Effective models for generalized Newtonian fluids through a thin porous medium following the Carreau law

E↵ective models for generalized Newtonian fluids through a thin porous medium following the Carreau law Mar´ıa ANGUIANO1, Matthieu BONNIVARD2, and Francisco J. SU´ AREZ-GRAU3 Abstract We consider the flow of a generalized Newtonian fluid through a thin porous medium of thickness ✏, perforated by periodically distributed solid cylinders of size ✏. We assume that the fluid is described by the 3D incompressible Stokes system, with a non-linear viscosity following the Carreau law of flow index 1 <r<+1, and scaled by a factor ✏,where2R. Generalizing (Anguiano et al., Q. J. Mech. Math., 75(1), 2022, 1-27), where the particular case r<2 and = 1 was addressed, we perform a new and complete study on the asymptotic behaviour of the fluid as ✏goes to zero. Depending on and the flow index r, using homogenization techniques, we derive and rigorously justify di↵erent e↵ective linear and non-linear lower-dimensional Darcy’s laws. Finally, using a finite element method, we study numerically the influence of the rheological parameters of the fluid and of the shape of the solid obstacles on the behaviour of the e↵ective systems. AMS classification numbers: 76-10, 76A05, 76M50, 76A20, 76S05, 35B27, 35Q35. Keywords: Homogenization, non-Newtonian fluid, Carreau law, thin porous media. 1 Introduction An incompressible generalized Newtonian fluid is a type of non-Newtonian fluid which is characterized by a viscosity depending on the principal invariants of the strain-rate tensor D[u]. If uis the velocity, p the pressure and Du the gradient velocity tensor, D[u]=(Du +Dtu)/2 denotes the strain-rate tensor and the stress tensor given by =pI +2⌘rD[u]. The viscosity ⌘ris constant for a Newtonian fluid but dependent of the shear rate, i.e. ⌘r=⌘r(D[u]), for generalized Newtonian fluids. The deviatoric stress tensor ⌧,i.e. the part of the total stress tensor that is zero at equilibrium, is then a nonlinear function of the shear rate D[u]: ⌧=⌘r(D[u])D[u] (see Barnes et al. [20], Bird et al. [21] and Mikeli´c [42] for more details). The power law or Ostwald-de Waele model (Ostwald, 1925; de Waele, 1923) is commonly used to describe the motion of a generalized Newtonian fluid. The corresponding viscosity formula is ⌘r(D[u]) = µ|D[u]|r2,1<r<+1,µ>0,(1.1) 1Departamento de An´alisis Matem´atico. Facultad de Matem´aticas. Universidad de Sevilla, 41012 Sevilla (Spain) [email protected] 2Ecole Centrale de Lyon, CNRS, INSA Lyon, Universite Claude Bernard Lyon 1, Universit´e Jean Monnet, ICJ UMR5208, 69130 Ecully, France, [email protected] 3Departamento de Ecuaciones Diferenciales y An´alisis Num´erico. Facultad de Matem´aticas. Universidad de Sevilla, 41012 Sevilla (Spain) [email protected] 1 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau where µ>0 is the consistency of the fluid and ris the flow index. The matrix norm |·|is defined by |⇠|2=Tr(⇠⇠t)with⇠2R3⇥3. We recall that the generalized Newtonian fluids are classified in two main categories (see Saramito [45, Chapter 2] for more details): –pseudoplastic or shear thinning fluids, where the viscosity decreases with the shear rate, which corresponds to the case of a flow index 1 <r<2; –dilatant or shear thickening fluids, where the viscosity increases with the shear rate, and r>2. We also recall that the case r= 2 corresponds to a Newtonian fluid. The power law was introduced to model polymer solutions at high shear rates, and by its simplicity, permits analytical calculations in simple geometries. However, it has the disadvantage of not describing a Newtonian plateau and even predicts an infinite viscosity as the shear rate goes to zero and 1 <r<2 (see Agassant et al. [2, p. 49]), whereas for real fluids it tends to some constant value ⌘0called the zero-shear-rate viscosity. For these reasons, other viscosity models are used, which better describe the real behaviour of pseudoplastic or dilatant fluids, but are more difficult to analyze mathematically. The Carreau law. Among those, an important model is the well-known Carreau law,whichwillbe considered in this paper and is defined by ⌘r(D[u]) = (⌘0⌘1)(1 + |D[u]|2)r 21+⌘1,1<r<+1,⌘ 0>⌘ 1>0,>0.(1.2) In this relation, ris the flow index of the fluid, ⌘0is the low-shear-rate limit of the viscosity, and for 1<r<2, ⌘1is the high-shear-rate limit of the viscosity. Parameter is a time constant and r2 describes the slope in the power law region. For r= 2, as in the power-law model, one recovers a Newtonian fluid model, with viscosity ⌘0. In this paper, we consider the flow of a Carreau fluid in a thin porous medium. In this context, the complexity of the fluid domain geometry makes it very costly to solve the equations from continuum mechanics directly. The homogenisation theory provides an alternative approach, which aims to derive an average model by analyzing the e↵ects of the micro-structure, i.e. the pore structure, on the solutions of these partial di↵erential equations. This mathematical method allows one to derive rigorously the equations describing the filtration of a generalized Newtonian fluid (see for instance Mikeli´c [42]). To derive the averaged law describing the generalized Newtonian fluid flow through a porous medium ⌦✏⇢R3, which is a domain with fixed height and periodically perforated by obstacles of size ✏, Bourgeat and Mikeli´c in [23] (see also Bourgeat et al. [24] and Kalousek [39]) used the two-scale convergence method. We also refer to Anguiano [6, 8] for nonlinear parabolic problems in porous medium. The domain without perforations is the bounded smooth domain ⌦⇢R3that is divided into two parts, the fluid part ⌦✏and the solid part ⌦\⌦✏. Moreover, assuming that the flow is sufficiently slow to neglect inertial e↵ects, the following stationary Stokes system with a non-linear viscosity following the Carreau law (1.2) was considered: 8 > > > < > > > : ✏div (⌘r(D[u✏])D[u✏]) + rp✏=fin ⌦✏, div u✏=0 in⌦ ✏, u✏= 0 on @⌦✏. (1.3) 2 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau In the momentum equation (1.3), the viscosity ⌘ris a↵ected by a factor ✏. This can be interpreted as a scaling of the low-shear-rate and high-shear-rate limit viscosities ⌘0,⌘ 1appearing in the Carreau law (1.2) by ✏, which allows one to characterize mathematically the impact of these quantities on the asymptotic behaviour of the system. Indeed, letting ✏tend to zero, di↵erent types of averaged momentum equations connecting the velocity and the pressure gradient were rigorously derived, depending on the value of and the flow index r. •If <1 and 1 <r<+1, the homogenized law is the classical 3D Darcy’s law for Newtonian fluids V(x)=K ⌘(f(x)rxp(x)) in ⌦,divxV(x)=0 in⌦,V(x)·n= 0 on @⌦,(1.4) where pis the limit pressure and the permeability tensor K2R3⇥3is obtained by solving 3D Stokes local problems posed in a reference cell which contains the information of the geometry of the obstacles. The viscosity ⌘is equal to ⌘0. •If = 1 and r6= 2, the mean global filtration velocity as a function of the pressure gradient is given by V(x)=U(f(x)rxp(x)) in ⌦,divxV(x)=0 in⌦,V(x)·n= 0 on @⌦,(1.5) where U:R3!R3is a permeability operator, not necessary linear, and is defined through the solutions of 3D local Stokes problems with non-linear viscosity following the Carreau law and posed in a reference cell. Linearity holds only if r= 2, where the classical 3D Darcy’s law for Newtonian fluids (1.4) is derived with ⌘=⌘0. •If >1, the following cases hold –For 1 <r2, the homogenized law is the classical 3D Darcy’s law for Newtonian fluids (1.4) with ⌘=⌘1if 1 <r<2 and ⌘=⌘0if r= 2. –For r>2, the mean global filtration velocity as a function of the pressure gradient is given by (1.5), where the permeability operator U:R3!R3is defined through the solutions of 3D local non-Newtonian Stokes problems with non-linear viscosity following the power law (1.1) with µ=(⌘0⌘1)(r/2)1and posed in a reference cell. On the other hand, in [22] Boughanim and Tapi´ero considered the generalized Newtonian fluids flow through a thin domain ⌦✏=!