Homogenization of the Darcy-Lapwood-Brinkman ow in a thin domain with highly oscillating boundaries

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) 23 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: 25