This paper deals with the homogenization of a nonlinear problem mod-elllng chemica! reactive flows through periodically perforated domains. The chemical reactions take place on the walls of the porous médium. The effective behavior of these reactive flows is described by a new elliptic boundary-value problem contalning an extra zero-order term which captures the effect of the chemical reactions

Conca, C.

Díaz, J.I.

Liñán Martínez, Amable

Timofte, C.

English

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

New Trenas in Continuum Mechantes, 99-107 
©Theta2005 
HOMOGENIZATION RESULTS FOR CHEMICAL REACTIVE FLOWS 
THROUGH POROUS MEDIA 
C. CONCA, J.L DÍAZ, A. LIÑÁN AND C. TIMOFTE 
ABSTRACT. This paper deals with the homogenization of a nonlinear problem mod-
elllng chemica! reactive flows through periodically perforated domains. The chemical 
reactions take place on the walls of the porous médium. The effective behavior of 
these reactive flows is described by a new elliptic boundary-value problem contalning 
an extra zero-order term which captures the effect of the chemical reactions. 
1. INTRODUCTION 
The aim of this paper is to study the homogenization of sorae chemical reactive 
flows through periodically perforated domains or porous media. We will focus our 
attention on a nonlinear problem which describes the motion of a fluid reacting on 
the boundary of a porous médium. 
Let ti be an open bounded set in K" and let us perfórate it by holes. As a result, 
we obtain an open set tie which will be referred to as being the perforated domain; e 
represents a small parameter related to the characteristic size of the perforations. 
The nonlinear problem studied in this paper concerns the stationary reactive flow 
of afluid confined in Cle, of concentration uc, reacting on the boundary of the perfora-
tions. A simplified versión of this problem can be written as follows: 
Í -Df&uE = f mtie, 
~Df^-= aeg{u£) onSE, 
u£=o on an. 
Here, v is the exterior unit normal to ti£, a > 0, / E L2{Cl) and SE is the boundary of 
our porous médium Q. \ ti£. Moreover, the fluid is assumed to be homogeneous and 
isotropic, with a constant diffusion coefficient Df > 0. 
The semilinear boundary condition on Se in problem (1.1) describes the chemical 
reactions which take place locally at the interface between the reactive fluid and the 
perforations. From strictly chemical point of view, this situation represents, equiva-
lently, the effective reaction onthe walls of the porous médium between the fluid filling 
íle and a rigid solid part filling the holes. 
The function g in (1.1) is assumed to be given. We shall address here the case of a 
single-valued maximal monotone graph with g(0) = 0, i.e. the case in which g is the 2000 Mathematics Subject Classificatioti. Primary: 35B27; Secondary: 35B40,35J65. Key wordsandphrases. Homogenization, reactive ñows, variational inequality, monotone graph. 
100 C. CONCA. J.I. DÍAZ, A. UÑAN AND C. TIMOFTE 
subdifferential of a convex lower semicontinuous function G. This situation is well 
illustrated by the following important practical example, left as an open case in [7]: 
g(v) -1 v\p~l v, 0 < p < 1 (Freundlich kinetics). 
