Homogeniza ion o he Da cy-Lapwood-B inkman low in a
hin domain wi h highly oscilla ing bounda ies
Igo PAˇ
ZANIN
Depa men o Ma hema ics
Facul y o Science, Uni e si y o Zag eb, Bijeniˇcka 30, 10000 Zag eb, C oa ia
[email p o ec ed]
F ancisco Ja ie SU´
AREZ-GRAU
Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico
Facul ad de Ma em´a icas, Uni e sidad de Se illa, 41012-Se illa (Spain)
[email p o ec ed]
Abs ac
In his pape we in es iga e he low h ough a hin co uga ed domain illed wi h luid sa u a ed
po ous medium. The po ous medium low is desc ibed by he nonlinea Da cy-Lapwood-B inkman
model acknowledging he iscous shea and he ine ial e ec s. The hickness o he domain is assumed
o be o he same small o de εas he pe iod o he oscilla ing bounda ies. Depending on he magni ude
o he pe meabili y wi h espec o ε, we igo ously de i e di e en asymp o ic models and compa e he
esul s wi h he non-oscilla o y case. We employ a homogeniza ion echnique based on he adap ion o
he un olding me hod and deduce he in luence o he po ous s uc u e and bounda y oscilla ions on he
e ec i e low.
AMS classi ica ion numbe s: 35B27, 35B40, 76S05.
Keywo ds: Da cy-Lapwood-B inkman equa ion; hin domain; highly oscilla ing bounda y; un olding
me hod.
1
1 In oduc ion
Nume ous models ha e been de eloped in he pas i een decades o desc ibe lows h ough po ous me-
dia. Da cy’s law [13] is, wi hou doub , he mos popula one exp essing ha he il a ion eloci y is
p opo ional o he d i ing p essu e g adien . Howe e , i is well known ha his simple law is only alid
i a a ie y o he condi ions a e being me and, he e o e, i is inapplicable in many physically ele an
se ings. Fo ins ance, as a i s o de PDE o he eloci y, Da cy’s equa ion canno sus ain he no-slip
bounda y condi ion imposed on an impe meable bounda y. Also, i he e ec s o ine ia a e impo an o
he p ocess (e.g. due o he cu ilinea i y o he low pa h), he Da cy’s law again canno be applied. Thus,
in such si ua ion when iscous shea and mac oscopic ine ial e ec s a e signi ican , i has been cus oma y
o use he so called Da cy-Lapwood-B inkman (DLB) equa ion o model he po ous medium low. This
model akes he o m (see e.g. Nield and Bejan [21]):
−µe∆u+∇p+µ
Ku= −ρ
φ2(u· ∇)u , (1.1)
di u= 0 ,(1.2)
whe e uand p ep esen s he il e eloci y and p essu e, µis he dynamic iscosi y coe icien , µedeno es
he e ec i e iscosi y o he luid in he po ous medium, ρis he luid densi y, φis he po osi y, is an
ex e io o ce, while Ks ands o he pe meabili y o he po ous medium. No e ha he second-o de DLB
equa ion (1.1) is capable o handling he p esence o he solid bounda y on which he no-slip condi ion o
he eloci y is imposed. Mo eo e , he e ec s o low ine ia a e also being inco po a ed making he model
(1.1)-(1.2) an impo an gene aliza ion o he Da cy law ha has sound physical basis.
Due o nonlinea i y o he equa ion (1.1), he Da cy-Lapwood-B inkman model has been mos ly ea ed
nume ically (see e.g. Chen e al. [11], Khalilli e al. [16], Umawa hi e al. [24]). Analy ical ea men s a e
spa se and add ess only simple 2D ac u es wi h plane-pa allel walls unde addi ional assump ions which
linea ize he momen um equa ion (1.1). We e e he eade o he pape s by Hamdan e al. [7, 15]. In
iew o ha , he goal o his pape is o analyze he 3D luid low h ough a hin laye o po ous medium
sandwiched be ween wo co uga ed walls and go e ned by (1.1)-(1.2). Fi s , in Sec ion 2, we s udy he
low in a hin cons ic ed ac u e wi hou bounda y oscilla ions, namely:
Ωε=(x0, x3)∈R2×R:x0∈ω, εh−x0< x3< εh+x0.(1.3)
He e ωis a smoo h bounded open se in R2, while h−and h+a e smoo h unc ions such ha h+> h−on ω.
Assuming ha pe meabili y K=Kεmay depend on he small pa ame e ε, we employ a homogeniza ion
echnique wi h espec o εand igo ously de i e h ee di e en e ec i e models (see Theo em 2.1):
- I Kε≈ε2, i.e. when he pe meabili y is o o de ε2, we ob ain a 2D Da cy law as an e ec i e model
including he e ec s o he domain’s geome y and he po ous s uc u e.
- I Kεε2, we ob ain a 2D p essu e-d i en Da cy law as an e ec i e model no accoun ing he e ec s
o he B inkman ( iscous) e m.
- I Kεε2, we ob ain no con ibu ion o he po ous s uc u e a he mac oscopic model. As a esul ,
we ob ain a solu ion in he o m o he Poiseuille low.
2
Mos ecen ly, Paˇzanin and Siddheshwa [22] conside ed he simila p oblem, bu in he case o he 2D
low. I is wo h men ioning ha he e ec i e exp ession in he c i ical case p o ided in Theo em 2.1 is
consis en wi h he one o mally ob ained in [22] ia wo-scale expansion me hod.
The Sec ion 3 is he cen al pa o he p esen wo k. He e we in oduce he oscilla ions a he op
and he bo om o he low domain, namely we assume ha he pe iod o he oscilla ions has he same
small o de as he domain hickness. In iew o ha , he domain o be conside ed is he ollowing:
Λε=(x0, x3)∈R2×R:x0∈ω, εh−x0
ε< x3< εh+x0
ε,(1.4)
o pe iodic unc ions h−and h+. As abo e, we aim o de e mine he asymp o ic beha io (as ε→0) o
he low go e ned by (1.1)-(1.2), now posed in Λε. The p oo o ou esul s is based on an adap a ion o
he un olding me hod (see A bogas e al. [6], and Cio anescu e al. [12]), which is s ongly ela ed o he
