Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions
NNewtonian fluid flow in a thin porous medium withnon-homogeneous slip boundary conditions
Mar´ıa ANGUIANODepartamento de An´alisis Matem´atico. Facultad de Matem´aticas.Universidad de Sevilla, P. O. Box 1160, 41080-Sevilla (Spain)[email protected] Javier SU ´AREZ-GRAUDepartamento de Ecuaciones Diferenciales y An´alisis Num´erico. Facultad de Matem´aticas.Universidad de Sevilla, 41012-Sevilla (Spain)[email protected]
Abstract
We consider the Stokes system in a thin porous medium Ω ε of thickness ε which is perforated by periodicallydistributed solid cylinders of size ε . On the boundary of the cylinders we prescribe non-homogeneous slipboundary conditions depending on a parameter γ . The aim is to give the asymptotic behavior of the velocityand the pressure of the fluid as ε goes to zero. Using an adaptation of the unfolding method, we give, followingthe values of γ , different limit systems. AMS classification numbers:
Keywords:
Homogenization; Stokes system; Darcy’s law; thin porous medium; Non-homogeneous slip boundarycondition. 1 a r X i v : . [ m a t h . A P ] O c t ar´ıa Anguiano and Francisco Javier Su´arez-Grau We consider a viscous fluid obeying the Stokes system in a thin porous medium Ω ε of thickness ε which is perforatedby periodically distributed cylinders (obstacles) of size ε . On the boundary of the obstacles, we prescribe aRobin-type condition depending on a parameter γ ∈ R . The aim of this work is to prove the convergence of thehomogenization process when ε goes to zero, depending on the different values of γ . The domain: the periodic porous medium is defined by a domain ω and an associated microstructure, or periodiccell Y (cid:48) = [ − / , / , which is made of two complementary parts: the fluid part Y (cid:48) f , and the obstacle part T (cid:48) ( Y (cid:48) f (cid:83) T (cid:48) = Y (cid:48) and Y (cid:48) f (cid:84) T (cid:48) = ∅ ). More precisely, we assume that ω is a smooth, bounded, connected set in R ,and that T (cid:48) is an open connected subset of Y (cid:48) with a smooth boundary ∂T (cid:48) , such that T (cid:48) is strictly included in Y (cid:48) .The microscale of a porous medium is a small positive number ε . The domain ω is covered by a regular mesh ofsquare of size ε : for k (cid:48) ∈ Z , each cell Y (cid:48) k (cid:48) ,ε = εk (cid:48) + εY (cid:48) is divided in a fluid part Y (cid:48) f k (cid:48) ,ε and an obstacle part T (cid:48) k (cid:48) ,ε ,i.e. is similar to the unit cell Y (cid:48) rescaled to size ε . We define Y = Y (cid:48) × (0 , ⊂ R , which is divided in a fluid part Y f = Y (cid:48) f × (0 ,
1) and an obstacle part T = T (cid:48) × (0 , Y k (cid:48) ,ε = Y (cid:48) k (cid:48) ,ε × (0 , ⊂ R , which is alsodivided in a fluid part Y f k (cid:48) ,ε and an obstacle part T k (cid:48) ,ε .We denote by τ ( T (cid:48) k (cid:48) ,ε ) the set of all translated images of T (cid:48) k (cid:48) ,ε . The set τ ( T (cid:48) k (cid:48) ,ε ) represents the obstacles in R . T k ," Y f k ," Y k ," Y k ," ⇥ (0 , " ) Y f k ," ⇥ (0 , " ) T k ," ⇥ (0 , " ) Figure 1: Views of a periodic cell in 2D (left) and 3D (right)The fluid part of the bottom ω ε ⊂ R of the porous medium is defined by ω ε = ω \ (cid:83) k (cid:48) ∈K ε T (cid:48) k (cid:48) ,ε , where K ε = { k (cid:48) ∈ Z : Y (cid:48) k (cid:48) ,ε ∩ ω (cid:54) = ∅} . The whole fluid part Ω ε ⊂ R in the thin porous medium is defined by (see Figures2 and 3) Ω ε = { ( x , x , x ) ∈ ω ε × R : 0 < x < ε } . (1.1)We make the following assumption:The obstacles τ ( T (cid:48) k (cid:48) ,ε ) do not intersect the boundary ∂ω. We define T εk (cid:48) ,ε = T (cid:48) k (cid:48) ,ε × (0 , ε ). Denote by S ε the set of the obstacles contained in Ω ε . Then, S ε is a finite unionof obstacles, i.e. S ε = (cid:91) k (cid:48) ∈K ε T εk (cid:48) ,ε . We define (cid:101) Ω ε = ω ε × (0 , , Ω = ω × (0 , , Λ ε = ω × (0 , ε ) . (1.2)We observe that (cid:101) Ω ε = Ω \ (cid:83) k (cid:48) ∈K ε T k (cid:48) ,ε , and we define T ε = (cid:83) k (cid:48) ∈K ε T k (cid:48) ,ε as the set of the obstacles contained in (cid:101) Ω ε . 2ar´ıa Anguiano and Francisco Javier Su´arez-Grau T k ," Y f k ," Figure 2: View from above S " "! " "! Figure 3: Views of the domain Ω ε (left) and Λ ε (right) The problem: let us consider the following Stokes system in Ω ε , with a Dirichlet boundary condition on theexterior boundary ∂ Λ ε and a non-homogeneous slip boundary condition on the cylinders ∂S ε : − µ ∆ u ε + ∇ p ε = f ε in Ω ε , div u ε = 0 in Ω ε ,u ε = 0 on ∂ Λ ε , − p ε · n + µ ∂u ε ∂n + αε γ u ε = g ε on ∂S ε , (1.3)where we denote by u ε = ( u ε, , u ε, , u ε, ) the velocity field, p ε is the (scalar) pressure, f ε = ( f ε, ( x , x ) , f ε, ( x , x ) , g ε = ( g ε, ( x , x ) , g ε, ( x , x ) ,
0) is the field of exterior surface force. Theconstants α and γ are given, with α > µ is the viscosity and n is the outward normal to S ε .This choice of f and g is usual when we deal with thin domains. Since the thickness of the domain, ε , is smallthen vertical component of the forces can be neglected and, moreover the force can be considered independent ofthe vertical variable.Problem (1.3) models in particular the flow of an incompressible viscous fluid through a porous medium underthe action of an exterior electric field. This system is derived from a physical model well detailed in the literature.As pointed out in Cioranescu et al. [1] and Sanchez-Palencia [2], it was observed experimentally in Reuss [3] thefollowing phenomenon: when a electrical field is applied on the boundary of a porous medium in equilibrium, amotion of the fluid appears. This motion is a consequence of the electrical field only. To describe such a motion,3ar´ıa Anguiano and Francisco Javier Su´arez-Grauit is usual to consider a modified Darcy’s law considering of including an additional term, the gradient of theelectrical field, or consider that the presence of this term is possible only if the electrical charges have a volumedistribution. However, this law contains implicitly a mistake, because if the solid and fluid parts are both dielectric,such a distribution does not occur, the electrical charges act only on the boundary between the solid and the fluidparts and so they have necessarily a surface distribution. If such hypothesis is done, we can describe the boundaryconditions in terms of the stress tensor σ ε as follows σ ε · n + αε γ u ε = g ε , which is precisely the non-homogeneous slip boundary condition (1.3) and means that the stress-vector σ ε · n induces a slowing effect on the motion of the fluid, expressed by the term αε γ . Moreover, if there are exterior forceslike for instance, an electrical field, then the non-homogeneity of the boundary condition on the ∂S ε is expressedin terms of surface charges contained in g ε .On the other hand, the behavior of the flow of Newtonian fluids through periodic arrays of cylinders has beenstudied extensively, mainly because of its importance in many applications in heat and mass transfer equipment.However, the literature on Newtonian thin film fluid flows through periodic arrays of cylinders is far less complete,although these problems have now become of great practical relevance because take place in a number of naturaland industrial processes. This includes flow during manufacturing of fibre reinforced polymer composites with liq-uid moulding processes (see Frishfelds et al. [4], Nordlund and Lundstrom [5], Tan and Pillai [6]), passive mixing inmicrofluidic systems (see Jeon [7]), paper making (see Lundstr¨om et al. [8], Singh et al. [9]), and block copolymersself-assemble on nanometer length scales (see Park et al. [10], Albert and Epps [11], Farrel et al. [12]).The Stokes problem in a periodically perforated domain with holes of the same size as the periodic has beentreated in the literature. More precisely, the case with Dirichlet conditions on the boundary of the holes was studiedby Ene and Sanchez-Palencia [13], where the model that describes the homogenized medium is a Darcy’s law. Thecase with non-homogeneous slip boundary conditions, depending on a parameter γ ∈ R , was studied by Cioranescu et al. [1], where using the variational method introduced by Tartar [14], a Darcy-type law, a Brinkmann-typeequation or a Stokes-type equation are obtained depending of the values of γ . The Stokes and Navier-Stokes equa-tions in a perforated domain with holes of size r ε , with r ε (cid:28) ε is considered by Allaire [15]. On the boundary ofthe holes, the normal component of the velocity is equal to zero and the tangential velocity is proportional to thetangential component of the normal stress. The type of the homogenized model is determined by the size r ε , i.e.by the geometry of the domain.The earlier results relate to a fixed height domain. For a thin domain, in [16] Anguiano and Su´arez-Grauconsider an incompressible non-Newtonian Stokes system, in a thin porous medium of thickness ε that is perforatedby periodically distributed solid cylinders of size a ε , with Dirichlet conditions on the boundary of the cylinders.Using a combination of the unfolding method (see Cioranescu et al. [17] and Cioranescu et al. [18] for perforateddomains) applied to the horizontal variables, with a rescaling on the height variable, and using monotonicityarguments to pass to the limit, three different Darcy’s laws are obtained rigorously depending on the relationbetween a ε and ε . We remark that an extension of the unfolding method to evolution problems in which theunfolding method is applied to the spatial variables and not on the time variable was introduced in Donato andYang [19] (see also [20]).The behavior observed when a ε ≈ ε in [16] has motivated the fact of considering non-homogeneous slip conditionson the boundary of the cylinders. In this sense, our aim in the present paper is to consider a Newtonian Stokessystem with the non-homogeneous slip boundary condition (1.3) in the thin porous medium described in (1.1) andwe prove the convergence of the homogenization process depending on the different values of γ . To do that, wehave to take into account that the normal component of the velocity on the cylinders is different to zero and theextension of the velocity is no longer obvious. If we consider the Stokes system with Dirichlet boundary conditionon the obstacles as in [16], the velocity can be easily extended by zero in the obstacles, however in this case weneed another kind of extension and adapt it to the case of a thin domain.4ar´ıa Anguiano and Francisco Javier Su´arez-GrauOne of the main difficulties in the present paper is to treat the surface integrals. The papers mentioned aboveabout problems with non-homogeneous boundary conditions use a generalization (see Cioranescu and Donato [21])of the technique introduced by Vanninathan [22] for the Steklov problem, which transforms the surface integrals intovolume integrals. In our opinion, an excellent alternative to this technique was made possible with the developmentof the unfolding method (see Cioranescu et al. [17]), which allows to treat easily the surface integrals. In the presentpaper, we extend some abstract results for thin domains, using an adaptation of the unfolding method, in orderto treat all the surface integrals and we obtain directly the corresponding homogenized surface terms. A similarapproach is made by Cioranescu et al. [23] and Zaki [24] with non-homogeneous slip boundary conditions, andCapatina and Ene [25] with non-homogeneous pure slip boundary conditions for a fixed height domain.In summary, we show that the asymptotic behavior of the system (1.3) depends on the values of γ :- for γ < −
1, we obtain a 2D Darcy type law as an homogenized model. The flow is only driven by the pressure.- for − ≤ γ <
1, we obtain a 2D Darcy type law but in this case the flow depends on the pressure, the externalbody force and the mean value of the external surface force.- for γ ≥
1, we obtain a 2D Darcy type law where the flow is only driven by the pressure with a permeabilitytensor obtained by means of two local 2D Stokes problems posed in the reference cell with homogeneousNeumann boundary condition on the reference cylinder.We observe that we have obtained the same three regimes as in Cioranescu et al. [23] (see Theorems 2.1 and 2.2),and Zaki [24] (see Theorems 14 and 16). Thus, we conclude that the fact of considering the thin domain does notchange the critical size of the parameter γ , but the thickness of the domain introduces a reduction of dimensionof the homogenized models and other consequences. More precisely, in the cases γ < − − ≤ γ <
1, weobtain the same Darcy type law as in [23, 24] with the vertical component of the velocity zero as consequence ofthe thickness of the domain. The main difference appears in the case γ ≥
1, in which a 3D Brinkmann or Stokestype law were derived in [23, 24] while a 2D Darcy type law is obtained in the present paper.We also remark the differences with the result obtained in [16] where Dirichlet boundary conditions are pres-cribed on the cylinders in the case a ε ≈ ε . In that case, a 2D Darcy law as an homogenized model with apermeability tensor was obtained through two Stokes local problems in the reference cell with Dirichlet boundaryconditions on the reference cylinder. Here, we obtain three different homogenized model depending on γ . The case γ ≥ γ < − − ≤ γ <
1, we obtain a 2D Darcytype law without microstructure.The paper is organized as follows. We introduce some notations in Section 2. In Section 3, we formulate theproblem and state our main result, which is proved in Section 4. The article closes with a few remarks in Section5.
