Lower-Dimensional Nonlinear Brinkman’s Law for Non-Newtonian Flows in a Thin Porous Medium

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ Esta es la versión aceptada del artículo publicado en: This is a accepted manuscript of a paper published in: Mediterranean Journal of Mathematics (2021): 01/06/2021 DOI: https://doi.org/10.1007/s00009-021-01814-5 Copyright: El acceso a la versión publicada del artículo puede requerir la suscripción de la revista. Access to the published version may require subscription. “This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s00009-021-01814-5” Lower-dimensional nonlinear Brinkman’s law for non-Newtonian flows in a thin porous medium Mar´ıa ANGUIANO1, Francisco J. SU´ AREZ-GRAU2 Abstract In this paper we study the stationary incompressible power law fluid flow in a thin porous medium. The media under consideration is a bounded perforated 3D domain confined between two parallel plates, where the distance between the plates is very small. The perforation consists in an array solid cylinders, which connect the plates in perpendicular direction, distributed periodically with diameters of small size compared to the period. For a specific choice of the thickness of the domain, we found that the homogenization of the power law Stokes system results a lower-dimensional nonlinear Brinkman type law. AMS classification numbers: 76A05, 35B27, 76M50. Keywords: Homogenization, non-Newtonian fluid, power law fluid, thin porous medium, Brinkman’s law. 1 Introduction Modeling non-Newtonian fluid flows in porous medium is very important due to its practical engineering applications, such as oil recovery, food processing, and materials processing. Non-Newtonian fluids exhibit a nonlinear behavior that is different from that of Newtonian fluids and the related equations are too complicated to be solved in the porous medium, due to its complex geometry. Thus, macroscopic laws obtained by homogenization techniques are used as a good substitute for the original physical laws. A relevant case of non-Newtonian fluid is the power law or Ostwald-de Waele model, which assumes that the viscosity satisfies a nonlinear power law. More precisely, if uis the velocity field and Du the gradient velocity tensor, denoting the shear rate by D[u] = 1 2(Du +Dtu), the viscosity as a function of the shear rate is given by ηr(D[u]) = µ|D[u]|r−2,1< r < +∞. The two parameters µ > 0 and rare called the consistency and the flow index, respectively. The matrix norm | · | is defined by |ξ|2=Tr(ξξt) with ξ∈R3, so the viscosity is expressed in term of the second invariant of the strain tensor DII [u] = D[u]D[u]tby |D[u]|r−2=|DII [u]|r 2−1. We recall that r= 2 yields the Newtonian fluid, for r∈(1,2) the fluid is a shear-thinning and for r∈(2,+∞) is a shear-thickening. The derivation of macroscopic law for power law fluids through periodic porous medium with a periodic arrangement of obstacles has been consider in Bourgeat et al. [17, 18]. Denoting εa small parameter related to the characteristic size of the obstacles and the period of the periodic structure and starting from the 3D power 1Departamento de An´alisis Matem´atico. Facultad de Matem´aticas. Universidad de Sevilla, 41012 Sevilla (Spain) [email protected] 2Departamento de Ecuaciones Diferenciales y An´alisis Num´erico. Facultad de Matem´aticas. Universidad de Sevilla, 41012 Sevilla (Spain) [email protected] 1 Mar´ıa Anguiano and Francisco J. Su´arez-Grau law Stokes system with body forces f, by means of the two-scale convergence method when εtends to zero, the following nonlinear Darcy type law for the pressure pis derived in an ε-independent domain Ω = ω×(0,1) ⊂R3      u=1 µU(f− ∇p) in Ω, div u= 0 in Ω, u ·n= 0 on ∂Ω. The nonlinear function U:R3→R3is called permeability function of the porous medium and is defined through the solutions of auxiliary problems of power law type depending on the geometrical structure of the domain. We refer to Mikeli´c [31, 32] for a complete review on the modeling of non-Newtonian flows in porous medium by using homogenization. A more general case of periodic porous medium is when the size of the solid obstacles is assumed to be much smaller than the period. Thus, denoting εthe period of the periodic porous structure and considering obstacles of size εδ with δεa parameter which depends on the period and changes simultaneously during the homogenization process, it is well-known that there exists a critical regime in which the