wo-scale con e gence me hod (see Allai e [2], Ngue seng [20] and also Ma uˇsi´c-Paloka e al. [18]). The
un olding me hod has been ex ensi ely used o s udy pe iodic homogeniza ion p oblems whe e he size
o he pe iodic cell ends o ze o. We e e he eade o a ecen wo ks by Anguiano and Su´a ez-G au
[3]-[5]. The basic idea is o in oduce sui able changes o a iables which ans o m e e y pe iodic cell in o
a simple e e ence se by using a supplemen a y a iable (mic oscopic a iable). In he p esen se ing, i
is necessa y o combine he un olding me hod wi h a escaling in he heigh a iable in o de o be able o
wo k wi h a domain o a ixed heigh . In pa icula , due o he bounda y oscilla ions, an ex ension ope a o
needs o be cons uc ed in o de o ex end he p essu e o an ε-independen domain. Consequen ly, we
manage o iden i y he c i ical size and la e on he e ec s o he mic os uc u e (bounda y oscilla ions) in
he co esponding e ec i e equa ions. I u ns ou ha he c i ical size is exac ly he same as he one we
ob ain o he non-oscilla o y case. Mo eo e , depending on he magni ude o he pe meabili y Kεwi h
espec o ε, we de i e h ee di e en cha ac e is ic cases (see Theo em 3.1):
- I Kε≈ε2, we ob ain a 2D Da cy law as an e ec i e model which includes bo h he e ec s o he
po ous s uc u e and he bounda y oscilla ions gi en by he local Da cy-B inkman p oblems in 3D.
- I Kεε2, we ob ain a 2D p essu e-d i en Da cy law as an e ec i e model no accoun ing he e ec s
o he B inkman ( iscous) e m, bu including he e ec s o he bounda y oscilla ions p o ided by
he local Hele-Shaw p oblems in 2D.
- I Kεε2, as in he non-oscilla o y case, we ob ain no con ibu ion o he po ous s uc u e a he
e ec i e model, bu he e he e ec s o he bounda y oscilla ions a e p esen h ough local S okes
p oblems in 3D.
To conclude, we belie e ha he analysis p esen ed in his pape is ins umen al o unde s anding he
e ec i e beha io o he po ous medium low in hin domains wi h highly oscilla ing bounda ies. As em-
phasized abo e, by conside ing he (nonlinea ) Da cy-Lapwood-B inkman equa ion, he impo an ea u es
ha e been aken in o accoun ha canno be cap u ed by a classical Da cy’s law. Consequen ly, he con-
side ed low na u ally inds applica ions bo h in indus y (chemical eac o s, hea exchange s, il e ing
equipmen , e c.) and in geophysical p oblems, see [21] and he e e ences he ein. By employing a ho-
mogeniza ion echnique, he a e aged e ec s o he bounda y oscilla ions and he po ous s uc u e ha e
been elegan ly deduced. I should be men ioned ha such homogenized sys em is o p ac ical in e es o
de eloping nume ical codes since i allows o il e ou he small scales o he bounda y, ha ing a high com-
pu a ional cos . In iew o ha , we hope ha ou esul s could ha e an impac on he known enginee ing
p ac ice.
3
2 Non-oscilla o y case
Th oughou he ex , he poin s x∈R3will be decomposed as x= (x0, x3) wi h x0∈R2,x3∈R.
Co espondingly, o he unc ions we use he same no a ion U= (U0, U3), U0∈R2.
In his sec ion we s udy he low o a iscous luid in he domain Ωεgi en by
Ωε=(x0, x3)∈R3:x0∈ω, εh−(x0)< x3< εh+(x0),
whe e h−, h+∈C1(ω)∩C(ω) such ha h+> h−on ω. We suppose ha he ac u e Ωεis illed by a
luid-sa u a ed spa sely-packed po ous medium. As explained in he In oduc ion, he low h ough he
po ous medium is modeled by he Da cy-Lapwood-B inkman (DLB) equa ion. In iew o ha , le us
conside a sequence (uε, pε)∈H1
0(Ωε)3×L2(Ωε) sa is ying
−µe∆uε+∇pε+µ
Kε
uε= −ρ
φ2(uε· ∇)uε,
di uε= 0 .
(2.5)
To comple e he p oblem, we impose a s anda d no-slip bounda y condi ion
uε= 0 on ∂Ωε.(2.6)
The igh -hand side is o he o m
(x)=( 0(x0),0),a.e. x∈ω,
whe e is assumed o be in L2(ω×(h−
min, h+
max))2. Such choice o is usual and jus i ied when we deal
wi h hin domains. Indeed, since he hickness o he domain is small, hen he e ical componen o he
o ce can be neglec ed and, mo eo e , he o ce can be conside ed independen o he e ical a iable.
Using s anda d echniques (see e.g. Galdi [14]), i can be es ablished ha (2.5)-(2.6) has a leas one
solu ion (uε, pε)∈H1
0(Ωε)3×L2
0(Ωε). The space L2
0(Ωε) is he space o unc ions o L2(Ωε) wi h null
in eg al. Ou aim is o s udy he asymp o ic beha io o uεand pεwhen he hickness ε ends o ze o,
aking in o accoun he magni ude o Kεwi h espec o ε. Fo his pu pose, we use he dila a ion in he
e ical a iable x3:
y3=x3
ε,(2.7)
in o de o ha e he unc ions de ined in an open se independen o εand wi h heigh o o de one:
Ω = (x0, y3)∈R3:x0∈ω, h−(x0)< y3< h+(x0).
We de ine ˜uε∈H1
0(Ω)3, ˜pε∈L2(Ω)/Rby
˜uε(x0, y3) = uε(x0, εy3),˜pε(x0, y3) = pε(x0, εy3), a.e. (x0, y3)∈Ω.
In iew o (2.7), he sys em (2.5) can be ew i en in Ω as
(−µe∆x0˜uε−ε−2µe∂2
y3˜uε+∇x0˜pε+ε−1∂y3˜pεe3+µ
Kε
˜uε= 0−ρ
φ2(˜uε· ∇ε)˜uε,
di x0˜u0
ε+ε−1∂y3˜uε,3= 0 (2.8)
wi h no-slip bounda y condi ion on ∂Ω, i.e.
˜uε= 0 on ∂Ω.(2.9)
The asymp o ic beha io o he sequence (˜uε, ˜pε) is p o ided in he ollowing esul :
4
Theo em 2.1. We dis inguish h ee cha ac e is ic cases:
i) i Kε≈ε2, wi h Kε/ε2→K,0< K < +∞, hen (˜uε/ε2,˜pε)con e ges weakly, as ε ends o ze o, in
H1(h−, h+;L2(ω)3)×L2
0(ω) o (˜u, ˜p), wi h ˜u3= 0 and ˜u= 0 on y3=h−, h+. Mo eo e , ˜p∈H1(ω)
and (e
U0(x0),˜p(x0)) is he solu ion o he e ec i e p oblem