Along this paper, the points x ∈ R will be decomposed as x = ( x (cid:48) , x ) with x (cid:48) ∈ R , x ∈ R . We also use thenotation x (cid:48) to denote a generic vector of R .In order to apply the unfolding method, we need the following notation: for k (cid:48) ∈ Z , we define κ : R → Z by κ ( x (cid:48) ) = k (cid:48) ⇐⇒ x (cid:48) ∈ Y (cid:48) k (cid:48) , . (2.4)Remark that κ is well defined up to a set of zero measure in R , which is given by ∪ k (cid:48) ∈ Z ∂Y (cid:48) k (cid:48) , . Moreover, forevery ε >
0, we have κ (cid:18) x (cid:48) ε (cid:19) = k (cid:48) ⇐⇒ x (cid:48) ∈ Y (cid:48) k (cid:48) ,ε . v = ( v (cid:48) , v ) and a scalar function w , we introduce the operators: D ε , ∇ ε and div ε , by( D ε v ) i,j = ∂ x j v i for i = 1 , , , j = 1 , , ( D ε v ) i, = 1 ε ∂ y v i for i = 1 , , , ∇ ε w = ( ∇ x (cid:48) w, ε ∂ y w ) t , div ε v = div x (cid:48) v (cid:48) + 1 ε ∂ y v . We denote by |O| the Lebesgue measure of |O| (3-dimensional if O is a 3-dimensional open set, 2-dimensionalof O is a curve).For every bounded set O and if ϕ ∈ L ( O ), we define the average of ϕ on O by M O [ ϕ ] = 1 |O| (cid:90) O ϕ dx . (2.5)Similarly, for every compact set K of Y , if ϕ ∈ L ( ∂K ) then M ∂K [ ϕ ] = 1 | ∂K | (cid:90) ∂K ϕ dσ , is the average of ϕ over ∂K .We denote by L (cid:93) ( Y ), H (cid:93) ( Y ), the functional spaces L (cid:93) ( Y ) = (cid:110) v ∈ L loc ( Y ) : (cid:90) Y | v | dy < + ∞ , v ( y (cid:48) + k (cid:48) , y ) = v ( y ) ∀ k (cid:48) ∈ Z , a.e. y ∈ Y (cid:111) , and H (cid:93) ( Y ) = (cid:110) v ∈ H loc ( Y ) ∩ L (cid:93) ( Y ) : (cid:90) Y |∇ y v | dy < + ∞ (cid:111) . We denote by : the full contraction of two matrices, i.e. for A = ( a i,j ) ≤ i,j ≤ and B = ( b i,j ) ≤ i,j ≤ , we have A : B = (cid:80) i,j =1 a ij b ij .Finally, we denote by O ε a generic real sequence, which tends to zero with ε and can change from line to line,and by C a generic positive constant which also can change from line to line. In this section we describe the asymptotic behavior of a viscous fluid obeying (1.3) in the geometry Ω ε describedin (1.1). The proof of the corresponding results will be given in the next section. The variational formulation: let us introduce the spaces H ε = (cid:8) ϕ ∈ H (Ω ε ) : ϕ = 0 on ∂ Λ ε (cid:9) , H ε = (cid:8) ϕ ∈ H (Ω ε ) : ϕ = 0 on ∂ Λ ε (cid:9) , and (cid:101) H ε = (cid:110) ˜ ϕ ∈ H ( (cid:101) Ω ε ) : ˜ ϕ = 0 on ∂ Ω (cid:111) , (cid:101) H ε = (cid:110) ˜ ϕ ∈ H ( (cid:101) Ω ε ) : ˜ ϕ = 0 on ∂ Ω (cid:111) . Then, the variational formulation of system (1.3) is the following one: µ (cid:90) Ω ε Du ε : Dϕ dx − (cid:90) Ω ε p ε div ϕ dx + αε γ (cid:90) ∂S ε u ε · ϕ dσ ( x ) = (cid:90) Ω ε f (cid:48) ε · ϕ (cid:48) dx + (cid:90) ∂S ε g (cid:48) ε · ϕ (cid:48) dσ ( x ) , ∀ ϕ ∈ H ε , (cid:90) Ω ε u ε · ∇ ψ dx = (cid:90) ∂S ε ( u ε · n ) ψ dσ ( x ) , ∀ ψ ∈ H ε . (3.6)6ar´ıa Anguiano and Francisco Javier Su´arez-GrauAssume that f ε ( x ) = ( f (cid:48) ε ( x (cid:48) ) , ∈ L ( ω ) and g ε ( x ) = g ( x (cid:48) /ε ), where g is a Y (cid:48) -periodic function in L ( ∂T ) .Under these assumptions, it is well known that (3.6) has a unique solution ( u ε , p ε ) ∈ H ε × L (Ω ε ) (see Theorem4.1 and Remark 4.1 in [26] for more details).Our aim is to study the asymptotic behavior of u ε and p ε when ε tends to zero. For this purpose, we use thedilatation in the variable x , i.e. y = x ε , (3.7)in order to have the functions defined in the open set with fixed height (cid:101) Ω ε defined by (1.2).Namely, we define ˜ u ε ∈ (cid:101) H ε , ˜ p ε ∈ L ( (cid:101) Ω ε ) by˜ u ε ( x (cid:48) , y ) = u ε ( x (cid:48) , εy ) , ˜ p ε ( x (cid:48) , y ) = p ε ( x (cid:48) , εy ) , a.e. ( x (cid:48) , y ) ∈ (cid:101) Ω ε . Using the transformation (3.7), the system (1.3) can be rewritten as − µ ∆ x (cid:48) ˜ u ε − ε − µ∂ y ˜ u ε + ∇ x (cid:48) ˜ p ε + ε − ∂ y ˜ p ε e = f ε in (cid:101) Ω ε , div x (cid:48) ˜ u (cid:48) ε + ε − ∂ y ˜ u ε, = 0 in (cid:101) Ω ε , ˜ u ε = 0 on ∂ Ω , (3.8)with the non-homogeneous slip boundary condition, − ˜ p ε · n + µ ∂ ˜ u ε ∂n + αε γ ˜ u ε = g ε on ∂T ε , (3.9)where e = (0 , , t .Taking in (3.6) as test function ˜ ϕ ( x (cid:48) , x /ε ) with ˜ ϕ ∈ (cid:101) H ε and ˜ ψ ( x (cid:48) , x /ε ) with ˜ ψ ∈ (cid:101) H ε , applying the change ofvariables (3.7) and taking into account that dσ ( x ) = εdσ ( x (cid:48) ) dy , the variational formulation of system (3.8)-(3.9)is then the following one: µ (cid:90) (cid:101) Ω ε D ε ˜ u ε : D ε ˜ ϕ dx (cid:48) dy − (cid:90) (cid:101) Ω ε ˜ p ε div ε ˜ ϕ dx (cid:48) dy + αε γ (cid:90) ∂T ε ˜ u ε · ˜ ϕ dσ ( x (cid:48) ) dy = (cid:90) (cid:101) Ω ε f (cid:48) ε · ˜ ϕ (cid:48) dx (cid:48) dy + (cid:90) ∂T ε g (cid:48) ε · ˜ ϕ (cid:48) dσ ( x (cid:48) ) dy , ∀ ˜ ϕ ∈ (cid:101) H ε , (cid:90) (cid:101) Ω ε ˜ u ε · ∇ ε ˜ ψ dx (cid:48) dy = (cid:90) ∂T ε (˜ u ε · n ) ˜ ψ dσ ( x (cid:48) ) dy , ∀ ˜ ψ ∈ (cid:101) H ε . (3.10)In the sequel, we assume that the data f (cid:48) ε satisfies that there exists f (cid:48) ∈ L ( ω ) such that εf (cid:48) ε (cid:42) f (cid:48) weakly in L ( ω ) . (3.11)Observe that, due to the periodicity of the obstacles, if f (cid:48) ε = f (cid:48) ε where f (cid:48) ∈ L ( ω ) , then χ Ω ε f (cid:48) ε = εf (cid:48) ε (cid:42) θf (cid:48) in L ( ω ) , assuming εf (cid:48) ε extended by zero outside of ω ε , where θ denotes the proportion of the material in the cell Y (cid:48) given by θ := | Y (cid:48) f || Y (cid:48) | . We also define the constant µ := | ∂T (cid:48) || Y (cid:48) | . 7ar´ıa Anguiano and Francisco Javier Su´arez-Grau Main result: our goal then is to describe the asymptotic behavior of this new sequence (˜ u ε , ˜ p ε ) when ε tends tozero. The sequence of solutions (˜ u ε , ˜ p ε ) ∈ (cid:101) H ε × L ( (cid:101) Ω ε ) is not defined in a fixed domain independent of ε but ratherin a varying set (cid:101) Ω ε . In order to pass the limit if ε tends to zero, convergences in fixed Sobolev spaces (defined inΩ) are used which requires first that (˜ u ε , ˜ p ε ) be extended to the whole domain Ω. For the velocity, we will denoteby ˜ U ε the zero extension of ˜ u ε to the whole Ω, and for the pressure we will denote by ˜ P ε the zero extension of ˜ p ε to the whole Ω.Our main result referred to the asymptotic behavior of the solution of (3.8)-(3.9) is given by the followingtheorem. Theorem 3.1.