macroscopic law describing the behavior of Newtonian fluids is the Brinkman’s law instead of the Darcy’s law, see Allaire [1, 2, 3] and Brillard [19]. This change of behavior occurs for power law fluids as well. More precisely, assuming that the relation σε=ε/δ 3−r r ε tends to a positive constant when εtends to zero, Fratrovi´c and Maruˇsi´c-Paloka [25] derived by means of Γ- convergence the following nonlinear Brinkman type law          −µdiv |D[u]|r−2D[u]+µG(u) + ∇p=fin Ω, div u= 0 in Ω, u= 0 on ∂Ω, where the nonlinear function G:R3→R3is called the drag force function of the porous medium and it provides the value of the drag force on the reference obstacle. Moreover, it is defined through the solution of auxiliary exterior problems of power law type depending on the geometrical structure of the obstacles. On the other hand, the derivation of macroscopic laws for fluid flows in a thin porous medium has recently become of great interest, see Anguiano [4, 5, 6, 7], Anguiano and Bunoiu [8, 9], Anguiano and Su´arez-Grau [11, 12, 13, 14], Fabricius et al. [22], Jouybari and T. S. Lundstr¨om [27], Prat and Aga¨esse [33], Su´arez-Grau [35], Yeghiazarian et al. [36] and Zhengan and Hongxing [37]. More precisely, a thin porous medium can be defined as a bounded perforated 3D domain confined between two parallel plates, where the distance between the plates is very small and the perforation consists of periodically distributed solid cylinders which connect the plates in perpendicular direction. In previous studies, the thin porous medium is described by two parameters: one parameter εdenoting the period of the periodic porous structure as well as the size of the diameter of the cylinders, and another parameter hεdenoting the thickness of the domain. Thus, the derivation of macroscopic laws through a thin porous medium depends on the relation between both parameters. Concerning power law fluids, in a recent article (see [10]) we derived different lower-dimensional forms of the nonlinear Darcy’s law in ω⊂R2by using an adaptation of the unfolding method depending on the parameter ε, which can be written as follows      u0=1 µU(f0− ∇x0p), u3= 0 in ω, divx0u0= 0 in ω, u0·n= 0 on ∂ω. Here u0= (u1, u2), f0= (f1, f2), x0= (x1, x2) and the permeability function U:R2→R2is defined through the solutions of lower-dimensional auxiliary problems of power law type. The goal of this paper is to study the homogenization of power law fluids in thin porous medium containing an array of periodically perforated cylinders with diameters much smaller than the period. As far as we know, this problem has never been considered in the previous literature. This domain is then defined by three small 2 Mar´ıa Anguiano and Francisco J. Su´arez-Grau parameters denoted by ε,δεand hε, where εdenotes the period of the periodic porous structure, εδεthe size of the diameter of the cylinders and hεthe thickness of the domain. For an appropriate choice (critical size) of hε, we derive in Ω the following lower-dimensional nonlinear Brinkman type law              −µ2−r 2∂x3(|∂x3u0|r−2∂x3u0) + µG(u0) + ∇x0p=f0, u3= 0 in Ω, divx0Z1 0 u0dx3= 0 in ω, Z1 0 u0dx3·n= 0 on ∂ω, u0(x0,0) = u0(x0,1) = 0 in ω, where the nonlinear drag force function G:R2→R2is defined through the solution of lower-dimensional auxiliary exterior problems of power law type depending on the geometrical structure of the obstacles. The result is obtained by using a combination of reduction of dimension techniques, monotonicity arguments and an adaptation of the unfolding method depending on parameters εand δε. We remark that the related estimates of the unfolding method are based on the Sobolev-Poincar´e-Wirtinger inequality, which requires that the dimension of the domain be larger than the exponent rfor functions in the space W1,r, see [20, Part IV]. Since we apply the adaptation of the unfolding method to the horizontal variables (dimension 2), then we have to restrict this study to r∈(1,2), i.e. to the case of shear thinning power law fluids. The structure of the paper is as follows. In Section 2 we introduce the domain and give the main result (Theorem 2.1). In Section 3, we establish some a priori estimates and in Section 4 we introduce the version of the unfolding method for a domain perforated by cylinders with small diameters. Some compactness results, which are the main keys when we will pass to the limit later, is addressed in Section 5. Finally, the proof of the Theorem 2.1 is given in Section 6. 