˜
U0(x0) = 2KAM(x0)
Mµ  0(x0)− ∇x0˜p(x0)in ω,
di x0˜
U0(x0) = 0 in ω,
˜
U0(x0)·n= 0 on ∂ω,
(2.10)
whe e ˜
U(x0) = Rh+(x0)
h−(x0)˜u(x0, y3)dy3,M=qµ
K µeand he unc ion AM(x0)is gi en by
AM(x0) = 2−eMh+(x0)−Mh−(x0)−eMh−(x0)−Mh+(x0)
eMh+(x0)−Mh−(x0)−eMh−(x0)−Mh+(x0)=1−ch(Mh+(x0)−Mh−(x0))
sh(Mh+(x0)−Mh−(x0)) .(2.11)
ii) i Kεε2, hen (˜uε/ε2,(Kε/ε2)˜pε)con e ges weakly, as ε ends o ze o, in H1(h−, h+;L2(ω)3)×
L2
0(ω) o (˜u, ˜p), wi h ˜u3= 0 and ˜u= 0 on y3=h−, h+. Mo eo e , ˜p∈H1(ω)and (˜
U, ˜p)is he
unique solu ion o he e ec i e p oblem

















˜
U0(x0, y3) = −A0(x0)
µ∇x0˜p(x0)in ω,
˜
U3(x0)=0 in ω,
di x0˜
U(x0) = 0 in ω,
˜
U(x0)·n= 0 on ∂ω ,
(2.12)
whe e ˜
U(x0) = Rh+(x0)
h−(x0)˜u(x0, y3)dy3and he unc ion A0(x0)is gi en by
A0(x0) = h+(x0)−h−(x0).
iii) i Kεε2, hen (˜uε/ε2,˜pε)con e ges weakly, as ε ends o ze o, in H1(h−, h+;L2(ω)3)×L2
0(ω) o
(˜u, ˜p), wi h ˜u3= 0 and ˜u= 0 on y3=h−, h+. Mo eo e , ˜p∈H1(ω)and (˜
U, ˜p)is he unique solu ion
o he e ec i e p oblem

















˜
U0(x0) = A∞(x0)
12µe 0(x0)− ∇x0˜p(x0)in ω,
˜
U3(x0) = 0 in ω,
di x0˜
U0(x0)=0 in ω,
˜
U0(x0)·n= 0 on ∂ω ,
(2.13)
whe e ˜
U(x0) = Rh+(x0)
h−(x0)˜u(x0, y3)dy3and he unc ion A∞(x0)is gi en by
A∞(x0) = h+(x0)3−3h−(x0)3−3h+(x0)2h−(x0)−h+(x0)h−(x0)2.
5

2.1 P oo o Theo em 2.1
Le us i s ix some no a ion.
We deno e by : he ull con ac ion o wo ma ices, namely o A= (ai,j)1≤i,j≤2and B= (bi,j)1≤i,j≤2,
we ha e A:B=P2
i,j=1 aijbij.
We deno e by Oεa gene ic eal sequence which ends o ze o wi h εand can change om line o line.
We deno e by Ca gene ic posi i e cons an which can change om line o line.
A p io i es ima es: i s we need o de i e he a p io i es ima es o uε. To accomplish his, we employ
some echnical esul s which can be e i ied s aigh o wa dly by a simple change o a iables ( o he p oo
see [18], Lemmas 8 and 11):
Lemma 2.2. The ollowing es ima es hold:
kϕεkL2(Ωε)3≤CεkDϕεkL2(Ωε)3×3,(2.14)
kϕεkL4(Ωε)3≤Cε1
2kDϕεkL2(Ωε)3×3,(2.15)
Now, we p o e he sha p a p io i es ima es o he eloci y uε.
P oposi ion 2.3. Fo uεsa is ying he sys em (2.5)-(2.6),
i) i Kε≈ε2, wi h Kε/ε2→K,0< K < +∞o Kεε2, i holds
kuεkL2(Ωε)3≤Cε5
2.(2.16)
ii) i Kεε2, i holds
kuεkL2(Ωε)3≤Cε3
2K
1
2
ε.(2.17)
Mo eo e , in e e y case i holds
kDuεkL2(Ωε)3×3≤Cε3
2.(2.18)
P oo . We conside uεas es unc ion in he weak o mula ion o p oblem (2.5), and so we ge
µeZΩε
|Duε|2dx +µ
KεZΩε
|uε|2dx =ZΩε
0(x0)·u0
εdx. (2.19)
Using he Cauchy-Schwa z inequali y, 0∈L2(ω)2and (2.14), we deduce
ZΩε
0(x0)·u0
εdx≤Cε3
2kDuεkL2(Ωε)3×3,
leading o
µeZΩε
|Duε|2dx +µ
KεZΩε
|uε|2dx ≤Cε3
2kDuεkL2(Ωε)3×3.(2.20)
6
On he one hand, his implies ha (2.18) holds. Consequen ly, using (2.14) we ge
kuεkL2(Ωε)3≤Cε5
2.(2.21)
On he o he hand, using (2.18) in (2.20), we also ob ain
kuεkL2(Ωε)3≤Cε3
2K
1
2
ε.(2.22)
F om (2.21) and (2.22), we ha e ha
kuεkL2(Ωε)3≤Cε5
2+ε3
2K
1
2
ε.
We compa e ε5
2wi h espec o ε3
2K
1
2
εand obse e ha he c i ical case is when Kε≈ε2which gi es he
es ima e (2.16). In he subc i ical case Kεε2we also deduce (2.16), while in he supe c i ical case we
deduce (2.17).
Co olla y 2.4. Fo ˜uεsa is ying he sys em (2.8)-(2.9),
i) i Kε≈ε2, wi h Kε/ε2→K,0< K < +∞, o Kεε2, he ollowing es ima e holds
k˜uεkL2(Ω)3≤Cε2.(2.23)
ii) i Kεε2, he ollowing es ima e holds
k˜uεkL2(Ω)3≤Cε K
1
2
ε.(2.24)
Mo eo e , in e e y cases i holds
kDx0˜uεkL2(Ω)2×3≤Cε , k∂y3˜uεkL2(Ω)3≤Cε2.(2.25)
P oo . Es ima es (2.23), (2.24) and (2.25) a e easily ob ained om (2.16), (2.17) and (2.18), espec i ely,
by applying he change o a iable (2.7).
Now, we p o e he a p io i es ima es o he p essu e pε. Fo his, we need one mo e echnical lemma
add essing he auxilia y di e gence p oblem (see Lemma 20 om [18]).
Lemma 2.5. The p oblem 


