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

Homogenization of the Darcy-Lapwood-Brinkman flow in a thin domain with highly oscillating boundaries Igor PAˇ ZANIN Department of Mathematics Faculty of Science, University of Zagreb, Bijeniˇcka 30, 10000 Zagreb, Croatia [email protected] Francisco Javier SU´ AREZ-GRAU Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Facultad de Matem´aticas, Universidad de Sevilla, 41012-Sevilla (Spain) [email protected] Abstract In this paper we investigate the flow through a thin corrugated domain filled with fluid saturated porous medium. The porous medium flow is described by the nonlinear Darcy-Lapwood-Brinkman model acknowledging the viscous shear and the inertial effects. The thickness of the domain is assumed to be of the same small order εas the period of the oscillating boundaries. Depending on the magnitude of the permeability with respect to ε, we rigorously derive different asymptotic models and compare the results with the non-oscillatory case. We employ a homogenization technique based on the adaption of the unfolding method and deduce the influence of the porous structure and boundary oscillations on the effective flow. AMS classification numbers: 35B27, 35B40, 76S05. Keywords: Darcy-Lapwood-Brinkman equation; thin domain; highly oscillating boundary; unfolding method. 1 1 Introduction Numerous models have been developed in the past fifteen decades to describe flows through porous media. Darcy’s law [13] is, without doubt, the most popular one expressing that the filtration velocity is proportional to the driving pressure gradient. However, it is well known that this simple law is only valid if a variety of the conditions are being met and, therefore, it is inapplicable in many physically relevant settings. For instance, as a first order PDE for the velocity, Darcy’s equation cannot sustain the no-slip boundary condition imposed on an impermeable boundary. Also, if the effects of inertia are important for the process (e.g. due to the curvilinearity of the flow path), the Darcy’s law again cannot be applied. Thus, in such situation when viscous shear and macroscopic inertial effects are significant, it has been customary to use the so called Darcy-Lapwood-Brinkman (DLB) equation to model the porous medium flow. This model takes the form (see e.g. Nield and Bejan [21]): −µe∆u+∇p+µ Ku=f−ρ φ2(u· ∇)u , (1.1) div u= 0 ,(1.2) where uand prepresents the filter velocity and pressure, µis the dynamic viscosity coefficient, µedenotes the effective viscosity of the fluid in the porous medium, ρis the fluid density, φis the porosity, fis an exterior force, while Kstands for the permeability of the porous medium. Note that the second-order DLB equation (1.1) is capable of handling the presence of the solid boundary on which the no-slip condition for the velocity is imposed. Moreover, the effects of flow inertia are also being incorporated making the model (1.1)-(1.2) an important generalization of the Darcy law that has sound physical basis. Due to nonlinearity of the equation (1.1), the Darcy-Lapwood-Brinkman model has been mostly treated numerically (see e.g. Chen et al. [11], Khalilli et al. [16], Umawathi et al. [24]). Analytical treatments are sparse and address only simple 2D fractures with plane-parallel walls under additional assumptions which linearize the momentum equation (1.1). We refer the reader to the papers by Hamdan et al. [7, 15]. In view of that, the goal of this paper is to analyze the 3D fluid flow through a thin layer of porous medium sandwiched between two corrugated walls and governed by (1.1)-(1.2). First, in Section 2, we study the flow in a thin constricted fracture without boundary oscillations, namely: Ωε=(x0, x3)∈R2×R:x0∈ω, εh−x0< x3< εh+x0.