2 Setting of the problem and main result Geometrical setting. A periodic porous medium is defined by a domain ωand an associated microstructure, or periodic cell Y0= [−1/2,1/2]2, which is made of two complementary parts: the fluid part Y0 f, and the solid part Y0 s, such that Y0 fSY0 s=Y0and Y0 fTY0 s=∅. More precisely, we assume that ωis a smooth, bounded, connected set in R2, and that Y0 sis an open connected subset of Y0with a smooth boundary ∂Y 0 s, such that Y0 s is strictly included in Y0. We denote Y=Y0×(0,1) ⊂R3and so Yf=Y0 f×(0,1) and Ys=Y0 s×(0,1). Let εand δbe positive parameters, smaller than one, where δ=δ(ε) is such that δ→0 as ε→0. Using both parameters, we define the parameter σε=ε δ2−r r .(1) We consider a thin porous medium Ωεδ of thickness σεwhich is perforated by solid cylinders with diameter of size εδ and distributed periodically with period ε. The choice of the thickness is justified by the Poincar´e-Korn inequality, see Remarks 3.2 and 3.3. In addition, to work within the thin domain framework, we assume lim ε→0σε= 0.(2) To define the microstructure of the domain ω, set Y0 δf =Y0\δY 0 s, such that the domain ωis covered by a regular mesh of size ε, i.e. for k0∈Z2, each cell Y0 k0,ε =εk0+εY 0is divided in a fluid part Y0 δfk0,ε and a solid part Y0 δsk0,ε, where Y0 δsk0,ε denotes the complement in Y0 k0,ε of the set Y0 δfk0,ε. We observe that Y0 k0,ε is similar to the unit cell Y0rescaled to size ε. Thus Yis divided in a fluid part Yδf and a solid part Yδs, and consequently Yk0,ε =Y0 k0,ε ×(0,1) ⊂R3, which is also divided in a fluid part Yδfk0,ε and a solid part Yδsk0,ε. 3 Mar´ıa Anguiano and Francisco J. Su´arez-Grau 1 1 Yf Ys Y0 f Y0 s  1 1 Figure 1: View of the 3D reference cells Y(left) and the 2D reference cell Y0(right). We denote by τ(Y0 δsk0,ε) the set of all translated images of Y0 δsk0,ε. The set τ(Y0 δsk0,ε) represents the solids in R2. The fluid part of the bottom ωεδ ⊂R2of the porous medium is defined by ωεδ =ω\Sk0∈KεY0 δsk0,ε, where Kε={k0∈Z2:Y0 k0,ε ∩ω6=∅}. Thus, the whole fluid part Ωεδ ⊂R3in the thin porous medium is defined by Ωεδ ={(x0, x3)∈R2×R:x0∈ωεδ ,0< x3< σε}.(3) We make the assumption that the solids τ(Y0 δsk0,ε) do not intersect the boundary ∂ω. We define Yσε δsk0,ε = Y0 δsk0,ε ×(0, σε). Denote by Sεδ the set of the solids contained in Ωεδ. Then, Sεδ is a finite union of solids, i.e. Sεδ =Sk0∈KεYσε δsk0,ε. " " Y" fk0," Y" sk0," Y0 fk0," Y0 sk0," " " " Figure 2: View of the 3D rescaled cell Yk0,ε (left) and the 2D rescaled cell Y0 k0,ε (right). We define e Ωεδ =ωεδ ×(0,1),Ω = ω×(0,1),and Qεδ =ω×(0, σε).We observe that e Ωεδ = Ω\Sk0∈KεYδsk0,ε, and we define Tεδ =Sk0∈KεYδsk0,ε as the set of the solids contained in e Ωεδ. " S" " " Figure 3: View of the thin porous medium Ωεδ (left) and the domain without microstructure Qεδ (right). Following previous notation, we remark that along the paper, the points x∈R3will be decomposed as x= (x0, x3) with x0= (x1, x2)∈R2,x3∈R. We also use the notation x0to denote a generic vector of R2. 4 Mar´ıa Anguiano and Francisco J. Su´arez-Grau Setting of the problem. In the thin porous medium Ωεδ described above we consider the following power law Stokes system          −µdiv |D[uεδ]|r−2D[uεδ]+∇pεδ =fin Ωεδ, div uεδ = 0 in Ωεδ, uεδ = 0 on ∂Ωεδ, (4) where uεδ is the velocity field and pεδ is the pressure. In system (4), we suppose that the right-hand side (body forces) fis of the form f(x)=(f0(x0),0),a.e. x∈Ω, where fis assumed in Lr0(ω)2, where r0=r/(r−1) is the conjugate exponent of r. We remark that the choice of fis usual when we deal with thin domains. Since the thickness of the domain is small, then the vertical component of fcan be neglected and moreover, it can be considered independent of the vertical variable. Therefore, the classical theory gives the existence of a unique solution (uεδ, pεδ)∈W1,r 0(Ωεδ)3×Lr0(Ωεδ) with 1 < r < +∞, see Lions [28]. This solution is unique up to an additive constant for pεδ, i.e. it is unique if we consider the corresponding equivalence class pεδ ∈Lr0(Ωεδ)/R. Our goal is to study the asymptotic behavior of uεδ and pεδ when εand δtend to zero. For this purpose, we use the dilatation in the variable x3 y3=x3 σε ,(5) in order to have the functions defined in the open set with fixed height e Ωεδ. Namely, we define ˜uεδ ∈W1,r 0(e Ωεδ)3, ˜pεδ ∈Lr0(e Ωεδ)/Rby ˜uεδ(x0, y3) = uεδ(x0, σεy3),˜pεδ(x0, y3) = pεδ(x0, σεy3), a.e. (x0, y3)∈e Ωεδ. Let us introduce some notation which will be useful