε + ∇q ^k _ε = 0 in C(r ε , ε 2 ), w ^k _ε = 0 if r = r ε , w ^k _ε = e ^(k) if r = ε

2 . Consequently, we define

v _ε ^k (x) =







0 if x ∈ T _ε ,

w _ε ^k (x − εi) if x ∈ C ⁱ (r _ε , ^ε ₂ ), i ∈ Z _ε , e ^(k) if x ∈ Ω ε \ ∪ _i∈Z

_ε

C ⁱ (r ε , ^ε ₂ ).

For ϕ ∈ D(Ω), we set v = ϕv ^k _ε in (9) and then using the energy method like in [1] we find the equation that the velocity field satisfies in H ⁻¹ (Ω):

−∆u + 6πγu = −∇p + aθe ⁽³⁾ in Ω. (70)

Finally, we summarize the results of Proposition 3.3, Corollary 4.4 together with the relation (70) into our main theorem.

Theorem 5.1 If (u ^ε , p ^ε ) is a solution of problem (12), then the following con- vergences hold on some subsequence

ˆ

u ^ε * u in H ₀ ¹ (Ω; R ³ ), θ ^ε * θ in H ₀ ¹ (Ω), θ ^ε dm ε

* ? τ dx in M b (Ω),

where (u, θ) ∈ H ₀ ¹ (Ω; R ³ ) × H ₀ ¹ (Ω), which stand for the macroscopic velocity and temperature of the fluid, and τ ∈ L ² (Ω), which stands for the macroscopic temperature of the vanished suspensions, form a solution of the following system:

divu = 0 in Ω,

−∆u + 6πγu = −∇p + aθe ⁽³⁾ in Ω, u∇θ − ∆θ + 4πγ(θ − τ ) = f in Ω,

4πγ(τ − θ) = 4πb

3 g in Ω.

Remark 5.2 In the present case, with suspensions of critical size, the Brinkman- Boussinesq equation was an expected result; nevertheless, our proof is different from that of [7], which treated the homogenization of the Navier-Stokes equa- tions for perforated domains in a similar case.

Remark 5.3 Our two-temperature model, with γ as transfer coefficient, is the

macroscopic effect of the assumption on the the ratio of the fluid/solid conduc-

tivities.

Remark 5.4 The appearance of the source term 4πb

3 g in the second macro- scopic heat equation is strictly the consequence of the assumption on the mi- croscopic radiation.

Acknowledgements. This work was done during the visit of Fadila Ben- talha and Dan Poli¸sevschi at the I.R.M.A.R.’s Department of Mechanics (Uni- versity of Rennes 1) whose support is gratefully acknowledged.

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* University of Batna, Department of Mathematics, Batna, Algeria,

** Universit´ e de Rennes1, I.R.M.A.R, Campus de Beaulieu, 35042 Rennes Cedex (France)

*** I.M.A.R., P.O. Box 1-764, Bucharest (Romania).