The exponent p is called the orderofthe reactíon. 
The existence and uniqueness of a weak solution of (1.1) can be settled by using the 
classical theory of semilinear monotone problems (see, for instance, [1] and [9]). As a 
result, we know that there exists a unique weak solution u£ e V£ n H2 {QE), where 
Ve = [veH1^) I v = 0 ondCl}. 
When we associate with Q.£ the following nonempty convex subset of VE: 
(1.2) KE = [vzVE\G{v)\s*eLl(SE)}, 
then uE is also known to be characterized as being the unique solution of the following 
variational problem: 
For all v£ e KE find ue e KE such that 
( L 3 )
 Dr[ DuED(vE-ue)dx-[ fívE-uE)áx + a(iistG(vc)-G{ue))^0, 
where f.tE is the linear form on M^1'1 (Í2) defined by 
{nE)<p) = e^ yda, V<pEW¿'\Q). 
From a georaetrical point of view, we shall just consider periodic structures ob-
tained by removing periodically from Q, with period eY (where 7 is a given hyper-
rectangle in IR"), an elementary hole T which has been appropriately rescaled and 
which isstrictly included in Y, le. Te: Y. 
We shall prove that the solution uE, properly extended to the whole of £2, converges 
to the unique solution of the following variational inequality: 
(1.4) f QDuD{v-u)dx*> ¡ f{V-u)áx~a~~ f (G(v) - G{u)) dx Jn Jn \Y\T\ Jo. 
for u e tfjcn) and for all v e JfJ(Q). Here, Q = (Gyy)) is the classical homogenized 
matrix, whose entries are defined as follows: 
(1.5) qn = DríSi i + - i — í & dy) 
*v f{ u \y\T\h\Tdyi y¡ 
in terms of the functions %} i = 1,..., n, solutions of the so-called cell problems 
(1.6) 
- A ^ = 0 in Y \ T, 
r =0 on oí , 
dv 
M F-periodic. 
Remark 1.1. For the case of a smooth function g, we refer to [4]. 
HOMOGENIZATION RESULTS FOR CHEMICAL REACTIVE FLOWS THROUGH POROUS MEDIA 101 
The structure of the paper is as follows: first, let us mention that we shall just focus 
on the case n 2* 3, which will be treated explicitly. The case n = 2 is much more simpler 
and we shall omit to treat it. Section 2 is devoted to the setting of our problem. In 
Section 3 we formúlate the main result of this paper. Section 4 contains some necessary 
preliminary results. In the last section we give the proof of our main result. 
Finally, notice that throughout the paper we denote by C a generic fixed strictly pos-
itive constant, whose valué can change from line to Une. 
2. SETTING OF THE PROBLEM 
Let £1 be a smooth bounded connected open subset of Un {n > 3) and let Y = 
[0, l\ [x • • • x [0, /„[ be the representative cell in R". Denote by T an open subset of Y 
with smooth boundary dT such that T c 7. We shall refer to T as being tiie elemen-
tary hole. 
Let e be a real parameter taking valúes in a sequence of positive numbers converging 
to zero. For each e and for any integer vector fceZ", set T£ the translated image of ET 
by the vector kl - (fci lj,.,., kn ln) 
T¡ = e(kl+T). 
The set T | represents the holes in R". Also, let us denote by TE the set of all the holes 
contained in O, i.e. 
r£=u{^|r|cn,fcez'1}. 
Set 
Henee, Q.E is a periodically perforated domain with holes of size of the same order as 
the period. Remark that the holes do not intersect the boundary dQ. 
Let 
SE = U{dTEk\TckcQ,k£Zn}. 
So 
dn£=dausE. 
We shall also use the following notations: 