in the following. For a vectorial function v= (v0, v3) and a scalar function w, we will denote Dx0[v] = 1 2(Dx0v+Dt x0v) and ∂y3[v] = 1 2(∂y3v+∂t y3v), where we denote ∂y3= (0,0,∂ ∂y3)t. Moreover, associated to the change of variables (5), we introduce the operators: Dσε,Dσε, divσεand ∇σε, by Dσε[v] = 1 2Dσεv+Dt σεv, (Dσεv)i,j =∂xjvifor i= 1,2,3, j = 1,2,(Dσεv)i,3=σ−1 ε∂y3vifor i= 1,2,3, divσεv= divx0v0+σ−1 ε∂y3v3,∇σεw= (∇x0w, σ−1 ε∂y3w)t. Using the transformation (5), system (4) can be rewritten as            −µdivσε|Dσε[˜uεδ]|p−2Dσε[˜uεδ]+∇σε˜pεδ =fin e Ωεδ, divσε˜uεδ = 0 in e Ωεδ, ˜uεδ = 0 on ∂e Ωεδ. (6) Our goal then is to describe the asymptotic behavior of the dilated sequence (˜uεδ, ˜pεδ), which is given by the theorem below. However, the sequence of solutions (˜uεδ, ˜pεδ)∈W1,r 0(e Ωεδ)3×Lr0(e Ωεδ)/Rof system (6) is not defined in a fixed domain independent of εand δ, but rather in a varying set e Ωεδ. Thus, in order to pass to the limit when εand δtend to zero, convergences in fixed Sobolev spaces (defined in Ω) are used. This previously requires to define an extension of (˜uεδ, ˜pεδ) to the whole domain Ω. In this sense, we consider the zero extensions to the whole Ω for both velocity and pressure, which coincide with the original functions in e Ωεδ. For simplicity the extensions will be denoted by the same symbol. Moreover, as noted in the introduction, we restrict the main result to shear thinning power law fluids, i.e. 1< r < 2. More precisely, this restriction of the flow index rcomes from the Sobolev-Poincar´e-Wirtinger inequality (Theorem 4.3-3) to apply the adaptation of the unfolding method on a 2Ddomain. 5 Mar´ıa Anguiano and Francisco J. Su´arez-Grau Theorem 2.1 (Main Theorem).Suppose 1< r < 2and let σεbe given by (1) satisfying (2). Then, there exist u= (u0,0) ∈W1,r(0,1; Lr(ω)3)with u= 0 on y3={0,1}and p∈Lr0(ω)independent of y3, such that the extension (˜uεδ,˜pεδ)of the solution of (6) satisfies the following convergences σ−r r−1 ε˜uεδ * u weakly in W1,r(0,1; Lp(ω)3),˜pεδ →pstrongly in Lr0(Ω), where (u0, p)is the unique solution of the lower-dimensional nonlinear Brinkman type equation                        −µ2−r 2∂y3|∂y3u0|r−2∂y3u0+µG(u0) + ∇x0p=f0in Ω, divx0Z1 0 u0dy3= 0 in ω, Z1 0 u0dy3·n= 0 on ∂ω, u0= 0 on y3={0,1}. (7) Here, the drag force function G:R2→R2is defined by G(ζ0)·τ0=ZR2\Y0 s |Dz0[wζ0]|r−2Dz0[wζ0] : Dz0[wτ0]dz0,∀τ0, ζ0∈R2,(8) where wξ0,ξ0=τ0, ζ0, is the unique solution to the auxiliary exterior problem                            −divz0|Dz0[wξ0]|r−2Dz0[wξ0]+∇z0πξ0= 0 in R2\Y0 s, divz0wξ0= 0 in R2\Y0 s, wξ0= 0 on ∂Y 0 s, lim |z0|→∞wξ0=ξ0, (wξ0, πξ0)∈D1,r(R2\Y0 s)2×Lr0(R2\Y0 s)/R. (9) The space D1,r denotes the homogeneous Sobolev space given by D1,r(O) = v∈L1 loc(O) : Dv ∈Lr(O). Remark 2.2 (Uniqueness of solution of the auxiliary exterior problem).For every ξ0∈R2, the exterior auxiliary problem (9) has a unique solution (wξ0, πξ0)∈D1,r(R2\Y0 s)2×Lr0(R2\Y0 s)/R, with 1< r < 2, see [29, Theorem 3]. For more details concerning the homogeneous Sobolev space D1,r we refer to [26, Chapter II.6]. Remark 2.3 (Properties of the drag force function).According to [24], the drag force function G:R2→R2is continuous, strictly monotone and satisfies the homogeneity condition G(λ ξ0) = |λ|r−2λG(ξ0),∀(λ, ξ0)∈R×R2. Moreover, there exists m, M > 0such that for every ξ0∈R2it holds m|ξ0|r−1≤ |G(ξ0)| ≤ M|ξ0|r−1. Remark 2.4. The mathematical derivation of lower-dimensional models describing power law fluid flows in a thin domain Ωε=ω×(0, ε)⊂R3has been considered in [30], see also [7, 13, 15, 16, 23, 34], obtaining the nonlinear Reynolds’ law      u0=1 2r0 2(r0+1)µr0−1|f0− ∇x0p|r0−2(f0− ∇x0p)in ω, divx0u0= 0 in ω, u0·n= 0 on ∂ω. (10) 6 Mar´ıa Anguiano and Francisco J. Su´arez-Grau The main result of this paper can be considered a generalization of previous result to the case of a thin porous medium. In fact, if we assume the drag force function G= 0 in the Brinkman’s law (7), i.e. in the absence of obstacles, we recover the homogenized model obtained in [30, Propositions 3.1 and 3.2] given by              −µ2−r 2∂x3(|∂x3u0|r−2∂x3u0) + ∇x0p=f0, u3= 0 in Ω = ω×(0,1), divx0Z1 0 u0dx3= 0 in ω, Z1 0 u0dx3·n= 0 on ∂ω, u0= 0 on y3={0,1}, which gives rise to the nonlinear Reynolds’ law (10), see [30, Propositions 3.4]. 