Analysis of the effects of a fissure for a non-Newtonian fluid flow in a porous medium

1 ANALYSIS OF THE EFFECTS OF A FISSURE FOR A NON-NEWTONIAN FLUID FLOW IN A POROUS MEDIUM MAR´ IA ANGUIANO†AND FRANCISCO JAVIER SU´ AREZ-GRAU ‡ Abstract. We study the solution of a non-Newtonian flow in a porous medium which characteristic size of the pores εand containing a fissure of width ηε. The flow is described by the incompressible Stokes system with a nonlinear viscosity, being a power of the shear rate (power law) of flow index 1< r < +∞. We consider the limit when size of the pores tends to zero and we obtain different models depending on the magnitude ηεwith respect to ε. Key words. Non-Newtonian flow; Stokes equation; Darcy’s law; porous medium; fissure AMS subject classifications. 76M50, 35B27 1. Introduction In this paper we consider an incompressible viscous non-Newtonian flow in a periodic porous medium with characteristic size of the pores εand containing a fissure {0≤xn≤ηε}of width ηεwith ε,ηεtwo small parameters devoted to tend to zero (see Figure 2.1). Modeling of non-Newtonian flow in fractured medium has encountered a renewed interest because it is essential in hydraulic fracturing operations, largely used for optimal exploitation of oil, gas and thermal reservoirs. Complex fluids interact with pre-existing rock fractures also during drilling operations, enhanced oil recovery, environmental remediation, and other natural phenomena such as magma and sand intrusions, and mud volcanoes. The aim of this work is to find the effective system corresponding to the limit when the size of the pores, and so the width of the fissure, tends to zero. Homogenization has been applied to the study of perforated materials for a long time. The question of a medium containing a fissure with properties different from those of the rest of the material has been the subject of many studies previously, see Ciarlet et al. [8], Panasenko [11] and Chapter 13 of Sanchez-Palencia [12] among others. A similar problem of the one considered in this paper, but for the Laplace’s equation, was studied in Bourgeat and Tapiero [4]. The peculiar behavior observed for the Laplace’s equation when ηε≈ε2 3 has motivated the analogous study for the Newtonian Stokes system in Bourgeat et al. [5] (see Zhao and Yao [15] for the Newtonian Navier-Stokes system). A lot of fluid used in industrial practice are modeled with a shear thinning law. For this reason, in this paper we extend the previous studies obtained for Newtonian fluids to the case of power law fluid, whose situation is completely different. The main reason is that the viscosity is a nonlinear function of the symmetrized gradient of the velocity. In this sense, we consider that the viscosity satisfies the non linear power law, which is widely used for melted polymers, oil, mud, etc. If uis the velocity and Du the gradient velocity tensor, denoting the shear rate by D[u]= 1 2(Du+Dtu), the viscosity ∗Mar´ıa Anguiano has been supported by Junta de Andaluc´ıa (Spain), Proyecto de Excelencia P12- FQM-2466. Francisco Javier Su´arez-Grau has been supported by Ministerio de Econom´ıa y Competitividad (Spain), Proyecto Excelencia MTM2014-53309-P. †Departamento de An´alisis Matem´atico. Facultad de Matem´aticas. Universidad de Sevilla. P. O. Box 1160, 41080-Sevilla (Spain), [email protected] ‡Departamento de Ecuaciones Diferenciales y An´alisis Num´erico. Facultad de Matem´aticas. Universidad de Sevilla, 41012-Sevilla (Spain), [email protected] 1 2The effects of a