⇥(0,✏)⇢R3,with!⇢R2,where✏is the small thickness of the domain. From the Stokes system (1.3), with external body force of the form f=(f0,0) such that f0=(f1,f 2), by using dimension reduction and homogenization techniques, they derived the limit when the thickness tends to zero. Depending on the value of and the flow index r, they obtained the following: •If <1, the following cases hold –For 1 <r2, the homogenization law is the classical linear 2D Reynolds law for Newtonian fluids 8 > < > : V0(x0)= 1 6µf0(x0)rx0p(x0),V 3(x0)=0 in!, divx0V(x0)=0 in!, V (x0)·n= 0 on @!, (1.6) where V0=(V1,V 2), x0=(x1,x 2). The viscosity µis equal to ⌘0. 3 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau –For r>2, the filtration velocity is zero, i.e. V⌘0. •If = 1 and r6= 2, the homogenization law corresponds to a non-linear 2D Reynolds law of Carreau type 8 > > < > > : V0(x0) = 2(f0(x0)rx0p(x0)) Z1 2 1 2 (1 2+⇠)⇠ (2|f0(x0)rx0p(x0)||⇠|)d⇠, V3(x0)=0 in!, divx0V0(x0)=0 in!, V 0(x0)·n= 0 on @!, where the function = (⌧), ⌧2R+, is the inverse of the equation ⌧= r2 ⇣ ⌘1 ⌘0⌘1⌘2 r21. Linearity holds only if r= 2, where the the classical linear 2D Reynolds law for Newtonian fluids (1.6) with ⌘=⌘0is derived. •If >1, the following cases hold –For 1 <r2, the homogenization law is the classical linear 2D Reynolds law for Newtonian fluids (1.6) with µ=⌘1if 1 <r<2 and µ=⌘0if r= 2. –For r>2, the homogenization law corresponds to a non-linear 2D Reynolds law of power type 8 > > > > > < > > > > > : V0(x0)= r0/21 (⌘0⌘1)r01 1 2r0/2(r0+ 1)|f0(x0)rx0p(x0)|r02(f0(x0)rx0p(x0)) in !, V3(x0)=0 in!, divx0V0(x0)=0 in!, V 0(x0)·n= 0 on @!. We remark that in [22] and [23] the Navier-Stokes equation is considered, which implies an upper limit on due to the contribution of the inertial term. However, as pointed out in the beginning of Section 1.4 in [23] (see also [24]), the corresponding results remain unchanged for the non-Newtonian Stokes systems. Furthermore, there are no upper limits on and the convergence of the pressure is stronger. This motivates our choice of studying the flow of a generalized Newtonian fluid governed by the stationary Stokes system (1.3), with a non-linear viscosity following the Carreau law (1.2) through a thin periodic medium. We now detail the geometry of the medium that we consider. The thin porous medium. The interest in the behaviour of generalized Newtonian fluids through thin porous media has increased recently, mainly because of their use in many industrial processes (see Prat and Aga¨esse [44] for more details). A thin porous medium is a microstructured thin domain, that can be represented as a domain of thickness ✏with 0 <✏⌧1, perforated by an array of periodically distributed solid cylinders of diameter a✏, where the parameter 0 <a ✏⌧1tendstozerowith✏. Recently, this problem has been analyzed in Anguiano and Su´arez-Grau [11, 13] and Frabricius et al. [31], where the behaviour of Newtonian or power law fluid flows through a thin porous medium is considered. Three classes of thin porous media were introduced. -The proportionally thin porous medium, corresponding to the critical case where the cylinder diameter is proportional to the interspatial distance, with ⌫the proportionality constant, i.e. a✏⇡✏,witha✏/✏!⌫,0<⌫<+1. 4 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau -The homogeneously thin porous medium, corresponding to the case where the cylinder diameter is much larger than the interspatial distance, i.e. a✏⌧✏which is equivalent to ⌫= 0. -The very thin porous medium, corresponding to the case where the cylinder diameter is much smaller than the interspatial distance, i.e. a✏✏which is equivalent to ⌫=+1. A lower-dimensional Darcy law is obtained in every case, but the permeability operator depends on the type of thin porous medium considered. More precisely, for a proportionally thin porous medium, the permeability operator is defined through the solutions of 3D local Stokes problems depending on ⌫. For a homogeneously thin porous medium, this operator is defined through the solutions of 2D local Stokes problems, whereas for a very thin porous medium, it is defined through the solutions of 2D local Hele-Shaw problems. We also refer to other recent studies by Anguiano [3, 4, 5], Anguiano and Su´arez-Grau [12, 15, 16, 17], Bunoiu and Timofte [25, 26], Fabricius et al. [32], Jouybari and Lundstr¨om [38], Su´arez-Grau [48], Yeghiazarian et al. [52] and Zhengan and Hongxing [53], where the behaviour of Newtonian or power law fluids through di↵erent types of thin porous medium is considered. For the case of a Bingham flow we refer to Anguiano and Bunoiu [10], and for the case of micropolar fluids we refer to Su´arezGrau [47]. The case of a fluid flow through a thin porous medium with slip boundary conditions on the cylinders is considered in Anguiano and Su´arez-Grau [14] and Fabricius and Gahn [33]. On the other hand, the first study about the reaction-di↵usion equation in a thin porous medium was done recently in Anguiano [7], the two-phase flow problem in thin porous domains of Brinkman type has been considered in Armiti-Juber [18], and an approach for e↵ective heat transport in thin porous media has been derived by Scholz and Bringedal [46]. However, the literature on Carreau fluid flows in this type of domains is far less complete, although these problems have now become of great practical relevance to Chemical Industry and Rheology, for instance in injection moulding of melted polymers, flow of oils, muds, etc. (see for example Pereire and Lecampion [43] and Wrobel et al. [49, 50]). In this paper, we consider a thin porous medium ⌦✏=!✏⇥(0,✏)⇢R3of small height ✏which is perforated by an array of periodically distributed solid cylinders of diameter of size ✏(see Figure 3). Observe that this corresponds to the case of a proportionally thin porous medium with ⌫= 1. Here, the bottom of the domain without perforations !⇢R2is made of two parts, the fluid part !✏ and the solid part !\!✏. Similarly to [22, 23], assuming that the flow is sufficiently slow to neglect inertial e↵ects, we consider that the generalized Newtonian fluid flow through the thin porous medium ⌦✏=!✏⇥(0,✏) is governed by the stationary Stokes system (1.3) with a non-linear viscosity following the Carreau law (1.2) and scaled by a factor ✏. This problem has been very recently considered in Anguiano et al. [9] in the particular case of a pseudoplastic fluid (1 <r<2) with = 1. Using homogenization techniques, it was proved that when ✏tends to zero, the mean global filtration velocity is given, as a function of the pressure gradient, by a non-linear 2D Darcy law of Carreau type 8 < : V0(x0)=Uf0(x0)rx0p(x0),V 3(x0)=0in!, divx0V0(x0)=0 in!, V 0(x0)·n= 0 on @!.