3A Priori Estimates We need some technical results to obtain a priori estimates of the solution (uεδ, pεδ) of problem (4). First, to estimate the velocity we need sharp Poincar´e and Korn inequalities in the thin porous medium necessary to derive uniform estimates for velocity uεδ. Next, to estimate the pressure we need a result of the inverse of the divergence operator in thin porous medium. Lemma 3.1. Suppose 1≤r < +∞and let σεbe given by σε=ε δ2−r r if 1≤r < 2, σε=ε|ln(δ)|1 2if r= 2,and σε=εif r > 2.(11) Then, there exists a constant Cindependent of εsuch that for any function ϕ∈W1,r 0(Ωεδ)3, one has kϕkLr(Ωεδ)3≤CσεkDϕkLr(Ωεδ)3×3,1≤r < +∞(Poincar´e’s inequality),(12) kDϕkLr(Ωεδ)3×3≤CkD[ϕ]kLr(Ωεδ)3×3,1< r < +∞(Korn’s inequality).(13) Proof. From [25, Lemma 4.1] applied to the 2Ddomain ωεδ, we have the sharp Poincar´e inequality kϕkr Lr(ωεδ)3≤Cσr εkDx0ϕkr Lr(ωεδ)3×2≤Cσr εkDϕkr Lr(ωεδ)3×3,1≤r < +∞, with σεgiven by (11). The desired result (12) follows by integrating previous estimates with respect to x3between 0 and σε. Finally, the classical Korn inequality in porous mediums implies that (13) holds for 1 < r < +∞. Remark 3.2 (Poincar´e–Korn’s inequality in Ωεδ).Estimates given in Lemma 3.1 can be expressed as follows kϕkLr(Ωεδ)3≤CσεkD[ϕ]kLr(Ωεδ )3×3,∀ϕ∈W1,r 0(Ωεδ)3,1< r < +∞.(14) Remark 3.3. As said in the introduction, the thin porous medium Ωεδ is generally described by three parameters. Thus, if εand δdescribe the perforation of the domain, a new parameter hεshould be introduced describing the thickness of the domain. In that case, it can be proved that the optimal Poincar´e-Korn inequality (14) depends on the relation between hεand σε. More precisely, it can be proved that if hεσεor hε≈σεthe Poincar´e-Korn inequality holds with Cσε, while if hεσεit holds with Chε. Therefore, the thickness of the domain considered in this paper can be viewed as the critical size of the periodic perforation with respect to the thickness of the domain, similarly to what happens in [10, Lemma 4.1]. Lemma 3.4 (Inverse of the divergence operator in Ωεδ).Suppose 1< r < +∞and let σεbe given by (11). Then, there exists a constant Cindependent of εsuch that for any g∈Lr(Ωεδ)there exists ϕ=ϕ(g)∈W1,r(Ωεδ)3 with ϕ= 0 on ∂Qεδ such that div ϕ=gin Ωεδ,(15) kϕkLr(Ωεδ)3≤CkgkLr(Ωεδ ),kDϕkLr(Ωεδ)3×3≤Cσ−1 εkgkLr(Ωεδ).(16) 7 Mar´ıa Anguiano and Francisco J. Su´arez-Grau Proof. Let g∈Lr(Ωεδ) be given. We extend ginside the cylinders by means of the function: G(x) =      g(x) if x∈Ωεδ, −1 |Qεδ \Ωεδ|ZΩεδ g(x)dx if x∈Qεδ \Ωεδ. It follows that G∈Lr 0(Qεδ) = nq∈Lr(Qεδ) : RQεδ q dx = 0oand kGkr Lr(Qεδ)=kgkr Lr(Ωεδ)+1 |Qεδ \Ωεδ|r−1ZΩεδ g(x)dx r .(17) Since |Qεδ \Ωεδ|is bounded from below by a positive number, it follows from (17) and H¨older’s inequality that kGkLr(Qεδ)≤CkgkLr(Ωεδ ).(18) Besides that, since G∈Lr(Qεδ), it follows from [21, Lemma 4] that we can find ϕ∈W1,r 0(Qεδ)3such that div ϕ=Gin Qεδ,(19) kϕkLr(Qεδ)3≤CkGkLr(Qεδ ),kDϕkLr(Qεδ)3×3≤Cσ−1 εkGkLr(Qεδ).(20) Let us consider ϕ|Ωεδ : it belongs to W1,r(Ωεδ) with ϕ= 0 on ∂Qεδ. Moreover, (15) follows from (19) and estimates given in (16) follow from (18) and (20). By means of previous technical results, we give the estimates of the solution (uεδ, pεδ) of problem (4) in Ωεδ. Lemma 3.5 (Estimates of velocity and pressure in Ωεδ).Suppose 1< r < +∞and let σεbe given by (11). Then, there exists a constant Cindependent of ε, such that if (uεδ, pεδ)∈W1,r 0(Ωεδ)3×Lr0(Ωεδ)/Ris the solution of the problem (4), one has kuεδkLr(Ωεδ)3≤Cσ 2r−1 r(r−1) +1 ε,kD[uεδ]kLr(Ωεδ)3×3≤Cσ 2r−1 r(r−1) ε,(21) kDuεδkLr(Ωεδ )3×3≤Cσ 2r−1 r(r−1) ε,(22) kpεδkLr0(Ωεδ)≤Cσ 1 r0 ε,(23) where r0=r r−1. Proof. We start with the velocity. Multiplying by uεδ in the first equation of (4) and integrating over Ωεδ, we have µZΩεδ |D[uεδ]|r−2D[uεδ] : D[uεδ]dx =ZΩεδ f·uεδ dx. (24) Using H¨older’s inequality and the assumption of f, we obtain that ZΩεδ f·uεδ dx0dy3≤Cσ r−1 r εkuεδkLr(Ωεδ)3, and by (24), we have kD[uεδ]kr Lr(Ωεδ)3×3≤Cσ r−1 r εkuεδkLr(Ωεδ)3.(25) Taking into account (14), we obtain the second estimate in (21) and consequently, from Korn’s inequality (13) we obtain (22). Now, from (14) and the second estimate in (21), we deduce the first estimate in (21). 