(1.3) Here ωis a smooth bounded open set in R2, while h−and h+are smooth functions such that h+> h−on ω. Assuming that permeability K=Kεmay depend on the small parameter ε, we employ a homogenization technique with respect to εand rigorously derive three different effective models (see Theorem 2.1): - If Kε≈ε2, i.e. when the permeability is of order ε2, we obtain a 2D Darcy law as an effective model including the effects of the domain’s geometry and the porous structure. - If Kεε2, we obtain a 2D pressure-driven Darcy law as an effective model not accounting the effects of the Brinkman (viscous) term. - If Kεε2, we obtain no contribution of the porous structure at the macroscopic model. As a result, we obtain a solution in the form of the Poiseuille flow. 2 Most recently, Paˇzanin and Siddheshwar [22] considered the similar problem, but in the case of the 2D flow. It is worth mentioning that the effective expression in the critical case provided in Theorem 2.1 is consistent with the one formally obtained in [22] via two-scale expansion method. The Section 3 is the central part of the present work. Here we introduce the oscillations at the top and the bottom of the flow domain, namely we assume that the period of the oscillations has the same small order as the domain thickness. In view of that, the domain to be considered is the following: Λε=(x0, x3)∈R2×R:x0∈ω, εh−x0 ε< x3< εh+x0 ε,(1.4) for periodic functions h−and h+. As above, we aim to determine the asymptotic behavior (as ε→0) of the flow governed by (1.1)-(1.2), now posed in Λε. The proof of our results is based on an adaptation of the unfolding method (see Arbogast et al. [6], and Cioranescu et al. [12]), which is strongly related to the two-scale convergence method (see Allaire [2], Nguetseng [20] and also Maruˇsi´c-Paloka et al. [18]). The unfolding method has been extensively used to study periodic homogenization problems where the size of the periodic cell tends to zero. We refer the reader to a recent works by Anguiano and Su´arez-Grau [3]-[5]. The basic idea is to introduce suitable changes of variables which transform every periodic cell into a simpler reference set by using a supplementary variable (microscopic variable). In the present setting, it is necessary to combine the unfolding method with a rescaling in the height variable in order to be able to work with a domain of a fixed height. In particular, due to the boundary oscillations, an extension operator needs to be constructed in order to extend the pressure to an ε-independent domain. Consequently, we manage to identify the critical size and later on the effects of the microstructure (boundary oscillations) in the corresponding effective equations. It turns out that the critical size is exactly the same as the one we obtain for the non-oscillatory case. Moreover, depending on the magnitude of the permeability Kεwith respect to ε, we derive three different characteristic cases (see Theorem 3.1): - If Kε≈ε2, we obtain a 2D Darcy law as an effective model which includes both the effects of the porous structure and the boundary oscillations given by the local Darcy-Brinkman problems in 3D. - If Kεε2, we obtain a 2D pressure-driven Darcy law as an effective model not accounting the effects of the Brinkman (viscous) term, but including the effects of the boundary oscillations provided by the local Hele-Shaw problems in 2D. - If Kεε2, as in the non-oscillatory case, we obtain no contribution of the porous structure at the effective model, but here the effects of the boundary oscillations are present through local Stokes problems in 3D. To conclude, we believe that the analysis presented in this paper is instrumental for understanding the effective behavior of the porous medium flow in thin domains with highly oscillating boundaries. As emphasized above, by considering the (nonlinear) Darcy-Lapwood-Brinkman equation, the important features have been taken into account that cannot be