fissure for a non-Newtonian fluid flow in a porous medium as a function of the shear rate is given by ηr(D[u])=µ|D[u]|r−2,1<r <+∞, where the two material parameters µ>0 and rare called the consistency and the flow index, respectively. Recall that r=2 yields the Newtonian fluid. For 1<r <2 the fluid is pseudoplastic (shear thinning), which is the characteristic of high polymers, polymer solutions, and many suspensions, whereas for r > 2 the fluid is dilatant (shear thickening), whose behavior is reported for certain slurries, like mud, clay, or cement, and implies an increased resistance to flow with intensified shearing. We consider fluids satisfying the non-Newtonian Stokes system in the domain described above, and our goal is to generalize the study of Bourgeat et al. [5] to the non-Newtonian case. We first establish a priori estimates in the framework of Sobolev spaces and variational formulations. To find these estimates and then the order of the limits, we use a variant of the Korn’s inequality for this type of domain. The results obtained here correspond to three characteristic situations depending on the parameter ηεwith respect to ε: •If ηεεr 2r−1the fissure is not giving any contribution. In this case, in order to find the limit, we use the theory developed by Allaire [2] and Nguesteng [10] of two-scale convergence and we obtain a nonlinear Darcy’s law. •If ηεεr 2r−1the fissure is dominant. We introduce a rescaling in the fissure in order to work with a domain with height one, and then we prove that the limit of the velocity is a Dirac measure concentrated on {xn=0}representing the corresponding tangential surface flow. Meanwhile in the porous medium the effective velocity is equal to zero. •If ηε≈εr 2r−1with ηε/ε r 2r−1→λ, 0<λ<+∞, it appears a coupling effect and the effective flow behaves as Darcy flow in the porous medium coupled with the tangential flow of the surface {xn=0}. Compared to the first case ηεεr 2r−1, the effective velocity has now an additional tangential component concentrated on {xn=0}. Moreover, the limit problem is now given by a new variational equation, in which appears the parameter λ, and consists of a nonlinear Darcy law in the porous medium and an additional Reynolds problem on the surface {xn=0}. 2. The domain and some notations Let Ω⊂Rn,n=2 or 3, be a bounded open domain and Ω+=Ω∩{xn>0},Ω−=Ω∩{xn<0},Σ=Ω∩{xn=0}. For some η0>0 we define the domain D=Ω−∪(η0en+Ω+)∪(Σ×[0,η0]). Let ε>0 and 0<ηε<η0be two small parameters devoted to tend to zero. With Ω we associate a microstructure through the periodic cell Y=(0,1)nmade of two complementary parts: the solid part A, which is a closed subset of Y, and the fluid part Y∗=Y\A. Defining Yk=k+Y,k∈Zn, we set Akand Y∗k=Yk\Akas the solid and fluid part in Ykrespectively. We also denote A−=[ k∈Zn − Ak, A+=[ k∈Zn + Ak, Mar´ıa Anguiano and Francisco Javier Su´arez-Grau 3 all the solid parts in Rn, where Zk −={k:k∈Zn, kn<0}and Zk +={k:k∈Zn, kn>0}. It is obvious that E∗=Rn\(A−∪A+) is an open subset in Rn. Following Allaire [1], we make the following assumptions on Y∗,E∗,Aand A∗= A+∪A−: i) Y∗is an open connected set of strictly positive measure, with a locally Lipschitz boundary. ii) Ahas strictly positive measure in Y. iii) E∗and the interior of A∗are open sets with boundaries of class C0,1and are locally located on one side of their boundaries. Moreover E∗is connected. We also define Y∗k ε=εY ∗k, k ∈Zn, A− ε=εA−, A+ εηε=ηεen+εA+, Sεηε=∂A− ε∪A+ εηε. We denote by Aεηε=A− ε∪A+ εηεthe solid part of the domain