di ϕε= ε∈L2
0(Ωε)in Ωε,
ϕε= 0 on ∂Ωε,
(2.26)
has a solu ion ϕε∈H1
0(Ωε)3such ha
kϕεkL2(Ωε)3≤Ck εkL2(Ωε),kDϕεkL2(Ωε)3×3≤C
εk εkL2(Ωε).(2.27)
P oposi ion 2.6. Fo pεsa is ying he sys em (2.5)-(2.6),
7
i) i Kε≈ε2, wi h Kε/ε2→K,0< K < +∞, o Kεε2, we ha e
kpεkL2(Ωε)≤Cε1
2,(2.28)
ii) i Kεε2, we ha e
kpεkL2(Ωε)≤Cε5
2
Kε
.(2.29)
P oo . We in oduce ϕεas he solu ion o he auxilia y p oblem
di ϕε=pε∈L2
0(Ωε) in Ωε, ϕε= 0 on ∂Ωε.
Acco ding o Lemma 2.5, such p oblem has a leas one solu ion such ha
kϕεkL2(Ωε)3≤CkpεkL2(Ωε),kDϕεkL2(Ωε)3×3≤C
εkpεkL2(Ωε).
We mul iply he sys em (2.5) by ϕεand in eg a ing o e Ωε, we ob ain
kpεk2
L2
0(Ωε)=ZΩε
pεdi ϕεdx≤µeZΩε
Duε:Dϕεdx+ZΩε
·ϕεdx
+
µ
φ2ZΩε
(uε· ∇)˜uεϕεdx+
µ
KεZΩε
uε·ϕεdx.
(2.30)
Taking in o accoun es ima e (2.18) and Lemma 2.5, we ha e
µeZΩε
Duε:Dϕεdx≤CkDuεkL2(Ωε)3×3kDϕεkL2(Ωε)3×3≤Cε1
2kpεkL2
0(Ωε).
Simila ly, we ob ain
ZΩε
·ϕεdx≤Cε1
2kpεkL2
0(Ωε).
Fo he con ec i e e m, om es ima e (2.18) and employing he inequali ies (2.14) and (2.15) and Lemma
2.5, we deduce

µ
φ2ZΩε
(uε· ∇)uεϕεdx≤CkDuεkL2(Ωε)3×3kuεkL4(Ωε)3kϕεkL4(Ωε)3
≤Cε5
2kDϕεkL2(Ωε)3≤Cε3
2kpεkL2
0(Ωε).
Finally, we ge 
µ
KεZΩε
uε·ϕεdx≤CkuεkL2(Ωε)3kϕεkL2(Ωε)3.
Depending on he magni ude o Kεwi h espec o ε, we conclude:
•i Kε≈ε2, es ima es (2.14), (2.16) and Lemma 2.5 yield

µ
KεZΩε
uε·ϕεdx≤Cε1
2kpεkL2(Ωε).
8
•i Kεε2, using es ima e (2.16) we ge

µ
KεZΩε
uε·ϕεdx≤Cε5
2
Kε
kpεkL2(Ωε).
•i Kεε2, using es ima e (2.17) we ge

µ
KεZΩε
uε·ϕεdx≤Cε3
2
K
1
2
ε
kpεkL2(Ωε).
Thus, in iew o (2.30), we deduce ha i Kε≈ε2o Kεε2, we ha e
kpεkL2(Ωε)≤Cε1
2.
Finally, i Kεε2, we ge
kpεkL2(Ωε)≤Cε5
2
Kε
.
Co olla y 2.7. Fo ˜pεsa is ying he sys em (2.8)-(2.9),
i) i Kε≈ε2, wi h Kε/ε2→K,0< K < +∞o Kεε2, we ha e
k˜pεkL2(Ω) ≤C , (2.31)
ii) i Kεε2, we ha e
k˜pεkL2(Ω) ≤Cε2
Kε
.(2.32)
P oo . Es ima es (2.31) a e (2.32) a e easily ob ained om (2.31) and (2.32) by applying he change o
a iables (2.7).
Some compac ness esul s: om he a p io i es ima es o (˜uε,˜pε), we can deduce he ollowing com-
pac ness esul s:
Lemma 2.8. Fo ˜uεsa is ying he sys em (2.8)-(2.9), he e exis s ˜u∈H1(h−, h+;L2(ω))3whe e ˜u3= 0
and ˜u= 0 on y3=h−, h+, such ha
˜uε
ε2*(˜u0,0) in H1(h−, h+;L2(ω))3as ε→0,(2.33)
di x0 Zh+(x0)
h−(x0)
˜u0(x0, y3)dy3!= 0 in ω, Zh+(x0)
h−(x0)
˜u0(x0, y3)dy3!·n= 0 on ∂ω. (2.34)
9
iii) i Kεε2, hen he ex ension (˜ ε/ε2,˜
Pε)con e ges weakly, as ε ends o ze o, in H1(h−
min, h+
max;L2(ω)3)×
L2
0(ω) o (˜ , ˜
P), wi h ˜ 3= 0 and ˜ 0= 0 on y3=h−
min, h+
max. Mo eo e , ˜
P∈H1(ω)and (e
V0(x0),˜
P(x0))
is he solu ion o he e ec i e p oblem

















˜
V0(x0) = A∞
µe 0(x0)− ∇x0˜
P(x0)in ω,
˜
V3(x0)=0
di x0˜
V0(x0, y3)=0 in ω,
˜
V0(x0, y3)·n= 0 on ∂ω .
(3.57)
whe e ˜
V(x0) = Rh+
max
h−
min
˜ (x0, y3)dy3and A∞∈R2×2→R2is symme ic, posi i e de ini e and de ined
by i s en ies:
(A∞)ij =ZY
Dwi(y) : Dywj(y)dy, ∀i, j = 1,2.(3.58)
He e wi(y)(i= 1,2) deno e he unique solu ions in H1
](Y)3o he local S okes p oblems in 3D