captured by a classical Darcy’s law. Consequently, the considered flow naturally finds applications both in industry (chemical reactors, heat exchangers, filtering equipment, etc.) and in geophysical problems, see [21] and the references therein. By employing a homogenization technique, the averaged effects of the boundary oscillations and the porous structure have been elegantly deduced. It should be mentioned that such homogenized system is of practical interest for developing numerical codes since it allows to filter out the small scales of the boundary, having a high computational cost. In view of that, we hope that our results could have an impact on the known engineering practice. 3 2 Non-oscillatory case Throughout the text, the points x∈R3will be decomposed as x= (x0, x3) with x0∈R2,x3∈R. Correspondingly, for the functions we use the same notation U= (U0, U3), U0∈R2. In this section we study the flow of a viscous fluid in the domain Ωεgiven by Ωε=(x0, x3)∈R3:x0∈ω, εh−(x0)< x3< εh+(x0), where h−, h+∈C1(ω)∩C(ω) such that h+> h−on ω. We suppose that the fracture Ωεis filled by a fluid-saturated sparsely-packed porous medium. As explained in the Introduction, the flow through the porous medium is modeled by the Darcy-Lapwood-Brinkman (DLB) equation. In view of that, let us consider a sequence (uε, pε)∈H1 0(Ωε)3×L2(Ωε) satisfying −µe∆uε+∇pε+µ Kε uε=f−ρ φ2(uε· ∇)uε, div uε= 0 . (2.5) To complete the problem, we impose a standard no-slip boundary condition uε= 0 on ∂Ωε.(2.6) The right-hand side fis of the form f(x)=(f0(x0),0),a.e. x∈ω, where fis assumed to be in L2(ω×(h− min, h+ max))2. Such choice of fis usual and justified when we deal with thin domains. Indeed, since the thickness of the domain is small, then the vertical component of the force can be neglected and, moreover, the force can be considered independent of the vertical variable. Using standard techniques (see e.g. Galdi [14]), it can be established that (2.5)-(2.6) has at least one solution (uε, pε)∈H1 0(Ωε)3×L2 0(Ωε). The space L2 0(Ωε) is the space of functions of L2(Ωε) with null integral. Our aim is to study the asymptotic behavior of uεand pεwhen the thickness εtends to zero, taking into account the magnitude of Kεwith respect to ε. For this purpose, we use the dilatation in the vertical variable x3: y3=x3 ε,(2.7) in order to have the functions defined in an open set independent of εand with height of order one: Ω = (x0, y3)∈R3:x0∈ω, h−(x0)< y3< h+(x0). We define ˜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 view of (2.7), the system (2.5) can be rewritten in Ω as (−µe∆x0˜uε−ε−2µe∂2 y3˜uε+∇x0˜pε+ε−1∂y3˜pεe3+µ Kε ˜uε=f0−ρ φ2(˜uε· ∇ε)˜uε, divx0˜u0 ε+ε−1∂y3˜uε,3= 0 (2.8) with no-slip boundary condition on ∂Ω, i.e. ˜uε= 0 on ∂Ω.(2.9) The asymptotic behavior of the sequence (˜uε, ˜pε) is provided in the following result: 4 Theorem 2.1. We distinguish three characteristic cases: i) if Kε≈ε2, with Kε/ε2→K,0< K < +∞, then (˜uε/ε2,˜pε)converges weakly, as εtends to zero, in H1(h−, h+;L2(ω)3)×L2 0(ω)to (˜u, ˜p), with ˜u3= 0 and ˜u= 0 on y3=h−, h+. Moreover, ˜p∈H1(ω) and (e U0(x0),˜p(x0)) is the solution of the effective problem            ˜ U0(x0) = 2KAM(x0) Mµ f0(x0)− ∇x0˜p(x0)in ω, divx0˜ U0(x0) = 0 in ω, ˜ U0(x0)·n= 0 on ∂ω, (2.10) where ˜ U(x0) = Rh+(x0) h−(x0)˜u(x0, y3)dy3,M=qµ K µeand the function AM(x0)is given 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) if Kεε2, then (˜uε/ε2,(Kε/ε2)˜pε)converges weakly, as εtends to zero, in H1(h−, h+;L2(ω)3)× L2 0(ω)to (˜u, ˜p), with ˜u3= 0 and ˜u= 0 on y3=h−, h+. Moreover, ˜p∈H1(ω)and (˜ U, ˜p)is the unique solution of the effective problem                  ˜ U0(x0, y3) = −A0(x0) µ∇x0˜p(x0)in ω, ˜ U3(x0)=0 in ω, divx0˜ U(x0) = 0 in ω, ˜ U(x0)·n= 0 on ∂ω , (2.12) where ˜ U(x0) = Rh+(x0) h−(x0)˜u(x0, y3)dy3and the function A0(x0)is given by A0(x0) = h+(x0)−h−(x0). iii) if Kεε2, then (˜uε/ε2,˜pε)converges weakly, as εtends to zero, in H1(h−, h+;L2(ω)3)×L2 0(ω)to (˜u, ˜p), with ˜u3= 0 and ˜u= 0 on y3=h−, h+. Moreover, ˜p∈H1(ω)and (˜ U, ˜p)is the unique solution of the effective problem                  ˜ U0(x0) = A∞(x0) 12µef0(x0)− ∇x0˜p(x0)in ω, ˜ U3(x0) = 0 in ω, divx0˜ U0(x0)=0 in ω, ˜ U0(x0)·n= 0 on ∂ω , (2.13) where ˜ U(x0) = Rh+(x0) h−(x0)˜u(x0, y3)dy3and the function A∞(x0)is given by A∞(x0) = h+(x0)3−3h−(x0)3−3h+(x0)2h−(x0)−h+(x0)h−(x0)2. 