D, Dεηε=D\Aεηεthe fluid part of the domain D(including the fissure), Iηε=Σ×(0,ηε) - the fissure in D, Ωεηε=Dεηε\Iηεthe fluid part of the porous medium, and Ω+ εηε=Dεηε∩{xn>ηε},Ω− εηε=Dεηε∩{xn<0},Γηε=∂Σ×(0,ηε). Finally we define D+=D∩{xn>0}, D−=Ω−. We denote by Oεa generic real sequence which tends to zero with εand can change from line to line. We denote by Ca generic positive constant which can change from line to line. 3. Setting and main results In the following, the points x∈Rnwill be decomposed as x=(x0,xn) with x0∈Rn−1, xn∈R. We use the notation ˜ ·to denote a generic function of Rn−1. In this section we describe the asymptotic behavior of an incompressible viscous non- Newtonian fluid in the porous medium with a fissure. The proof of the corresponding results will be given in the next sections. Our results are referred to the non-Newtonian Stokes system. Namely, for f∈ C(D)nlet us consider a sequence (uεηε,pεηε)∈W1,r 0(Dεηε)n×Lr0 0(Dεηε), 1<r <+∞, which satisfies (−div µ|D[uεηε]|r−2D[uεηε]+∇pεηε=fin Dεηε, divuεηε= 0 in Dεηε,(3.1) where µ>0 is the consistency, r0=r/(r−1) is the conjugate exponent of rand Lr0 0(Dεηε) is the space of functions of Lr0(Dεηε) with null integral. We may consider Dirichlet boundary conditions without altering the generality of the problem under consideration, uεηε=0 on ∂Dεηε.(3.2) 4The effects of a fissure for a non-Newtonian fluid flow in a porous medium ⌦+ "⌘" ⌦ "⌘" I⌘" " " xn=⌘" xn=0 Figure 2.1.View of the domain Dεηε It is well known that (3.1)-(3.2) has a unique solution (uεηε,pεηε)∈W1,r 0(Dεηε)n× Lr0 0(Dεηε) for every ε,ηε>0 (see the classical theory [14] for more details). Our aim is to study the asymptotic behavior of uεηεand pεηεwhen εtends to zero. As usual, in order to study the behavior of uεηε,pεηεin the fissure we rewrite our equations in the unit cylinder I1=Σ×(0,1) by introducing the change of variable z=xn ηε ,(3.3) which transform Iηεin a fixed domain I1. We define the new functions Uεηε(x0,z)=uεηε(x0,ηεz), P εηε(x0,z)=pεηε(x0,ηεz)−cεηε,(3.4) and ˜ Uεηε=(Uεηε 1,Uεηε 2,...,Uεηε n−1), with cεηε=1 |Iηε|ZIηε pεηεdx. (3.5) Let us introduce some notation which will be useful in the following. For a vectorial function v=(˜v,vn), we will denote Dx0[v]= 1 2(Dx0v+Dt x0v) and ∂z[v]= 1 2(∂zv+∂t zv), where we denote ∂z=(0,0,∂ ∂z )t, and associated to the change of variables (3.3), we introduce the operators: Dηε,Dηεand divηε, by Dηε[v]= 1 2Dηεv+Dt ηεv,divηεv= divx0˜v+1 ηε ∂zvn, (Dηεv)i,j =∂xjvifor i=1,...,n, j =1,...,n−1,(Dηεv)i,n =1 ηε ∂zvifor i=1,...,n. Mar´ıa Anguiano and Francisco Javier Su´arez-Grau 5 Using the transformation (3.3), the system (3.1) can be rewritten as (−divηεµ|Dηε[Uεηε]|r−2Dηε[Uεηε]+∇ηεPεηε=f(x0,ηεz) in I1, divηεUεηε= 0 in I1,(3.6) with Dirichlet boundary condition, Uεηε=0 on ∂Σ×(0,1).(3.7) In order to simplify the notation, we define Sras the r-Laplace operator Sr(ξ)=|ξ|r−2ξ, ∀ξ∈Rn×n sym ,1<r <+∞, and K:Rn→Rnas a function defined by K(ξ)=ZY∗ wξ(y)dy, ∀ξ∈Rn,(3.8) where wξ(y), for every ξ∈Rn, denote the unique solution in W1,r #(Y∗)n(# denotes Y-periodicity) of the local problem        −divySD[wξ]+∇yπξ=ξin Y∗, divywξ= 0 in Y∗, wξ= 0 in ∂A, wξ,πξY−periodic. (3.9) Our main result referred to the asymptotic behavior of the solution of (3.1)-(3.2) is given by the following theorem. Theorem 3.1. Let ηε≈εr 2r−1, with ηε/ε r 2r−1→λ,0<λ<+∞,1<r <+∞and let (uεηε,pεηε)be the solution of problem (3.1)-(3.2). Then there exist a Darcy velocity v∈Lr(D)n, a Reynolds velocity V ∈Lr(Σ)n, with Vn= 0, and a pressure p∈W1,r0(D) such that ε−r