Let (˜ u ε , ˜ p ε ) be the solution of (3.8)-(3.9). Then there exists an extension operator (cid:101) Π ε ∈ L ( (cid:101) H ε ; H (Ω) ) such thati) if γ < − , then ˜ Π ε ˜ u ε (cid:42) in H (Ω) . Moreover, ( ε − ˜ U ε , ε − γ ˜ P ε ) is bounded in H (0 , L ( ω ) ) × L ( ω ) and any weak-limit point (˜ u ( x (cid:48) , y ) , ˜ p ( x (cid:48) )) of this sequence satisfies ˜ u (cid:48) = 0 on y = { , } , ˜ u = 0 and the following Darcy type law: ˜ v (cid:48) ( x (cid:48) ) = − θα µ ∇ x (cid:48) ˜ p ( x (cid:48) )˜ v ( x (cid:48) ) = 0 , in ω , (3.12) where ˜ v ( x (cid:48) ) = (cid:82) ˜ u ( x (cid:48) , y ) dy ,ii) if − ≤ γ < , then ε γ +12 ˜ Π ε ˜ u ε (cid:42) in H (Ω) . Moreover, ( ε γ ˜ U ε , ε ˜ P ε ) is bounded in H (0 , L ( ω ) ) × L ( ω ) and any weak-limit point (˜ u ( x (cid:48) , y ) , ˜ p ( x (cid:48) )) ofthis sequence satisfies ˜ u (cid:48) = 0 on y = { , } , ˜ u = 0 and the following Darcy type law: ˜ v (cid:48) ( x (cid:48) ) = θα µ ( f (cid:48) − ∇ x (cid:48) ˜ p ( x (cid:48) ) + µ M ∂T (cid:48) [ g (cid:48) ])˜ v ( x (cid:48) ) = 0 , in ω , (3.13) where ˜ v ( x (cid:48) ) = (cid:82) ˜ u ( x (cid:48) , y ) dy ,iii) if γ ≥ , then ε ˜ Π ε ˜ u ε (cid:42) in H (Ω) . Moreover, ( ˜ U ε , ε ˜ P ε ) is bounded in H (0 , L ( ω ) ) × L ( ω ) and any weak-limit point (˜ u ( x (cid:48) , y ) , ˜ p ( x (cid:48) )) of thissequence satisfies ˜ u = 0 on y = { , } and the following Darcy type law: ˜ v (cid:48) ( x (cid:48) ) = − θµ A ∇ x (cid:48) ˜ p ( x (cid:48) )˜ v ( x (cid:48) ) = 0 , in ω , (3.14) where ˜ v ( x (cid:48) ) = (cid:82) ˜ u ( x (cid:48) , y ) dy , and the symmetric and positive permeability tensor A ∈ R × is defined by itsentries A ij = 1 | Y (cid:48) f | (cid:90) Y (cid:48) f Dw i ( y (cid:48) ) : Dw j ( y (cid:48) ) dy, i, j = 1 , . For i = 1 , , w i ( y (cid:48) ) , denote the unique solutions in H (cid:93) ( Y (cid:48) f ) of the local Stokes problems with homogeneousNeumann boundary conditions in 2D: − ∆ y (cid:48) w i + ∇ y (cid:48) q i = e i in Y (cid:48) f div y (cid:48) ˆ w i = 0 in Y (cid:48) f ∂w i ∂n = 0 on ∂T (cid:48) ,w i , q i Y (cid:48) − periodic , M Y f [ w i ] = 0 . (3.15) Remark 3.2.
We observe that in the homogenized problems related to system (3.8)-(3.9), the limit functions do notsatisfy any incompressibility condition, so (3.12), (3.13) and (3.14) do not identify in a unique way (˜ v, ˜ p ) . This isa consequence of the fact that the normal component of ˜ u ε does not vanish on the boundary of the cylinders, so theaverage fluid flow, given by ˜ v , is (eventually) represented by the motion of a compressible fluid. As pointed out inConca [26] (see Remark 2.1) and Cioranescu et al. [1] (see Remarks 2.3 and 2.6), this result, which at first glanceseems unexpected, can be justified as follows: the boundary condition (3.9) implies that the normal component ˜ u ε · n of ˜ u ε is not necessarily zero on ∂T ε so we can not expect that the extension ˜ U ε will be a zero-divergence function.In fact, from the second variational formulation in (3.10), we have (cid:90) Ω ˜ U ε · ∇ ε ˜ ψ dx (cid:48) dy = (cid:90) ∂T ε (˜ u ε · n ) ˜ ψ dσ ( x (cid:48) ) dy , ∀ ψ ∈ ˜ H ε , and the term on the right-hand side is not necessarily zero. Therefore, by weak continuity, it is not possible toobtain an incompressibility condition of the form div x (cid:48) ˜ v (cid:48) ( x (cid:48) ) = 0 in ω as it is obtained in the case with Dirichletcondition considered in [16]. Roughly speaking, ˜ u ε · n (cid:54) = 0 on ∂T ε means that some fluid “dissapear” through thecylinders, and this fact implies that the incompressibility condition is not necessary verified at the limit in ω . In the context of homogenization of flow through porous media Arbogast et al. [27] use a L dilation operator toresolve oscillations on a prescribed scale of weakly converging sequences. It was observed in Bourgeat et al. [28]that this operator yields a characterization of two-scale convergence (see Allaire [29] and Nguetseng [30]). Later,Cioranescu et al. [17, 23] introduced the periodic unfolding method as a systematic approach to homogenizationwhich can be used for H functions and take into account the boundary of the holes by using the so-called boundaryunfolding operator. In this section we prove our main result. In particular, Theorem 3.1, is proved in Subsection4.3 by means of a combination of the unfolding method applied to the horizontal variables, with a rescaling onthe vertical variable. One of the main difficulties is to treat the surface integrals using an adaptation of theboundary unfolding operator. To apply this method, a priori estimates are established in Subsection 4.1 and somecompactness results are proved in Subsection 4.2. a priori estimates The a priori estimates independent of ε for ˜ u ε and ˜ p ε will be obtained by using an adaptation of the unfoldingmethod. Some abstract results for thin domains: let us introduce the adaption of the unfolding method in which wedivide the domain (cid:101) Ω ε in cubes of lateral lengths ε and vertical length 1. For this purpose, given ˜ ϕ ∈ L p ( (cid:101) Ω ε ) ,1 ≤ p < + ∞ , (assuming ˜ ϕ extended by zero outside of ω ε ), we define ˆ ϕ ε ∈ L p ( R × Y f ) byˆ ϕ ε ( x (cid:48) , y ) = ˜ ϕ (cid:18) εκ (cid:18) x (cid:48) ε (cid:19) + εy (cid:48) , y (cid:19) , a.e. ( x (cid:48) , y ) ∈ R × Y f , (4.16)9ar´ıa Anguiano and Francisco Javier Su´arez-Grauwhere the function κ is defined in (2.4). Remark 4.1.
The restriction of ˆ ϕ ε , to Y (cid:48) f (cid:48) k ,ε × Y f does not depend on x (cid:48) , whereas as a function of y it is obtainedfrom ˜ v ε , by using the change of variables y (cid:48) = x (cid:48) − εk (cid:48) ε , (4.17) which transforms Y f k (cid:48) ,ε into Y f . Proposition 4.2.
We have the following estimates:i) for every ˜ ϕ ∈ L p ( (cid:101) Ω ε ) , ≤ p + ∞ , we have (cid:107) ˆ ϕ ε (cid:107) L p ( R × Y f ) = | Y (cid:48) | p (cid:107) ˜ ϕ (cid:107) L p ( (cid:101) Ω ε ) , (4.18) where ˆ ϕ ε is given by (4.16),ii) for every ˜ ϕ ∈ W ,p ( (cid:101) Ω ε ) , ≤ p < + ∞ , we have that ˆ ϕ ε , given by (4.16), belongs to L p ( R ; W ,p ( Y f ) ) , and (cid:107) D y ˆ ϕ ε (cid:107) L p ( R × Y f ) × = ε | Y (cid:48) | p (cid:107) D ε ˜ ϕ (cid:107) L p ( (cid:101) Ω ε ) × . (4.19) Proof.
Let us prove i ). Using Remark 4.1 and definition (4.16), we have (cid:90) R × Y f | ˆ ϕ ε ( x (cid:48) , y ) | p dx (cid:48) dy = (cid:88) k (cid:48) ∈ Z (cid:90) Y (cid:48) k (cid:48) ,ε (cid:90) Y f | ˆ ϕ ε ( x (cid:48) , y ) | p dx (cid:48) dy = (cid:88) k (cid:48) ∈ Z (cid:90) Y (cid:48) k (cid:48) ,ε (cid:90) Y f | ˜ ϕ ( εk (cid:48) + εy (cid:48) , y ) | p dx (cid:48) dy. We observe that ˜ ϕ does not depend on x (cid:48) , then we can deduce (cid:90) R × Y f | ˆ ϕ ε ( x (cid:48) , y ) | p dx (cid:48) dy = ε | Y (cid:48) | (cid:88) k (cid:48) ∈ Z (cid:90) Y f | ˜ ϕ ( εk (cid:48) + εy (cid:48) , y ) | p dy. For every k (cid:48) ∈ Z , by the change of variable (4.17), we have k (cid:48) + y (cid:48) = x (cid:48) ε , dy (cid:48) = dx (cid:48) ε ∂ y (cid:48) = ε∂ x (cid:48) , (4.20)and we obtain (cid:90) R × Y f | ˆ ϕ ε ( x (cid:48) , y ) | p dx (cid:48) dy = | Y (cid:48) | (cid:90) ω ε × (0 , | ˜ ϕ ( x (cid:48) , y ) | p dx (cid:48) dy which gives (4.18).Let us prove ii ). Taking into account the definition (4.16) of ˆ ϕ ε and observing that ˜ ϕ does not depend on x (cid:48) ,then we can deduce (cid:90) R × Y f | D y (cid:48) ˆ ϕ ε ( x (cid:48) , y ) | p dx (cid:48) dy = ε | Y (cid:48) | (cid:88) k (cid:48) ∈ Z (cid:90) Y f | D y (cid:48) ˜ ϕ ( εk (cid:48) + εy (cid:48) , y ) | p dy. By (4.20), we obtain (cid:90) R × Y f | D y (cid:48) ˆ ϕ ε ( x (cid:48) , y ) | p dx (cid:48) dy = ε p | Y (cid:48) | (cid:88) k (cid:48) ∈ Z (cid:90) Y (cid:48) fk (cid:48) ,ε (cid:90) | D x (cid:48) ˜ ϕ ( x (cid:48) , y ) | p dx (cid:48) dy = ε p | Y (cid:48) | (cid:90) ω ε × (0 , | D x (cid:48) ˜ ϕ ( x (cid:48) , y ) | p dx (cid:48) dy . (4.21)10ar´ıa Anguiano and Francisco Javier Su´arez-GrauFor the partial of the vertical variable, proceeding similarly to (4.18), we obtain (cid:90) R × Y f | ∂ y ˆ ϕ ε ( x (cid:48) , y ) | p dx (cid:48) dy = | Y (cid:48) | (cid:90) ω ε × (0 , | ∂ y ˜ ϕ ( x (cid:48) , y ) | p dx (cid:48) dy = ε p | Y (cid:48) | (cid:90) ω ε × (0 , (cid:12)(cid:12)(cid:12)(cid:12) ε ∂ y ˜ ϕ ( x (cid:48) , y ) (cid:12)(cid:12)(cid:12)(cid:12) p dx (cid:48) dy , which together with (4.21) gives (4.19).In a similar way, let us introduce the adaption of the unfolding method on the boundary of the obstacles ∂T ε (see Cioranescu et al. [23] for more details). For this purpose, given ˜ ϕ ∈ L p ( ∂T ε ) , 1 ≤ p < + ∞ , we defineˆ ϕ bε ∈ L p ( R × ∂T ) by ˆ ϕ bε ( x (cid:48) , y ) = ˜ ϕ (cid:18) εκ (cid:18) x (cid:48) ε (cid:19) + εy (cid:48) , y (cid:19) , a.e. ( x (cid:48) , y ) ∈ R × ∂T, (4.22)where the function κ is defined in (2.4). Remark 4.3.