(1.7) In the above system, nis the outward normal to @!,V0=(V1,V 2), x0=(x1,x 2), and the permeability operator U:R2!R2is defined through the solutions of 3D local non-Newtonian Stokes problems with non-linear viscosity following the Carreau law (1.2) and posed in a reference cell. 5 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau In this paper, we perform a new and complete study on the asymptotic behaviour of Carreau fluids, modeled by the equations of motion (1.3) in the thin porous medium ⌦✏, depending on the type of fluid and the value of . We generalize the results obtained in [9] by considering system (1.3) not only for pseudoplastic fluids, but also for dilatant fluids and Newtonian fluids, and moreover, for any exponent 2R. Starting from problem (1.3) and using homogenization techniques, we derive di↵erent e↵ective problems depending on the type of fluid and the value of , describing the asymptotic behaviour of the model as ✏tends to zero. The approach that we use relies strongly on an adaptation of the periodic unfolding method introduced by Cioranescu et al. [27, 29]. In order to give a taste of the kind of arguments that will allow us to distinguish between the di↵erent regimes related to rand , let us give the heuristics of the obtention of the e↵ective system in the pseudoplastic case 1 <r<2. The so-called unfolded velocity and pressure (ˆu✏,ˆ P✏)(definedin Section 3.2) satisfy for any admissible test function 'the following inequality: (⌘0⌘1)Z!⇥Z (1 + ✏2(1)|Dz[']|2)r 21Dz[']:Dz['✏2ˆu✏]dx0dz +⌘1Z!⇥Z Dz[']:Dz['✏2ˆu✏]dx0dz Z!⇥Z ˆ P✏divx0('0✏2ˆu0 ✏)dx0dz Z!⇥Z f0·('0✏2ˆu0 ✏)dx0dz +O✏, where O✏tends to zero with ✏. Also, (✏2ˆu✏,ˆ P✏) converges in appropriate Sobolev spaces to a pair of functions called (ˆu, ˜ P). We refer to Sections 3.3 and 3.4 (in particular equation (3.56)) for more details. Then, we observe that if <1, 2(1 )>0so✏2(1)|Dy[']|2tends to zero, whereas for >1, 2(1 )<0so(1+✏2(1)|Dy[']|2)r 21tends to zero. As a consequence, the sum of the two first terms in the previous formulation converges to the linear term ⌘Z!⇥Z Dz[']:Dz['ˆu]dx0dz, with ⌘=⌘0if <1 and ⌘=⌘1if >1. In case = 1, the critical case that couples the nonlinear term and the linear one, their sum converges to (⌘0⌘1)Z!⇥Z (1 + |Dz[']|2)r 21Dz[']:Dz['ˆu]dx0dz +⌘1Z!⇥Z Dz[']:Dz['ˆu]dx0dz. Thus, we obtain three di↵erent asymptotic behaviours depending on whether the value of is smaller, equal or greater than 1. For dilatant fluids, there exist three di↵erent convergences of the unfolding velocity depending on the value of and, as consequence, three di↵erent homogenized models are derived. Summary of the di↵erent asymptotic regimes. In summary, we have the following asymptotic behaviours of Carreau fluids depending on the the value of and the type of fluid: •If <1, regardless of the value of r, the e↵ective problem is the linear 2D Darcy law 8 > < > : V0(x0)=1 ⌘Af0(x0)rx0p(x0),V 3(x0)=0 in!, divx0V0(x0)=0 in!, V 0(x0)·n= 0 on @!, (1.8) 6 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau where the permeability tensor A2R2⇥2is obtained by solving 3D local Newtonian Stokes problems, posed in a reference cell containing the information on the obstacles’ geometry. The viscosity ⌘is equal to ⌘0. •If = 1, the asymptotic behaviour of the model depends on the fact that the fluid is Newtonian of not. –For 1 <r<1with r6= 2, the e↵ective problem is the non-linear 2D Darcy law of Carreau type (1.7), which is obtained in [9]. –For r= 2, the e↵ective problem is the linear 2D Darcy law (1.8) with viscosity ⌘=⌘0. •If >1, then pseudoplastic, Newtonian and dilatant fluid flows have distinct asymptotic properties. –For r2(1,2), the e↵ective problem is the linear 2D Darcy law (1.8) with viscosity ⌘=⌘1. –For r= 2, the e↵ective problem is the linear 2D Darcy law (1.8) with viscosity ⌘=⌘0. –For r>2, the e↵ective problem is a non-linear 2D Darcy law of power law type 8 > > < > > : V0(x0)= r0/21 (⌘0⌘1)r01U⇣f0(x0)rx0˜ P(x0)⌘,V 3(x0)=0in!, divx0V0(x0)=0 in!, V 0(x0)·n= 0 on @!, where the permeability operator U:R2!R2is defined through the solutions of 3D local non-Newtonian Stokes problems with non-linear viscosity following the power law (1.1) and posed in a reference cell. In Table 1, we summarize every asymptotic behaviour of the Carreau flow governed by (1.3) depending on the type of fluid and the value of : 1<r<2r=2 r>2 <1Linear 2D Darcy’s law (viscosity ⌘0)Linear Linear 2D Darcy’s law (viscosity ⌘0) =1 Non-linear 2D Darcy’s law (Carreau type) 2D Darcy’s law Non-linear 2D Darcy’s law (Carreau type) >1Linear 2D Darcy’s law (viscosity ⌘1)(viscosity ⌘0)Non-linear 2D Darcy’s law (power law type) Table 1: Asymptotic behaviours of Carreau fluids depending on the values of rand . How can one determine the exponent in practice? As revealed by the asymptotic analysis carried out in [23] and in the present contribution, the computation of the exponent is crucial to characterize the flow of pseudoplastic and dilatant fluids through porous media. The presence of the scaling factor ✏expresses the modelling assumption that the viscosities ⌘0,⌘ 1are the only parameters from the Carreau law that are a↵ected by the small geometrical scale ✏(and that the e↵ect is the same on both viscosities), while the other parameters rand are independent on ✏. Under these 7 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau assumptions, one possible strategy to compute the exponent associated to a specific problem could be the following. When dealing with a quasi-Newtonian fluid, start by fitting experimentally the parameters ⌘0,⌘1, r,appearing in the Carreau law (1.2). We refer to [21, Table 4.1-1] and [51, Table 3] for examples of experimental results related to the shear flow of polysterene solutions. After obtaining these values, one has to apply the same technique to the flow of the fluid under study in a thin porous domain of characteristic size ✏. If the parameters rand are indeed invariant, the adjusted values of the rescaled viscosities ✏⌘0and ✏⌘1give then access to the value of ✏, hence to itself. The structure of the paper is as follows. In Section 2 we introduce the domain, make the statement of the problem and give the main results (Theorems 2.1, 2.3 and 2.5). The proofs of the main results are provided in Section 3. Finally, we perform in Section 4 a numerical study of the di↵erent e↵ective systems described in Theorems 2.1, 2.3 and 2.5, based on the computation of permeability tensors A and permeability operators Uusing a finite element method. A list of references completes the paper. 2 Setting of the problem and main result Geometrical setting. The periodic porous medium is defined by a domain !and an associated microstructure, or periodic cell Z0=(1/2,1/2)2, which is made of two complementary parts: the fluid part Z0 f, and the solid part T0(Z0 fST0=Z0and Z0 fTT0=;). More precisely, we assume that !is a smooth, bounded, connected set in R2with smooth enough boundary @!, that nis the outward normal to @!, and that T0is an open connected subset of Z0with a smooth boundary @T0, such that T0is strictly included in Z0. The microscale of the porous medium is a small positive number ✏. The domain !is covered by a regular mesh of squares of size ✏: for k02Z2, each cell Z0 k0,✏=✏k0+✏Z0is divided in a fluid part Z0 fk0,✏ and a solid part T0 k0,✏,i.e. is similar to the unit cell Z0rescaled to size ✏.WedefineZ=Z0⇥(0,1) ⇢R3, which is divided in a fluid part Zf=Z0 f⇥(0,1) and a solid part T=T0⇥(0,1), and consequently Zk0,✏=Z0 k0,✏⇥(0,1) ⇢R3, which is also divided in a fluid part Zfk0,✏and a solid part Tk0,✏(see Figures 1 and 2). We denote by ⌧(T0 k0,✏) the set of all translated images of T0 k0,✏.Theset⌧(T0 k0,✏)representsthe obstacles in R2. 