r−1uεηε ? *v+λ2r−1 r−1VδΣin M(D), pεηε→pin Lr0(D),(3.10) where δΣis the Dirac measure concentrated on Σ, and M(D)nis the space of Radon measures on D. The velocities vand Vare given by v(x)= 1 µK(f(x)−∇p(x)),in D, (3.11) ˜ V(x0)= 1 2r0 2(r+1)µr0−1Sr0˜ f(x0,0)−∇x0P(x0),V=(˜ V,0),in Σ,(3.12) where the pressure P∈W1,r0(Σ) is connected with the pressure pby the relation p(x0,0)=P(x0)+ e C, e C∈R. Moreover, the pressure p∈VΣ={ϕ∈W1,r0(D) : ϕ(·,0) ∈W1,r0(Σ)}is the unique solution of the variational problem ZD v(x)·∇ϕ(x)dx+λ2r−1 r−1ZΣ ˜ V(x0)·∇x0ϕ(x0,0)dx0=0,∀ϕ∈VΣ.(3.13) 6The effects of a fissure for a non-Newtonian fluid flow in a porous medium Remark 3.2. Formally, (3.13) is the weak formulation of the following boundary value problem (−divv(x)−λ2r−1 r−1divx0˜ V(x0)δΣ= 0 in D, v(x)·ν+λ2r−1 r−1˜ V(x0)δ∂Σ·˜ν= 0 on ∂D, (3.14) where νis the outward normal to ∂D and ˜νis the outward normal to ∂Σ. In the case λ=0, i.e. ηεεr 2r−1, then the fissure is not giving any contribution. In fact, if λtends to zero in (3.14) we obtain the following Darcy’s law on D −divv(x)=0 in D, v(x)·ν= 0 on ∂D, (3.15) where vis given by (3.11). On the other hand, in the case λ= +∞, i.e. ηεεr 2r−1, then the fissure is dominant. In fact, multiplying (3.14) by λ−2r−1 r−1and tending λto +∞, we obtain the following Reynolds problem on Σ −divx0˜ V(x0)=0 in Σ, ˜ V(x0)·˜ν=0 on ∂Σ,(3.16) where ˜ Vis given by (3.12). Remark 3.3. The monotonicity and coerciveness properties of the permeability function Kgiven by (3.8) can be found in sections 2 and 4 in [7], which implies that (3.15) is well posed. On the other hand, the r0-Laplace operator is well know that is monotone and coercive (see [9] for more details), which implies that (3.16) is well posed. Therefore, the problem (3.13) is also well posed. In Section 4 we establish a priori estimates of the velocity and the pressure. Section 5 is devoted to prove Theorem 3.1, whose proof is divided in three subsections. In Subsection 5.1 we analyze the problem in the porous part (ηεεr 2r−1) while in Subsection 5.2 the problem in the fissure part (ηεεr 2r−1) is analyzed, which give the rigorously proof of (3.15) and (3.16), respectively. Finally, in Subsection 5.3 we prove that there is a balanced interaction between the fissure and the porous medium giving Theorem 3.1. 4. A Priori Estimates Let us begin with the following variant of the Korn’s inequality in the porous medium Ωεηε, which will be very useful (see for example Bourgeat and Mikelic in [6]). Lemma 4.1. There exists a constant Cindependent of ε, such that, for any function v∈W1,r(Dεηε)nand v=0 on Sεηε, one has kvkLr(Ωεηε)n≤CεkD[v]kLr(Ωεηε)n×n,1<r <+∞.(4.1) Next, we give an useful estimate in the fissure Iηε. Lemma 4.2. There exists a constant Cindependent of ε, such that, for any function v∈W1,r(Dεηε)nand v=0 on Sεηε, one has kvkLr(Iηε)n≤Cηε 1 2(ηε+ε)1 2kD[v]kLr(Dεηε)n×n,1<r < +∞.(4.2) Mar´ıa Anguiano and Francisco Javier Su´arez-Grau 7 Proof. Because the thickness of Iηεis ηε, we have, by the classical Poincar´e inequality, kvkLr(Iηε)n≤CηεkDvkLr(Iηε)n×n.