Observe that from this definition, if we consider ˜ ϕ ∈ L p ( ∂T ) , ≤ p < + ∞ , a Y (cid:48) -periodic function,and we define ˜ ϕ ε ( x (cid:48) , y ) = ˜ ϕ ( x (cid:48) /ε, y ) , it follows that ˆ ϕ bε ( x (cid:48) , y ) = ˜ ϕ ( y ) .Observe that for ˜ ϕ ∈ W ,p ( (cid:101) Ω ε ) , ˆ ϕ bε is the trace on ∂T of ˆ ϕ ε . Therefore ˆ ϕ bε has a similar properties as ˆ ϕ ε . We have the following property.
Proposition 4.4. If ˜ ϕ ∈ L p ( ∂T ε ) , ≤ p < + ∞ , then (cid:107) ˆ ϕ bε (cid:107) L p ( R × ∂T ) = ε p | Y (cid:48) | p (cid:107) ˜ ϕ (cid:107) L p ( ∂T ε ) , (4.23) where ˆ ϕ bε is given by (4.22).Proof. We take ( x (cid:48) , y ) ∈ ∂T k (cid:48) ,ε . Taking into account (4.22), we obtain (cid:90) R × ∂T | ˆ ϕ bε ( x (cid:48) , y ) | p dx (cid:48) dσ ( y ) = ε | Y (cid:48) | (cid:88) k (cid:48) ∈ Z (cid:90) ∂T | ˜ ϕ ( εk (cid:48) + εy (cid:48) , y ) | p dσ ( y ) . For every k (cid:48) ∈ Z , by taking x (cid:48) = ε ( k (cid:48) + y (cid:48) ), we have dσ ( x (cid:48) ) = εdσ ( y (cid:48) ). Since the thickness of the obstacles is one,we have that dσ ( x (cid:48) ) dy = εdσ ( y ). Hence (cid:90) R × ∂T | ˆ ϕ bε ( x (cid:48) , y ) | p dx (cid:48) dσ ( y ) = ε | Y (cid:48) | (cid:90) ∂T ε | ˜ ϕ ( x (cid:48) , y ) | p dσ ( x (cid:48) ) dy , which gives (4.23).Now, let us give two results which will be useful for obtaining a priori estimates of the solution (˜ u ε , ˜ p ε ) ofproblem (3.8)-(3.9). These results are an extension of Cioranescu et al. (Proposition 5.3 and Corollary 5.4 in [31])to the thin domain case. Proposition 4.5.
Let g ∈ L ( ∂T (cid:48) ) and ˜ ϕ ∈ H ( (cid:101) Ω ε ) , extended by zero in outside of w ε . Let ˆ ϕ ε be given by(4.16). Then, there exists a positive constant C , independent of ε , such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × ∂T g ( y (cid:48) ) · ˆ ϕ ε ( x (cid:48) , y ) dx (cid:48) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C |M ∂T (cid:48) [ g ] | (cid:16) (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) + ε (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × (cid:17) . (4.24) In particular, if g = 1 , there exists a positive constant C , independent of ε , such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × ∂T ˆ ϕ ε ( x (cid:48) , y ) dx (cid:48) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) + ε (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × (cid:17) . (4.25)11ar´ıa Anguiano and Francisco Javier Su´arez-Grau Proof.
Due to density properties, it is enough to prove this estimate for functions in D ( R ) . Let ˜ ϕ ∈ D ( R ) , onehas (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × ∂T g ( y (cid:48) ) · ˆ ϕ ε ( x (cid:48) , y ) dx (cid:48) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × ∂T g ( y (cid:48) ) · ˜ ϕ (cid:18) εκ (cid:18) x (cid:48) ε (cid:19) + εy (cid:48) , y (cid:19) dx (cid:48) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × ∂T g ( y (cid:48) ) · ˜ ϕ (cid:18) εκ (cid:18) x (cid:48) ε (cid:19) , y (cid:19) dx (cid:48) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × ∂T g ( y (cid:48) ) · (cid:18) ˜ ϕ (cid:18) εκ (cid:18) x (cid:48) ε (cid:19) + εy (cid:48) , y (cid:19) − ˜ ϕ (cid:18) εκ (cid:18) x (cid:48) ε (cid:19) , y (cid:19)(cid:19) dx (cid:48) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C |M ∂T (cid:48) [ g ] | (cid:16) (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) + ε (cid:107) D x (cid:48) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × (cid:17) ≤ C |M ∂T (cid:48) [ g ] | (cid:16) (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) + ε (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × (cid:17) , which implies (4.24). In particular, if g = 1, proceeding as above, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × ∂T ˆ ϕ ε ( x (cid:48) , y ) dx (cid:48) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × ∂T ˜ ϕ (cid:18) εκ (cid:18) x (cid:48) ε (cid:19) + εy (cid:48) , y (cid:19) dx (cid:48) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × ∂T ˜ ϕ (cid:18) εκ (cid:18) x (cid:48) ε (cid:19) , y (cid:19) dx (cid:48) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × ∂T (cid:18) ˜ ϕ (cid:18) εκ (cid:18) x (cid:48) ε (cid:19) + εy (cid:48) , y (cid:19) − ˜ ϕ (cid:18) εκ (cid:18) x (cid:48) ε (cid:19) , y (cid:19)(cid:19) dx (cid:48) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) + ε (cid:107) D x (cid:48) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × (cid:17) ≤ C (cid:16) (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) + ε (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × (cid:17) , which implies (4.25). Corollary 4.6.
Let g ∈ L ( ∂T ) be a Y (cid:48) -periodic function. Then, for every ˜ ϕ ∈ H ( (cid:101) Ω ε ) , we have that there existsa positive constant C , independent of ε , such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂T ε g ( x (cid:48) /ε ) · ˜ ϕ ( x (cid:48) , y ) dσ ( x (cid:48) ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε (cid:16) (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) + ε (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × (cid:17) . (4.26) In particular, if g = 1 , there exists a positive constant C , independent of ε , such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂T ε ˜ ϕ ( x (cid:48) , y ) dσ ( x (cid:48) ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε (cid:16) (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) + ε (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × (cid:17) . (4.27) Proof.
Since ˜ ϕ ∈ H ( (cid:101) Ω ε ) , then ˆ ϕ bε has similar properties as ˆ ϕ ε . By using Proposition 4.4 with p = 1 and Remark4.3, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂T ε g ( x (cid:48) /ε ) · ˜ ϕ ( x (cid:48) , y ) dσ ( x (cid:48) ) dy (cid:12)(cid:12)(cid:12)(cid:12) = 1 ε | Y (cid:48) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) R × ∂T g ( y (cid:48) ) · ˆ ϕ ε ( x (cid:48) , y ) dx (cid:48) dσ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) , and by Proposition 4.5, we can deduce estimates (4.26) and (4.27).12ar´ıa Anguiano and Francisco Javier Su´arez-GrauMoreover, for the proof of the a priori estimates for the velocity, we need the following lemma due to Conca[26] generalized to a thin domain Ω ε . Lemma 4.7.
There exists a constant C independent of ε , such that, for any function ϕ ∈ H ε , one has (cid:107) ϕ (cid:107) L (Ω ε ) ≤ C (cid:16) ε (cid:107) Dϕ (cid:107) L (Ω ε ) × + ε (cid:107) ϕ (cid:107) L ( ∂S ε ) (cid:17) . (4.28) Proof.
We observe that the microscale of the porous medium ε is similar than the thickness of the domain ε , whichlead us to divide the domain Ω ε in small cubes of lateral length ε and vertical length ε . We consider the periodiccell Y . The function ϕ → (cid:16) (cid:107) Dϕ (cid:107) L ( Y f ) × + (cid:107) ϕ (cid:107) L ( ∂T ) (cid:17) / is a norm in H ( Y f ) , equivalent to the H ( Y f ) -norm(see Neˇcas [32]). Therefore, for any function ϕ ( z ) ∈ H ( Y f ) , we have (cid:90) Y f | ϕ | dz ≤ C (cid:32)(cid:90) Y f | D z ϕ | dz + (cid:90) ∂T | ϕ | dσ ( z ) (cid:33) , (4.29)where the constant C depends only on Y f .Then, for every k (cid:48) ∈ Z , by the change of variable k (cid:48) + z (cid:48) = x (cid:48) ε , z = x ε , dz = dxε , ∂ z = ε∂ x , dσ ( x ) = ε dσ ( z ) , (4.30)we rescale (4.29) from Y f to Q f k (cid:48) ,ε = Y (cid:48) f k (cid:48) ,ε × (0 , ε ). This yields that, for any function ϕ ( x ) ∈ H ( Q f k (cid:48) ,ε ) , one has (cid:90) Q fk (cid:48) ,ε | ϕ | dx ≤ C (cid:32) ε (cid:90) Q fk (cid:48) ,ε | D x ϕ | dx + ε (cid:90) T (cid:48) k (cid:48) ,ε × (0 ,ε ) | ϕ | dσ ( x ) (cid:33) , (4.31)with the same constant C as in (4.29). Summing the inequality (4.31) for all the periods Q f k (cid:48) ,ε and T (cid:48) k (cid:48) ,ε × (0 , ε ),gives (cid:90) Ω ε | ϕ | dx ≤ C (cid:18) ε (cid:90) Ω ε | D x ϕ | dx + ε (cid:90) ∂S ε | ϕ | dσ ( x ) (cid:19) . In fact, we must consider separately the periods containing a portion of ∂ω , but they yield at a distance O ( ε ) of ∂ω ,where ϕ is zero. Therefore, using Poincare’s inequality one can easily verify that in this part (4.28) holds withoutconsidering the boundary term occuring in (4.28).Considering the change of variables given in (3.7) and taking into account that dσ ( x ) = εdσ ( x (cid:48) ) dy , we obtainthe following result for the domain (cid:101) Ω ε . Corollary 4.8.
There exists a constant C independent of ε , such that, for any function ˜ ϕ ∈ (cid:101) H ε , one has (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) ≤ C (cid:16) ε (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × + ε (cid:107) ˜ ϕ (cid:107) L ( ∂T ε ) (cid:17) . (4.32)The presence in (1.3) of the stress tensor in the boundary condition implies that the extension of the velocityis no longer obvious. If we consider the Stokes system with Dirichlet boundary condition on the obstacles, thevelocity would be extended by zero in the obstacles. However, in this case, we need another kind of extension forthe case in which the velocity is non-zero on the obstacles. Since in the extension required, the vertical variableis not concerned, therefore the proof of the required statement is basically the extension of the result given inCioranescu and Saint-Jean Paulin [33, 34] to the time-depending case given in Cioranescu and Donato [35], so weomit the proof. We remark that the extension is not divergence free, so we cannot expect the homogenized solutionto be divergence free. Hence we cannot use test functions that are divergence free in the variational formulation,which implies that the pressure has to be included. 13ar´ıa Anguiano and Francisco Javier Su´arez-Grau Lemma 4.9.
There exists an extension operator Π ε ∈ L ( H ε ; H (Λ ε ) ) and a positive constant C , independent of ε , such that Π ε ϕ ( x ) = ϕ ( x ) , if x ∈ Ω ε , (cid:107) DΠ ε ϕ (cid:107) L (Λ ε ) × ≤ C (cid:107) Dϕ (cid:107) L (Ω ε ) × , ∀ ϕ ∈ H ε . Considering the change of variables given in (3.7), we obtain the following result for the domain (cid:101) Ω ε . Corollary 4.10.
There exists an extension operator (cid:101) Π ε ∈ L ( (cid:101) H ε ; H (Ω) ) and a positive constant C , independentof ε , such that (cid:101) Π ε ˜ ϕ ( x (cid:48) , y ) = ˜ ϕ ( x (cid:48) , y ) , if ( x (cid:48) , y ) ∈ (cid:101) Ω ε , (cid:107) D ε (cid:101) Π ε ˜ ϕ (cid:107) L (Ω) × ≤ C (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × , ∀ ˜ ϕ ∈ (cid:101) H ε . Using Corollary 4.10, we obtain a Poincar´e inequality in (cid:101) H ε . Corollary 4.11.