−∆ywi+∇yqi=eiin Y ,
di ywi= 0 in Y ,
wi= 0 in y3=h−(y0), h+(y0),
wi, πiY0−pe iodic.
(3.59)
3.1 P oo o he main esul
A p io i es ima es: Using he same a gumen s as in Sec ion 2, we de i e he a p io i es ima es o ˜uε
and ˜pεin e
Λε.
Lemma 3.2. Fo uεsa is ying he sys em (3.50)-(3.51),
i) i Kε≈ε2, wi h Kε/ε2→K,0< K < +∞o Kεε2, he ollowing es ima e holds:
k˜uεkL2(
e
Λε)3≤Cε2.(3.60)
ii) i Kεε2, he ollowing es ima e holds:
k˜uεkL2(
e
Λε)3≤CεK
1
2
ε.(3.61)
Mo eo e , in e e y cases i holds
kDx0˜uεkL2(
e
Λε)3×2≤Cε, k∂y3˜uεkL2(
e
Λε)3≤Cε2.(3.62)
Now, we u n ou a en ion o he p essu e. As in he p e ious sec ion, om equa ion (3.50) we can
ob ain he es ima e o he p essu e pε. Howe e , now he cons an Cappea ing in (2.27) depends on he
domain Λε, and, hus, he es ima e o he co esponding p essu e ˜pεmay no be uni o mly bounded when
ε→0. Fo ha eason, he idea is o ex end he p essu e ˜pε o he ε-independen domain e
Λ.
16

The Ex ension o (˜uε,˜pε) o he domain e
Λ : i is easy o ex end he eloci y by ze o in e
Λ e
Λε( his
is compa ible wi h i s Di ichle bounda y condi ion on ∂e
Λε). We will deno e by ˜ ε he con inua ion o ˜uε
in e
Λ. I is well known ha ex ension by ze o p ese es L2and H1
0no ms. We no e ha he ex ension ˜ ε
belongs o H1
0(e
Λ)3.
Howe e , ex ending he p essu e is a much mo e di icul ask. Ta a [23] in oduced a con inua ion o
he p essu e o a low in po ous media. This cons uc ion applies o pe iodic holes in a domain e
Λεwhen
each hole is s ic ly con ained in o he pe iodic cell. In his con ex , we can no use di ec ly his esul
because he “holes” a e along he op and bo om bounda ies o Λε, and mo eo e he scale o he e ical
di ec ion is smalle han he scales o he ho izon al di ec ions. This ac will induce se e al limi a ions
in he esul s ob ained by using he me hod, especially in iew o he con e gence o he p essu e. In
his sense, o he case o New onian luids in a domain wi h a op bounda y wi h oughness, Bayada
and Chamba [8] and Mikeli´c [19] in oduced an ope a o Rεgene alizing he esul s o Ta a [23] o his
con ex . In ou case, we need an ope a o Rεbe ween H1
0(Qε)3and H1
0(Λε)3wi h simila p ope ies, whe e
Qε=ω×(εh−
min, εh+
max).
Following [8], we make a ew mo e assump ions on he geome ical s uc u e:
H1 The su ace oughness is made o de ached smoo h humps pe iodically gi en on he uppe ( esp. he
lowe ) pa o he gap.
H2 We conside ha he domain ωis co e ed by a ini e numbe o pe iodic cells Y0
k0,ε, o size ε, whe e o
k0∈Z2, each cell Y0
k0,ε =εk0+εY 0, wi h Y0= (−1/2,1/2)2. We de ine Tε=nk0∈Z2:ω∩Y0
k0,ε 6=∅o.
We conside a smoo h su ace included in Yand su ounding he hump (in he op) such ha Yis
spli in o wo a eas Y+
and Y+
m(see Figu e 1 o mo e de ails).
H3 ∂Y +
mis a C1mani old.
y0
y3
h+
max
y3=h+(y0)
y3=h(y0)
Y
Y+
m
Y+
S+
+
Figu e 1: Basic cell Y
We no e
Π+=Y0×(h−(y0), h+
max), S+=∂Y +
m∩∂Y +
.
We ob ain he ollowing esul .
Lemma 3.3. Fo gi en ˜ϕ∈H1(Π+)3such ha ˜ϕ= 0 on Γ+, he e exis s ˜w+∈H1(Y+
m)3such ha :
˜w+
|S+= ˜ϕ|S+and ˜w+
|∂Y +
m S+.
17
Mo eo e , he e exis s a cons an Cwhich does no depend on ˜ϕsuch ha :
(k˜w+kH1(Y+
m)3≤Ck˜ϕkH1(Π+)3,
di ε˜ϕ= 0 ⇒di ε˜w+= 0 .(3.63)
P oo . I is analogous o he p oo o Lemma 3.1 in [8].
Lemma 3.4. The e exis s an ope a o R+
ε:H1
0(Q+
ε)→H1
0(Λε)such ha
1. ϕ∈H1
0(Λε)3⇒R+
ε(ϕ) = ϕ,
2. di ϕ= 0 ⇒di R+
ε(ϕ) = 0 .
3. Fo any ϕ∈H1
0(Qε)3, we ha e
kR+
ε(ϕ)kL2(Λε)3≤CkϕkL2(Q+
ε)3+εkDϕkL2(Q+
ε)3×3,
kDR+
ε(ϕ)kL2(Λε)3×3≤C1
εkϕkL2(Q+
ε)3+kDϕkL2(Q+
ε)3×3,
wi h cons an Cindependen o ϕand ε.
P oo . Fo any ˜ϕ∈H1
0(Π+)3such ha ˜ϕ= 0 on Γ+, Lemma 3.5 allows us o de ine R+( ˜ϕ)∈H1(Π+)3by
R+( ˜ϕ) = 