5 2.1 Proof of Theorem 2.1 Let us first fix some notation. We denote by : the full contraction of two matrices, namely for A= (ai,j)1≤i,j≤2and B= (bi,j)1≤i,j≤2, we have A:B=P2 i,j=1 aijbij. We denote by Oεa generic real sequence which tends to zero with εand can change from line to line. We denote by Ca generic positive constant which can change from line to line. A priori estimates: first we need to derive the a priori estimates for uε. To accomplish this, we employ some technical results which can be verified straightforwardly by a simple change of variables (for the proof see [18], Lemmas 8 and 11): Lemma 2.2. The following estimates hold: kϕεkL2(Ωε)3≤CεkDϕεkL2(Ωε)3×3,(2.14) kϕεkL4(Ωε)3≤Cε1 2kDϕεkL2(Ωε)3×3,(2.15) Now, we prove the sharp a priori estimates for the velocity uε. Proposition 2.3. For uεsatisfying the system (2.5)-(2.6), i) if Kε≈ε2, with Kε/ε2→K,0< K < +∞or Kεε2, it holds kuεkL2(Ωε)3≤Cε5 2.(2.16) ii) if Kεε2, it holds kuεkL2(Ωε)3≤Cε3 2K 1 2 ε.(2.17) Moreover, in every case it holds kDuεkL2(Ωε)3×3≤Cε3 2.(2.18) Proof. We consider uεas test function in the weak formulation of problem (2.5), and so we get µeZΩε |Duε|2dx +µ KεZΩε |uε|2dx =ZΩε f0(x0)·u0 εdx. (2.19) Using the Cauchy-Schwarz inequality, f0∈L2(ω)2and (2.14), we deduce ZΩε f0(x0)·u0 εdx≤Cε3 2kDuεkL2(Ωε)3×3, leading to µeZΩε |Duε|2dx +µ KεZΩε |uε|2dx ≤Cε3 2kDuεkL2(Ωε)3×3.(2.20) 6 On the one hand, this implies that (2.18) holds. Consequently, using (2.14) we get kuεkL2(Ωε)3≤Cε5 2.(2.21) On the other hand, using (2.18) in (2.20), we also obtain kuεkL2(Ωε)3≤Cε3 2K 1 2 ε.(2.22) From (2.21) and (2.22), we have that kuεkL2(Ωε)3≤Cε5 2+ε3 2K 1 2 ε. We compare ε5 2with respect to ε3 2K 1 2 εand observe that the critical case is when Kε≈ε2which gives the estimate (2.16). In the subcritical case Kεε2we also deduce (2.16), while in the supercritical case we deduce (2.17). Corollary 2.4. For ˜uεsatisfying the system (2.8)-(2.9), i) if Kε≈ε2, with Kε/ε2→K,0< K < +∞, or Kεε2, the following estimate holds k˜uεkL2(Ω)3≤Cε2.(2.23) ii) if Kεε2, the following estimate holds k˜uεkL2(Ω)3≤Cε K 1 2 ε.(2.24) Moreover, in every cases it holds kDx0˜uεkL2(Ω)2×3≤Cε , k∂y3˜uεkL2(Ω)3≤Cε2.(2.25) Proof. Estimates (2.23), (2.24) and (2.25) are easily obtained from (2.16), (2.17) and (2.18), respectively, by applying the change of variable (2.7). Now, we prove the a priori estimates for the pressure pε. For this, we need one more technical lemma addressing the auxiliary divergence problem (see Lemma 20 from [18]). Lemma 2.5. The problem    div ϕε=fε∈L2 0(Ωε)in Ωε, ϕε= 0 on ∂Ωε, (2.26) has a solution ϕε∈H1 0(Ωε)3such that kϕεkL2(Ωε)3≤CkfεkL2(Ωε),kDϕεkL2(Ωε)3×3≤C εkfεkL2(Ωε).(2.27) Proposition 2.6. For pεsatisfying the system (2.5)-(2.6), 7 i) if Kε≈ε2, with Kε/ε2→K,0< K < +∞, or Kεε2, we have kpεkL2(Ωε)≤Cε1 2,(2.28) ii) if Kεε2, we have kpεkL2(Ωε)≤Cε5 2 Kε .(2.29) Proof. We introduce ϕεas the solution of the auxiliary problem div ϕε=pε∈L2 0(Ωε) in Ωε, ϕε= 0 on ∂Ωε. According to Lemma 2.5, such problem has at least one solution such that kϕεkL2(Ωε)3≤CkpεkL2(Ωε),kDϕεkL2(Ωε)3×3≤C εkpεkL2(Ωε). We multiply the system (2.5) by ϕεand integrating over Ωε, we obtain kpεk2 L2 0(Ωε)=ZΩε pεdiv ϕεdx≤µeZΩε Duε:Dϕεdx+ZΩε f·ϕεdx + µ φ2ZΩε (uε· ∇)˜uεϕεdx+ µ KεZΩε uε·ϕεdx. (2.30) Taking into account estimate (2.18) and Lemma 2.5, we have µeZΩε Duε:Dϕεdx≤CkDuεkL2(Ωε)3×3kDϕεkL2(Ωε)3×3≤Cε1 2kpεkL2 0(Ωε). Similarly, we obtain ZΩε f·ϕεdx≤Cε1 2kpεkL2 0(Ωε). For the convective term, from estimate (2.18) and employing the inequalities (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 get  µ KεZΩε uε·ϕεdx≤CkuεkL2(Ωε)3kϕεkL2(Ωε)3. Depending on the magnitude of Kεwith respect to ε, we conclude: •if Kε≈ε2, estimates (2.14), (2.16) and Lemma 2.5 yield  µ KεZΩε uε·ϕεdx≤Cε1 2kpεkL2(Ωε). 