There exists a constant C independent of ε , such that, for any function ˜ ϕ ∈ (cid:101) H ε , one has (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) ≤ C (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × . (4.33) Proof.
We observe that (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) ≤ (cid:107) (cid:101) Π ε ˜ ϕ (cid:107) L (Ω) , ∀ ˜ ϕ ∈ (cid:101) H ε . (4.34)Since (cid:101) Π ε ˜ ϕ ∈ H (Ω) , we can apply the Poincar´e inequality in H (Ω) and then taking into account Corollary 4.10,we get (cid:107) (cid:101) Π ε ˜ ϕ (cid:107) L (Ω) ≤ C (cid:107) D (cid:101) Π ε ˜ ϕ (cid:107) L (Ω) × ≤ C (cid:107) D ε (cid:101) Π ε ˜ ϕ (cid:107) L (Ω) × ≤ C (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × . This together with (4.34) gives (4.33).Now, for the proof of the a priori estimates for the pressure, we also need the following lemma due to Conca[26] generalized to a thin domain Ω ε . Lemma 4.12.
There exists a constant C independent of ε , such that, for each q ∈ L (Ω ε ) , there exists ϕ = ϕ ( q ) ∈ H ε , such that div ϕ = q in Ω ε , (4.35) (cid:107) ϕ (cid:107) L (Ω ε ) ≤ C (cid:107) q (cid:107) L (Ω ε ) , (cid:107) Dϕ (cid:107) L (Ω ε ) × ≤ Cε (cid:107) q (cid:107) L (Ω ε ) . (4.36) Proof.
Let q ∈ L (Ω ε ) be given. We extend q inside the cylinders by means of the function: Q ( x ) = q ( x ) if x ∈ Ω ε − | Λ ε − Ω ε | (cid:90) Ω ε q ( x ) dx if x ∈ Λ ε − Ω ε . It is follows that Q ∈ L (Λ ε ) = { p ∈ L (Λ ε ) : (cid:82) Λ ε p dx = 0 } and (cid:107) Q (cid:107) L (Λ ε ) = (cid:107) q (cid:107) L (Ω ε ) + 1 | Λ ε − Ω ε | (cid:18)(cid:90) Ω ε q ( x ) dx (cid:19) . (4.37)Since | Λ ε − Ω ε | is bounded from below by a positive number, it follows from (4.37) and Cauchy-Schwartz inequalitythat (cid:107) Q (cid:107) L (Λ ε ) ≤ C (cid:107) q (cid:107) L (Ω ε ) . (4.38)14ar´ıa Anguiano and Francisco Javier Su´arez-GrauBesides that, since Q ∈ L (Λ ε ), it follows from Maruˇsi´c and Maruˇsi´c-Paloka (Lemma 20 in [36]) that we can find ϕ ∈ H (Λ ε ) such that div ϕ = Q in Λ ε , (4.39) (cid:107) ϕ (cid:107) L (Λ ε ) ≤ C (cid:107) Q (cid:107) L (Λ ε ) , (cid:107) Dϕ (cid:107) L (Λ ε ) × ≤ Cε (cid:107) Q (cid:107) L (Λ ε ) . (4.40)Let us consider ϕ | Ω ε : it belongs to H ε . Moreover, (4.35) follows from (4.39) and the estimates (4.36) follows from(4.40) and (4.38).Considering the change of variables given in (3.7), we obtain the following result for the domain (cid:101) Ω ε . Corollary 4.13.
There exists a constant C independent of ε , such that, for each ˜ q ∈ L ( (cid:101) Ω ε ) , there exists ˜ ϕ =˜ ϕ (˜ q ) ∈ (cid:101) H ε , such that div ε ˜ ϕ = ˜ q in (cid:101) Ω ε , (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) ≤ C (cid:107) ˜ q (cid:107) L ( (cid:101) Ω ε ) , (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × ≤ Cε (cid:107) ˜ q (cid:107) L ( (cid:101) Ω ε ) . A priori estimates for ( ˜ u ε , ˜ p ε ) in (cid:101) Ω ε : first, let us obtain some a priori estimates for ˜ u ε for different values of γ . Lemma 4.14.
We distinguish three cases:i) for γ < − , then there exists a constant C independent of ε , such that (cid:107) ˜ u ε (cid:107) L ( (cid:101) Ω ε ) ≤ Cε, (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × ≤ C . (4.41) ii) for − ≤ γ < , then there exists a constant C independent of ε , such that (cid:107) ˜ u ε (cid:107) L ( (cid:101) Ω ε ) ≤ Cε − γ , (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × ≤ Cε − γ . (4.42) iii) for γ ≥ , then there exists a constant C independent of ε , such that (cid:107) ˜ u ε (cid:107) L ( (cid:101) Ω ε ) ≤ Cε − , (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × ≤ Cε − . (4.43) Proof.
Taking ˜ u ε ∈ (cid:101) H ε as function test in (3.10), we have µ (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × + αε γ (cid:107) ˜ u ε (cid:107) L ( ∂T ε ) = (cid:90) (cid:101) Ω ε f (cid:48) ε · ˜ u (cid:48) ε dx (cid:48) dy + (cid:90) ∂T ε g (cid:48) ε · ˜ u (cid:48) ε dσ ( x (cid:48) ) dy . (4.44)Using Cauchy-Schwarz’s inequality and f (cid:48) ε ∈ L (Ω) , we obtain that (cid:90) (cid:101) Ω ε f (cid:48) ε · ˜ u (cid:48) ε dx (cid:48) dy ≤ C (cid:107) ˜ u ε (cid:107) L ( (cid:101) Ω ε ) , and by using that g (cid:48) ∈ L ( ∂T ) is a Y (cid:48) -periodic function and estimate (4.26), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂T ε g (cid:48) ε · ˜ u (cid:48) ε dσ ( x (cid:48) ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cε (cid:16) (cid:107) ˜ u ε (cid:107) L ( (cid:101) Ω ε ) + ε (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × (cid:17) . Putting these estimates in (4.44), we get µ (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × + αε γ (cid:107) ˜ u ε (cid:107) L ( ∂T ε ) ≤ C (cid:16) (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × + ε − (cid:107) ˜ u ε (cid:107) L ( (cid:101) Ω ε ) (cid:17) . (4.45)15ar´ıa Anguiano and Francisco Javier Su´arez-GrauIn particular, if we use the Poincar´e inequality (4.33) in (4.45), we have (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × ≤ Cε , (4.46)therefore (independently of γ ∈ R ), using again (4.33), we get (cid:107) ˜ u ε (cid:107) L ( (cid:101) Ω ε ) ≤ Cε . (4.47)These estimates can be refined following the different values of γ . To do so, observe that from estimate (4.32) wehave ε − (cid:107) ˜ u ε (cid:107) L ( (cid:101) Ω ε ) ≤ C (cid:16) (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × + ε − (cid:107) ˜ u ε (cid:107) L ( ∂T ε ) (cid:17) . Using Young’s inequality, we get ε − (cid:107) ˜ u ε (cid:107) L ( ∂T ε ) ≤ ε − γ ε γ (cid:107) ˜ u ε (cid:107) L ( ∂T ε ) ≤ α ε − − γ + α ε γ (cid:107) ˜ u ε (cid:107) L ( ∂T ε ) . Consequently, from (4.45), we get µ (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × + α ε γ (cid:107) ˜ u ε (cid:107) L ( ∂T ε ) ≤ C (cid:16) (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × + ε − − γ (cid:17) , which applying in a suitable way the Young inequality gives µ (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × + αε γ (cid:107) ˜ u ε (cid:107) L ( ∂T ε ) ≤ C (cid:0) ε − − γ (cid:1) . (4.48)For the case when γ < −
1, estimate (4.48) reads (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × ≤ C, (cid:107) ˜ u ε (cid:107) L ( ∂T ε ) ≤ Cε − γ . Then, estimate (4.32) gives (cid:107) ˜ u ε (cid:107) L ( (cid:101) Ω ε ) ≤ C ( ε + ε − γ ) ≤ Cε, since 1 ≤ (1 − γ ) /
2, and so, we have proved (4.41).For γ ≥ −
1, estimate (4.48) reads (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × ≤ Cε − γ , (cid:107) ˜ u ε (cid:107) L ( ∂T ε ) ≤ Cε − − γ . Applying estimate (4.32), we get (cid:107) ˜ u ε (cid:107) L ( (cid:101) Ω ε ) ≤ C ( ε − γ + ε − γ ) ≤ Cε − γ since − γ ≤ (1 − γ ) /
2. Then, we have proved (4.42) for − ≤ γ <
1. Observe that for γ ≥
1, the estimates(4.46)-(4.47) are the optimal ones, so we have (4.43).We will prove now a priori estimates for the pressure ˜ p ε for different values of γ . Lemma 4.15.
We distinguish three cases:i) for γ < − , then there exists a constant C independent of ε , such that (cid:107) ˜ p ε (cid:107) L ( (cid:101) Ω ε ) ≤ C ε γ . (4.49)16ar´ıa Anguiano and Francisco Javier Su´arez-Grau ii) for − ≤ γ < , then there exists a constant C independent of ε , such that (cid:107) ˜ p ε (cid:107) L ( (cid:101) Ω ε ) ≤ C ε − . (4.50) iii) for γ ≥ , then there exists a constant C independent of ε , such that (cid:107) ˜ p ε (cid:107) L ( (cid:101) Ω ε ) ≤ C ε − . (4.51) Proof.
Let ˜ Φ ∈ L ( (cid:101) Ω ε ). From Corollary 4.13, there exists ˜ ϕ ∈ (cid:101) H ε such thatdiv ε ˜ ϕ = ˜ Φ in (cid:101) Ω ε , (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) ≤ C (cid:107) ˜ Φ (cid:107) L ( (cid:101) Ω ε ) , (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × ≤ Cε (cid:107) ˜ Φ (cid:107) L ( (cid:101) Ω ε ) . (4.52)Taking ˜ ϕ ∈ (cid:101) H ε as function test in (3.10), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:101) Ω ε ˜ p ε ˜ Φ dx (cid:48) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ µ (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × + αε γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂T ε ˜ u ε · ˜ ϕ dσ ( x (cid:48) ) dy (cid:12)(cid:12)(cid:12)(cid:12) + C (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂T ε g (cid:48) ε · ˜ ϕ (cid:48) dσ ( x (cid:48) ) dy (cid:12)(cid:12)(cid:12)(cid:12) . (4.53)By using that g ∈ L ( ∂T ) is a Y (cid:48) -periodic function and estimate (4.26), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂T ε g (cid:48) ε · ˜ ϕ (cid:48) dσ ( x (cid:48) ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:16) ε − (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) + (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) × (cid:17) . Analogously, using estimate (4.27) and the Cauchy- Schwarz inequality, a simple computation gives αε γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂T ε ˜ u ε · ˜ ϕ dσ ( x (cid:48) ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε γ − C (cid:107) ˜ u ε (cid:107) L ( (cid:101) Ω ε ) (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) + ε γ C (cid:107) ˜ u ε (cid:107) L ( (cid:101) Ω ε ) (cid:107) D ε ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) + ε γ C (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) (cid:107) ˜ ϕ (cid:107) L ( (cid:101) Ω ε ) . Then, turning back to (4.53) and using (4.52), one has (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:101) Ω ε ˜ p ε ˜ Φ dx (cid:48) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:0) ε − + ε γ (cid:1) (cid:107) D ε ˜ u ε (cid:107) L ( (cid:101) Ω ε ) × (cid:107) ˜ Φ (cid:107) L ( (cid:101) Ω ε ) + C (cid:16) ε γ − (cid:107) ˜ u ε (cid:107) L ( (cid:101) Ω ε ) + ε − (cid:17) (cid:107) ˜ Φ (cid:107) L ( (cid:101) Ω ε ) . (4.54)The a priori estimates for the pressure follow now from (4.54) and estimates (4.41)-(4.42) and (4.43), correspondingto the different values of γ . A priori estimates of the unfolding functions (ˆ u ε , ˆ p ε ) : let us obtain some a priori estimates for the sequences(ˆ u ε , ˆ p ε ) where ˆ u ε and ˆ p ε are obtained by applying the change of variable (4.16) to (˜ u ε , ˜ p ε ). Lemma 4.16.