\ü)\- the Lebesgue measure of any measurable subset o) of Rn, 
X = the characteristic function of the set a), (2.1) "(O 
\Y*\ Y* = Y\T and p = m 
Moreover, for an arbitrary function yr e I2(0£), we shall denote by y/ its extensión by 
zero inside the holes. 
As already mentioned, we are interested in studyingme behavior of the solution, in 
such a perforated domain, of the following problem: 
(2.2) 
~D¡kuE -f in H£, 
~ &uE ,
 P, „F 
-Df—— = aegiu") on SE, 
J
 av 
u
E
 = 0 on dü. 
102 C. CONCA, J.I. DÍAZ, A. UÑAN AND C. T1M0FTE 
We shall treat the case in which the function g appearing in (2,2) is a single-valued 
maximal monotone graph in IR x R, satisfying the condition g(0) = 0. Moreover, if we 
denote by D{g) the domain of g, i.e. D{g) = íí e R I g(<í) 7* 0}, then we suppose that 
D{g) = R. We also assume that g is continuous and there exist C > 0 and an exponent 
4, with0 ^ q <nl[n~2), suchthat 
(2.3) \gív)\<C{\ + \v\% 
This situation is weü illustrated by the following important practical example: 
g(y) = \vf~l v, O < p < 1 (Freundlich kinetics). 
We know that in this case títere exists a lower semicontinuous convex function G 
from U to ] - oo,+oo], G proper, i.e. G ?á +oo such that g is the subdifferential of G, 
g = dG{G is an indefinite "integral" of g). Let G(v) = /0" g(s)ds. 
Let us introduce the functional space VE = {v e Jí1 (Í2e) | y = 0 on di}}, with || v\\Vc = 
|j V y ||L2(ílf). Define the convex set 
(2.4) K£={veVE | G(i/)|s« el^CS*)}. 
For a given function / e l 2 (Q) the weak solution ofthe problem (2.2) is also the unique 
solution ofthe following variational inequality: 
For all vE e K£find ue e KE such that 
í2
'
5)
 Df[ DuED(uE-u£)dx~ [ f{ve-u£)áx+a(u£,G{.vE)-G{uE))^Q. 
Notice that there exists a unique weak solution uE e Ve n /f2(0E) ofthe above varia-
tional inequality (see [1]). 
3 . T H E MAIN RESULT 
First, let us notice that it is well-known that the solution u£ of the variational in-
equality (2.5) is also the unique solution ofthe minimization problem: 
u
£
eK£t J£{uE)= inf f(v), 
ve.K£ 
where 
(v) = -Df[ \Dv\2áx + aUE,G{v))-{ fváx. 2 Jn* Jn* 
Introduce the following functional defined on H¿ (£1): 
l\v) = \ ( QDvDváx+a^¡ [ Gtv)dx- [ fváx, 
2 Ja. \Y*\Ja Jn 
The main result of this paper is the following: 
Theorem 3.1. One can construct an extensión PE uE ofthe solution u£ ofthe variational 
inequality (2.5) such that 
P£uE — u weaklyin HQ(P), 
where u is the unique solution ofthe minimization problem: 
(3.1) Find ueJíJ(Q) such that J°(u)= inf f{v). 
veH¡[£l) 
HOMOGENIZATION RESUUFS FOR CHEMICAL REACTIVE FLOWS THROUGH POROUS MEDIA 103 
Moreover, G{u) e I 1 (Q). Here, Q = (q^) is the classical homogenized matrix, whose en~ 
tríes are deflned asfollows: 
(3.2,
 qiJ,Df(Sí¡ + ±.fy^iy] 
in termsofthefunctionsx.,i = l,...,n, solutionsoftheso-calledcellproblems 
(3.3) 
-A^O inY\ 
=0 on dT, 
dv 
.Xi Y-periodic. 
4. PRELIMINARY RESULTS 
4.1. An extensión result. The solution u£ of problem (2.2) beingdefined only on D.s, 
we need to extend it to the whole of Q to be able to state the convergence result. In 
orderto do that, letus recall the following well-known extensión result (see [3]): 
Lemraa4.1.1. There exists P£ e £e(.L2{nEy,L2íO))n&(yE;Hl(Q.)) a linear continuous 