1 1 Yf T <latexit sha1_base64="jgGdGYmS94g60FG/9SbBo2voxfM=">AAAB5HicbZDLTgIxFIZP8YZ4Q126aSQmrsiMIeqS6MYlRrlEmJBOOQMNnUvajgmZ8Aa6MurOJ/IFfBsLzkLBf/X1/H+T8x8/kUIbx/kihZXVtfWN4mZpa3tnd6+8f9DScao4NnksY9XxmUYpImwaYSR2EoUs9CW2/fH1zG8/otIiju7NJEEvZMNIBIIzY0d3D/2gX644VWcuugxuDhXI1eiXP3uDmKchRoZLpnXXdRLjZUwZwSVOS71UY8L4mA2xazFiIWovm686pSdBrKgZIZ2/f2czFmo9CX2bCZkZ6UVvNvzP66YmuPQyESWpwYjbiPWCVFIT01ljOhAKuZETC4wrYbekfMQU48bepWTru4tll6F1VnXPq7XbWqV+lR+iCEdwDKfgwgXU4QYa0AQOQ3iGN3gnAXkiL+T1J1og+Z9D+CPy8Q0iSItV</latexit> Zf <latexit 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1 <latexit sha1_base64="XoWlZtusfOssYBKVPwvkLH4zY34=">AAAB4nicbZDLTgJBEEVr8IX4Ql266UhMXJEZY9Ql0Y1LSOSRwIT0NDXQoeeR7hoTQvgBXRl15yf5A/6NDc5Cwbs6Xfd2UreCVElDrvvlFNbWNza3itulnd29/YPy4VHLJJkW2BSJSnQn4AaVjLFJkhR2Uo08ChS2g/Hd3G8/ojYyiR9okqIf8WEsQyk42VHD65crbtVdiK2Cl0MFctX75c/eIBFZhDEJxY3pem5K/pRrkkLhrNTLDKZcjPkQuxZjHqHxp4tFZ+wsTDSjEbLF+3d2yiNjJlFgMxGnkVn25sP/vG5G4Y0/lXGaEcbCRqwXZopRwuZ92UBqFKQmFrjQ0m7JxIhrLshepWTre8tlV6F1UfWuqpeNy0rtNj9EEU7gFM7Bg2uowT3UoQkCEJ7hDd6dgfPkvDivP9GCk/85hj9yPr4BdDWKUw==</latexit> 1 Figure 1: View of the 3D reference cells Z(left) and the 2D reference cell Z0(right). The fluid part of the bottom !✏⇢R2of a porous medium is defined by !✏=!\Sk02K✏T0 k0,✏,where K✏={k02Z2:Z0 k0,✏\!6=;}. The whole fluid part ⌦✏⇢R3in the thin porous medium is defined by (see Figure 3) ⌦✏={(x1,x 2,x 3)2!✏⇥R:0<x 3<✏}.(2.1) 8 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau   T k,⇥(0,) Y fk,⇥(0,) <latexit sha1_base64="c8T6QG5fmSSygXb4YzVaykT6X10=">AAACCnicbVBLSwMxGMzWV62vVY9egkVaQcquFPVY9OKxgn1gd1my6bdtaPZBkhXK0n+gf0ZPot48+gf8N6Z1Ea3OaTIzIZnxE86ksqwPo7CwuLS8Ulwtra1vbG6Z2zttGaeCQovGPBZdn0jgLIKWYopDNxFAQp9Dxx9dTP3OLQjJ4uhajRNwQzKIWMAoUVryzMpNxcsCLxtVJkcOJJLxOJo4ioUgcdX6lg49s2zVrBnwX2LnpIxyND3z3enHNA0hUpQTKXu2lSg3I0IxymFSclIJCaEjMoCephHRL7rZrNAEHwSxwGoIeHb+mc1IKOU49HUmJGoo572p+J/XS1Vw5mYsSlIFEdUR7QUpxyrG011wnwmgio81IVQw/UtMh0QQqvR6JV3fni/7l7SPa/ZJrX5VLzfO8yGKaA/toyqy0SlqoEvURC1E0T16RC/o1bgzHown4/krWjDyO7voF4y3T4mImhk=</latexit> Z fk,⇥(0,✏) <latexit sha1_base64="w7yBNjnFLAz8g0PzWoOrqoA7HM4=">AAAB6XicbZDNTgIxFIXv4B/iH+rSTSMxcUVmDFGXRDcuMZGfBCakU+5AQ6edtB0TQngIXRl15+v4Ar6NBWeh4Fl9vec0uedGqeDG+v6XV1hb39jcKm6Xdnb39g/Kh0ctozLNsMmUULoTUYOCS2xabgV2Uo00iQS2o/Ht3G8/ojZcyQc7STFM6FDymDNq3ajTw9RwoWS/XPGr/kJkFYIcKpCr0S9/9gaKZQlKywQ1phv4qQ2nVFvOBM5KvcxgStmYDrHrUNIETThd7DsjZ7HSxI6QLN6/s1OaGDNJIpdJqB2ZZW8+/M/rZja+DqdcpplFyVzEeXEmiFVkXpsMuEZmxcQBZZq7LQkbUU2ZdccpufrBctlVaF1Ug8tq7b5Wqd/khyjCCZzCOQRwBXW4gwY0gYGAZ3iDd2/sPXkv3utPtODlf47hj7yPbzNhjb4=</latexit> ✏ <latexit sha1_base64="0+UB7XWbwzigfqdl3zVx5rcAyZY=">AAACBnicbVDNSgMxGMzWv1r/qh69BIu0gpRdKeqx6MVjhf5BdynZ9Ns2NJtdkqxQlt71ZfQk6s0H8AV8G9O6iFbnNJmZkMz4MWdK2/aHlVtaXlldy68XNja3tneKu3ttFSWSQotGPJJdnyjgTEBLM82hG0sgoc+h44+vZn7nFqRikWjqSQxeSIaCBYwSbaR+sdQs99Nx+cSFWDEeiamrWQgKV+xv6dik7Ko9B/5LnIyUUIZGv/juDiKahCA05USpnmPH2kuJ1IxymBbcREFM6JgMoWeoIOZFL52XmeKjIJJYjwDPzz+zKQmVmoS+yYREj9SiNxP/83qJDi68lIk40SCoiRgvSDjWEZ5tggdMAtV8YgihkplfYjoiklBtliuY+s5i2b+kfVp1zqq1m1qpfpkNkUcH6BBVkIPOUR1dowZqIYru0SN6Qa/WnfVgPVnPX9Gcld3ZR79gvX0CFTOYLg==</latexit> T k,⇥(0,✏) <latexit sha1_base64="w7yBNjnFLAz8g0PzWoOrqoA7HM4=">AAAB6XicbZDNTgIxFIXv4B/iH+rSTSMxcUVmDFGXRDcuMZGfBCakU+5AQ6edtB0TQngIXRl15+v4Ar6NBWeh4Fl9vec0uedGqeDG+v6XV1hb39jcKm6Xdnb39g/Kh0ctozLNsMmUULoTUYOCS2xabgV2Uo00iQS2o/Ht3G8/ojZcyQc7STFM6FDymDNq3ajTw9RwoWS/XPGr/kJkFYIcKpCr0S9/9gaKZQlKywQ1phv4qQ2nVFvOBM5KvcxgStmYDrHrUNIETThd7DsjZ7HSxI6QLN6/s1OaGDNJIpdJqB2ZZW8+/M/rZja+DqdcpplFyVzEeXEmiFVkXpsMuEZmxcQBZZq7LQkbUU2ZdccpufrBctlVaF1Ug8tq7b5Wqd/khyjCCZzCOQRwBXW4gwY0gYGAZ3iDd2/sPXkv3utPtODlf47hj7yPbzNhjb4=</latexit> ✏   Y fk, T k, <latexit sha1_base64="e1CDEWmSwnYejZ4lhxK422RKQbk=">AAAB93icbZC7TsMwFIadcivlFi4bS0SFyoCqBFXAWMHCWCR6EW0UOe5Ja9VxIttBClGeBSYEbLwHL8Db4JYM0PJPn8//Wzrn92NGpbLtL6O0tLyyulZer2xsbm3vmLt7HRklgkCbRCwSPR9LYJRDW1HFoBcLwKHPoOtPrqd+9wGEpBG/U2kMbohHnAaUYKVHnnlwX/OywMsmtfx0ALGkLOK5Z1btuj2TtQhOAVVUqOWZn4NhRJIQuCIMS9l37Fi5GRaKEgZ5ZZBIiDGZ4BH0NXIcgnSz2fa5dRxEwlJjsGbv39kMh1Kmoa8zIVZjOe9Nh/95/UQFl25GeZwo4ERHtBckzFKRNS3BGlIBRLFUAyaC6i0tMsYCE6WrqujznfljF6FzVnfO643bRrV5VRRRRofoCJ0gB12gJrpBLdRGBD2iZ/SG3o3UeDJejNefaMko/uyjPzI+vgEUN5K6</latexit> Z fk, <latexit sha1_base64="w7yBNjnFLAz8g0PzWoOrqoA7HM4=">AAAB6XicbZDNTgIxFIXv4B/iH+rSTSMxcUVmDFGXRDcuMZGfBCakU+5AQ6edtB0TQngIXRl15+v4Ar6NBWeh4Fl9vec0uedGqeDG+v6XV1hb39jcKm6Xdnb39g/Kh0ctozLNsMmUULoTUYOCS2xabgV2Uo00iQS2o/Ht3G8/ojZcyQc7STFM6FDymDNq3ajTw9RwoWS/XPGr/kJkFYIcKpCr0S9/9gaKZQlKywQ1phv4qQ2nVFvOBM5KvcxgStmYDrHrUNIETThd7DsjZ7HSxI6QLN6/s1OaGDNJIpdJqB2ZZW8+/M/rZja+DqdcpplFyVzEeXEmiFVkXpsMuEZmxcQBZZq7LQkbUU2ZdccpufrBctlVaF1Ug8tq7b5Wqd/khyjCCZzCOQRwBXW4gwY0gYGAZ3iDd2/sPXkv3utPtODlf47hj7yPbzNhjb4=</latexit> ✏ <latexit sha1_base64="w7yBNjnFLAz8g0PzWoOrqoA7HM4=">AAAB6XicbZDNTgIxFIXv4B/iH+rSTSMxcUVmDFGXRDcuMZGfBCakU+5AQ6edtB0TQngIXRl15+v4Ar6NBWeh4Fl9vec0uedGqeDG+v6XV1hb39jcKm6Xdnb39g/Kh0ctozLNsMmUULoTUYOCS2xabgV2Uo00iQS2o/Ht3G8/ojZcyQc7STFM6FDymDNq3ajTw9RwoWS/XPGr/kJkFYIcKpCr0S9/9gaKZQlKywQ1phv4qQ2nVFvOBM5KvcxgStmYDrHrUNIETThd7DsjZ7HSxI6QLN6/s1OaGDNJIpdJqB2ZZW8+/M/rZja+DqdcpplFyVzEeXEmiFVkXpsMuEZmxcQBZZq7LQkbUU2ZdccpufrBctlVaF1Ug8tq7b5Wqd/khyjCCZzCOQRwBXW4gwY0gYGAZ3iDd2/sPXkv3utPtODlf47hj7yPbzNhjb4=</latexit> ✏ <latexit sha1_base64="e1CDEWmSwnYejZ4lhxK422RKQbk=">AAAB93icbZC7TsMwFIadcivlFi4bS0SFyoCqBFXAWMHCWCR6EW0UOe5Ja9VxIttBClGeBSYEbLwHL8Db4JYM0PJPn8//Wzrn92NGpbLtL6O0tLyyulZer2xsbm3vmLt7HRklgkCbRCwSPR9LYJRDW1HFoBcLwKHPoOtPrqd+9wGEpBG/U2kMbohHnAaUYKVHnnlwX/OywMsmtfx0ALGkLOK5Z1btuj2TtQhOAVVUqOWZn4NhRJIQuCIMS9l37Fi5GRaKEgZ5ZZBIiDGZ4BH0NXIcgnSz2fa5dRxEwlJjsGbv39kMh1Kmoa8zIVZjOe9Nh/95/UQFl25GeZwo4ERHtBckzFKRNS3BGlIBRLFUAyaC6i0tMsYCE6WrqujznfljF6FzVnfO643bRrV5VRRRRofoCJ0gB12gJrpBLdRGBD2iZ/SG3o3UeDJejNefaMko/uyjPzI+vgEUN5K6</latexit> Z fk, <latexit sha1_base64="1dFjHOc8h0UeKuVcGFw/ZA1CeN4=">AAAB8nicbZDNSgMxFIUz9a/Wn466dBMsogspM1LUZdGNywr9g3YYMultG5pJhiQjlKEvoitRdz6KL+DbmNZZaOtZfbnnBO65UcKZNp735RTW1jc2t4rbpZ3dvf2ye3DY1jJVFFpUcqm6EdHAmYCWYYZDN1FA4ohDJ5rczf3OIyjNpGiaaQJBTEaCDRklxo5Ct9wMs8nZRR8SzbgUs9CteFVvIbwKfg4VlKsRup/9gaRpDMJQTrTu+V5igowowyiHWamfakgInZAR9CwKEoMOssXiM3w6lAqbMeDF+3c2I7HW0ziymZiYsV725sP/vF5qhjdBxkSSGhDURqw3TDk2Es/74wFTQA2fWiBUMbslpmOiCDX2SiVb318uuwrty6p/Va091Cr12/wQRXSMTtA58tE1qqN71EAtRFGKntEbeneM8+S8OK8/0YKT/zlCf+R8fANgI5Ce</latexit> Tk, Figure 2: View of the 3D reference cells Zk0,✏(left) and the 2D reference cell Z0 k0,✏(right). We assume that the obstacles ⌧(T0 k0,✏) do not intersect the boundary @! and we denote by S✏the set of the solid cylinders contained in ⌦✏,i.e. S✏=Sk02K✏T0 k0,✏⇥(0,✏).   <latexit sha1_base64="w7yBNjnFLAz8g0PzWoOrqoA7HM4=">AAAB6XicbZDNTgIxFIXv4B/iH+rSTSMxcUVmDFGXRDcuMZGfBCakU+5AQ6edtB0TQngIXRl15+v4Ar6NBWeh4Fl9vec0uedGqeDG+v6XV1hb39jcKm6Xdnb39g/Kh0ctozLNsMmUULoTUYOCS2xabgV2Uo00iQS2o/Ht3G8/ojZcyQc7STFM6FDymDNq3ajTw9RwoWS/XPGr/kJkFYIcKpCr0S9/9gaKZQlKywQ1phv4qQ2nVFvOBM5KvcxgStmYDrHrUNIETThd7DsjZ7HSxI6QLN6/s1OaGDNJIpdJqB2ZZW8+/M/rZja+DqdcpplFyVzEeXEmiFVkXpsMuEZmxcQBZZq7LQkbUU2ZdccpufrBctlVaF1Ug8tq7b5Wqd/khyjCCZzCOQRwBXW4gwY0gYGAZ3iDd2/sPXkv3utPtODlf47hj7yPbzNhjb4=</latexit> ✏ <latexit sha1_base64="w7yBNjnFLAz8g0PzWoOrqoA7HM4=">AAAB6XicbZDNTgIxFIXv4B/iH+rSTSMxcUVmDFGXRDcuMZGfBCakU+5AQ6edtB0TQngIXRl15+v4Ar6NBWeh4Fl9vec0uedGqeDG+v6XV1hb39jcKm6Xdnb39g/Kh0ctozLNsMmUULoTUYOCS2xabgV2Uo00iQS2o/Ht3G8/ojZcyQc7STFM6FDymDNq3ajTw9RwoWS/XPGr/kJkFYIcKpCr0S9/9gaKZQlKywQ1phv4qQ2nVFvOBM5KvcxgStmYDrHrUNIETThd7DsjZ7HSxI6QLN6/s1OaGDNJIpdJqB2ZZW8+/M/rZja+DqdcpplFyVzEeXEmiFVkXpsMuEZmxcQBZZq7LQkbUU2ZdccpufrBctlVaF1Ug8tq7b5Wqd/khyjCCZzCOQRwBXW4gwY0gYGAZ3iDd2/sPXkv3utPtODlf47hj7yPbzNhjb4=</latexit> ✏   Figure 3: View of the thin porous media ⌦✏(left) and domain without perforations Q✏(right). We define e ⌦✏=!✏⇥(0,1),⌦=!⇥(0,1),Q ✏=!⇥(0,✏).