˜ϕi y∈Y+
,
˜w+i y∈Y+
m,
0 i y∈Y+
s,
which sa is ies ZΠ+
|R+( ˜ϕ)|2dy +ZΠ+
|DyR+( ˜ϕ)|2dy ≤CZΠ+
|˜ϕ|2dy +ZΠ+
|Dy˜ϕ|2dy.(3.64)
Fo e e y k0∈Tε, by he change o a iables
k0+y0=x0
ε, y3=x3
ε, dy =dx
ε3, ∂y=ε ∂x,(3.65)
we escale (3.68) om Π+ o Q+
k0,ε. This yields ha , o e e y unc ion ϕ∈H1(Q+
k0,ε)3, one has
ZQ+
k0,ε
|R+(ϕ)|2dx +ε2ZQ+
k0,ε
|DxR+(ϕ)|2dx ≤C ZQ+
k0,ε
|ϕ|2dx +ε2ZQ+
k0,ε
|Dx0ϕ|2dx.!
We de ine R+
εby applying R+ o each pe iod Q+
k0,ε. Summing he p e ious inequali ies o all he pe iods
Qk0,ε, and aking in o accoun ha om (H2) we ha e Qε=∪k0∈TεQk0,ε, gi es
ZQ+
ε
|R+
ε(ϕ)|2dx +ε2ZQ+
ε
|DxR+
ε(ϕ)|2dx ≤CZQ+
ε
|ϕ|2dx +ε2ZQ+
ε
|Dxϕ|2dx.(3.66)
Ob iously R+
ε(ϕ) lies in H1
0(Λε)3and is equal o ϕi ϕis ze o on Q+
ε Λε, so we ge he es ima es in 3.
Mo eo e , he second i em is ob ious om (3.63)2and he de ini ion o R+
ε.
18
y0
y3
h+
max
y3=h(y0)
h
min
⇧+
S
Y
Y
m

Figu e 2: Basic cell Π+
We make a ew mo e assump ions on he geome ical s uc u e. Thus, we conside a smoo h su ace
included in Π+and su ounding he hump such ha Π+is spli in o wo a eas Y−
and Y−
m(see Figu e 2
o mo e de ails). We also assume ha ∂Y −
mis a C1mani old.
We no e
Π = Y0×(h−
min, h+
max), S−=∂Y −
m∩∂Y −
.
Analogously, we ha e he ollowing esul .
Lemma 3.5. Fo gi en ˜ϕ∈H1(Π)3such ha ˜ϕ= 0 on Γ−, he e exis s ˜w−∈H1(Y−
m)3such ha :
˜w−
|S−= ˜ϕ|S−and ˜w−
|∂Y −
m S−.
Mo eo e , he e exis s a cons an Cwhich does no depend on ˜ϕsuch ha :
(k˜w−kH1(Y−
m)3≤Ck˜ϕkH1(Π)3,
di ε˜ϕ= 0 ⇒di ε˜w−= 0 .(3.67)
Finally, we gi e he p ope ies o he ope a o Rε.
Lemma 3.6. The e exis s an ope a o Rε:H1
0(Qε)→H1
0(Λε)such ha
1. ϕ∈H1
0(Λε)3⇒Rε(ϕ) = ϕ,
2. di ϕ= 0 ⇒di Rε(ϕ)=0.
3. Fo any ϕ∈H1
0(Qε)3, we ha e
kRε(ϕ)kL2(Λε)3≤CkϕkL2(Qε)3+εkDϕkL2(Qε)3×3,
kDRε(ϕ)kL2(Λε)3×3≤C1
εkϕkL2(Qε)3+kDϕkL2(Qε)3×3,
wi h cons an Cindependen o ϕand ε.
19
P oo . Fo any ˜ϕ∈H1
0(Π)3such ha ˜ϕ= 0 on Γ−, Lemma 3.5 allows us o de ine R( ˜ϕ)∈H1(Π)3by
R( ˜ϕ) = 