extensión operatorandapositiveconstante, independentofe, such that 
\\pEv\\mm^c^vhHan 
and 
pPe4mn)^c\[Vv\\L2m, 
foranyve VE. 
An immediate consequence of the previous lemma is the following Poincaré's in-
equalityin VE: 
Lemma 4.1.2. There exists a positive constant C, independent ofe, such that 
| | i / | | L 2 ( n c ) « C | | V y ! | L 2 ( n e ) , 
foranyve Ve. 
4.2. Aconvergenceresult. Inorderto getthe effectivebehaviorofour solution uEwe 
have to pass to the limit in (2.5). Let us introduce therefore, for any h e Ls> {dT), 1 «s s' «s 
oo, the linear form ¡iEh on W^ífí) defined by 
(!%,¥) = ejth[±)<pdff, VcpEW¡>síCl), 
with l/s+ lis' = 1. Itis proved in [2] that 
(4.1) fj?h - \ih strongly in (W¿'sm', 
where (i¿¡v(p) = fj.hfa(pdx, with ¡xn = (.l/\Y\)fdTh(y)da. In the particular case when 
h e L°°(dT) or even h is constant, we have 
fjFh - \ih strongly in W~lfi°iCl). 
104 C. CONCA, J.I. DÍAZ, A. UÑAN AND C. TIMOFTE 
In what follows, we denote by \iE the above introduced measure in the particular case 
inwhich/2 = L Notice that in this case ^ becomes \i\ ~ \dT\l\Y\. 
Let F be a continuously differentiable function, monotonously non-decreasing and 
such that F(y) = 0 if and only if v - 0. We suppose that there exist a positive constant 
C and an exponent q, with 0 *s q < nl(n - 2), such that 
dF (4.2) 
dv 
Let us prove now that for any q> e 0(0) and for any vE — v weakly in H¿ (O), we get 
(4.3) ipF{ve) - (pF{v) weakly in W¡'*{&), 
where 
_ 2n q- • 
q{n~2) + n 
In order to prove (4.3), let us first note that 
(4.4) sup||VF(z/)||L?(íl)<oo. 
Indeed, from the growth condition (4.2) imposed to F, we get 
^— dx 
ax¡ I Jn\dxi j Jnx } 
:(i+(/Qi„rH"r)(/niv"fSd*) 
where we have taken y and 6 such that qS -2, l/j+1/5 = 1 and q~qy = 2nt[n -2). 
Notice that from here we get q — 2nl(q[n- 2) + n). We also have q > 1 since 0 «é q < 
nKn-2). Now, as 
suplí/II 2lL <oo, 
we immediately get (4.4). Henee, to obtain (4.3), it only remains to prove that 
(4.5) F{vE) - F[v) strongly in L* (íl). 
But this is just a consequence of the following well-known result (see [9]): 
Theorem 4.2.1. LetH:£lxU~*Ubea Carathéodory function, le. 
(a) For euery v the function H( -, v) is measurable with résped toxeCl. 
(b) For every (a.e.) xeQ, the function H(x,-) is continuous with respect to v. 
Moreover, if we assume that there exists a positive constant C such that 
\H(x,v)\^Ca + \v\rlt), 
with r s* 1 and t<oo, then the map v £ Lr(Ü) -* H{x, v(x)) e I f(ü) is continuous in the 
strong topologies. 
Indeed, since 
\F{v)\*zC(l + \v\1+l), 
then by applying the above theorem for H{x, v)=F{v), t = q and r = (2n/ (n - 2)) - r', 
with r' > O such that q + 1 <rlt and using the compact injectíon Hl (ft) >— Lr(¡Q) we 
easily get (4.5). 
HOMOGENIZATION BESULTS FOR CHEMICAL REACTIVE FLOWS THHOUGH POROUS MEDIA 105 
5 . PROOF OF THE MAIN RESULT 
This section is devoted to the proof of Theorem 3.1. 
Proof. Let u£ be the solution of the variational inequality (2.5) and let Pe ue be the ex-
tensión of uE given by Lemma 4.1.1. It is not difficult to see that PE uF is bounded in 
HQ (Q.). So by extracting a subsequence, one has 
(5.1) Peu£-u weaklyin JÍ(J(Q). 
Let ip e@(Í3). Classical regularity results imply x¡ e L°°, The boundedness of %i and (p 