(2.2) We observe that e ⌦✏=⌦\Sk02K✏Tk0,✏,and we define T✏=Sk02K✏Tk0,✏as the set of the solid cylinders contained in e ⌦✏. To finish, we introduce some notation that will be useful throughout the paper. The points x2R3 will be decomposed as x=(x0,x 3)withx0=(x1,x 2)2R2,x32R. We also use the notation x0to denote a generic vector of R2. Let C1 #(Z) be the space of infinitely di↵erentiable functions in R3that are Z0-periodic. By Lq #(Z) (resp. W1,q #(Z)), 1 <q<+1, we denote its completion in the norm Lq(Z)(resp. W1,q(Z)) and by Lq 0,#(Z) the space of functions in Lq #(Z) with zero mean value. We denote by W1,q 0,#(Zf) the subspace of W1,q(Z) composed of functions vanishing in T, with zero trace on Z0⇥{0,1}. For q= 2, we set H1(Z)=W1,2(Z) and H1 0,#(Zf)=W1,2 0,#(Zf). For a vectorial function '=('0,' 3) and a scalar function , we will denote Dx0[']=1 2(Dx0'+Dt x0') and @z3[']=1 2(@z3'+@t z3'), where @z3=(0,0,@ @z3)t. Finally, we denote by O✏a generic real sequence, which tends to zero with ✏and can change from line to line, and by Ca generic positive constant which also can change from line to line. Statement of the problem. We consider the following stationary Stokes system with non-linear viscosity following the Carreau law (1.2) in ⌦✏, with a zero boundary condition on the exterior boundary 9 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau 1. R✏ q'=',if'2W1,q 0(⌦✏)3(elements of W1,q 0(⌦✏)are extended by 0to Q✏). 2. divR✏ q'=0in ⌦✏,ifdiv '=0in Q✏. 3. There exists a positive constant C, independent of ✏, such that for every '2W1,q 0(Q✏)3, kR✏ q'kLq(⌦✏)3+✏kDR✏ q'kLq(⌦✏)3⇥3Ck'kLq(Q✏)3+✏kD'kLq(Q✏)3⇥3.(3.10) In the next result, using the restriction operator defined in Lemma 3.4, we extend the pressure gradient rp✏by duality in W1,q0(Q✏)3. Then, by means of the dilatation, we extend ˜p✏to ⌦and derive the corresponding estimates. Lemma 3.5. Let ˜p✏the pressure solution of (2.6). (i)(Pseudoplastic and Newtonian fluid.) If 1<r2, there exist an extension ˜ P✏2L2 0(⌦)of ˜p✏and a positive constant C, independent of ✏, such that k˜ P✏kL2(⌦)C, kr✏˜ P✏kH1(⌦)3C. (3.11) (ii)(Dilatant fluid.) If r>2, there exist an extension ˜ P✏2Lr0 0(⌦)of ˜p✏and a positive constant C, independent of ✏, such that k˜ P✏kLr0(⌦)C, kr✏˜ P✏kW1,r0(⌦)3C, (3.12) where r0is the conjugate exponent of r. Proof. We divide the proof in three steps. First, we extend the pressure in all cases (pseudoplastic, Newtonian and dilatant). Then, we obtain the estimates for pseudoplastic and Newtonian fluids, before deriving the estimate for dilatant fluids. Step 1. Extension of the pressure. Let q= max{2,r}and q0be the conjugate exponent of q. Using the restriction operator R✏ qgiven in Lemma 3.4, we define the linear functional F✏on W1,q 0(Q✏)3by F✏(')=hrp✏,R ✏ q'iW1,q0(⌦✏)3,W1,q 0(⌦✏)3,for any '2W1,q 0(Q✏)3.(3.13) Using the variational formulation of problem (2.3), the right hand side of (3.13) can be rephrased as follows: F✏(')= ✏(⌘0⌘1)Z⌦✏ (1 + |D[u✏]|2)r 21D[u✏]:DR✏ q'dx ✏⌘1Z⌦✏ D[u✏]:DR✏ q'dx+Z⌦✏ f0·(R✏ q')0dx . (3.14) Using Lemma 3.2 for fixed ✏, we see that F✏2W1,q0(Q✏)3. Moreover, div '= 0 implies F✏(')=0, hence De Rham theorem gives the existence of P✏in Lq0 0(Q✏) such that F✏=rP✏. 16 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau Now, we define ˜ P✏2Lq0 0(⌦) by ˜ P✏(x0,z 3)=P✏(x0,✏z 3), and take ˜'2W1,q 0(⌦)3and the corresponding function '2W1,q 0(Q✏)3satisfying ˜'(x0,z 3)='(x0,✏z 3). Using the change of variables (2.5) and the identification (3.14) of F✏, we see that hr✏˜ P✏,˜'iW1,q0(⌦)3,W1,q 0(⌦)3=Z⌦ ˜ P✏div✏˜'dx 0dz3 =✏1ZQ✏ P✏div 'dx =✏1hrP✏,'iW1,q0(Q✏)3,W 1,q 0(Q✏)3 =✏1F✏(') =✏1✓✏(⌘0⌘1)Z⌦✏ (1 + |D[u✏]|2)r 21D[u✏]:DR✏ q'dx ✏⌘1Z⌦✏ D[u✏]:DR✏ q'dx+Z⌦✏ f0·(R✏ q')0dx◆ =✏(⌘0⌘1)Ze ⌦✏ (1 + |D✏[˜u✏]|2)r 21D✏[˜u✏]:D✏˜ R✏ q˜'dx 0dz3 ✏⌘1Ze ⌦✏ D✏[˜u✏]:D✏˜ R✏ q˜'dx 0dz3+Ze ⌦✏ f0(x0)·(˜ R✏ q˜')0dx0dz3,(3.15) where ˜ R✏ qis defined by ( ˜ R✏ q˜')(x0,z 3)=(R✏ q')(x0,✏z 3). Step 2.Estimates of the extended pressure for pseudoplastic fluids and Newtonian fluids. Applying the dilatation in (3.10) for q= 2, we have that ˜ R✏ 2˜'satisfies the following estimate k˜ R✏ 2˜'kL2( e ⌦✏)3+✏kD✏˜ R✏ 2˜'kL2( e ⌦✏)3⇥3Ck˜'kL2(⌦)3+✏kD✏˜'kL2(⌦)3⇥3,(3.16) and since ✏⌧1, we deduce k˜ R✏ 2˜'kL2( e ⌦✏)3Ck˜'kH1 0(⌦)3,kD✏˜ R✏ 2˜'kL2( e ⌦✏)3⇥3C ✏k˜'kH1 0(⌦)3.(3.17) Taking into account that 1 <r2, we notice that (1 + |D✏[˜u✏]|2)r 211, so that Cauchy-Schwarz inequality yields Ze ⌦✏ (1 + |D✏[˜u✏]|2)r 21D✏[˜u✏]:D✏˜ R✏ 2˜'dx 0dz3Ze ⌦✏|D✏[˜u✏]||D✏˜ R✏ 2˜'|dx0dz3 kD✏[˜u✏]kL2( e ⌦✏)3⇥3kD✏˜ R✏ 2˜'kL2( e ⌦✏)3⇥3. Using last estimate in (3.3) and last estimate on the dilated restricted operator given in (3.17), we obtain ✏(⌘0⌘1)Ze ⌦✏ (1 + |D✏[˜u✏]|2)r 21D✏[˜u✏]:D✏˜ R✏ 2˜'dx 0dz3Ck˜'kH1 0(⌦)3.(3.18) 17 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau and ✏⌘1Ze ⌦✏ D✏[˜u✏]:D✏˜ R✏ 2˜'dx 0dz3C✏kD✏[˜u✏]kL2( e ⌦✏)3⇥3kD✏˜ R✏ 2˜'kL2( e ⌦✏)3⇥3Ck˜'kH1 0(⌦)3.(3.19) Since f0=f0(x0)isinL1(!), we also get by the first estimate in (3.17) that Ze ⌦✏ f0·(˜ R✏ 2˜')0dx0dz3Ck˜ R✏ 2˜vkL2( e ⌦✏)3Ck˜'kH1 0(⌦)3.(3.20) Coming back to the expression (3.15) of hr✏˜ P✏,˜'i, we deduce from (3.18)–(3.20) the second estimate in (3.11). Finally, by Neˇcas inequality, there exists a representative ˜ P✏2L2 0(⌦) such that k˜ P✏kL2(⌦)Ckr˜ P✏kH1(⌦)3Ckr✏˜ P✏kH1(⌦)3, which implies the first estimate in (3.11). Step 3.Estimates of the extended pressure for dilatant fluids. Applying the dilatation in (3.10) for q=r, we get that ˜ R✏ r˜'satisfies the following estimate: k˜ R✏ r˜'kLr( e ⌦✏)3+✏kD✏˜ R✏ r˜'kLr( e ⌦✏)3⇥3Ck˜'kLr(⌦)3+✏kD✏˜'kLr(⌦)3⇥3,(3.21) and since ✏⌧1, this yields k˜ R✏ r˜'kLr( e ⌦✏)3Ck˜'kW1,r 0(⌦)3,kD✏˜ R✏ r˜'kLr( e ⌦✏)3⇥3C ✏k˜'kW1,r 0(⌦)3.(3.22) Since r>2, the embedding Lr(e ⌦✏),!L2(e ⌦✏) is continuous, so we can deduce from H¨older inequality and from the inequality (1 + X)↵C(1 + X↵) that holds true for X0,↵>0: Ze ⌦✏(1 + |D✏[˜u✏]|2)r 21D✏[˜u✏]:D✏˜ R✏ r˜'dx0dz3 C✓Ze ⌦✏|D✏[˜u✏]||D✏˜ R✏ r˜'|dx0dz3+Ze ⌦✏|D✏[˜u✏]|r1|D✏˜ R✏ r˜'|dx0dz3◆ C⇣kD✏[˜u✏]kL2( e ⌦✏)3⇥3kD✏˜ R✏ r˜'kL2( e ⌦✏)3⇥3+kD✏[˜u✏]kr1 Lr( e ⌦✏)3⇥3kD✏˜ R✏ r˜'kLr( e ⌦✏)3⇥3⌘ C⇣kD✏[˜u✏]kL2( e ⌦✏)3⇥3+kD✏[˜u✏]kr1 Lr( e ⌦✏)3⇥3⌘kD✏˜ R✏ r˜'kLr( e ⌦✏)3⇥3. Observe that if <1, taking into account that 2 r(1) >1 (r1), using the last estimates in (3.3) and (3.4), and the last estimate of the dilated restricted operator given in (3.22), we obtain ✏(⌘0⌘1)Ze ⌦✏ (1 + |D✏[˜u✏]|2)r 21D✏[˜u✏]:D✏˜ R✏ r˜'dx 0dz3 C✏(⌘0⌘1)⇣✏1+✏2 r(1)(r1)⌘✏1k˜'kW1,r 0( e ⌦✏)3 C✏(⌘0⌘1)⇣✏1+✏1 r1(r1)⌘✏1k˜'kW1,r 0( e ⌦✏)3 Ck˜'kW1,r 0( e ⌦✏)3. 18 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau If 1, by the last estimate in (3.5) and (3.6), kD✏[˜u✏]kLr( e ⌦✏)3⇥3kC✏1 r1, so a similar argument proves that the estimate ✏(⌘0⌘1)Ze ⌦✏ (1 + |D✏[˜u✏]|2)r 21D✏[˜u✏]:D✏˜ R✏ r˜'dx 0dz3Ck˜'kW1,r 0(⌦)3(3.23) remains valid for any 2R. Moreover, from Cauchy-Schwarz inequality, last estimate in (3.3), the continuous embedding Lr(e ⌦✏),! L2(e ⌦✏), the assumption on f0given in (2.4) and estimates (3.22), we deduce the upper bounds ✏⌘1Ze ⌦✏ D✏[˜u✏]:D✏˜ R✏ r˜'dx 0dz3C✏kD✏[˜u✏]kL2( e ⌦✏)3⇥3kD✏˜ R✏ r˜'kLr( e ⌦✏)3⇥3Ck˜'kW1,r 0(⌦)3, (3.24) Ze ⌦✏ f0·(˜ R✏ r˜')0dx0dz3Ck˜ R✏ r˜'kLr( e ⌦✏)3Ck˜'kW1,r 0(⌦)3.(3.25) Taking into account the above estimates (3.23)–(3.25), the relation (3.15) (with q=r) yields hr✏˜ P✏,˜'iW1,r0(⌦)3,W1,r 0(⌦)3Ck˜'kW1,r 0(⌦)3. This implies the second estimate in (3.12) and, by Ne˘cas inequality, the existence of a representative ˜ P✏2Lr0 0(⌦) such that k˜ P✏kLr0(⌦)Ckr˜ P✏kW1,r0(⌦)3Ckr✏˜ P✏kW1,r0(⌦)3, which provides the first estimate in (3.12). 3.2 Adaptation of the unfolding method The change of variables (2.5) does not provide the information we need about the behaviour of ˜u✏in the microstructure associated to e ⌦✏. To solve this difficulty, we use an adaptation introduced in [11] of the unfolding method from [27]. Let us recall that this adaptation of the unfolding method divides the domain e ⌦✏in cubes of lateral length ✏and vertical length 1. Thus, given ( ˜'✏,˜ ✏)2Lq(⌦)3⇥Lq0(⌦), 1 <q<+1and 1/q +1/q0= 1, we define ( ˆ'✏,ˆ ✏)2Lq(!