We distinguish three cases:i) for γ < − , then there exists a constant C independent of ε , such that (cid:107) ˆ u ε (cid:107) L ( R × Y f ) ≤ Cε, (cid:107) D y ˆ u ε (cid:107) L ( R × Y f ) × ≤ Cε, (4.55) (cid:107) ˆ p ε (cid:107) L ( R × Y f ) ≤ Cε γ . (4.56)17ar´ıa Anguiano and Francisco Javier Su´arez-Grau ii) for − ≤ γ < , then there exists a constant C independent of ε , such that (cid:107) ˆ u ε (cid:107) L ( R × Y f ) ≤ Cε − γ , (cid:107) D y ˆ u ε (cid:107) L ( R × Y f ) × ≤ Cε − γ , (4.57) (cid:107) ˆ p ε (cid:107) L ( R × Y f ) ≤ Cε − . (4.58) iii) for γ ≥ , then there exists a constant C independent of ε , such that (cid:107) ˆ u ε (cid:107) L ( R × Y f ) ≤ Cε − , (cid:107) D y ˆ u ε (cid:107) L ( R × Y f ) × ≤ C, (4.59) (cid:107) ˆ p ε (cid:107) L ( R × Y f ) ≤ Cε − . (4.60) Proof.
Using properties (4.18) and (4.19) with p = 2 and the a priori estimates given in Lemma 4.14 and Lemma4.15, we have the desired result. Let us remember that, for the velocity, we denote by ˜ U ε the zero extension of ˜ u ε to the whole Ω, and for thepressure we denote by ˜ P ε the zero extension of ˜ p ε to the whole Ω. In this subsection we obtain some compactnessresults about the behavior of the sequences ˜ Π ε ˜ u ε , where ˜Π ε is given in Corollary 4.10, ( ˜ U ε , ˜ P ε ) and (ˆ u ε , ˆ p ε ). Lemma 4.17.
There exists an extension operator ˜Π ε , given in Corollary 4.10, such thati) for γ < − , then ˜ Π ε ˜ u ε (cid:42) in H (Ω) . (4.61) Moreover, ( ε − ˜ U ε , ε − γ ˜ P ε ) is bounded in H (0 , L ( ω ) ) × L (Ω) and any weak-limit point (˜ u, ˜ p ) of thissequence satisfies ε − ˜ U ε (cid:42) ˜ u in H (0 , L ( ω ) ) , (4.62) ε − γ ˜ P ε (cid:42) ˜ p in L (Ω) , (4.63) ii) for − ≤ γ < , then ε γ +12 ˜ Π ε ˜ u ε (cid:42) in H (Ω) . (4.64) Moreover, ( ε γ ˜ U ε , ε ˜ P ε ) is bounded in H (0 , L ( ω ) ) × L (Ω) and any weak-limit point (˜ u, ˜ p ) of this sequencesatisfies ε γ ˜ U ε (cid:42) ˜ u in H (0 , L ( ω ) ) , (4.65) ε ˜ P ε (cid:42) ˜ p in L (Ω) , (4.66) iii) for γ ≥ , then ε ˜ Π ε ˜ u ε (cid:42) in H (Ω) . (4.67) Moreover, ( ˜ U ε , ε ˜ P ε ) is bounded in H (0 , L ( ω ) ) × L (Ω) and any weak-limit point (˜ u, ˜ p ) of this sequencesatisfies ˜ U ε (cid:42) ˜ u in H (0 , L ( ω ) ) , (4.68) ε ˜ P ε (cid:42) ˜ p in L (Ω) . (4.69)18ar´ıa Anguiano and Francisco Javier Su´arez-Grau Proof.
We proceed in three steps:
Step 1.
For γ < −
1: from estimates (4.41) and (4.49), we have immediately the convergences, after eventualextraction of subsequences, (4.62) and (4.63).Moreover, we have || ˜ u ε || L ( (cid:101) Ω ε ) ≤ C, and we can apply Corollary 4.10 to ˜ u ε and we deduce the existence of u ∗ ∈ H (Ω) such that ˜ Π ε ˜ u ε convergesweakly to u ∗ in H (Ω) . Consequently, by Rellich theorem, ˜ Π ε ˜ u ε converges strongly to u ∗ in L (Ω) .On the other side, we have the following indentity: χ (cid:101) Ω ε (cid:16) ˜ Π ε ˜ u ε (cid:17) = ε ε − ˜ U ε in Ω . Due the periodicity of the domain (cid:101) Ω ε we have that χ (cid:101) Ω ε converges weakly- (cid:63) to | Y (cid:48) f || Y (cid:48) | in L ∞ (Ω), and we can pass tothe limit in the term of the left hand side. Thus, χ (cid:101) Ω ε (cid:16) ˜ Π ε ˜ u ε (cid:17) converges weakly to | Y (cid:48) f || Y (cid:48) | u ∗ in L (Ω) . In the righthand side, ˜ U ε converges to zero, so we obtain (4.61). Step 2.
For − ≤ γ <
1: from the estimates (4.42) and (4.50), we deduce convergences (4.65) and (4.66).Moreover, as − ≤ γ <
1, we have (cid:107) ε γ +12 ˜ u ε (cid:107) L ( (cid:101) Ω ε ) ≤ C, and using Corollary 4.10, we have ε γ +12 ˜ Π ε ˜ u ε (cid:42) u ∗ in H (Ω) . Consequently, ε γ +12 ˜ Π ε ˜ u ε → u ∗ in L (Ω) , and passing to the limit in the identity χ (cid:101) Ω ε (cid:16) ε γ +12 ˜ Π ε ˜ u ε (cid:17) = ε − γ ε γ ˜ U ε in Ω , we deduce that u ∗ = 0, and so (4.64) is proved. Step 3.
For γ ≥
1: from estimate (4.43) and Dirichlet boundary condition, we deduce that || ˜ U ε || L (Ω) ≤ || ∂ y ˜ U ε || L (Ω) ≤ C, and we have immediately, after eventual extraction of subsequences, the convergence (4.68). From estimate (4.51),we have immediately, after eventual extraction of subsequences, the convergence (4.69).Moreover, we can apply Corollary 4.10 to ˜ u ε and we deduce the existence of u ∗ ∈ H (Ω) such that ε ˜ Π ε ˜ u ε converges weakly to u ∗ in H (Ω) . Consequently, by Rellich theorem, ε ˜ Π ε ˜ u ε converges strongly to u ∗ in L (Ω) .On the other side, we have the following indentity: χ (cid:101) Ω ε (cid:16) ε ˜ Π ε ˜ u ε (cid:17) = ε ˜ U ε in Ω . We can pass to the limit in the term of the left hand side. Thus, χ (cid:101) Ω ε (cid:16) ε ˜ Π ε ˜ u ε (cid:17) converges weakly to | Y (cid:48) f || Y (cid:48) | u ∗ in L (Ω) .In the right hand side, ε ˜ U ε converges to zero, so we obtain (4.67).Finally, we give a convergence result for ˆ u ε . Lemma 4.18.
We distinguish three cases: i) for γ < − , then for a subsequence of ε still denote by ε , there exists ˆ u ∈ L ( R ; H (cid:93) ( Y f ) ) , such that ε − ˆ u ε (cid:42) ˆ u in L ( R ; H ( Y f ) ) , (4.70) ε − ˆ u ε (cid:42) ˆ u in L ( R ; H ( ∂T ) ) , (4.71) | Y (cid:48) f || Y (cid:48) | M Y (cid:48) f [ˆ u ] = ˜ u a.e. in Ω , (4.72) ii) for − ≤ γ < , then for a subsequence of ε still denote by ε , there exists there exists ˆ u ∈ L ( R ; H (cid:93) ( Y f ) ) ,independent of y , such that ε γ ˆ u ε (cid:42) ˆ u in L ( R ; H ( Y f ) ) , (4.73) ε γ ˆ u ε (cid:42) ˆ u in L ( R ; H ( ∂T ) ) , (4.74) | Y (cid:48) f || Y (cid:48) | ˆ u = ˜ u a.e. in Ω , (4.75) iii) for γ ≥ , then for a subsequence of ε still denote by ε , there exists there exists ˆ u ∈ L ( R ; H (cid:93) ( Y f ) ) suchthat ˆ u ε − M Y f [ˆ u ε ] (cid:42) ˆ u in L ( R ; H ( ∂T ) ) . (4.76) D y ˆ u ε (cid:42) D y ˆ u in L ( R × Y f ) × . (4.77) | Y (cid:48) f || Y (cid:48) | M Y (cid:48) f [ˆ u ] = ˜ u a.e. in Ω , (4.78)div y ˆ u = 0 in ω × Y f . (4.79) Proof.
We proceed in three steps:
Step 1.
For γ < −
1, using (4.55), there exists ˆ u : R × Y f → R , such that convergence (4.70) holds. Passingto the limit by semicontinuity and using estimates (4.55), we get (cid:90) R × Y f | ˆ u | dx (cid:48) dy ≤ C, (cid:90) R × Y f | D y ˆ u | dx (cid:48) dy ≤ C, which, once we prove the Y (cid:48) -periodicity of ˆ u in y (cid:48) , shows that ˆ u ∈ L ( R ; H (cid:93) ( Y f ) ).It remains to prove the Y (cid:48) -periodicity of ˆ u in y (cid:48) . To do this, we observe that by definition of ˆ u ε given by (4.16)applied to ˜ u ε , we haveˆ u ε ( x + ε, x , − / , y , y ) = ˆ u ε ( x (cid:48) , / , y , y ) a.e. ( x (cid:48) , y , y ) ∈ R × ( − / , / × (0 , . Multiplying by ε − and passing to the limit by (4.70), we getˆ u ( x (cid:48) , − / , y , y ) = ˆ u ( x (cid:48) , / , y , y ) a.e. ( x (cid:48) , y , y ) ∈ R × ( − / , / × (0 , . Analogously, we can proveˆ u ( x (cid:48) , y , − / , y ) = ˆ u ( x (cid:48) , y , / , y ) a.e. ( x (cid:48) , y , y ) ∈ ω × ( − / , / × (0 , . These equalities prove that ˆ u is periodic with respect to Y (cid:48) . Convergence (4.71) is straightforward from thedefinition (4.22) and the Sobolev injections.Finally, using Proposition 4.2, we can deduce1 | Y (cid:48) | (cid:90) R × Y f ˆ u ε ( x (cid:48) , y ) dx (cid:48) dy = (cid:90) (cid:101) Ω ε ˜ u ε ( x (cid:48) , y ) dx (cid:48) dy , ε − and taking into account that ˜ u ε is extended by zero to the whole Ω, we have1 ε | Y (cid:48) | (cid:90) R × Y f ˆ u ε ( x (cid:48) , y ) dx (cid:48) dy = 1 ε (cid:90) Ω ˜ U ε ( x (cid:48) , y ) dx (cid:48) dy . Using convergences (4.62) and (4.70), we have (4.72).
Step 2.