8 •if Kεε2, using estimate (2.16) we get  µ KεZΩε uε·ϕεdx≤Cε5 2 Kε kpεkL2(Ωε). •if Kεε2, using estimate (2.17) we get  µ KεZΩε uε·ϕεdx≤Cε3 2 K 1 2 ε kpεkL2(Ωε). Thus, in view of (2.30), we deduce that if Kε≈ε2or Kεε2, we have kpεkL2(Ωε)≤Cε1 2. Finally, if Kεε2, we get kpεkL2(Ωε)≤Cε5 2 Kε . Corollary 2.7. For ˜pεsatisfying the system (2.8)-(2.9), i) if Kε≈ε2, with Kε/ε2→K,0< K < +∞or Kεε2, we have k˜pεkL2(Ω) ≤C , (2.31) ii) if Kεε2, we have k˜pεkL2(Ω) ≤Cε2 Kε .(2.32) Proof. Estimates (2.31) are (2.32) are easily obtained from (2.31) and (2.32) by applying the change of variables (2.7). Some compactness results: from the a priori estimates of (˜uε,˜pε), we can deduce the following compactness results: Lemma 2.8. For ˜uεsatisfying the system (2.8)-(2.9), there exists ˜u∈H1(h−, h+;L2(ω))3where ˜u3= 0 and ˜u= 0 on y3=h−, h+, such that ˜uε ε2*(˜u0,0) in H1(h−, h+;L2(ω))3as ε→0,(2.33) divx0 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) if Kεε2, then the extension (˜vε/ε2,˜ Pε)converges weakly, as εtends to zero, in H1(h− min, h+ max;L2(ω)3)× L2 0(ω)to (˜v, ˜ P), with ˜v3= 0 and ˜v0= 0 on y3=h− min, h+ max. Moreover, ˜ P∈H1(ω)and (e V0(x0),˜ P(x0)) is the solution of the effective problem                  ˜ V0(x0) = A∞ µef0(x0)− ∇x0˜ P(x0)in ω, ˜ V3(x0)=0 divx0˜ V0(x0, y3)=0 in ω, ˜ V0(x0, y3)·n= 0 on ∂ω . (3.57) where ˜ V(x0) = Rh+ max h− min ˜v(x0, y3)dy3and A∞∈R2×2→R2is symmetric, positive definite and defined by its entries: (A∞)ij =ZY Dwi(y) : Dywj(y)dy, ∀i, j = 1,2.(3.58) Here wi(y)(i= 1,2) denote the unique solutions in H1 ](Y)3of the local Stokes problems in 3D        −∆ywi+∇yqi=eiin Y , divywi= 0 in Y , wi= 0 in y3=h−(y0), h+(y0), wi, πiY0−periodic. (3.59) 3.1 Proof of the main result A priori estimates: Using the same arguments as in Section 2, we derive the a priori estimates for ˜uε and ˜pεin e Λε. Lemma 3.2. For uεsatisfying the system (3.50)-(3.51), i) if Kε≈ε2, with Kε/ε2→K,0< K < +∞or Kεε2, the following estimate holds: k˜uεkL2( e Λε)3≤Cε2.(3.60) ii) if Kεε2, the following estimate holds: k˜uεkL2( e Λε)3≤CεK 1 2 ε.(3.61) Moreover, in every cases it holds kDx0˜uεkL2( e Λε)3×2≤Cε, k∂y3˜uεkL2( e Λε)3≤Cε2.(3.62) Now, we turn our attention to the pressure. As in the previous section, from equation (3.50) we can obtain the estimate for the pressure pε. However, now the constant Cappearing in (2.27) depends on the domain Λε, and, thus, the estimate for the corresponding pressure ˜pεmay not be uniformly bounded when ε→0. For that reason, the idea is to extend the pressure ˜pεto the ε-independent domain e Λ. 16 The Extension of (˜uε,˜pε) to the domain e Λ : it is easy to extend the velocity by zero in e Λ\e Λε(this is compatible with its Dirichlet boundary condition on ∂e Λε). We will denote by ˜vεthe continuation of ˜uε in e Λ. It is well known that extension by zero preserves L2and H1 0norms. We note that the extension ˜vε belongs to H1 0(e Λ)3. However, extending the pressure is a much more difficult task. Tartar [23] introduced a continuation of the pressure for a flow in porous media. This construction applies to periodic holes in a domain e Λεwhen each hole is strictly contained into the periodic cell. In this context, we can not use directly this result because the “holes” are along the top and bottom boundaries of Λε, and moreover the scale of the vertical direction is smaller than the scales of the horizontal directions. This fact will induce several limitations in the results obtained by using the method, especially in view of the convergence