8 Mar´ıa Anguiano and Francisco J. Su´arez-Grau From this convergence and (49), we get that σ−r r−1 ε˜uεδ,3tends to zero and so, by uniqueness of the limit, u3= 0 which finishes the proof of (46). Moreover, ˜uεδ = 0 on y3={0,1}and the continuity of the trace applications from the space of functions usuch that kukLrand k∂y3ukLrto Lr(ω× {y3}) with y3={0,1}imply that u= 0 on y3={0,1}and so u∈W1,r 0(0,1; Lr(ω)3) with u3= 0. Next, we prove the divergence condition (47). To do this, we consider ϕ∈C1(ω) as test function in divσε˜uεδ = 0 in Ω and taking into account that ˜uεδ = 0 on ∂Ω, we get 0 = ZΩσ−r r−1 εdivx0˜u0 εδ +σ−2r−1 r−1 ε∂y3˜uεδ,3ϕ(x0)dx0dy3=−ZΩ σ−r r−1 ε˜u0 εδ · ∇x0ϕ(x0)dx0dy3. From convergence (46), we get ZΩ u0· ∇x0ϕ(x0)dx0dy3= 0, which implies (47). Last, we focus on the extension of the pressure by proving (48). From estimate (30) we deduce that there exists p∈Lr0(Ω)/Rsuch that, up to a subsequence, it holds ˜pεδ * p weakly in Lr0(Ω)/R.(52) To prove that pis independent of y3, we multiply system (6) by σεϕwith ϕ∈C∞ 0(Ω). Integrating by parts and taking into account estimates of the extension of the velocity (28), assumption of force fand convergence (52), we pass to the limit and obtain ZΩ p ∂y3ϕ dx0dy3= 0. This implies that pdoes not depend on y3. Moreover, if we argue similarly as in [17, Lemma 4.4], we have that the convergence of the pressure ˜pεδ is in fact strong, which concludes the proof. Next, we give a compactness result for the unfolded function ˆuεδ. To do this, following [20, Chapters 9 and 10], we consider the homogeneous Sobolev space of weakly differentiable functions defined locally on R2having a gradient in Lr(R2)2and zero value on the obstacle Y0 s, which is given by KY0 s=nΦ(z0)∈W1,r loc (R2) : ∇z0Φ∈Lr(R2)2and Φ= 0 on Y0 so.(53) We remark that if Φ∈KY0 sthen it has a limit at infinity denoted Φ∞, i.e. there exists Φ∞∈Rsuch that lim|z0|→+∞Φ(z0) = Φ∞. In addition, to relate the value at infinity of the limit of ˆuεδ with the limit of ˜uεδ, we consider a more general space LY0 s=nΦ(x0, z0, y3)∈Lr(Ω; KY0 s) : Φ∞=Φ(·,∞,·)∈W1,r 0(0,1; Lr(ω))o.(54) Lemma 5.2 (Compactness results for unfolded velocity).Suppose 1< r < 2, let σεbe given by (1) satisfying (2) and u∈W1,r 0(0,1; Lr(ω)3)be given in Lemma 5.1. Then, for a subsequence of ε, still denoted by ε, there exists U∈L3 Y0 swhere U∞=uand U3is independent of y3, such that σ−r r−1 εˆuεδ * U weakly in Lr(Ω; Lr loc(R2)3),(55) σ−r r−1 εDz0ˆuεδ11 δY0* Dz0Uweakly in Lr(ω×R2×(0,1))3×2,(56) divx0Z1 0 U0 ∞dy3= 0 in ω, (57) divz0U0= 0 in ω×R2×(0,1) .(58) 15 Mar´ıa Anguiano and Francisco J. Su´arez-Grau Proof. For the sake of completeness, we give a sketch of a proof, following, for instance, the proof of [20, Proposition 10.1]. By estimate (41), there exists U∈Lr(Ω; Lr loc(R2)3) such that, up to a subsequence, convergence (55) holds. Using the continuous embedding of W1,r(0,1; Lr(ω)) into Lr(Ω), the definition of the local average (38) and according to [20, Proposition 1.25(ii)], we deduce from (46) the following convergence σ−r r−1 ε¯uεδ * u weakly in Lr(Ω)3, so that σ−r r−1 ε¯uεδ11 δY0* u weakly in Lr(Ω; Lr loc(R2)3).(59) From the first estimate in (43) and estimate (44), there exists W∈Lr(Ω; Lr∗(R2)3) with Dz0Win Lr(ω×R2× (0,1))3×2such that σ−r r−1 ε(ˆuεδ −¯uεδ)11 δY0* W weakly in Lr(Ω; Lr∗(R2)3),(60) and σ−r r−1 εDz0ˆuεδ11 δY0* Dz0Wweakly in Lr(ω×R2×(0,1))3×2.(61) From (55), (59) and (60), one concludes U=W+uand Dz0U=Dz0W, (62) and so, from convergence (61), we have that (56) holds. Since ˜uεδ = 0 in Tεδ, we have ˆuεδ = 0 in ω×Ysand then, using convergence (55), we deduce U= 0 in ω×Ys.(63) Due to (60)-(63) and [20, Proposition 9.2], we have that U∈L3 Y0 sand U∞=u. Since U∞=u, then (57) holds from the divergence condition (47). Next, from divergence condition divσε˜uεδ = 0 in e Ωεδ and the change of variables (32) and (39), we deduce (εδ)−1divz0ˆu0 εδ 11 δY0+σ−1 ε∂y3ˆuεδ,311 δY0= 0 in ω×R2×(0,1).(64) Multiplying by σ−r r−1 ε(εδ), we get σ−r r−1 εdivz0ˆu0 εδ 11 δY0+δ2 rσ−r r−1 ε∂y3ˆuεδ,311 δY0= 0 in ω×R2×(0,1).(65) From the second estimate in (43) and convergence (55), we deduce that δ2 rσ−r r−1 ε∂y3ˆuεδ,3tends to zero, and so passing to the limit in (65) we get (58). Finally, we prove that U3is independent of y3. To do this, consider ϕ∈C1 c(Ω) as test function in (64), which gives Zω×1 δY0×(0,1) ∂y3ˆuεδ,3ϕ(x0, y3)dx0dz0dy3= 0. Multiplying by σ−r r−1 ε, we have Zω×1 δY0×(0,1) σ−r r−1 εˆuεδ,3∂y3ϕ(x0, y3)dx0dz0dy3= 0, and from convergence (55), we get U3is independent of y3. 16 Mar´ıa Anguiano and Francisco J. Su´arez-Grau 6 Homogenized model: proof of the main Theorem In this section, we use previous compactness results to prove the main result of the paper. To do this, we need the following version of [20, Lemma 10.4] in order to choose an appropriate test function in the variational formulation of system (6) and then pass to the limit. Lemma 6.1. Suppose 1≤r < +∞. Let ϕbe in D(Ω; W1,r loc (R2)) such that ∇z0ϕis in D(Ω; Lr(R2)2)and has a compact support. We set ϕεδ(x0, y3) = ϕ x0,1 δ x0−εκ(x0 ε) ε, y3!in (x0, y3)∈Ω.