For − ≤ γ <
1, using (4.57) and taking into account that ε − γ ≤ ε − γ , then there exists ˆ u : R × Y f → R , such that convergence (4.73) holds. Convergence (4.74) is straightforward from the definition (4.22) and theSobolev injections.On the other hand, since ε γ D y ˆ u ε = ε γ +12 ε γ − D y ˆ u ε and ε γ − D y ˆ u ε is bounded in L ( R × Y f ) × , using (4.73)we can deduce that D y ˆ u = 0. This implies that ˆ u is independent of y . Finally, (4.75) is obtained similarly to theStep 1. Step 3.
For γ ≥
1, using the second a priori estimate in (4.59) and the Poincar´e-Wirtinger inequality (cid:90) Y f (cid:12)(cid:12) ˆ u ε − M Y f [ˆ u ε ] (cid:12)(cid:12) dy ≤ C (cid:90) Y f | D y ˆ u ε | dy, a.e. in ω, we deduce that there exists ˆ u ∈ L ( R ; H ( Y f ) ) such thatˆ U ε = ˆ u ε − M Y f [ˆ u ε ] (cid:42) ˆ u in L ( R ; H ( Y f ) ) , and (4.77) holds. Convergence (4.76) is straightforward from the definition (4.22) and the Sobolev injections.It remains to prove the Y (cid:48) -periodicity of ˆ u in y (cid:48) . To do this, we observe that by definition of ˆ u ε given by (4.16)applied to ˜ u ε , we haveˆ u ε ( x + ε, x , − / , y , y ) = ˆ u ε ( x (cid:48) , / , y , y ) a.e. ( x (cid:48) , y , y ) ∈ R × ( − / , / × (0 , , which implies ˆ U ε ( x (cid:48) , − / , y , y ) − ˆ U ε ( x (cid:48) , / , y , y ) = −M Y f [ˆ u ε ]( x (cid:48) + ε e ) + M Y [ˆ u ε ]( x (cid:48) ) , which tends to zero (see the proof of Proposition 2.8 in [31]), and soˆ u ( x (cid:48) , − / , y , y ) = ˆ u ( x (cid:48) , / , y , y ) a.e. ( x (cid:48) , y , y ) ∈ R × ( − / , / × (0 , . Analogously, we can proveˆ u ( x (cid:48) , y , − / , y ) = ˆ u ( x (cid:48) , y , / , y ) a.e. ( x (cid:48) , y , y ) ∈ ω × ( − / , / × (0 , . These equalities prove that ˆ u is periodic with respect to Y (cid:48) . Step 4.
From the second variational formulation in (3.10), by Proposition 4.4, we have that (cid:90) (cid:101) Ω ε (cid:16) ˜ u (cid:48) ε · ∇ x (cid:48) ˜ ψ + ε − ˜ u ε, ∂ y ˜ ψ (cid:17) dx (cid:48) dy = ε − | Y (cid:48) | (cid:90) ω × ∂T (ˆ u ε · n ) ˆ ψ ε dx (cid:48) dσ ( y (cid:48) ) dy , ∀ ˜ ψ ∈ (cid:101) H ε , (4.80)and using the extension of the velocity, we obtain (cid:90) Ω (cid:16) ˜ U (cid:48) ε · ∇ x (cid:48) ˜ ψ + ε − ˜ U ε, ∂ y ˜ ψ (cid:17) dx (cid:48) dy = ε − | Y (cid:48) | (cid:90) ω × ∂T (ˆ u ε · n ) ˆ ψ ε dx (cid:48) dσ ( y (cid:48) ) dy , ∀ ˜ ψ ∈ (cid:101) H ε . We remark that the second term in the left-hand side and the one in the right-hand side have the same order, so inevery cases when passing to the limit after multiplying by a suitable power of ε and using that ˆ ψ ε converges stronglyto ˜ ψ in L ( ω × ∂T ) (see Proposition 2.6 in [31] for more details), it is not possible to find the usual incompressibilitycondition in thin domains given by div x (cid:48) (cid:18)(cid:90) ˜ u (cid:48) ( x (cid:48) , y ) dy (cid:19) = 0 on ω. ε − | Y (cid:48) | (cid:90) ω × Y f ˆ u ε · ∇ y ˆ ψ ε dx (cid:48) dy = ε − | Y (cid:48) | (cid:90) ω × ∂T ˆ u ε · ˆ ψ ε dx (cid:48) dσ ( y (cid:48) ) dy , (4.81)which, multiplying by ε | Y (cid:48) | and since M Y f [ˆ u ε ] does not depend on the variable y , is equivalent to (cid:90) ω × Y f (ˆ u ε − M Y f [ˆ u ε ]) · ∇ y ˆ ψ ε dx (cid:48) dy = (cid:90) ω × ∂T [(ˆ u ε − M Y f [ˆ u ε ]) · n ] · ˆ ψ ε dx (cid:48) dσ ( y (cid:48) ) dy (4.82)Thus, passing to the limit using convergences (4.77), we get condition (4.79). In this section, we will multiply system (3.10) by a test function having the form of the limit ˆ u (as explained inLemma 4.18), and we will use the convergences given in the previous section in order to identify the homogenizedmodel in every cases. Proof of Theorem 3.1:
We proceed in three steps:
Step 1.
For γ < −
1. Let ˜ ϕ ∈ D (Ω) and ˜ ψ ∈ D (Ω) be test functions in (3.10). By Proposition 4.4, one has µ (cid:90) (cid:101) Ω ε D ε ˜ u ε : D ε ˜ ϕ dx (cid:48) dy − (cid:90) (cid:101) Ω ε ˜ p ε div ε ˜ ϕ dx (cid:48) dy + α ε γ − | Y (cid:48) | (cid:90) ω × ∂T ˆ u ε · ˆ ϕ ε dx (cid:48) dσ ( y )= (cid:90) (cid:101) Ω ε f (cid:48) ε · ˜ ϕ (cid:48) dx (cid:48) dy + ε − | Y (cid:48) | (cid:90) ω × ∂T ˜ g (cid:48) · ˆ ϕ (cid:48) ε dx (cid:48) dσ ( y ) , i.e., µ (cid:90) (cid:101) Ω ε D x (cid:48) ˜ u ε : D x (cid:48) ˜ ϕ dx (cid:48) dy + µε (cid:90) (cid:101) Ω ε ∂ y ˜ u ε · ∂ y ˜ ϕ dx (cid:48) dy − (cid:90) (cid:101) Ω ε ˜ p ε div x (cid:48) ˜ ϕ (cid:48) dx (cid:48) dy − ε (cid:90) (cid:101) Ω ε ˜ p ε ∂ y ˜ ϕ dx (cid:48) dy + α ε γ − | Y (cid:48) | (cid:90) ω × ∂T ˆ u ε · ˆ ϕ ε dx (cid:48) dσ ( y )= (cid:90) (cid:101) Ω ε f (cid:48) ε · ˜ ϕ (cid:48) dx (cid:48) dy + ε − | Y (cid:48) | (cid:90) ω × ∂T ˜ g (cid:48) · ˆ ϕ (cid:48) ε dx (cid:48) dσ ( y ) , (4.83)Next, we prove that ˜ p does not depend on y . Let ˜ ϕ = (0 , ε − γ +1 ˜ ϕ ) ∈ D (Ω) be a test function in (4.83), wehave µ ε − γ +1 (cid:90) (cid:101) Ω ε ∇ x (cid:48) ˜ u ε, · ∇ x (cid:48) ˜ ϕ dx (cid:48) dy + µ ε − γ − (cid:90) (cid:101) Ω ε ∂ y ˜ u ε, ∂ y ˜ ϕ dx (cid:48) dy − ε − γ (cid:90) (cid:101) Ω ε ˜ p ε ∂ y ˜ ϕ dx (cid:48) dy + α | Y (cid:48) | (cid:90) ω × ∂T ˆ u ε, · ˆ ϕ ε, dx (cid:48) dσ ( y ) = 0 . Taking into account that ˜ P ε is zero extension of ˜ p ε to the whole Ω, we have (cid:90) (cid:101) Ω ε ˜ p ε ∂ y ˜ ϕ dx (cid:48) dy = (cid:90) Ω ˜ P ε ∂ y ˜ ϕ dx (cid:48) dy , and by the second a priori estimate (4.41), the convergences (4.63) and (4.71), passing to the limit we have (cid:90) Ω ˜ p ∂ y ˜ ϕ dx (cid:48) dy = 0 , so ˜ p does not depend on y . 22ar´ıa Anguiano and Francisco Javier Su´arez-GrauLet ˜ ϕ = ( ε − γ ϕ (cid:48) ( x (cid:48) , y ) , ε − γ ˜ ϕ ( x (cid:48) )) ∈ D (Ω) be a test function in (4.83), we have µε − γ (cid:90) (cid:101) Ω ε D x (cid:48) ˜ u (cid:48) ε : D x (cid:48) ˜ ϕ (cid:48) dx (cid:48) dy + µε − γ − (cid:90) (cid:101) Ω ε ∂ y ˜ u (cid:48) ε · ∂ y ˜ ϕ (cid:48) dx (cid:48) dy − ε − γ (cid:90) (cid:101) Ω ε ˜ p ε div x (cid:48) ˜ ϕ (cid:48) dx (cid:48) dy + α ε − | Y (cid:48) | (cid:90) ω × ∂T ˆ u (cid:48) ε · ˆ ϕ (cid:48) ε dx (cid:48) dσ ( y ) = ε − γ (cid:90) (cid:101) Ω ε f (cid:48) ε · ˜ ϕ (cid:48) dx (cid:48) dy + ε − γ − | Y (cid:48) | (cid:90) ω × ∂T ˜ g (cid:48) · ˆ ϕ (cid:48) ε dx (cid:48) dσ ( y ) , and µ ε − γ (cid:90) (cid:101) Ω ε ∇ x (cid:48) ˜ u ε, · ∇ x (cid:48) ˜ ϕ dx (cid:48) dy + α ε − | Y (cid:48) | (cid:90) ω × ∂T ˆ u ε, ˆ ϕ ε, dx (cid:48) dσ ( y ) = 0 . Taking into account that ˜ P ε is zero extension of ˜ p ε to the whole Ω, we have (cid:90) (cid:101) Ω ε ˜ p ε div x (cid:48) ˜ ϕ (cid:48) dx (cid:48) dy = (cid:90) Ω ˜ P ε div x (cid:48) ˜ ϕ (cid:48) dx (cid:48) dy . Using that ˆ ϕ ε converges strongly to ˜ ϕ in L ( ω × ∂T ) (see Proposition 2.6 in [31] for more details) and by thesecond a priori estimate (4.41), the convergences (4.63) and (4.71), passing to the limit we have − (cid:90) Ω ˜ p ( x (cid:48) ) div x (cid:48) ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dy + α | Y (cid:48) | (cid:90) ω × ∂T (cid:48) (cid:90) ˆ u (cid:48) ( x (cid:48) , y ) · ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dσ ( y (cid:48) ) dy = 0 , and α | Y (cid:48) | (cid:90) ω × ∂T ˆ u ( x (cid:48) , y ) ˜ ϕ ( x (cid:48) ) dx (cid:48) dσ ( y ) = 0 , which implies that M ∂T (cid:48) [ˆ u ] = 0.Taking into account that (cid:90) ω × ∂T (cid:48) (cid:90) ˆ u (cid:48) ( x (cid:48) , y ) · ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dσ ( y (cid:48) ) dy = | ∂T (cid:48) | (cid:90) Ω M ∂T (cid:48) [ˆ u (cid:48) ]( x (cid:48) , y ) · ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dy , implies that (cid:90) Ω ∇ x (cid:48) ˜ p ( x (cid:48) ) · ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dy = − α | ∂T (cid:48) || Y (cid:48) | (cid:90) Ω M ∂T (cid:48) [ˆ u (cid:48) ]( x (cid:48) , y ) · ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dy . (4.84)In order to obtain the homogenized system (3.12), we introduce the auxiliary problem − ∆ y (cid:48) χ ( y (cid:48) ) = − | ∂T (cid:48) || Y (cid:48) f | M Y (cid:48) f [ˆ u ]( x (cid:48) , y ) , in Y (cid:48) f ,∂χ∂n = ˆ u, on ∂T (cid:48) , M Y (cid:48) f [ χ ] = 0 ,χ ( y ) Y (cid:48) − periodic , for a.e. ( x (cid:48) , y ) ∈ Ω, which has a unique solution in H ( Y (cid:48) f ) (see Chapter 2, Section 7.3 in Lions and Magenes [37]).Using this auxiliary problem, we conclude that (cid:90) Ω M ∂T (cid:48) [ˆ u ] · ˜ ϕ dx (cid:48) dy = (cid:90) Ω M Y (cid:48) f [ˆ u ] · ˜ ϕ dx (cid:48) dy , (4.85)which together with (4.84) and M ∂T (cid:48) [ˆ u ] = 0 gives M Y (cid:48) f [ˆ u (cid:48) ]( x (cid:48) , y ) = − | Y (cid:48) | α | ∂T (cid:48) | ∇ x (cid:48) ˜ p ( x (cid:48) ) , M Y (cid:48) f [ˆ u ] = 0 , which together with (4.72) gives ˜ u (cid:48) ( x (cid:48) , y ) = − | Y (cid:48) f | α | ∂T (cid:48) | ∇ x (cid:48) ˜ p ( x (cid:48) ) , ˜ u ( x (cid:48) , y ) = 0 . This together with the definition of θ and µ , implies (3.12). Step 2.