for the pressure. In this sense, for the case of Newtonian fluids in a domain with a top boundary with roughness, Bayada and Chambat [8] and Mikeli´c [19] introduced an operator Rεgeneralizing the results of Tartar [23] to this context. In our case, we need an operator Rεbetween H1 0(Qε)3and H1 0(Λε)3with similar properties, where Qε=ω×(εh− min, εh+ max). Following [8], we make a few more assumptions on the geometrical structure: H1 The surface roughness is made of detached smooth humps periodically given on the upper (resp. the lower) part of the gap. H2 We consider that the domain ωis covered by a finite number of periodic cells Y0 k0,ε, of size ε, where for k0∈Z2, each cell Y0 k0,ε =εk0+εY 0, with Y0= (−1/2,1/2)2. We define Tε=nk0∈Z2:ω∩Y0 k0,ε 6=∅o. We consider a smooth surface included in Yand surrounding the hump (in the top) such that Yis split into two areas Y+ fand Y+ m(see Figure 1 for more details). H3 ∂Y + mis a C1manifold. y0 y3 h+ max y3=h+(y0) y3=h(y0) Y Y+ m Y+ f S+ + Figure 1: Basic cell Y We note Π+=Y0×(h−(y0), h+ max), S+=∂Y + m∩∂Y + f. We obtain the following result. Lemma 3.3. For given ˜ϕ∈H1(Π+)3such that ˜ϕ= 0 on Γ+, there exists ˜w+∈H1(Y+ m)3such that: ˜w+ |S+= ˜ϕ|S+and ˜w+ |∂Y + m\S+. 17 Moreover, there exists a constant Cwhich does not depend on ˜ϕsuch that: (k˜w+kH1(Y+ m)3≤Ck˜ϕkH1(Π+)3, divε˜ϕ= 0 ⇒divε˜w+= 0 .(3.63) Proof. It is analogous to the proof of Lemma 3.1 in [8]. Lemma 3.4. There exists an operator R+ ε:H1 0(Q+ ε)→H1 0(Λε)such that 1. ϕ∈H1 0(Λε)3⇒R+ ε(ϕ) = ϕ, 2. div ϕ= 0 ⇒divR+ ε(ϕ) = 0 . 3. For any ϕ∈H1 0(Qε)3, we have 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, with constant Cindependent of ϕand ε. Proof. For any ˜ϕ∈H1 0(Π+)3such that ˜ϕ= 0 on Γ+, Lemma 3.5 allows us to define R+( ˜ϕ)∈H1(Π+)3by R+( ˜ϕ) =    ˜ϕif y∈Y+ f, ˜w+if y∈Y+ m, 0 if y∈Y+ s, which satisfies ZΠ+ |R+( ˜ϕ)|2dy +ZΠ+ |DyR+( ˜ϕ)|2dy ≤CZΠ+ |˜ϕ|2dy +ZΠ+ |Dy˜ϕ|2dy.(3.64) For every k0∈Tε, by the change of variables k0+y0=x0 ε, y3=x3 ε, dy =dx ε3, ∂y=ε ∂x,(3.65) we rescale (3.68) from Π+to Q+ k0,ε. This yields that, for every function 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 define R+ εby applying R+to each period Q+ k0,ε. Summing the previous inequalities for all the periods Qk0,ε, and taking into account that from (H2) we have Qε=∪k0∈TεQk0,ε, gives ZQ+ ε |R+ ε(ϕ)|2dx +ε2ZQ+ ε |DxR+ ε(ϕ)|2dx ≤CZQ+ ε |ϕ|2dx +ε2ZQ+ ε |Dxϕ|2dx.(3.66) Obviously R+ ε(ϕ) lies in H1 0(Λε)3and is equal to ϕif ϕis zero on Q+ ε\Λε, so we get the estimates in 3. Moreover, the second item is obvious from (3.63)2and the definition of R+ ε. 18 y0 y3 h+ max y3=h(y0) h min ⇧+ S Y f Y m  Figure 2: Basic cell Π+ We make a few more assumptions on the geometrical structure. Thus, we consider a smooth surface included in Π+and surrounding the hump such that Π+is split into two areas Y− fand Y− m(see Figure 2 for more details). We also assume that ∂Y − mis a C1manifold. We note Π = Y0×(h− min, h+ max), S−=∂Y − m∩∂Y − f. Analogously, we have the following result. Lemma 3.5. For given ˜ϕ∈H1(Π)3such that ˜ϕ= 0 on Γ−, there exists ˜w−∈H1(Y− m)3such that: ˜w− |S−= ˜ϕ|S−and ˜w− |∂Y − m\S−. Moreover, there exists a constant Cwhich does not depend on ˜ϕsuch that: (k˜w−kH1(Y− m)3≤Ck˜ϕkH1(Π)3, divε˜ϕ= 0 ⇒divε˜w−= 0 .(3.67) Finally, we give the properties of the operator Rε. Lemma 3.6. There exists an operator Rε:H1 0(Qε)→H1 0(Λε)such that 1. ϕ∈H1 0(Λε)3⇒Rε(ϕ) = ϕ, 2. div ϕ= 0 ⇒divRε(ϕ)=0. 3. For any ϕ∈H1 0(Qε)3, we have 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, with constant Cindependent of ϕand ε. 19 Proof. For any ˜ϕ∈H1 0(Π)3such that ˜ϕ= 0 on Γ−, Lemma 3.5 allows us to define R( ˜ϕ)∈H1(Π)3by R( ˜ϕ) =    R+( ˜ϕ) if y∈Y− f, ˜w−if y∈Y− m, 0 if y∈Y− s, which satisfies ZΠ |R( ˜ϕ)|2dy +ZΠ |DyR( ˜ϕ)|2dy ≤CZΠ |R+( ˜ϕ)|2dy +ZΠ |DyR+( ˜ϕ)|2dy.