R+( ˜ϕ) i y∈Y−
,
˜w−i y∈Y−
m,
0 i y∈Y−
s,
which sa is ies
ZΠ
|R( ˜ϕ)|2dy +ZΠ
|DyR( ˜ϕ)|2dy ≤CZΠ
|R+( ˜ϕ)|2dy +ZΠ
|DyR+( ˜ϕ)|2dy.(3.68)
Fo e e y k0∈Tε, by he change o a iables (3.65), we escale (3.68) om Π o Qk0,ε. This yields ha ,
o e e y unc ion ϕ∈H1(Qk0,ε)3, one has
ZQk0,ε
|R(ϕ)|2dx +ε2ZQk0,ε
|DxR(ϕ)|2dx ≤C ZQk0,ε
|R+(ϕ)|2dx +ε2ZQk0,ε
|DxR+(ϕ)|2dx.!
We de ine Rεby applying R o each pe iod Qk0,ε. Summing he p e ious inequali ies o all he pe iods
Qk0,ε, and aking in o accoun ha om (H2) we ha e Qε=∪k0∈TεQk0,ε, gi es
ZQε
|Rε(ϕ)|2dx +ε2ZQε
|DxRε(ϕ)|2dx ≤CZQε
|R+
ε(ϕ)|2dx +ε2ZQε
|DxR+
ε(ϕ)|2dx,
which hanks o (3.66) gi es
ZQε
|Rε(ϕ)|2dx +ε2ZQε
|DxRε(ϕ)|2dx ≤CZQε
|ϕ|2dx +ε2ZQε
|Dxϕ|2dx,
Ob iously Rε(ϕ) lies in H1
0(Λε)3and is equal o ϕi ϕis ze o on Qε Λε, so we ge he es ima es in 3.
Mo eo e , he second i em is ob ious om (3.67)2and he de ini ion o Rε.
We ob ain he ollowing a p io i es ima es o he ex ension ( ε, Pε) in he domain Qε.
Lemma 3.7. The e exis s a cons an Cindependen o ε, such ha he ex ension ( ε, Pε)∈H1
0(Qε)3×
L2
0(Qε)o a solu ion (uε, pε)o p oblem (3.48)-(3.49) sa is ies
i) i Kε≈ε2, wi h Kε/ε2→K,0< K < +∞, he ollowing es ima es hold
k εkL2(Qε)3≤Cε5
2,(3.69)
kPεkL2(Qε)≤Cε1
2.(3.70)
ii) i Kεε2, he ollowing es ima es hold
k εkL2(Qε)3≤Cε5
2,(3.71)
kPεkL2(Qε)≤Cε5
2
Kε
.(3.72)
20
iii) i Kεε2, he ollowing es ima es hold
k εkL2(Qε)3≤Cε3
2K
1
2
ε,(3.73)
kPεkL2(Qε)≤Cε1
2.(3.74)
Mo eo e , in e e y case i holds
kD εkL2(Qε)3×3≤Cε3
2.(3.75)
P oo . We i s es ima e he eloci y. Taking in o accoun Lemma 3.2, i is clea ha , a e ex ension,
(3.69), (3.71), (3.73) and (3.75) hold.
The mapping Rεde ined in Lemma 3.4 allows us o ex end he p essu e pε o Qεin oducing Fεin
H−1(Qε)3:
hFε, ϕiQε=h∇pε, Rε(ϕ)iΛε, o any ϕ∈H1
0(Qε)3.(3.76)
We calcula e he igh hand side o (3.76) by using (3.48) o ob ain
hFε, ϕiQε=−µeZΛε
Duε:DRε(ϕ)dx −µ
KεZΛε
uε·Rε(ϕ)dx
+ZΛε
0·Rε
p(ϕ)dx −ρ
φ2ZΛε
(uε· ∇)uεRε(ϕ)dx .
(3.77)
Mo eo e , di ϕ= 0 implies
hFε, ϕiQε= 0 ,
and he DeRham heo em (see e.g. [14]) gi es he exis ence o Pεin L2
0(Qε) wi h Fε=∇Pε.
We in oduce ϕεas he unc ion solu ion o he auxilia y p oblem
di ϕε=Pε∈L2
0(Qε) in Qε, ϕε= 0 on ∂Qε.
Acco ding o Lemma 2.5, such p oblem has a leas one solu ion such ha
kϕεkL2(Ωε)3≤CkPεkL2(Qε),kDϕεkL2(Ωε)3×3≤C
εkPεkL2(Qε).
Thus, we ge
kPεkL2(Qε)=ZQε
Pεdi ϕεdx≤µeZΛε
Duε:DRε(ϕ)dx+
µ
KεZΛε
uε·Rε(ϕε)dx
+ZΛε
0·Rε(ϕ)dx+
ρ
φ2ZΛε
(uε· ∇)uεRε(ϕ)dx.
(3.78)
Taking in o accoun Lemma 3.4 iii) and Lemma 2.5 applied o he domain Qε, we conclude
kRε(ϕε)kL2(Λε)3≤CkϕεkL2(Qε)3+εkDϕεkL2(Qε)3×3≤CkPεkL2(Qε),
kDRε(ϕε)kL2(Λε)3×3≤C1
εkϕεkL2(Qε)3+kDϕεkL2(Qε)3×3≤C
εkPεkL2(Qε).
Finally, p oceeding as in he p oo o Lemma 2.6, we deduce he desi ed es ima es o he p essu e in e e y
case.
21