(4.3) Next, if we choose a point x1∈Aεηε, which is close to the point x∈Iηε, then we have v(x)−v(x1)= Dv(ξ)(x−x1)≤(ε+ηε)|Dv|. Since v(x1)= 0 because x1∈Aεηε, we have kv(x)kLr(Iηε)n≤C(ε+ηε)kDvkLr(Iηε)n×n. Multiplying the above inequality with (4.3) we obtain kvkLr(Iηε)n≤Cηε 1 2(ηε+ε)1 2kDvkLr(Iηε)n×n≤Cηε 1 2(ηε+ε)1 2kDvkLr(Dεηε)n×n,(4.4) and from the classical Korn inequality we obtain (4.2). Let us give the classical estimate [3], for the a function in Lrwhen we deal with a fissure. Lemma 4.3. There exists a constant Cindependent of ε, such that, for any function v∈Lr(Iηε)with RIηεvdx=0, one has kvkLr(Iηε)≤C ηε k∇vkW−1,r(Iηε)n,1< r <+∞. Now, we are in position to obtain some a priori estimates for uεηε. Lemma 4.4. There exists a constant Cindependent of ε, such that if uεηε∈ W1,r 0(Dεηε)n, with 1<r < +∞, is the solution of the problem (3.1)-(3.2), one has kuεηεkLr(Ωεηε)n≤C(ηε 2r−1 rεr−1+εr)1 r−1,(4.5) kuεηεkLr(Iηε)n≤Cηεr−1ηε 2r−1 r+εηεr−11 r−1+ηε 1 2εr+1 2(r−1) ,(4.6) kD[uεηε]kLr(Dεηε)n×n≤C(ηε 2r−1 r+ε)1 r−1,(4.7) kDuεηεkLr(Dεηε)n×n≤C(ηε 2r−1 r+ε)1 r−1.(4.8) Proof. Multiplying by uεηεin the first equation of (3.1) and integrating over Dεηε, we have µkD[uεηε]kr Lr(Dεηε)n×n=ZDεηε f·uεηεdx. (4.9) Using H¨older’s inequality and the assumption of f, we obtain that there exists a constant Csuch that ZDεηε f·uεηεdx≤Cηε 1 r0kfkL∞(Iηε)nkuεηεkLr(Iηε)n+kfkLr0(Ωεηε)nkuεηεkLr(Ωεηε)n, 8The effects of a fissure for a non-Newtonian fluid flow in a porous medium and by inequalities (4.1) and (4.2), we have ZDεηε f·uεηεdx≤Cηε 1 r0ηε 1 2(ε+ηε)1 2+εkD[uεηε]kLr(Dεηε)n×n ≤Cηε 1 r0ηε+ηε 1 r0ηε 1 2ε1 2+εkD[uεηε]kLr(Dεηε)n×n. Therefore, from (4.9) we get kD[uεηε]kLr(Dεηε)n×n≤Cηε 1 r0ηε+ηε 1 r0ηε 1 2ε1 2+ε1 r−1. Since ηε 1 r0ηε 1 2ε1 2<ηε 1 r0ηεif ε<ηεand η 1 r0 εη1 2 εε1 2≤η 1 r0 εε<ε if ηε<ε, the term ηε 1 r0ηε 1 2ε1 2 can be dropped. Taking into account that 1/r0+1= (2r−1)/r, this gives (4.7) and from the classical Korn inequality we have (4.8). Applying (4.1) together with (4.7) we obtain (4.5). Finally, applying (4.2) and (4.7) we get kuεηεkLr(Iηε)n≤C(ηε+ηε 1 2ε1 2)(ηε 2r−1 r+ε)1 r−1 ≤Cηεr−1ηε 2r−1 r+εηεr−11 r−1+ηε r−1 2ηε 2r−1 rεr−1 2+ηε r−1 2εr+1 21 r−1. Since ηε r−1 2ηε 2r−1 rεr−1 2<ηεr−1ηε 2r−1 rif ηε>ε and ηε r−1 2ηε 2r−1 rεr−1 2<ηε r−1 2εr+1 2if ηε<ε, the term ηε r−1 2ηε 2r−1 rεr−1 2can be dropped, and (4.6) holds. In order to investigate the behavior of solutions to (3.1)-(3.2), as ε→0, we need to extend the pressure pεηεto the whole of D. Extending the pressure is a difficult task. The extension is closely related to the construction of a restriction operator. Such extension for the case of a porous medium without fissure is given in Tartar [13] for the case r=2. We need a restriction operator, Rε r, between W1,r 0(D)ninto W1,r 0(Dεηε)n with similar properties, which is given in [6]. Since the construction of the operator is local, having no obstacles in Iηεmeans that we do not have to use the extension in that part. Next, we give the properties of the restriction operator Rε r(see Lemma 1.2. in [6] for more details). Lemma 4.5. There exists a linear continuous operator Rε racting from W1,r 0(D)ninto W1,r 0(Dεηε)n,1<r < +∞, such that 1. Rε rv=v, if v∈W1,r 0(Dεηε)n 2. div(Rε rv)=0, if divv= 0 3. For any v∈W1,r 0(D)n(the constant Cis independent of vand ε), kRε rvkLr(Dεηε)n≤CkvkLr(D)n+CεkDvkLr(D)n×n, kDRε rvkLr(Dεηε)n×n≤C εkvkLr(D)n+CkDvkLr(D)n×n. In order to extend the pressure to the whole domain D, we define a function Fεηε∈ W−1,r0(D)nby the following formula (brackets are for the duality products between W−1,r0and W1,r 0): hFεηε,viD=h∇pεηε,Rε rviDεηε,for any v∈W1,r 0(D)n,(4.10) Mar´ıa Anguiano and Francisco Javier Su´arez-Grau 9 where Rε ris the operator defined in Lemma 4.5. We calcule the right hand side of (4.10) by using (3.1) and we have hFεηε,viD=Ddivµ|D[uεηε]|r−2D[uεηε],Rε rvEDεηε +hf,Rε rviDεηε,(4.11) and by using the third point in Lemma 4.5, for fixed ε,ηεwe see that it is a bounded functional on W1,r 0(D)3, and in fact Fεηε∈W−1,r0(D)n. Moreover, if