⇥Z)3⇥Lq0(!⇥Z)by ˆ'✏(x0,z)= ˜'✏✓✏ ✓x0 ✏◆+✏z0,z 3◆,ˆ ✏(x0,z)= ˜ ✏✓✏ ✓x0 ✏◆+✏z0,z 3◆,a.e. (x0,z)2!⇥Z, (3.26) assuming ˜'✏and ˜ ✏are extended by zero outside !, where the function :R2!Z2is defined by (x0)=k0() x02Z0 k0,1,8k02Z2. Remark 3.6. We make the following comments: 19 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau - The function is well defined up to a set of zero measure in R2(the set [k02Z2@Z0 k0,1). Moreover, for every ✏>0, we have ✓x0 ✏◆=k0() x02Z0 k0,✏. -For k02K ✏, the restriction of (ˆu✏,ˆ P✏)to Z0 k0,✏⇥Zdoes not depend on x0, whereas as a function of zit is obtained from (˜u✏,˜ P✏)by using the change of variables z0=x0✏k0 ✏,(3.27) which transforms Zk0,✏into Z. Following the proof of [11, Lemma 4.9], the following estimates relate ( ˆ'✏,ˆ ✏)to(˜'✏,˜ ✏). Lemma 3.7. We have the following estimates: (i)For every ˜'✏2Lq(e ⌦✏)3,1q<+1, kˆ'✏kLq(!⇥Z)3k˜'✏kLq(⌦)3, where ˆ'✏is given by (3.26)1. Similarly, for every ˜ 2Lq0(e ⌦✏), the function ˆ ✏, given by (3.26)2 satisfies kˆ ✏kLq(!⇥Z)k˜ ✏kLq(⌦). (ii)For every ˜'2W1,q(e ⌦✏)3,1q<+1, the function ˆ'✏given by (3.26)1belongs to Lq(!;W1,q(Z)3), and kDz0ˆ'✏kLq(!⇥Z)3⇥2✏kDx0˜'✏kLq(⌦)3⇥2,k@z3ˆ'✏kLq(!⇥Z)3k@z3˜'✏kLq(⌦)3, kDz0[ˆ'✏]kLq(!⇥Z)3⇥2✏kDx0[˜'✏]kLq(⌦)3⇥2,k@z3[ˆ'✏]kLq(!⇥Z)3k@z3[˜'✏]kLq(⌦)3. Definition 3.8 (Unfolded velocity and pressure).Let us define the unfolded velocity and pressure (ˆu✏,ˆ P✏)from (˜u✏,˜ P✏)depending on the type of fluid: – (Pseudoplastic fluids and Newtonian fluids.) From (˜u✏,˜ P✏)2H1 0(⌦)3⇥L2 0(⌦), we define (ˆu✏,ˆ P✏) by (3.26) with ˜'✏=˜u✏,˜ ✏=˜ P✏and q=2. – (Dilatant fluids.) From (˜u✏,˜ P✏)2W1,r 0(⌦)3⇥Lr0 0(⌦), we define (ˆu✏,ˆ P✏)by (3.26) with ˜'✏=˜u✏, ˜ ✏=˜ P✏and q=r. Now, combining estimates on the extended velocity (3.3)-(3.6) and pressure (3.11)-(3.12) with Lemma 3.7, we deduce the following estimates on (ˆu✏,ˆ P✏). Lemma 3.9. The unfolded velocity/pressure pair (ˆu✏,ˆ P✏)satisfies the following estimates, depending on the type of fluid. (i)(Pseudoplastic fluids and Newtonian fluids.) Consider 1<r2. There exists a constant C>0 independent of ✏, such that, for every value of , kˆu✏kL2(!⇥Z)3C✏2,kDzˆu✏kL2(!⇥Z)3⇥3C✏2,kDz[ˆu✏]kL2(!⇥Z)3⇥3C✏2,(3.28) kˆ P✏kL2(!⇥Z)C. (3.29) 20 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau (ii)(Dilatant fluids.) Consider r>2. There exists a constant C>0independent of ✏, such that estimates (3.28) hold true, and also, depending on the value of , we have: –If <1, kˆu✏kLr(!⇥Z)3C✏2 r(1)+1,kDzˆu✏kLr(!⇥Z)3⇥3C✏2 r(1)+1, kDz[ˆu✏]kLr(!⇥Z)3⇥3C✏2 r(1)+1. (3.30) –If >1, kˆu✏kLr(!⇥Z)3C✏1 r1+1,kDzˆu✏kLr(!⇥Z)3⇥3C✏1 r1+1, kDz[ˆu✏]kLr(!⇥Z)3⇥3C✏1 r1+1. (3.31) –If =1, kˆu✏kLr(!⇥Z)3C✏,kDzˆu✏kLr(!⇥Z)3⇥3C✏, kDz[ˆu✏]kLr(!⇥Z)3⇥3C✏. (3.32) Moreover, the pressure satisfies kˆ P✏kLr0(!⇥Z)C, (3.33) where r0is the conjugate exponent of r. 3.3 Compactness results. In this section, we analyze the asymptotic behaviour of extended functions (˜u✏,˜ P✏) and the corresponding unfolded functions (ˆu✏,ˆ P✏), when ✏tends to zero. Lemma 3.10. The velocities ˜u✏,ˆu✏satisfy the following convergence results. (i)(Pseudoplastic fluids and Newtonian fluids.) Consider 1<r2, then there exist –˜u2H1(0,1; L2(!)3)where ˜u=0on !⇥{0,1}and ˜u3⌘0, such that, up to a subsequence, ✏2˜u✏*(˜u0,0) weakly in H1(0,1; L2(!)3),(3.34) –ˆu2L2(!;H1 0,#(Z)3),withˆu=0on !⇥Z0⇥{0,1}, such that, up to a subsequence, ✏2ˆu✏*ˆuweakly in L2(!;H1(Z)3).(3.35) In addition, the following relation holds between ˜uand ˆu: ˜u(x0,z 3)=ZZ0 ˆu(x0,z)dz0with ZZ0 ˆu3(x0,z)dz0=0,(3.36) and so, Z1 0 ˜u(x0,z 3)dz3=ZZ ˆu(x0,z)dz with ZZ ˆu3(x0,z)dz =0.(3.37) 21 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau (ii)(Dilatant fluids.) Consider r>2, then •if <1, there exist –˜u2H1(0,1; L2(!)3)where ˜u=0on !⇥{0,1}and ˜u3⌘0, such that, up to a subsequence, ✏2˜u✏*˜u=(˜u0,0) weakly in H1(0,1; L2(!)3),(3.38) –ˆu2L2(!;H1 0,#(Z)3),withˆu=0on !⇥Z0⇥{0,1}, such that, up to a subsequence, ✏2ˆu✏*ˆuweakly in L2(!;H1(Z)3),(3.39) where the relations between ˜uand ˆugiven by (3.36) and (3.37) hold; •if 1, there exist –˜u2W1,r(0,1; Lr(!)3)where ˜u=0on !⇥{0,1}and ˜u3⌘0, such that, up to a subsequence, in the case >1, ✏r r1˜u✏*˜u=(˜u0,0) weakly in W1,r(0,1; Lr(!)3),(3.40) and in the case =1, ✏1˜u✏*(˜u0,0) weakly in W1,r(0,1; Lr(!)3),(3.41) –ˆu2Lr(!;W1,r 0,#(Z)3),withˆu=0on !⇥Z0⇥{0,1}, such that, up to a subsequence, in the case >1, ✏r r1ˆu✏*ˆuweakly in Lr(!;W1,r(Z)3),(3.42) and in the case =1, ✏1ˆu✏*ˆuweakly in Lr(!;W1,r(Z)3),(3.43) where the relation between ˜uand ˆugiven by (3.36) and (3.37) hold. Moreover, ˜uand ˆusatisfy the following divergence conditions for any r>1: divx0✓Z1 0 ˜u0(x0,z 3)dz3◆=0 in !,✓Z1 0 ˜u0(x0,z 3)dz3◆·n=0 in @!,(3.44) divzˆu(x0,z)=0 in !⇥Zf,divx0 ZZf ˆu0(x0,z)dz!=0 in !, ZZf ˆu0(x0,z)dz!·n=0 on @!. (3.45) Proof. The proof is based on compactness results given in [11]: (i) Arguing as in [11, Lemma 5.2.-(i)], we obtain convergence (3.34) and divergence condition (3.44). Moreover, proceeding similarly as in [11, Lemma 5.4.-(i)] we deduce convergence (3.35), properties (3.36) and (3.37), and divergence conditions (3.45). (ii) The proof also follows the lines of (i), just taking into account the estimates of ˜u✏and ˆu✏in each case. Nevertheless, for r>2, the choice of the relevant estimates depends on the value of . 22 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau - In case <1, by Lemma 3.2-(ii), the velocity satisfies two types of estimates: estimates (3.3) in L2(e ⌦✏) and (3.4) in Lr(e ⌦✏). Noticing that 2 >2 r(1), estimates (3.3) are in fact the optimal ones, so we proceed as in (i) to finish the proof. - In case >1, if we used the L2-estimates given in Lemmas 3.2-(ii) and 3.9-(ii), we would obtain convergences (3.38) and (3.39), respectively. However, we would not be able to pass to the limit in the formulation (3.56) because the term (1 + ✏2(1)|Dy[']|2)r 21diverges. For that reason, we apply the Lr-estimates of the velocities ˜u✏and ˆu✏given in Lemmas 3.2-(ii) and 3.9-(ii) respectively. We argue as in (i) to conclude. - In case = 1, we proceed as in case >1 by considering the Lr-estimates of the velocities ˜u✏and ˆu✏given in Lemmas 3.2-(ii) and 3.9-(ii)respectively. Lemma 3.11. The extended pressure ˜ P✏and the corresponding unfolding function ˆ P✏satisfy the following convergence results. (i)(Pseudoplastic fluids and Newtonian fluids.) Consider 1<r2. There exists ˜ P2L2 0(!)such that ˜ P✏!˜ Pstrongly in L2(⌦),(3.46) ˆ P✏!˜ Pstrongly in L2(!⇥Z).(3.47) (ii)(Dilatant fluids.) Consider r>2. There exists ˜ P2Lr0 0(!)such that ˜ P✏!˜ Pstrongly in Lr0(⌦),(3.48) ˆ P✏!˜ Pstrongly in Lr0(!⇥Z),(3.49) where r0is the conjugate exponent of r. Proof. We give some remarks concerning case (i), case (ii) being similar. The first estimate in (3.11) implies, up to a subsequence, the existence of ˜ P2L2 0(⌦) such that ˜ P✏*˜ Pweakly in L2(⌦).(3.50) Also, from the second estimate in (3.11), since @z3˜ P✏/✏also converges weakly in H1(⌦), we obtain @z3˜ P= 0 and so ˜ Pis independent of z3. Moreover, arguing in [23, Lemma 4.4], we deduce that the convergence (3.50) of the pressure ˜ P✏is in fact strong. Since ˜ P✏has null mean value in ⌦, then ˜ Phas null mean value in !, which concludes the proof of (3.46). Finally, the strong convergence of ˆ P✏given in (3.47) follows from [29, Proposition 1.9-(ii)] and the strong convergence of ˜ P✏given in (3.46). 3.4 Proof of Theorems 2.1, 2.3 and 2.5. Using monotonicity arguments together with Minty’s lemma (see for instance [30, 22]), we derive a variational inequality that will be useful in the proofs. 