(66) By [20, Proposition 9.2], it has a limit at infinity denoted by ϕ∞∈ D(Ω). If δis small enough, the function ϕεδ belongs to D(0,1; W1,r(ω)) and ϕεδ →ϕ∞strongly in Lr(Ω).(67) Remark 6.2 (Properties of test function).From the definition of ϕεδ given in (66), we have ∇x0ϕεδ(x0, y3) = ∇x0ϕ x0,1 δ x0−εκ(x0 ε) ε, y3!+ (εδ)−1∇z0ϕ x0,1 δ x0−εκ(x0 ε) ε, y3!, ∂y3ϕεδ(x0, y3) = ∂y3ϕ x0,1 δ x0−εκ(x0 ε) ε, y3!. (68) Moreover, applying the unfolding operator (31), we have ˆϕεδ(x0, z0, y3) =    ϕ(x0, z0, y3) + Θεδ(x0, z0, y3),if (x0, z0, y3)∈ω×1 δY0×(0,1), 0otherwise, with Θεδ(x0, z0, y3) = ϕ(εκ(x0 ε) + εδz0, z0, y3)−ϕ(x0, z0, y3). Consequently ∇z0ˆϕεδ(x0, z0, y3) = ∇z0ϕ(x0, z0, y3) + ∇z0Θεδ(x0, z0, y3)in ω×1 δY0×(0,1),(69) where, from the mean value theorem applied to ∇z0Θεδ, the fact that |εκ(x0 ε) + εδz0−x0|< ε for x0∈Y0 k0,ε, k0∈ Kε, and ∇z0ϕ∈ D(Ω; Lr(R2)2), it holds k∇z0ΘεδkLr(ω×R2×(0,1))2≤Cε. (70) Proof of Theorem 2.1. The proof of the main result will be divided in two steps. Step 1. Suppose 1 < r < 2. We set W=       (v0, V 0)∈W1,r(0,1; Lr(ω)2)×L2 Y0 s:V0 ∞(x0, y3) = v0(x0, y3) a.e. in (x0, y3)∈Ω, divz0V0= 0 in ω×R2×(0,1),divx0Z1 0 v0dy3= 0 in ω, Z1 0 v0dy3·n= 0 on ∂ω,        . To simplify the notation, we define the operator S:R3 sym →R3 sym by S(ξ) = |ξ|r−2ξ, ∀ξ∈R3×3 sym, and denote by Oεa generic real sequence, which tends to zero with εand δand can change from line to line. 17 Mar´ıa Anguiano and Francisco J. Su´arez-Grau In this step, we prove that the pair of functions (u, U)∈Wgiven in Lemma 5.2 satisfies the homogenized problem µZΩ S(∂y3[u0]) : ∂y3[v0]dx0dy3+µZω×(R2\Y0 s)×(0,1) S(Dz0[U0]) : Dz0[V0]dx0dz0dy3 =ZΩ f0·v0dx0dy3,∀(v0, V 0)∈W. (71) To prove this, we consider ϕ(x0, z0, y3)∈ D(Ω; K3 Y0 s) such that Dz0ϕhas a compact support, ϕ3is independent of y3and we define ϕεδ by (66). From Lemma 6.1, if δis small enough the function ϕεδ belongs to D(0,1; W1,r(ω)3) and it has a limit at infinity ϕ∞∈ D(Ω)3satisfying convergence (67). Multiplying (6) by ϕεδ, taking into account the extensions of the velocity and pressure, integrating by parts and using (68), we have µZΩ S(Dσε[˜uεδ]) : Dx0[ϕ] + σ−1 ε∂y3[ϕ]dx0dy3+µ(εδ)−1ZΩ S(Dσε[˜uεδ]) : Dz0[ϕ]dx0dy3 −ZΩ ˜pεδ divx0ϕ0dx0dy3−(εδ)−1ZΩ ˜pεδ divz0ϕ0dx0dy3=Ze Ωεδ f0·ϕ0dx0dy3. (72) To simplify, from now on we use the following notation: •ϕ=ϕ(x0,1 δ x0−εκ(x0 ε) ε, y3) in the integrals in Ω, •ϕ=ϕ(x0, z0, y3) in the integrals in ω×1 δY0×(0,1) obtained after applying the changes of variables (32) and (39). First, we analyze the first term in (72). Thus, µZΩ SDx0[˜uεδ] + σ−1 ε∂y3[˜uεδ]:Dx0[ϕ] + σ−1 ε∂y3[ϕ]dx0dy3 =µZΩ SσεDx0hσ−r r−1 ε˜uεδi+∂y3hσ−r r−1 ε˜uεδi: (σεDx0[ϕ] + ∂y3[ϕ]) dx0dy3. (73) Next, we analyze the second term in (72). By the changes of variables (32) and (39) and taking into account (69), we deduce µ(εδ)−1ZΩ SDx0[˜uεδ] + σ−1 ε∂y3[˜uεδ]:Dz0[ϕ]dx0dy3 =µ(εδ)−1δ2Zω×1 δY0×(0,1) S(εδ)−1Dz0[ˆuεδ] + σ−1 ε∂y3[ˆuεδ]: (Dz0[ϕ] + Dz0[Θεδ]) dx0dz0dy3(74) =µZω×1 δY0×(0,1) SDz0hσ−r r−1 εˆuεδi+δ2 r∂y3hσ−r r−1 εˆuεδi: (Dz0[ϕ] + Dz0[Θεδ]) dx0dz0dy3. By H¨older’s inequality, estimates given in (42) and taking into account (70), we have Zω×1 δY0×(0,1) SDz0hσ−r r−1 εˆuεδi+δ2 r∂y3hσ−r r−1 εˆuεδi:Dz0[Θεδ]dx0dz0dy3≤Cε, 18 Mar´ıa Anguiano and Francisco J. Su´arez-Grau and then, (74) reads µ(εδ)−1ZΩ SDx0[˜uεδ] + σ−1 ε∂y3[˜uεδ]:Dz0[ϕ]dx0dy3 =µZω×1 δY0×(0,1) SDz0hσ−r r−1 εˆuεδi+δ2 r∂y3hσ−r r−1 εˆuεδi:Dz0[ϕ]dx0dz0dy3+Oε =µZω×1 δY0×(0,1) SDz0hσ−r r−1 εˆuεδi+δ2 r∂y3hσ−r r−1 εˆuεδi:Dz0[ϕ] + δ2 r∂y3[ϕ]dx0dz0dy3(75) −µ δ 2 rZω×1 δY0×(0,1) SDz0hσ−r r−1 εˆuεδi+δ2 r∂y3hσ−r r−1 εˆuεδi:∂y3[ϕ]dx0dz0dy3+Oε. Since ϕbelongs to D(Ω; W1,r loc (R2)3), by H¨older’s inequality and estimates in (42), we have δ2 rZω×1 δY0×(0,1) SDz0hσ−r r−1 εˆuεδi+δ2 r∂y3hσ−r r−1 εˆuεδi:∂y3[ϕ]dx0dz0dy3≤Cδ 2 r. Then, (75) reads µ(εδ)−1ZΩ SDx0[˜uεδ] + σ−1 ε∂y3[˜uεδ]:Dz0[ϕ]dx0dy3(76) =µZω×1 δY0×(0,1) SDz0hσ−r r−1 εˆuεδi+δ2 r∂y3hσ−r r−1 εˆuεδi:Dz0[ϕ] + δ2 r∂y3[ϕ]dx0dz0dy3+Oε. Now, we analyze the terms which involve the pressure. By the changes of variables (32) and (39) in the third and fourth terms in (72) and taking into account (69), we deduce −ZΩ ˜pεδ divx0ϕ0dx0dy3−(εδ)−1ZΩ ˜pεδ divz0ϕ0dx0dy3 =−ZΩ ˜pεδ divx0ϕ0dx0dy3−δ2(εδ)−1Zω×R2×(0,1) ˆpεδ (divz0ϕ0+ divz0Θ0 εδ)dx0dz0dy3. (77) By H¨older’s inequality, estimate (45) and taking into account (70), we have δ2(εδ)−1Zω×R2×(0,1) ˆpεδ divz0Θ0 εδ dx0dz0dy3≤Cδ 2−r r, and then, we have that (77) reads −ZΩ ˜pεδ divx0ϕ0dx0dy3−(εδ)−1ZΩ ˜pεδ divz0ϕ0dx0dy3 =−ZΩ ˜pεδ divx0ϕ0dx0dy3−δ2(εδ)−1Zω×R2×(0,1) ˆpεδ divz0ϕ0dx0dz0dy3+Oε. (78) Then, taking into account (73), (76) and (78) in (72), we obtain µZΩ SσεDx0hσ−r r−1 ε˜uεδi+∂y3hσ−r r−1 ε˜uεδi: (σεDx0[ϕ] + ∂y3[ϕ]) dx0dy3 +µZω×1 δY0×(0,1) SDz0hσ−r r−1 εˆuεδi+δ2 r∂y3hσ−r r−1 εˆuεδi:Dz0[ϕ] + δ2 r∂y3[ϕ]dx0dz0dy3 −ZΩ ˜pεδ divx0ϕ0dx0dy3−δ2(εδ)−1Zω×R2×(0,1) ˆpεδ divz0ϕ0dx0dz0dy3 =Ze Ωεδ f0·ϕ0dx0dy3+Oε. (79) 19 Mar´ıa Anguiano and Francisco J. Su´arez-Grau Now, we consider v= (v0,0) ∈ D(Ω)3which satisfies the divergence condition divx0R1 0v0dy3= 0 in ω. Also, we consider V∈ D(Ω; K3 Y0 s) such that Dz0Vhas a compact support, V3is independent of y3and divz0V0= 0 in ω×R2×(0,1). We define Vεδ by (66), which has a limit at infinity denoted by V∞∈ D(Ω)3, and suppose V∞(x0, y3) = v(x0, y3) a.e. in Ω. Then, we choose as test function ϕin (79) the following functions •ϕεδ =Vx0,1 δ x0−εκ(x0 ε) ε, y3−σ−r r−1 ε˜uεδ in the integrals in Ω and e Ωεδ, •ϕεδ =V(x0, z0, y3)−σ−r r−1 εˆuεδ in the integrals in ω×1 δY0×(0,1) and ω×R2×(0,1). Taking into account (64), applying H¨older’s inequality and using the second estimate in (43) and estimate (45), the fourth term in (79) satisfies δ2(εδ)−1Zω×R2×(0,1) ˆpεδ divz0ϕ0 εδ dx0dz0dy3=δ2(εδ)−1Zω×R2×(0,1) ˆpεδ divz0ˆu0 εδ dx0dz0dy3 =δ2σ−1 εZω×R2×(0,1) ˆpεδ ∂y3ˆuεδ,3dx0dz0dy3≤Cσ 1 r−1 ε, and so (79) reads µZΩ SσεDx0hσ−r r−1 ε˜uεδi+∂y3hσ−r r−1 ε˜uεδi: (σεDx0[ϕεδ] + ∂y3[ϕεδ]) dx0dy3 +µZω×1 δY0×(0,1) SDz0hσ−r r−1 εˆuεδi+δ2 r∂y3hσ−r r−1 εˆuεδi:Dz0[ϕεδ] + δ2 r∂y3[ϕεδ]dx0dz0dy3 −ZΩ ˜pεδ divx0ϕ0 εδ dx0dy3=Ze Ωεδ f0·ϕ0 εδ dx0dy3+Oε. From this, we deduce −µZΩSσεDx0hσ−r r−1 ε˜uεδi+∂y3hσ−r r−1 ε˜uεδi−S(σεDx0[V] + ∂y3[V]): (σεDx0[ϕεδ] + ∂y3[ϕεδ]) dx0dy3 −µZΩ S(σεDx0[V] + ∂y3[V]) : (σεDx0[ϕεδ] + ∂y3[ϕεδ]) dx0dy3 −µZω×1 δY0×(0,1) SDz0hσ−r r−1 εˆuεδi+δ2 r∂y3hσ−r r−1 εˆuεδi−SDz0[V] + δ2 r∂y3[V] :Dz0[ϕεδ] + δ2 r∂y3[ϕεδ]dx0dz0dy3 −µZω×1 δY0×(0,1) SDz0[V] + δ2 r∂y3[V]:Dz0[ϕεδ] + δ2 r∂y3[ϕεδ]dx0dz0dy3 +ZΩ ˜pεδ divx0ϕ0 εδ dx0dy3=−Ze Ωεδ f0·ϕ0 εδ dx0dy3+Oε. Since the operator Sis monotone, i.e. (S(ξ)−S(ζ)) : (ξ−ζ)≥0 for every ξ, ζ ∈R3, we have µZΩ S(σεDx0[V] + ∂y3[V]) : (σεDx0[ϕεδ] + ∂y3[ϕεδ]) dx0dy3 +µZω×1 δY0×(0,1) SDz0[V] + δ2 r∂y3[V]:Dz0[ϕεδ] + δ2 r∂y3[ϕεδ]dx0dz0dy3 −ZΩ ˜pεδ divx0ϕ0 εδ dx0dy3≥Ze Ωεδ f0·ϕ0 εδ dx0dy3+Oε. (80) 20 Mar´ıa Anguiano and Francisco J. Su´arez-Grau Then, passing to the limit by taking into account that σεand δ2 rtend to 0 and using convergences (46), (48), (56) and (67), we obtain µZΩ S(∂y3[V0 ∞]) : (∂y3[V0 ∞−u0]) dx0dy3+µZω×(R2\Y0 s)×(0,1) S(Dz0[V]) : Dz0[V−U]dx0dz0dy3 −ZΩ pdivx0(V0 ∞−u0)dx0dy3≥ZΩ f0·(V0 ∞−u0)dx0dy3. Since pdoes not depend on y3, according to V0 ∞=v0in Ω and divx0R1 0v0dy3= 0 in ω, we have ZΩ pdivx0(V0 ∞−u0)dx0dy3=ZΩ pdivx0(v0−u0)dx0dy3=Zω pdivx0Z1 0 (v0−u0)dy3dx0= 0, so we get µZΩ S(∂y3[V0 ∞]) : (∂y3[V0 ∞−u0]) dx0dy3+µZω×(R2\Y0 s)×(0,1) S(Dz0[V]) : Dz0[V−U]dx0dz0dy3 ≥ZΩ f0·(V0 ∞−u0)dx0dy3, which, due to Minty Lemma [28, Chapter 3, Lemma 1.2], is equivalent to µZΩ S(∂y3[u0]) : ∂y3[V0 ∞]dx0dy3+µZω×(R2\Y0 s)×(0,1) S(Dz0[U]) : Dz0[V]dx0dz0dy3=ZΩ f0·V0 ∞dx0dy3.(81) Taking in the previous variational equation W= (0,0, U3) as test function, we have ZR2\Y0 s |Dz0[U]|r−2Dz0[U] : Dz0[W]dz0= 0,a.e. in Ω, and since U3∈LY0 s, from [20, Proposition 9.2], we have kU3−U∞,3kr Lr∗(R2)≤CkDz0[U3]kr Lr(R2)2a.e. in Ω. From Lemma 5.2, we know that U∞,3= 0 in Ω and so, we deduce kU3kr Lr∗(R2)≤CkDz0[U3]kr Lr(R2)2≤CZR2\Y0 s |Dz0[U]|r−2Dz0[U] : Dz0[W]dz0= 0, which implies U3= 0. Then, the variational equation (81) reads µZΩ S(∂y3[u0]) : ∂y3[v0]dx0dy3+µZω×(R2\Y0 s)×(0,1) S(Dz0[U0]) : Dz0[V0]dx0dz0dy3=ZΩ f0·v0dx0dy3, which by density gives (71) for every (v0, V 0)∈W. Step 2. Let us obtain a problem for u0identifying U0in (71). For this purpose, for every ξ0∈R2we consider the unique solution of the auxiliary problem (9) denoted by (wξ0, πξ0)∈D1,r(R2\Y0 s)2×Lr0(R2\Y0 s)/Rand the drag force function G(ξ0) given by (8). We remark that then wξ0∈W1,r loc (R2\Y0 s)2, see [26, Lemma II.6.1]. Thus, we can take in (71) the pairs of functions (u0, U0),(v0, V 0)∈Win the following way U0(x0, z0, y3) = wu0(x0,y3)(z0), V 0(x0, z0, y3) = wv0(x0,y3)(z0) a.e. ω×(R2\Y0 s)×(0,1), 21 Mar´ıa Anguiano and Francisco J. Su´arez-Grau and then, we deduce µZΩ S(∂y3[u0]) : ∂y3[v0]dx0dy3+µZΩ G(u0)·v0dx0dy3=ZΩ f0·v0dx0dy3,∀v0∈ V,(82) where V=v0∈W1,r(0,1; Lr(ω)2) : divx0Z1 0 v0(x0, y3)dy3= 0 in ω, Z1 0 v0(x0, y3)dy3·n= 0 on ∂ω. 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