v∈W1,r 0(Dεηε)nand we continue it by zero out of Dεηε, we see from (4.10) and the first point in Lemma 4.5 that Fεηε|Dεηε=∇pεηε. On the other hand, if divv=0 by the second point in Lemma 4.5 and (4.10), we have that hFεηε,viΩ= 0 and this implies that Fεηεis the gradient of some function in Lr0(D). This means that Fεηεis a continuation of ∇pεηεto D, and that this continuation is a gradient. We also may say that pεηεhas been continuated to D. We denote the extended pressure again by pεηεand since it is defined up to a constant we take pεηε such that RDpεηεdx= 0. Moreover, we have Fεηε≡∇pεηε. For such extended pressure we obtain the following result. Lemma 4.6. There exists a constant Cindependent of ε, such that if pεηε∈Lr0 0(D), with r0the conjugate exponent of rand 1<r <+∞, is the extended pressure to the whole domain D, one has kpεηεkLr0(D)≤C ηε r0+1 r0 ε+1!,(4.12) kpεηε−cεηεkLr0(Iηε)≤Cηε 1 r0+ε ηε,(4.13) where cεηεis given by (3.5). Proof. Let us first estimate ∇pεηε. To do this we estimate the right side of (4.11). Using H¨older’s inequality and from (4.7) we have Ddivµ|D[uεηε]|r−2D[uεηε],Rε rvEDεηε≤µkD[uεηε]kr−1 Lr(Dεηε)n×nkDRε rvkLr(Dεηε)n×n ≤Cηε 2r−1 r+εkDRε rvkLr(Dεηε)n×n. Using the assumption of f, we obtain that there exists a constant Csuch that hf,Rε rviDεηε≤CkRε rvkLr(Dεηε)n. Then, from (4.11), we deduce h∇pεηε,viD≤Cηε 2r−1 r+εkDRε rvkLr(Dεηε)n×n+CkRε rvkLr(Dεηε)n. Taking into account the third point in Lemma 4.5, we have h∇pεηε,viD≤Cηε 2r−1 r+ε1 εkvkLr(D)n+kDvkLr(D)n×n +CkvkLr(D)n+εkDvkLr(D)n×n. 16 The effects of a fissure for a non-Newtonian fluid flow in a porous medium which finishes (5.16). Finally, we need to indentify the effective system and compute the expression of the solution. For this, thanks to Propositions 3.1 and 3.2 in Mikeli´c and Tapiero [9] we have that the effective system is given by −∂z|∂z˜ U|r−2∂z˜ U=2r 2µ˜ f(x0,0)−∇x0P(x0),in I1, divx0R1 0˜ U(x0,z)dz= 0 in Σ,R1 0˜ U(x0,z)dz·˜ν= 0 on ∂Σ, taking into account Proposition 3.3. in [9] we have that P=P(x0)∈W1,r0(Σ) and thanks to Proposition 3.4 in [9] we have that the expression of the solution is given by (5.17). It remains to prove the convergence of the whole velocity ηε−2r−1 r−1uεηε ? *VδΣin M(D)n,(5.22) where V(x0)= R1 0U(x0,z)dz is given by (3.12), and also prove that the pressure Pis the unique solution of the Reynolds problem (3.16). Taking as test function ϕ∈C∞(D) in divuεηε=0 in D, we obtain ZD divuεηεϕdx =−ZD vεηε·∇ϕdx−ηεZI1 Uεηε·∇ϕ(x0,ηεz)dx0dz =0, so that multiplying by ηε−2r−1 r−1, ZI1 ηε−r r−1˜ Uεηε·∇x0ϕ(x0,ηεz)dx0dz (5.23) =−ZD ηε−2r−1 r−1vεηε·∇ϕdx−ZI1 ηε−r r−1Uεηε n∂nϕ(x0,ηεz)dx0dz. Using (4.5) and taking into account ηεεr 2r−1, we obtain ηε−2r−1 r−1kvεηεkLr(D)n≤C ε ηε 2r−1 r +εr r−1 ηε 2r−1 r−1!→0 as ε→0.(5.24) Taking the limit in (5.23) as ε→0, using convergence (5.16) and Un=0, we have ZI1 ˜ U ·∇x0ϕ(x0,0)dx0dz =0, and by definition (3.12), we get the Reynolds problem (3.16). Consequently, Pis the unique solution of (3.16) (see Proposition 3.4 in Mikelic and Tapiero [9] for more details). Finally, we consider ϕ∈C0(D)nand so we have ZD ηε−2r−1 r−1uεηεϕdx=ZD ηε−2r−1 r−1vεηεϕdx+ZI1 ηε−r r−1Uεηεϕ(x0,ηεz)dx0dz. Using (5.24) and convergence (5.16) and Un=0, we obtain ZD ηε−2r−1 r−1uεηεϕdx→ZI1 ˜ U(x0,z) ˜ϕ(x0,0)dx0dz =ZΣ ˜ V(x0) ˜ϕ(x0,0)dx0=hV(x0)δΣ,ϕiM(D)n,C0(D)n, which implies (5.22). Mar´ıa Anguiano and Francisco Javier Su´arez-Grau 17 5.3. Effects of coupling ηε≈εr 2r−1 The conclusion of the previous two subsections is that for any sequence of solutions (vεηε,pεηε) with ηεεr 2r−1and (Uεηε,Pεηε) with ηεεr 2r−1, and letting ε→0, we can extract subsequences still denoted by vεηε,pεηε,Uεηε,Pεηεand find functions v∈Lr(D)n,p∈W1,r0(D), ˜ U ∈Lr(I1)n−1,P∈W1,r0(Σ) such that ε−r r−1vεηε*v in Lr(D)n, pεηε→pin Lr0(D), ηε−r r−1Uεηε*Uin Lr(I1)nwith U=( ˜ U,0), Pεηε* P in Lr0(I1).