23 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau We choose a test function v(x0,z)2D(!;C1 #(Z)3)withv(x0,z)=0in!⇥Tand on !⇥Z0⇥{0,1}. Multiplying (2.6) by v(x0,x 0/✏, z3), integrating by parts, and taking into account the extension of ˜u✏ and ˜ P✏,wehave ✏(⌘0⌘1)Z⌦ (1 + |D✏[˜u✏]|2)r 21D✏[˜u✏]:Dx0[v]+✏1Dz[v]dx0dz3 +✏⌘1Z⌦ D✏[˜u✏]:Dx0[v]+✏1Dz[v]dx0dz3 Z⌦ ˜ P✏divx0v0+✏1divzvdx0dz3=Z⌦ f0·v0dx0dz3+O✏, where O✏is a generic real sequence depending on ✏that can change from line to line. By the change of variables given in Remark 3.6, we obtain ✏1(⌘0⌘1)Z!⇥Z (1 + |✏1Dz[ˆu✏]|2)r 21✏1Dz[ˆu✏]:Dz[v]dx0dz +✏1⌘1Z!⇥Z ✏1Dz[ˆu✏]:Dz[v]dx0dz Z!⇥Z ˆ P✏divx0v0dx0dz ✏1Z!⇥Z ˆ P✏divzvdx 0dz =Z!⇥Z f0·v0dx0dz +O✏, (3.51) with |O✏|C✏for every 2R. Now, let us define the functional Jrby Jr(v)=⌘0⌘1 rZ!⇥Z (1 + |Dz[v]|2)r 2dx0dz +⌘1 2Z!⇥Z|Dz[v]|2dx0dz. Observe that Jris convex and Gateaux di↵erentiable on Lq(!;W1,q #(Z)3)withq= max{2,r}, (see [19, Proposition 2.1 and Section 3] for more details) and Ar=J0 ris given by (Ar(w),v)=(⌘0⌘1)Z!⇥Z (1 + |Dz[w]|2)r 21Dz[w]:Dz[v]dx0dz +⌘1Z!⇥Z Dz[w]:Dz[v]dx0dz. Applying [40, Proposition 1.1., p.158], Aris monotone, i.e. (Ar(w)Ar(v),wv)0,8w,v 2Lq(!;W1,q #(Z)3).(3.52) On the other hand, for all '2D(!;C1 #(Z)3)with'=0in!⇥Tand on !⇥Z0⇥{0,1}, satisfying the divergence conditions divx0RZ'0dz =0in!,RZf'(x0,z)dz ·n= 0 on @!, and divz'=0in!⇥Z, we choose v✏defined by v✏='✏1ˆu✏, as a test function in (3.51). Taking into account that div✏˜u✏= 0, we get that ✏1divzˆu✏= 0, and then we obtain ✏1(Ar(✏1ˆu✏),v ✏)Z!⇥Z ˆ P✏divx0v0 ✏dx0dz =Z!⇥Z f0·v0 ✏dx0dz +O✏, 24 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau which is equivalent to ✏1(Ar(')Ar(✏1ˆu✏),v ✏)✏1(Ar('),v ✏)+Z!⇥Z ˆ P✏divx0v0 ✏dx0dz =Z!⇥Z f0·v0 ✏dx0dz +O✏. Due to (3.52), we can deduce ✏1(Ar('),v ✏)Z!⇥Z ˆ P✏divx0v0 ✏dx0dz Z!⇥Z f0·v0 ✏dx0dz +O✏, i.e. ✏1(⌘0⌘1)Z!⇥Z (1 + |Dz[']|2)r 21Dz[']:Dz[v✏]dx0dz +✏1⌘1Z!⇥Z Dz[']:Dz[v✏]dx0dz Z!⇥Z ˆ P✏divx0v0 ✏dx0dz Z!⇥Z f0·v0 ✏dx0dz +O✏. (3.53) In the above inequality, in case 1 <r2, |O✏|C✏↵,with↵=1if1 and ↵=2if >1. In case r>2, |O✏|C✏↵,with↵=+2 r(1 )if1 and ↵=11 r1if >1. Proof of Theorem 2.1. We recall that 1 <r<2. The case = 1 is developed in [9], so we omit it and consider that 6= 1. The proof will be divided in two steps. In the first step, we obtain the homogenized behaviour given by a coupled system, with a constant macroviscosity, and in the second step we decouple it to obtain the macroscopic law. Step 1. Using Lemmas 3.10 and 3.11, in this step we will prove that the sequence (✏2ˆu✏,˜ P✏) converges to (ˆu, ˜ P)2L2(!;H1 #(Zf)3)⇥(L2 0(!)\H1(!)), characterized as the unique solutions of the following two-pressures Newtonian Stokes problem with the linear viscosity ⌘equal to ⌘0if <1 and ⌘1if >1: 8 > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > : ⌘divzDz[ˆu]+rzˆ⇡=f0rx0˜ Pin !⇥Zf, divzˆu=0 in !⇥Zf, divx0 ZZf ˆu0dz!=0 in!, ZZf ˆu0dz!·n= 0 on @!, ˆu=0 in!⇥T, ˆu=0 in!⇥Z0⇥{0,1}, ˆ⇡2L2(!;L2 0,#(Zf)). (3.54) Divergence conditions (3.54)2,3,4and condition (3.54)5,6follow from Lemma 3.10. To prove that (ˆu, ˜ P) satisfies the momentum equation given in (3.54), we follow the lines of the proof to obtain (3.53) but choose now v✏and 'such that v✏=✏1'✏1ˆu✏with '2D(!;C1 #(Z)3) satisfying '=0in!⇥T 25 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau with non-linear Carreau viscosity (1.2) 8 > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > : divy(⌘r(Dz[ˆu])Dz[ˆu]) + rzˆ⇡=f0rx0˜ Pin !⇥Zf, divzˆu=0 in !⇥Zf, divx0 ZZf ˆu0dz!=0 in!, ZZf ˆu0dz!·n= 0 on @!, ˆu=0 in!⇥T, ˆu= 0 on !⇥Z0⇥{0,1}, ˆ⇡2Lr0(!;Lr0 0,#(Zf)). Proceeding as in Step 3, we deduce the non-linear 2D Darcy’s law of Carreau type (2.10), where the permeability operator U:R2!R2is defined by (2.11). Here, for ⇠02R2,(w⇠0,⇡ ⇠0)2W1,r 0,#(Zf)3⇥ Lr0 0,#(Zf) is the unique solution of the local Stokes system (2.12) with nonlinear viscosity given by the Carreau law (1.2). Proof of Theorem 2.5. We recall that in this case r= 2, the Carreau law (1.2) reduces to ⌘0.Thus,by linearity, the variational formulation (3.51) can be written as follows: ✏2⌘0Z!⇥Z Dz[ˆu✏]:Dz[']dx0dy Z!⇥Z ˆ P✏divx0'0dx0dz =Z!⇥Z f0·'0dx0dz +O✏,(3.67) for all '2D(!;C1 #(Z)3)with'=0in!⇥Tand on !⇥Z0⇥{0,1}, satisfying the divergence conditions divx0RZ'0dz =0in!,RZ'0dz ·n= 0 on @!, and divz'=0in!⇥Z. Passing to the limit in (3.67) using convergences (3.35) and (3.47), we take into account that the second term converges to Z!⇥Z ˜ Pdivx0'0dx0dz. Since ˜ Pdoes not depend on z, by the divergence condition divx0RZ'0dz = 0, we have Z!⇥Z ˜ Pdivx0'0dx0dz =Z! ˜ Pdivx0✓ZZ '0dz◆dx0=0. Therefore, the following relation holds true: ⌘0Z!⇥Z Dz[ˆu]:Dz[']dx0dz =Z!⇥Z f0·'0dx0dz, and by a density argument, remains valid for every '2Vwith Vgiven by (3.58). 32 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau Proceeding similarly as the end of Step 1 of the proof of Theorem 2.1, this variational formulation is equivalent to the system (3.54) with viscosity ⌘0, which admits a unique solution (ˆu, ˆ⇡,˜ P)2 L2(!;H1 #(Zf)3)⇥L2(!;L2 0,#(Zf)) ⇥(L2 0(!)\H1(!)). This establishes the convergence of the whole sequence (˜u✏,˜ P✏). Finally, reasoning as in Step 2 of the proof of Theorem 2.1, we get the linear e↵ective 2D Darcy’s law (2.14), which concludes the proof of Theorem 2.5. 4 Numerical simulations of the e↵ective models In this section, we perform a numerical study of the asymptotic behaviour of a flow of a Carreau fluid between two parallel plates, separated by a thin layer of porous medium, as described in Section 2. We assume that the flow is driven by a constant body force f=(f0,0) with f02R2,whichisa realistic assumption that is used in many applications such as enhanced oil recovery [21, Chapter 4]. For simplicity, we also assume that !is the unit square !=(1,1)2and impose periodic boundary conditions on @Q✏=@! ⇥(0,✏). System (2.3) is thus rewritten 8 > > > > > > > < > > > > > > > : ✏div (⌘r(D[u✏])D[u✏]) + rp✏=fin ⌦✏, div u✏=0 in⌦ ✏, u✏= 0 on @S✏, u✏(x1,1) = u✏(x1,1),x 12(1,1), u✏(1,x 2)=u✏(1,x 2),x 22(1,1), (4.1) As observed in [9, Section 4], Lemma 3.10 needs to be slightly modified in that case: conditions (3.44) and (3.45) are respectively replaced by divx0✓Z1 0 ˜u0(x0,z 3)dz3◆=0 in!,(4.2) divzˆu(x0,z)=0 in!⇥Zf,divx0 ZZf ˆu0(x0,z)dz!=0 in!.(4.3) As a result, by the periodicity hypothesis on the flow, the boundary condition ˜ V0·n0= 0 does not hold anymore on @!. Hence, in this particular configuration, we will discuss numerical simulations of Darcy’s laws of the form as (2.7), (2.10), (2.13) and (2.14), but without the aforementioned boundary condition on ˜ V0. Since f0is constant, one gets that ˜ P⌘0 and ˜ V0is also a constant vector in R2. Choice of rheological parameters. Let us spectify the range of parameters that we use in the numerical tests. In addition to the exponent 2R, the model (4.1) depends on four rheological parameters: ⌘0,⌘ 1,and r. Since in many applications (see for instance [21]), ⌘1is very small compared with ⌘0, we arbitrarily fix ⌘0= 1 and ⌘1= 103. As regards , we take 2{1,10,100}.In the case where the e↵ective model is nonlinear with respect to , the possibility of multiplying by a factor 10 from one simulation to another will give us access to a large panel of behaviours for the e↵ective models. Finally, we consider a pseudoplastic case r=1.7, the Newtonian case r= 2 and two dilatant cases r=2.3 and r=2.6. 