For − ≤ γ <
1. First, we prove that ˜ p does not depend on y . Let ˜ ϕ = (0 , ε ˜ ϕ ) ∈ D (Ω) be a testfunction in (4.83). Reasoning as Step 1 and by the second a priori estimate (4.42), the convergence (4.66) and(4.74), passing to the limit we deduce that ˜ p does not depend on y .Let ˜ ϕ = ( εϕ (cid:48) ( x (cid:48) , y ) , ε ˜ ϕ ( x (cid:48) )) ∈ D (Ω) be a test function in (4.83). Reasoning as Step 1 and by the second apriori estimate (4.42), the convergences (3.11), (4.66) and (4.74), passing to the limit we have − (cid:90) Ω ˜ p ( x (cid:48) ) div x (cid:48) ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dy + α | Y (cid:48) | (cid:90) ω × ∂T (cid:48) (cid:90) ˆ u (cid:48) ( x (cid:48) ) · ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dσ ( y (cid:48) ) dy = (cid:90) Ω f (cid:48) ( x (cid:48) ) · ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dy + 1 | Y (cid:48) | (cid:90) ω × ∂T (cid:48) (cid:90) g (cid:48) ( y (cid:48) ) · ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dσ ( y (cid:48) ) dy , and α | Y (cid:48) | (cid:90) ω × ∂T ˆ u ( x (cid:48) ) ˜ ϕ ( x (cid:48) ) dx (cid:48) dσ ( y ) = 0 , which implies that ˆ u = 0.Taking into account that (cid:90) ω × ∂T (cid:48) (cid:90) ˆ u (cid:48) ( x (cid:48) ) · ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dσ ( y (cid:48) ) dy = | ∂T (cid:48) | (cid:90) Ω ˆ u (cid:48) ( x (cid:48) ) · ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dy , implies that (cid:90) Ω ∇ x (cid:48) ˜ p ( x (cid:48) ) · ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dy + α | ∂T (cid:48) || Y (cid:48) | (cid:90) Ω ˆ u (cid:48) ( x (cid:48) ) · ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dy = (cid:90) Ω f (cid:48) ( x (cid:48) ) · ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dy + | ∂T (cid:48) || Y (cid:48) | (cid:90) Ω M ∂T (cid:48) [ g (cid:48) ] · ˜ ϕ (cid:48) ( x (cid:48) , y ) dx (cid:48) dy , which together with (4.75) gives (3.13) after integrating the vertical variable y between 0 and 1. Step 3.
For γ ≥
1. For all ˆ ϕ ( x (cid:48) , y ) ∈ D ( ω ; C ∞ (cid:93) ( Y ) ), we choose ˆ ϕ ε ( x ) = ˆ ϕ ( x (cid:48) , x (cid:48) /ε, y ) as test function in (4.83).Then we get the following variational formulation: µε (cid:90) ω × Y f D y ˆ u ε : D y (cid:48) ˆ ϕ dx (cid:48) dy − (cid:90) ω × Y f ˆ p ε div x (cid:48) ˆ ϕ (cid:48) dx (cid:48) dy − ε − (cid:90) ω × Y f ˆ p ε div y ˆ ϕ dx (cid:48) dy + αε γ − (cid:90) ω × ∂T ˆ u ε · ˆ ϕ dx (cid:48) dσ ( y ) = (cid:90) ω × Y f f (cid:48) ε · ˆ ϕ (cid:48) dx (cid:48) dy + ε − (cid:90) ω × ∂T ˜ g (cid:48) · ˆ ϕ (cid:48) dx (cid:48) dσ ( y ) + O ε . (4.86)First, we remark that thanks to (4.60), there exists ˆ p ∈ L ( ω × Y f ) such that ε ˆ p ε converges weakly to ˆ p in L ( ω × Y f ). Now, we prove that ˆ p does not depend on y . For that, we consider ε ˆ ϕ ε in the previous formulation,and passing to the limit by (4.76) and (4.77), we get (cid:90) ω × Y f ˆ p div y ˆ ϕ dx (cid:48) dy = 0 , which shows that ˆ p does not depend on y . 24ar´ıa Anguiano and Francisco Javier Su´arez-GrauNow, we consider ˆ ϕ with div y ˆ ϕ = 0 in ω × Y f (which is necessary because ˆ u satisfies (4.79)), and similarly wedefine ˆ ϕ ε . Then, taking ε ˆ ϕ ε , the variational formulation (4.86) is the following: µ (cid:90) ω × Y f D y ˆ u ε : D y (cid:48) ˆ ϕ dx (cid:48) dy − ε (cid:90) ω × Y f ˆ p ε div x (cid:48) ˆ ϕ (cid:48) dx (cid:48) dy + αε γ +1 (cid:90) ω × ∂T ˆ u ε · ˆ ϕ dx (cid:48) dσ ( y ) = ε (cid:90) ω × Y f f (cid:48) ε · ˆ ϕ (cid:48) dx (cid:48) dy + ε (cid:90) ω × ∂T ˜ g (cid:48) · ˆ ϕ (cid:48) dx (cid:48) dσ ( y ) + O ε . (4.87)Reasoning as Step 1 , and using the convergences (3.11), (4.76), (4.77) and the convergence of ˆ p ε , passing to thelimit we have µ (cid:90) ω × Y f D y ˆ u : D y ˆ ϕ dx (cid:48) dy − (cid:90) ω × Y f ˆ p ( x (cid:48) ) div x (cid:48) ˆ ϕ (cid:48) dx (cid:48) dy = 0 . (4.88)By density, this equation holds for every function ˆ ϕ ( x (cid:48) , y ) ∈ L ( ω ; H (cid:93) ( Y ) ) such that div y ˆ ϕ = 0. This implies thatthere exists ˆ q ( x (cid:48) , y ) ∈ L (cid:93) ( ω × Y f ) such that (4.88) is equivalent to the following problem: − µ ∆ y ˆ u + ∇ y ˆ q = −∇ x (cid:48) ˆ p in ω × Y f , div y ˆ u = 0 in ω × Y f ,∂ ˆ u∂n = 0 on ω × ∂T, ˆ u = 0 on y = 0 , ,y (cid:48) → ˆ u ( x (cid:48) , y ) , ˆ q ( x (cid:48) , y ) Y (cid:48) − periodic . We remark that ˆ p is already the pressure ˜ p . This can be easily proved by multiplying equation (4.83) by a testfunction ε ϕ (cid:48) ( x (cid:48) , y ) and identifying limits.Finally, we will eliminate the microscopic variable y in the effective problem. Observe that we can easily deducethat ˆ u = 0 and ˆ q = ˆ q ( x (cid:48) , y (cid:48) ) and moreover, the derivation of (3.14) from the previous effective problem is an easyalgebra exercise. In fact, we can write (cid:82) ˆ u ( x (cid:48) , y ) dy = µ (cid:80) i =1 ∂ x i ˜ p ( x (cid:48) ) w i ( y (cid:48) ) and ˆ q ( x (cid:48) , y (cid:48) ) = µ (cid:80) i =1 ∂ x i ˜ p ( x (cid:48) ) q i ( y (cid:48) )with ( w i , q i ), i = 1 ,
2, the solutions of the local problems (3.15), and use property (4.78) which involves the functions (cid:82) ˆ u ( x (cid:48) , y ) dy and (cid:82) ˜ u ( x (cid:48) , y ) dy . As well-known, the local problems (3.15) are well-posed with periodic boundaryconditions, and it is easily checked, by integration by parts, that A ij = 1 | Y f | (cid:90) Y f D y w i ( y ) : D y w j ( y ) dy = (cid:90) Y f w i ( y ) e j dy, i, j = 1 , . By definition A is symmetric and positive definite. The behavior of the flow of Newtonian fluids through periodic arrays of cylinders has been studied extensively,mainly because of its importance in many applications in heat and mass transfer equipment. However, the literatureon Newtonian thin film fluid flows through periodic arrays of cylinders is far less complete, although these problemshave now become of great practical relevance because take place in a number of natural and industrial processes.This paper deals with the modelization by means of homogenization techniques of a thin film fluid flow governedby the Stokes system in a thin perforated domain Ω ε which depends on a small parameter ε . More precisely, Ω ε has thickness ε and is perforated by a periodic array of cylindrical obstacles of period ε .The main novelty here are the combination of the mixed boundary condition considered on the obstacles andthe thin thickness of the domain. Namely, a standard (no-slip) condition is imposed on the exterior boundary,whereas a non-standard boundary condition of Robin type which depends on a parameter γ is imposed on theinterior boundary. This type of boundary condition is motivated by the phenomenon in which a motion of the25ar´ıa Anguiano and Francisco Javier Su´arez-Graufluid appears when a electrical field is applied on the boundary of a porous medium in equilibrium. The mainmathematical difficulties of this work are to treat the surface integrals and extend the solution to a fixed domainin order to pass to the limit with respect to the parameter ε . We overcome the first difficulty by using a versionof the unfolding method which let us treat the surface integrals quite easily. Moreover, we need to develop someextension abstract results and adapt them to the case of thin domain.By means of a combination of homogenization and reduction of dimension techniques, depending on the pa-rameter γ , we obtain three modified 2D Darcy type laws which model the behavior of the fluid and include theeffect of the surface forces and the measure of the obstacles. We remark that we are not able to prove a divergencecondition for the limit averaged fluid flow as obtained if we had considered Dirichlet boundary conditions, whichfrom the mechanical point of view means that some fluid “dissapear” through the cylinders and so, it is representedby the motion of a compressible fluid. To conclude, it is our firm belief that our results will prove useful in theengineering practice, in particular in those industrial applications where the flow is affected by the effects of thesurface forces, the fluid microstructure and the thickness of the domain. Acknowledgments
We would like to thank the referees for their comments and suggestions. Mar´ıa Anguiano has been supportedby Junta de Andaluc´ıa (Spain), Proyecto de Excelencia P12-FQM-2466. Francisco Javier Su´arez-Grau has beensupported by Ministerio de Econom´ıa y Competitividad (Spain), Proyecto Excelencia MTM2014-53309-P.