(3.68) For every k0∈Tε, by the change of variables (3.65), we rescale (3.68) from Π to Qk0,ε. This yields that, for every function 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 define Rεby applying Rto each period Qk0,ε. Summing the previous inequalities for all the periods Qk0,ε, and taking into account that from (H2) we have Qε=∪k0∈TεQk0,ε, gives ZQε |Rε(ϕ)|2dx +ε2ZQε |DxRε(ϕ)|2dx ≤CZQε |R+ ε(ϕ)|2dx +ε2ZQε |DxR+ ε(ϕ)|2dx, which thanks to (3.66) gives ZQε |Rε(ϕ)|2dx +ε2ZQε |DxRε(ϕ)|2dx ≤CZQε |ϕ|2dx +ε2ZQε |Dxϕ|2dx, Obviously Rε(ϕ) lies in H1 0(Λε)3and is equal to ϕif ϕis zero on Qε\Λε, so we get the estimates in 3. Moreover, the second item is obvious from (3.67)2and the definition of Rε. We obtain the following a priori estimates for the extension (vε, Pε) in the domain Qε. Lemma 3.7. There exists a constant Cindependent of ε, such that the extension (vε, Pε)∈H1 0(Qε)3× L2 0(Qε)of a solution (uε, pε)of problem (3.48)-(3.49) satisfies i) if Kε≈ε2, with Kε/ε2→K,0< K < +∞, the following estimates hold kvεkL2(Qε)3≤Cε5 2,(3.69) kPεkL2(Qε)≤Cε1 2.(3.70) ii) if Kεε2, the following estimates hold kvεkL2(Qε)3≤Cε5 2,(3.71) kPεkL2(Qε)≤Cε5 2 Kε .(3.72) 20 iii) if Kεε2, the following estimates hold kvεkL2(Qε)3≤Cε3 2K 1 2 ε,(3.73) kPεkL2(Qε)≤Cε1 2.(3.74) Moreover, in every case it holds kDvεkL2(Qε)3×3≤Cε3 2.(3.75) Proof. We first estimate the velocity. Taking into account Lemma 3.2, it is clear that, after extension, (3.69), (3.71), (3.73) and (3.75) hold. The mapping Rεdefined in Lemma 3.4 allows us to extend the pressure pεto Qεintroducing Fεin H−1(Qε)3: hFε, ϕiQε=h∇pε, Rε(ϕ)iΛε,for any ϕ∈H1 0(Qε)3.(3.76) We calculate the right hand side of (3.76) by using (3.48) to obtain hFε, ϕiQε=−µeZΛε Duε:DRε(ϕ)dx −µ KεZΛε uε·Rε(ϕ)dx +ZΛε f0·Rε p(ϕ)dx −ρ φ2ZΛε (uε· ∇)uεRε(ϕ)dx . (3.77) Moreover, divϕ= 0 implies hFε, ϕiQε= 0 , and the DeRham theorem (see e.g. [14]) gives the existence of Pεin L2 0(Qε) with Fε=∇Pε. We introduce ϕεas the function solution of the auxiliary problem div ϕε=Pε∈L2 0(Qε) in Qε, ϕε= 0 on ∂Qε. According to Lemma 2.5, such problem has at least one solution such that kϕεkL2(Ωε)3≤CkPεkL2(Qε),kDϕεkL2(Ωε)3×3≤C εkPεkL2(Qε). Thus, we get kPεkL2(Qε)=ZQε Pεdiv ϕεdx≤µeZΛε Duε:DRε(ϕ)dx+ µ KεZΛε uε·Rε(ϕε)dx +ZΛε f0·Rε(ϕ)dx+ ρ φ2ZΛε (uε· ∇)uεRε(ϕ)dx. (3.78) Taking into account Lemma 3.4 iii) and Lemma 2.5 applied to the 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, proceeding as in the proof of Lemma 2.6, we deduce the desired estimates of the pressure in every case. 21 Applying the dilatation (2.7) we obtain the following a priori estimates for the extension (˜vε,˜ Pε) in e Λ. Corollary 3.8. For the extension (˜vε,˜ Pε)satisfying the system (3.50)-(3.51), we have i) if Kε≈ε2, with Kε/ε2→K,0< K < +∞, the following estimates hold k˜vεkL2( e Λ)3≤Cε2,(3.79) k˜ PεkL2( e Λ) ≤C . (3.80) ii) if Kεε2, the following estimates hold k˜vεkL2( e Λ)3≤Cε2,(3.81) k˜ PεkL2( e Λ) ≤Cε2 Kε .(3.82) iii) if Kεε2, the following estimates hold k˜vεkL2( e Λ)3≤CεK 1 2 ε,(3.83) k˜ PεkL2( e Λ) ≤C . (3.84) Moreover, in every case it holds kDx0˜vεkL2( e Λ)3×2≤Cε, k∂y3˜vεkL2( e Λ)3≤Cε2.(3.85) Adaptation of the Unfolding Method: the change of variable (2.7) does not provide the information we need about the behavior of ˜uεin the microstructure associated to e Λε. To solve this difficulty, we introduce an adaptation of the unfolding method (see [6, 12] for more details). First, we explain the notation used in the sequel. Recalling that Y0= (−1/2,1/2)2,Yk0,ε =εk0+εY 0, for every k0∈Tε, and that the basic cell is given by Y=y∈R3:y0∈Y0, h−(y0)< y3< h+(y0), we define Yk0,ε =Y0 k0,ε ×(h−(y0), h+(y0)) for every k0∈Tε. We also define the extension of the basic cell by Π = Y0×(h− min, h+ max). The corresponding cubes of size εand height ε(h+ max −h− min) are given by Qk0,ε =Y0 k0,ε ×(εh− min, εh+ max) and e Qk0,ε =Y0 k0,ε ×(h− min, h+ max). Given ˜uε∈H1 0(e Λε)3a solution of the rescaled system (3.50), extended by zero outside of e Λε, we define ˆuε, by ˆuε(x0, y) = ˜uεεκ x0 ε+εy0, y3,a.e. (x0, y)∈ω×Y. (3.86) 22 Here the function κis defined as follows: for k0∈Z2, we define κ:R2→Z2by κ(x0) = k0⇐⇒ x0∈Y0 k0,1. Note that κis well defined up to a set of zero measure in R2(the set ∪k0∈Z2∂Y 0 k0,1). Moreover, for every ε > 0, we have κx0 ε=k0⇐⇒ x0∈Y0 k0,ε . In the same sense, given the extension of the pressure ˜ Pε∈L2 0(e Λ), we define ˆ Pεby ˆ Pε(x0, y) = ˜ Pεεκ x0 ε+εy0, y3,a.e. (x0, y)∈ω×Π.