(5.25) Moreover such limit functions v,p,U,Pnecessarily satisfy the equations v=1 µK(f(x)−∇p(x)) in D, v·ν=0 on ∂D, ˜ U=2r0 2 r01 2r0 −1 2−zr0|˜ f(x0,0)−∇x0P(x0)|r0−2(˜ f(x0,0)−∇x0P(x0)) in I1. (5.26) We are going to find the connection between the functions pand P, i.e. to find the coupling effects between the solution in the porous part and in the fissure. Lemma 5.5. Let ηε≈εr 2r−1, with ηε/ε r 2r−1→λ,0<λ<+∞,1<r < +∞, and let {pεηε}⊂ Lr0(D),p∈W1,r0(D),P∈W1,r0(Σ) be such that (5.25) and (5.26) hold. Then, ZD 1 µK(f(x)−∇p(x))·∇ϕ(x)dx +λ2r−1 r−1ZΣ˜ f(x0,0)−∇x0P(x0) r0−2 2r0 2(r+1)µr0−1(˜ f(x0,0)−∇x0P(x0))·∇x0ϕ(x0,0)dx0= 0, (5.27) for every ϕ∈VΣ. Proof. Let ϕ∈VΣ. Taking into account the definitions (5.1) of vεηεand (3.4) of Uεηε, and from divuεηε= 0 in Dwe have ZD ε−r r−1uεηε·∇ϕdx=ZD ε−r r−1vεηε·∇ϕdx +ηε εr 2r−12r−1 r−1ZI1 ηε−r r−1Uεηε·∇ϕ(x0,ηεz)dx0dz =0. Taking the limit as ε→0, using (5.25), Un= 0 and ηε/ε r 2r−1→λ, we obtain ZD v(x)·∇ϕ(x)dx+λ2r−1 r−1ZI1 ˜ U(x0,z)·∇x0ϕ(x0,0)dx0dz =0, and taking into account expressions (5.26) and (3.12), we get (5.27). In the following result, we are going to prove the relation between the pressures p and P. Lemma 5.6. Let ηε≈εr 2r−1,ηε/ε r 2r−1→λ,0<λ<+∞,1<r < +∞and let p,P be the limit pressures from (5.25). Then, there exists e C∈Rsuch that p(x0,0)= P(x0)+ e C, (5.28) and p∈VΣis the unique solution of the variational problem (3.13). Proof. We need to extend the test functions considered in the proof of Lemma 5.3 to the fissure Iηε. To do this, we define Bηε=D−∪Σ∪Iηεand Y0=Y∗∩{xn= 0}, and 18 The effects of a fissure for a non-Newtonian fluid flow in a porous medium we consider φ(y)∈C∞ #(Bηε)nbe such that φ(y)=0 in Y\Y∗and divyφ(y)=0 in Y∗. We define φε(x)=    φx εin D−, Knenin Iηε,where Kn=ZY0 φn(y0,0)dy0. Let ϕ∈C∞ 0(B1), with B1=D−∪Σ∪I1be such that ZΣ ϕ(x0,0)dx0=0.(5.29) We define ϕηε(x)= (ϕ(x) in D− ϕx0,xn ηεin Iηε. Taking in (3.1) as test function wε(x)= (ϕ(x)φx ε−ε−r r−1vεηεin D−, ϕx0,xn ηεKnenin Iηε, we obtain ZBηε Sr(D[uεηε]): Dwεdx =ZBηε f·wεdx−ZBηε pεηεdivwεdx. (5.30) Taking into account that KnZIηε f·ϕx0,xn ηεendx=ηεKnZI1 f·ϕ(x0,z)endx0dz →0 as ε→0, and by using estimates (5.19), (5.20), that KnZIηε Sr(D[Uεηε])∂xnϕ(x0,xn ηε )dx=KnZI1 Sr(Dηε[Uεηε])∂zϕ(x0,z)dx0dz ≤Cηε 1 r−1→0,as ε→0, from (5.30), we obtain ZD− Sr(D[vεηε]): Dwεdx (5.31) =ZD− f·wεdx+ZD− pεηεdivwεdx+KnZIηε pεηε∂xnϕ(x0,xn ηε )dx+Oε. For the last term on the right hand side, we have KnZIηε pεηε∂xnϕ(x0,xn ηε )dx =KnZIηε (pεηε−cεηε)∂xnϕ(x0,xn ηε )dx+KnZIηε cεηε∂xnϕ(x0,xn ηε )dx, Mar´ıa Anguiano and Francisco Javier Su´arez-Grau 19 where cεηεis defined in (3.5). Using (5.25) and (5.29) we obtain KnZIηε (pεηε−cεηε)∂xnϕ(x0,xn ηε )dx=KnZI1 Pεηε∂zϕ(x0,z)dx0dz →KnZI1 P(x0)∂zϕ(x0,z)dx0dz =−KnZΣ P(x0)ϕ(x0,0)dx0,as ε→0, (5.32) where Pεηεis given by (3.4), and KncεηεZIηε ∂xnϕ(x0,xn ηε )dx=KncεηεZI1 ∂zϕ(x0,z))dx0dz =0. Passing to the limit in (5.31) similarly as in the proof of Lemma 5.3, we know that ˆvand pare related by the variational formulation of problem (5.9)-(5.11), and taking into account (5.32) and ZD×Y p(x)divx(ϕ(x)φ(y))dxdy =−ZD×Y ∇xp(x)ϕ(x)φ(y)dxdy+ZΣ×Y0 p(x0,0)ϕ(x0,0)φn(y0,0)dx0dy0 =−ZD×Y ∇xp(x)ϕ(x)φ(y)dxdy+KnZΣ p(x0,0)ϕ(x0,0)dx0, then we have ZΣ (p(x0,0)−P(x0))ϕ(x0,0)dx0=0, so that