ensure that there exists Aí>0 such that 
dxi LJXtfo-<M. 
Let 
(5.2) ^
 + ££g (^.(í). 
Then vE £ K£ and we can take it as a test function in (2.5). Moreover, vE — cp strongly 
inl 2 (n) . 
Let us compute DvE: 
So 
where e,-, 1 ** / < n, are the elements of the canonical basis in Un. 
Using v£ as a test function in (2.5) we can write 
Df[ Du£Dvsdx>{ f{ve~uE)dx+Df[ DuEDuEdx-a(ff,G{ve)~G{ue)). 
Jn* Jw Jnc 
In fact, we have 
Df [ DP£ue(DvE)dx 
(5.3) j f i 
>í f(vE~uE)dx+Df( DueDuEdx-a(fi£,G(ve)~G{u£)). 
Denote 
(5.4) pQej = Í ^ J D ; f^ {DXj + ey) dy, 
where p = |7* | / |7 | . Neglecting the term eY.iD^{x)%.{fi which actually tends 
strongly to zero, we can immediately pass to the limit in the left-hand side of (5.3). 
Henee 
(5.5) Df ¡ DP£ueDv£dx^ I pQDuDípdx. 
Jn Jn 
106 C. CONCA, J.I, DÍAZ, A. UÑAN AND C. TIMOFTE 
It is not difficult to pass to the limit in the first term of the right-hand side of (5.3). 
Indeed, since v£ —• <p strongly in L2(Q), we get 
(5.6) í f{v£-ue)dx = [ fxnAv£-PeuE)dx-> [ fp{<p-u)dx. Jn£ Ja Jn 
For the third term of the right-hand side of (5.3), by assuming the growth condition 
(2.4) for the single-valued maximal monotone graph g and by using the preliminary 
results from Section 4.2 (more precisely (4.3) written for G and for v£ = PEue), we get 
G(PEuE) - G{u) weakly in W^{Q) 
which in cornbination withthe convergence (4.1) written for h = 1, leads to 
< ^ , G ( P V ) ) ^ ^ f Gíiddx 
\¿ I Jn 
In a similar manner, we obtain 
{^MvE))~*l^~faGícp)dx 
and henee we get 
(5.7) a(fj.E,GlvE)~G(P£u£))^a-^ f (G(<p)-G(u))dx 
It remains to pass to the limit in the second term of the right-hand side of (5.3) only. 
We start by writting down the subdifferential inequality 
(5.8) DFÍ Du£Du£dx>Dr[ DwEDwEdx + 2Df [ DwE(DuE-DwE)dx, 
JaE J Jn* JnE 
for any wE e H¿ (£1). We followthe same procedure as above and by choosing 
where lp enjoys similar properties as the corresponding <p, we may pass to the limit in 
the right-hand side of inequality (5.8) and we get 
liminfDf i Du£Du£dx** ¡ pQDlpDlpdx + Z \ pQD7p{Du-Dw)dx} 
«-o JQE Jn Jn 
for any Tp e ©(Í2). But since u e H¿ (O), by density, we conclude 
(5.9) liminfDf / DueDuedx& \ pQDuDudx. 
e-0 Jne JQ 
We put together (5.5)-(5.7) and (5.9) and obtain 
l pQDuD<pdx> I fp((p-u)dx+ \ pQDuDudx~a~-+ \ (G(tp)-G(u))dx, 
Jn Jn Jn \Y\ Ja 
for any </? e @(0) and henee by density for any v e HQ (Q). 
So, finally, we get 
I QDuD(v-u)dx> f f(v-u)dx-a—J- [ (G{<p)-G{u))dx, 
Jn Jn \Y *|Jn 
which gives exactly the limit problem (3.1). This ends the proof of Theorem 3.1. • 
HOMOGENIZATION RESU1TS FOR CHEMICAL REACTIVE FLOWS THROUGH POROUS MEDIA 107 
Remark 5.1. We can treat in a similar manner the case of a multi-valued maximal 
monotone graph, which includes various semilinear classical boundary-value prob-
lems such as Dirichlet or Neumann problems, Robín boundary conditions, Signorini's 
unilateral conditions, climatization problems (see, for instance, [1,2, 6]). For the non-
stationary case, see also [5] and [8]. 
ACKNOWLEDGEMENTS 
This work has been partially supported by Fondap through its Programme on Math-
ematical Mechanics. 
The first author gratefuily acknowledges Chilean and French Governments support 