(3.87) Remark 3.9. For k0∈Tε, the restrictions of ˆuεto Y0 k0,ε ×Yand ˆ Pεto Y0 k0,ε ×Πdo not depend on x0. As a function of y, it is obtained from (˜uε,˜ Pε)by using the change of variables y0=x0−εk0 ε,(3.88) transforming Yk0,ε into Yand e Qk0,ε into Π, respectively. Lemma 3.10. There exists a constant Cindependent of ε, such that (ˆuε,ˆ Pε)defined by (3.86)-(3.87) satisfies i) if Kε≈ε2, with Kε/ε2→K,0< K < +∞, the following estimates hold kˆuεkL2(ω×Y)3≤Cε2,(3.89) kˆ PεkL2(ω×Π) ≤C . (3.90) ii) if Kεε2, the following estimates hold kˆuεkL2(ω×Y)3≤Cε2,(3.91) kˆ PεkL2(ω×Π) ≤Cε2 Kε .(3.92) iii) if Kεε2, the following estimates hold kˆuεkL2(ω×Y)3≤CεK 1 2 ε,(3.93) kˆ PεkL2(ω×Π) ≤C . (3.94) Moreover, in every case it holds kDy0ˆuεkL2(ω×Y)3×2≤Cε2,k∂y3ˆuεkL2(ω×Y)3≤Cε2.(3.95) 23 Proof. Let us first derive some estimates for the sequence ˆuεdefined by (3.86). We obtain 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 observe that ˜uεdoes not depend on x0so we deduce Zω×YDy0ˆuε(x0, y)2dx0dy =ε2X k0∈TεZY0Zh+(y0) h−(y0)Dy0˜uε(εk0+εy0, y3)2dy0dy3. Using the change of variables (3.88) and the Y0-periodicity of h−and h+, we get 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 the estimate (3.62)1, we deduce the (3.95)1. Similarly, using Remark 3.9 and definition (3.86), we have Zω×Y∂y3ˆuε(x0, y)2dx0dy ≤ε2X k0∈TεZY∂y3˜uε(εk0+εy0, y3)2dy. Using the change of variables (3.88) and the estimate (3.62)2, we obtain Zω×Y∂y3ˆuε(x0, y)2dx0dy ≤Ze Λε∂y3˜uε(x0, y3)2dx0dy3≤Cε4, proving (3.95)2. Similarly, using the definition (3.86), the change of variables (3.88) and the estimates (3.60) and (3.61), we have in the cases Kε≈ε2and Kεε2that Zω×Yˆuε(x0, y)2dx0dy ≤Cε4, whereas, in the case Kεε2, it holds Zω×Yˆuε(x0, y)2dx0dy ≤Cε2Kε, 24 implying (3.89), (3.91) and (3.93). Finally, let us obtain some estimates for the sequence ˆ Pεdefined by (3.87). We observe that using the definition (3.87) of ˆ Pε, we obtain Zω×Πˆ Pε(x0, y) p0 dx0dy ≤X k0∈TεZY0 k0,ε ZY0Zh+ max h− min ˜ Pε(εk0+εy0, y3) 2dx0dy. We also note that ˜ Pεdoes not depend on x0so we have Zω×Πˆ Pε(x0, y) 2dx0dy ≤ε2X k0∈TεZY0Zh+ max h− min ˜ Pε(εk0+εy0, y3) 2dy0dy3. By the change of variables (3.88), we get Zω×Πˆ Pε(x0, y) 2dx0dy ≤Ze Λ˜ Pε(x0, y3) 2dx0dy3. Taking into account (3.80), (3.82) and (3.84), we deduce (3.90), (3.92) and (3.94), respectively. Some compactness results: from the a priori estimates of the extension (˜vε,˜ Pε), we can deduce the following compactness results. Lemma 3.11. Consider the extension ˜vεof euεsatisfying the system (3.50)-(3.51). Then, there exists ˜v∈H1(h− min, h+ max;L2(ω))3where ˜u3= 0 and ˜u= 0 on y3=h− min, h+ max, such that ˜vε ε2*(˜v0,0) in H1(h− min, h+ max;L2(ω))3,as ε→0,(3.96) divx0 Zh+ max h− min ˜v0(x0, y3)dy3!= 0 in ω, Zh+ max h− min ˜v0(x0, y3)dy3!·n= 0 on ∂ω. (3.97) We omit the proof since it is similar to the proof of Lemma 2.8 considering the domain e Λ instead of Ω. Lemma 3.12. Consider the extension ˜ Pεof ˜pεsatisfying the system (3.50)-(3.51). Then, i) if Kε≈ε2, with Kε/ε2→K,0< K < +∞of Kεε2, then there exists ˜ P∈L2 0(e Λ) such that ˜ Pε*˜ Pin L2(e Λ),as ε→0.(3.98) ii) if Kεε2, then there exists ˜ P∈L2 0(e Λ) such that Kε ε2˜ Pε*˜ Pin L2(e Λ),as ε→0.(3.99) Again we omit the proof since it is similar to the proof of Lemma 2.9 considering the domain e Λ instead of Ω. Next, from the a priori estimates of (ˆuε,ˆ Pε), we can prove the following compactness results: 25