ZΣ (p(x0,0)−P(x0))ψ(x0)dx0=0, for every ψ∈C∞ 0(Σ) such that RΣψdx0= 0. Finally we conclude that there exists a constant e C∈Rsuch that (5.28) holds and p(·,0)∈W1,r0(Σ), i.e. p∈VΣ. Using (5.28) and (5.27), we obtain the variational formulation for the limit pressure pin the space VΣin the form ZD 1 µK(f(x)−∇p(x))·∇ϕ(x)dx +λ2r−1 r−1ZΣ˜ f(x0,0)−∇x0p(x0,0) r0−2 2r0 2(r+1)µr0−1(˜ f(x0,0)−∇x0p(x0,0))·∇x0ϕ(x0,0)dx0= 0, (5.33) for every ϕ∈VΣ. Since Kand Sr0are coercive and monotone (see Remark 3.3 for more details), it can be proved that (5.33) has a unique solution in the Banach space VΣ/Requipped with the norm |v|VΣ=|v|W1,r0(D)+|v(·,0)|W1,r0(Σ), by direct application of Lax-Milgram Theorem. Therefore the whole sequence converges to p, the unique solution of the problem (3.13). Proof. [Proof of Theorem 3.1] It remains to prove the convergence (3.10) of the whole velocity. 20 The effects of a fissure for a non-Newtonian fluid flow in a porous medium Let ϕ∈C0(D)n. Then ZD ε−r r−1uεηε·ϕdx =ZD ε−r r−1vεηε·ϕdx+ηε εr 2r−12r−1 r−1ZI1 ηε−r r−1Uεηε·ϕ(x0,ηεz)dx0dz. Taking the limit as ε→0, using (5.25), Un= 0 and ηε/ε r 2r−1→λ, we obtain ZD ε−r r−1uεηε·ϕdx→ZD v·ϕdx+λ2r−1 r−1ZI1 ˜ U(x0,z)ϕ(x0,0)dx0dz. Taking into account that ZI1 ˜ U(x0,z)ϕ(x0,0)dx0dz =ZΣ V(x0)ϕ(x0,0)dx0=hVδΣ,ϕiM(D)n,C0(D)n, where V(x0) is given by (3.12), we get (3.10). REFERENCES [1] G. Allaire, Homogenization of the Stokes flow in a connected porous medium, Asymptotic Analysis, 2, 203-22, 1989. [2] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23, 1482-1518, 1992. [3] A. Bourgeat, H. ElAmri, R. Tapiero, Existence d’une taille critique pour une fissure dans un milieu poreux, Second Colloque Franco Chilien de Mathematiques Appliqu´ees, Cepadu´es Edts, Tolouse, 67-80, 1991. [4] A. Bourgeat, R. Tapiero, Homogenization in a perforated domain including a thin full interlayer, Int. Ser. Num. Math., 114, 25-36, 1993. [5] A. Bourgeat, E. Maruˇsic-Paloka, A. Mikeli´c, Effective fluid flow in a porous medium containing a thin fissure, Asymptot. Anal., 11, 241-262, 1995. [6] A. Bourgeat, A. Mikeli´c, Homogenization of a polymer flow through a porous medium, Nonlinear Analysis, 26, 1221-1253, 1996. [7] A. Bourgeat, O. Gipouloux, E. Maruˇsi´c-Paloka, Filtration law for polymer flow through porous media, Multiscale Model. Simul., 1 no. 3, 432-457, 2003. [8] P.G. Ciarlet, H. Ledret, R. Nzwenga, Mod´elisation de la jonction entre un corps ´elastique tridimensionnel et une plaque, C. R. Acad. Sci., Paris, S´erie I, 305, 55-58, 1987. [9] A. Mikeli´c, R. Tapi´ero, Mathematical derivation of the power law describing polymer flow through a thin slab, RAIRO-Model. Math. Anal. Num., (1) 29, 3-21, 1995. [10] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20, 608-623, 1989. [11] G.P. Panasenko, Higher order asymptotics of solutions of problems on the contact of periodic structures, Math. U.S.S.R. Sbornik, 38, 465-494, 1981. [12] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Springer Lecture Notes in Physics, 127, 398 pp, 1980). [13] L. Tartar, Incompressible fluid flow in a porous medium convergence of the homogenization process. In: Appendix to Lecture Notes in Physics, 127. Berlin: Springer-Velag, 1980. [14] R. Temam, Navier-Stokes equations and nonlinear functional analysis, in: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 41, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1983. [15] H. Zhao, Z. Yao, Effective models of the Navier-Stokes flow in porous media with a thin fissure, J. Math. Anal. Appl., 387, 542-555, 2012.