through the Scientific Committee Ecos-Conicyt. 
The research of J.I. Díaz was partially supported by project REN2000-0766 of the 
DGES (Spain). J.I. Díaz and A. Liñán are members of the RTN HPRN-CT-2002-00274 
oftheEC. 
The work of the fourth author is part of the European Research Training Network 
HMS 2000, under contract HPRN-2000-00109. 
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auverts perfores, AsymptoticAnal. 1(1988), 115-138. 
[31 D, CIOUANESCU, J. SAINT JEAN PAULIN, Homogenization in open sets witli lióles, J. Math. Anal. Appl. 
71(1979), 590-607. 
[41 C. CONCA, J.I. DÍAZ, A. LIÑAN, C. TIMOFTE, Homogenization in chemical reactive flows, Electron. J. 
Differential Bquations 40(2004), 1-22. 
[5] C. CONCA, J.I. DIAZ, C. TIMOFTE, Effective chemical processes in porous media, Math. ModelsMethods 
Appl. Sel 13(2003), 1437-14S2. 
[6] C. CONCA,F. MURAT.C.TIMOFTE,AgeneralizedstrangcterminSignorini'stypeproblems,M2ANMath. 
ModelNumer.Anal. 37(2003), 773-806. 
[7] U. HORNUNG, Homogenization and Poroiis Media, Spríngei; New York, 1997. 
[8] U. HORNUNG, W. JAGER, Diffusion, convection, adsorption and reaction of Chemicals in porous media, 
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París, 1969; Dunod, 2002. 
DEPARTAMENTO DE INGENIERÍA MATEMÁTICA AND CENTRO DE MODELAMIENTO MATEMÁTICO, UMR 2071 
CNRS-UCHILE, FACULTAD DE CIENCIAS FÍSICAS V MATEMÁTICAS, UNIVERSIDAD DE CHILE, CASILLA 170/3, 
SANTIAGO, CHILE; E-raali: cconca@dim.uchile.cl 
DEPARTAMENTO DE MATEMÁTICAAPLICADA, FACULTAD DE MATEMÁTICAS, UNIVERSIDAD COMPLUTENSE, 
28040 MADRID, SPAIN; E-mall: jidiaz@ucm.es 
ESCUELA T.S. DE INGENIEROS AERONÁUTICOS, UNIVERSIDAD POLITÉCNICADE MADRID, MADRID, SPAIN; 
E-mail: linan@tupi.dmt.upm.es 
DEPARTMENT OF MATHEMATICS, FACULTY OF PHYSICS, UNIVERSITY OF BUCHAREST, P.O. Box MG-il , 
BUCHAREST-MÁGURELE, ROMANÍA; E-mail: claudiatimofte@hotmail.com 


Homogenization results for chemical reactive flows through porous media

Proceedings of the international conference New Trends in Continuum Mechanics,  Constanta, Romania September 8-12, 2003.This paper deals with the homogenization of a nonlinear problem modelllng chemical reactive flows through periodically perforated domains. The chemical reactions take place on the walls of the porous médium. The effective behavior of these reactive flows is described by a new elliptic boundary-value problem contalning an extra zero-order term which captures the effect of the chemical reactions.Fondap-Programme on Mathematical MechanicsChilean and French Governments-Scientific Committee Ecos-ConicyDGES (Spain)ECEuropean Research Training Network HMS 2000Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu

Díaz Díaz, Jesús Ildefonso

LIñán, A.

Docta Complutense

Archivo Digital UPM

convection, adsorption and reaction of Chemicals in porous media,

Effective chemical processes in porous media,

Homogénéisation du probléme de Neumann non homogéne dans des auverts perfores,

Homogenization and Poroiis Media, Spríngei;

Problémes unilatéraux,

http://oa.upm.es/1534/1/LINAN_P2005_01.pdf

Homogenization results for chemical reactive flows through porous media

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