Applying he dila a ion (2.7) we ob ain he ollowing a p io i es ima es o he ex ension (˜ ε,˜
Pε) in e
Λ.
Co olla y 3.8. Fo he ex ension (˜ ε,˜
Pε)sa is ying he sys em (3.50)-(3.51), we ha e
i) i Kε≈ε2, wi h Kε/ε2→K,0< K < +∞, he ollowing es ima es hold
k˜ εkL2(
e
Λ)3≤Cε2,(3.79)
k˜
PεkL2(
e
Λ) ≤C . (3.80)
ii) i Kεε2, he ollowing es ima es hold
k˜ εkL2(
e
Λ)3≤Cε2,(3.81)
k˜
PεkL2(
e
Λ) ≤Cε2
Kε
.(3.82)
iii) i Kεε2, he ollowing es ima es hold
k˜ εkL2(
e
Λ)3≤CεK
1
2
ε,(3.83)
k˜
PεkL2(
e
Λ) ≤C . (3.84)
Mo eo e , in e e y case i holds
kDx0˜ εkL2(
e
Λ)3×2≤Cε, k∂y3˜ εkL2(
e
Λ)3≤Cε2.(3.85)
Adap a ion o he Un olding Me hod: he change o a iable (2.7) does no p o ide he in o ma ion
we need abou he beha io o ˜uεin he mic os uc u e associa ed o e
Λε. To sol e his di icul y, we
in oduce an adap a ion o he un olding me hod (see [6, 12] o mo e de ails). Fi s , we explain he
no a ion used in he sequel. Recalling ha Y0= (−1/2,1/2)2,Yk0,ε =εk0+εY 0, o e e y k0∈Tε, and
ha he basic cell is gi en by
Y=y∈R3:y0∈Y0, h−(y0)< y3< h+(y0),
we de ine Yk0,ε =Y0
k0,ε ×(h−(y0), h+(y0)) o e e y k0∈Tε. We also de ine he ex ension o he basic cell by
Π = Y0×(h−
min, h+
max).
The co esponding cubes o size εand heigh ε(h+
max −h−
min) a e gi en by Qk0,ε =Y0
k0,ε ×(εh−
min, εh+
max)
and e
Qk0,ε =Y0
k0,ε ×(h−
min, h+
max).
Gi en ˜uε∈H1
0(e
Λε)3a solu ion o he escaled sys em (3.50), ex ended by ze o ou side o e
Λε, we de ine
ˆuε, by
ˆuε(x0, y) = ˜uεεκ x0
ε+εy0, y3,a.e. (x0, y)∈ω×Y. (3.86)
22
He e he unc ion κis de ined as ollows: o k0∈Z2, we de ine κ:R2→Z2by
κ(x0) = k0⇐⇒ x0∈Y0
k0,1.
No e ha κis well de ined up o a se o ze o measu e in R2( he se ∪k0∈Z2∂Y 0
k0,1). Mo eo e , o e e y
ε > 0, we ha e
κx0
ε=k0⇐⇒ x0∈Y0
k0,ε .
In he same sense, gi en he ex ension o he p essu e ˜
Pε∈L2
0(e
Λ), we de ine ˆ
Pεby
ˆ
Pε(x0, y) = ˜
Pεεκ x0
ε+εy0, y3,a.e. (x0, y)∈ω×Π.(3.87)
Rema k 3.9. Fo k0∈Tε, he es ic ions o ˆuε o Y0
k0,ε ×Yand ˆ
Pε o Y0
k0,ε ×Πdo no depend on x0. As
a unc ion o y, i is ob ained om (˜uε,˜
Pε)by using he change o a iables
y0=x0−εk0
ε,(3.88)
ans o ming Yk0,ε in o Yand e
Qk0,ε in o Π, espec i ely.
Lemma 3.10. The e exis s a cons an Cindependen o ε, such ha (ˆuε,ˆ
Pε)de ined by (3.86)-(3.87)
sa is ies
i) i Kε≈ε2, wi h Kε/ε2→K,0< K < +∞, he ollowing es ima es hold
kˆuεkL2(ω×Y)3≤Cε2,(3.89)
kˆ
PεkL2(ω×Π) ≤C . (3.90)
ii) i Kεε2, he ollowing es ima es hold
kˆuεkL2(ω×Y)3≤Cε2,(3.91)
kˆ
PεkL2(ω×Π) ≤Cε2
Kε
.(3.92)
iii) i Kεε2, he ollowing es ima es hold
kˆuεkL2(ω×Y)3≤CεK
1
2
ε,(3.93)
kˆ
PεkL2(ω×Π) ≤C . (3.94)
Mo eo e , in e e y case i holds
kDy0ˆuεkL2(ω×Y)3×2≤Cε2,k∂y3ˆuεkL2(ω×Y)3≤Cε2.(3.95)
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P oo . Le us i s de i e some es ima es o he sequence ˆuεde ined by (3.86). We ob ain
Zω×YDy0ˆuε(x0, y)2dx0dy =X
k0∈TεZY0
k0,ε ZYDy0ˆuε(x0, y)2dx0dy
=X
k0∈TεZY0
k0,ε ZY0Zh+(y0)
h−(y0)Dy0˜uε(εk0+εy0, y3)2dx0dy0dy3.
We obse e ha ˜uεdoes no depend on x0so we deduce
Zω×YDy0ˆuε(x0, y)2dx0dy
=ε2X
k0∈TεZY0Zh+(y0)
h−(y0)Dy0˜uε(εk0+εy0, y3)2dy0dy3.
Using he change o a iables (3.88) and he Y0-pe iodici y o h−and h+, we ge
Zω×YDy0ˆuε(x0, y)2dx0dy
=ε2X
k0∈TεZY0
k0,ε Zh+(x0
ε−k0)
h−(x0
ε−k0)Dx0˜uε(x0, y3)2dx0dy3
=ε2X
k0∈TεZY0
k0,ε Zh+(x0
ε)
h−(x0
ε)Dx0˜uε(x0, y3)2dx0dy3
=ε2Ze
ΛεDx0˜uε(x0, y3)2dx0dy3.
Employing he es ima e (3.62)1, we deduce he (3.95)1.
Simila ly, using Rema k 3.9 and de ini ion (3.86), we ha e
Zω×Y∂y3ˆuε(x0, y)2dx0dy ≤ε2X
k0∈TεZY∂y3˜uε(εk0+εy0, y3)2dy.
Using he change o a iables (3.88) and he es ima e (3.62)2, we ob ain
Zω×Y∂y3ˆuε(x0, y)2dx0dy ≤Ze
Λε∂y3˜uε(x0, y3)2dx0dy3≤Cε4,
p o ing (3.95)2.
Simila ly, using he de ini ion (3.86), he change o a iables (3.88) and he es ima es (3.60) and (3.61),
we ha e in he cases Kε≈ε2and Kεε2 ha
Zω×Yˆuε(x0, y)2dx0dy ≤Cε4,
whe eas, in he case Kεε2, i holds
Zω×Yˆuε(x0, y)2dx0dy ≤Cε2Kε,
24
implying (3.89), (3.91) and (3.93).
Finally, le us ob ain some es ima es o he sequence ˆ
Pεde ined by (3.87). We obse e ha using he
de ini ion (3.87) o ˆ
Pε, we ob ain
Zω×Πˆ
Pε(x0, y)
p0
dx0dy ≤X
k0∈TεZY0
k0,ε ZY0Zh+
max
h−
min ˜
Pε(εk0+εy0, y3)
2dx0dy.
We also no e ha ˜
Pεdoes no depend on x0so we ha e
Zω×Πˆ
Pε(x0, y)
2dx0dy ≤ε2X
k0∈TεZY0Zh+
max
h−
min ˜
Pε(εk0+εy0, y3)
2dy0dy3.
By he change o a iables (3.88), we ge
Zω×Πˆ
Pε(x0, y)
2dx0dy ≤Ze
Λ˜
Pε(x0, y3)
2dx0dy3.
Taking in o accoun (3.80), (3.82) and (3.84), we deduce (3.90), (3.92) and (3.94), espec i ely.
Some compac ness esul s: om he a p io i es ima es o he ex ension (˜ ε,˜
Pε), we can deduce he
ollowing compac ness esul s.
Lemma 3.11. Conside he ex ension ˜ εo euεsa is ying he sys em (3.50)-(3.51). Then, he e exis s
˜ ∈H1(h−
min, h+
max;L2(ω))3whe e ˜u3= 0 and ˜u= 0 on y3=h−
min, h+
max, such ha
˜ ε
ε2*(˜ 0,0) in H1(h−
min, h+
max;L2(ω))3,as ε→0,(3.96)
di x0 Zh+
max
h−
min
˜ 0(x0, y3)dy3!= 0 in ω, Zh+
max
h−
min
˜ 0(x0, y3)dy3!·n= 0 on ∂ω. (3.97)
We omi he p oo since i is simila o he p oo o Lemma 2.8 conside ing he domain e
Λ ins ead o Ω.
Lemma 3.12. Conside he ex ension ˜
Pεo ˜pεsa is ying he sys em (3.50)-(3.51). Then,
i) i Kε≈ε2, wi h Kε/ε2→K,0< K < +∞o Kεε2, hen he e exis s ˜
P∈L2
0(e
Λ) such ha
˜
Pε*˜
Pin L2(e
Λ),as ε→0.(3.98)
ii) i Kεε2, hen he e exis s ˜
P∈L2
0(e
Λ) such ha
Kε
ε2˜
Pε*˜
Pin L2(e
Λ),as ε→0.(3.99)
Again we omi he p oo since i is simila o he p oo o Lemma 2.9 conside ing he domain e
Λ ins ead
o Ω.
Nex , om he a p io i es ima es o (ˆuε,ˆ
Pε), we can p o e he ollowing compac ness esul s:
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