33 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau Shape E1Shape E2 Shape E3Shape E4 Figure 4: Representation of the 2Dreference cells Z0used in the numerical simulations of the e↵ective models. The inclusion T0(in grey) is surrounded by Z0 f(in blue). From left to right and top to bottom: disk or radius 0.1, ellipses of semi-major axis 0.3 and semi-minor axis 0.1, oriented respectively along the xand ydirections, and disk of radius 0.3. These shapes are numbered E1,E 2,E 3and E4. Shapes of inclusions T.In order to illustrate the e↵ect of a change of volume of the inclusion T, and the e↵ect of anisotropy, we will consider four possible shapes for T0: two disks of respective radius 0.1 and 0.3, and the ellipse of semi-major axis 0.3 and semi-minor axis 0.1, parallel to the xor the y axis (see Fig. 4). These shapes will be numbered E1,E 2,E 3and E4in the rest of this section. Numerical resolution of the cell problems. The solution of each cell problem of the form (2.9) or (2.12) is computed using a mixed formulation, that we solve by a finite element method, using FreeFem++ software [37]. In the nonlinear cases (2.12) where the viscosity ⌘rfollows a Carreau law or a power law, we rely on a fixed point algorithm (see for instance [45, Section 2.8]). We consider the Taylor-Hood approximation for the velocity-pressure pair, namely P2elements for the velocity field and P1elements for the pressure. This choice is well known to be compatible with the BabuˇskaBrezzi condition [35]. Each three-dimensional mesh of a cell Zfis obtained by constrained Delaunay tetrahedralization, and contains approximately 8000 tetrahedra. 4.1 Permeability tensor A In the cases where the e↵ective system is described by a linear 2DDarcy law, the response of the fluid to a constant pressure gradient f0takes the form ˜ V0=1 ⌘Af0, where the constant viscosity ⌘is either 34 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau Inclusion Permeability tensor A E1✓0.0697955 3.17061 ⇥106 3.17061 ⇥1060.0697947 ◆ E2✓0.054708 2.3436 ⇥106 2.3436 ⇥1060.0210978 ◆ E3✓0.0211078 5.69438 ⇥107 5.69438 ⇥1070.0547038 ◆ E4✓0.0153292 3.7408 ⇥107 3.7408 ⇥1070.0153284 ◆ Figure 5: Permeability tensor Acomputed numerically in the case of an elliptic inclusion (E1to E4). equal to ⌘0or ⌘1, depending on the values of rand (see Table 1). Hence, the asymptotic behaviour of the fluid is encoded in the permeability tensor A. In order to highlight certain properties of A, we have summarized in Figure 5 the coefficients that we obtain numerically, for the di↵erent shape geometries E1to E4. We first notice that Ais perfectly symmetric, which comes from its very definition in the continuous setting. Indeed, testing against wj in system (2.9) satisfied by wi, or against wiin the same system but satisfied by wj, we obtain ZZf wj i(z)dz =ZZf Dzwi(z):Dzwj(z)dz =ZZf Dzwj(z):Dzwi(z)dz =ZZf wi j(z)dz , hence Ai,j =Aj,i. One can also observe that, for isotropic inclusions E1and E4, up to numerical errors, the tensor A is a diagonal matrix of the form aI. This means that, as expected, the filtration velocity ˜ V0is simply given by the product between f0and the positive constant afor such geometries. Also, the value of a appears to be a decreasing function of the area of the obstacle, which is quite intuitive as well. In the case of anisotropic geometries E2and E3,Ais still diagonal, but its diagonal coefficients are not equal. Since E3is obtained by applying a rotation of angle ⇡/2toE2, by symmetry, the associated matrix A is the transposed of the one associated with shape E2. However, for more general geometries of inclusions, the permeability tensor Ais no longer symmetric. For instance, we have given in Fig. 6 the coefficients of Acomputed when E2is rotated by an angle ✓2{⇡/16,⇡/8,⇡/4}. As expected, the diagonal coefficients A1,2,A2,1are not equal to zero in such configurations. 4.2 Carreau law (=1) In case = 1, the behaviour of the e↵ective model associated with a pseudoplastic fluid has been studied numerically in [9]. In this subsection, we complete these numerical results by considering dilatant fluids as well. In order to perform comparisons, we use the same parameters as in [9], namely ⌘0= 1, ⌘1= 103, and 2{1,10,100}. We consider dilatant fluids with r2{2.3,2.6}, a Newtonian fluid (r= 2) and a pseudoplastic fluid (r=1.7). We explore numerically the influence of the amplitude of f0and of its orientation, on the computed value of the filtration velocity ˜ V0. 35 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau Rotation angle ✓Permeability tensor A ⇡/16 ✓0.0534164 0.00341334 0.00341334 0.0225729 ◆ ⇡/8✓0.0498291 0.00653147 0.00653147 0.0266649 ◆ ⇡/4✓0.0385604 0.00963438 0.00963438 0.0385636 ◆ Figure 6: Permeability tensor Acomputed numerically in the case where the E2inclusion is rotated of an angle ✓with respect to the x1axis. 4.2.1 Influence of the amplitude of the pressure gradient f0 We impose an exterior force f0directed by e1,i.e. of the form f0=(f1,0), with f12[0,1]. In that case, the computed filtration velocity ˜ V0is also directed by e1and reads ˜ V0=(˜ V1,0). Fig. 7 represents ˜ V1as a function of f1, for the di↵erent obstacle shapes E1to E4and the choice of parameters r, specified above. We observe that, for any choice of rand ,˜ V1is an increasing function of f1. However, for r=1.7 (pseudoplastic case), ˜ V1appears as a convex function of f1, whereas for r>2 (dilatant case), it is a concave function of f1. For r= 2 (Newtonian case), the dependency on f1is linear, as expected. Also, the separation between the curves gets more pronounced as increases, which comes from the fact that is the coefficient in front of the nonlinear term |D(u)|r2in the definition of the viscosity following the Carreau law (1.2). For any values of , and any geometry of obstacle shape E1to E4, for a given pressure gradient f0, the amplitude of the filtration velocity ˜ V0diminishes as the exponent rincreases. This is consistent with the fact that, for high values of the shear rate, the viscosity of the dilatant fluid increases, whereas the behaviour of pseudoplastic fluids is the opposite. 4.2.2 Influence of the orientation of f0 In order to test the impact of a rototion of f0on the behaviour of the e↵ective system, we consider the anisotropic shape E2and a family of pressure gradients f0= (cos ✓,sin ✓), with the angle ✓2[0,⇡/2]. The results that we obtain are represented in Fig. 8. We notice that the orientation of V0, which is generally not parallel to the pressure gradient f0due to the anisotropy of the obstacle, does not appear to depend on the rheological parameters r, .In all the simulated configurations, the orientation of V0remains very close to what is observed in the Newtonian case r= 2. As regards the amplitude of V0, as observed in the previous paragraph, there is almost no observable e↵ect for = 1. This can be explained by the fact that, for small values of , and an imposed pressure gradient of fixed size |f0|= 1, the viscosity described by the Carreau law is close to the constant value ⌘r=⌘0corresponding to the Newtonian case r= 2. On the contrary, for = 100, the amplitude of V0is noticeably reduced as rincreases: for instance, for r=1.7, the maximal filtration velocity is about 0.085 while it reaches only about 0.035 for r=2.6. 36 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau 0.2 0.4 0.6 0.8 1 2 4 6 8·102 f1 V1 =1 r=1.7 r=2 r=2.3 r=2.6 0.2 0.4 0.6 0.8 1 2 4 6 8 ·102 f1 = 10 0.2 0.4 0.6 0.8 1 0 2·102 4·102 6·102 8·102 0.1 0.12 f1 = 100 0.2 0.4 0.6 0.8 1 2 4 6·102 f1 V1 r=1.7 r=2 r=2.3 r=2.6 0.2 0.4 0.6 0.8 1 2 4 6 ·102 f1 0.2 0.4 0.6 0.8 1 0 2 4 6 8 ·102 f1 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 ·102 f1 V1 r=1.7 r=2 r=2.3 r=2.6 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5·102 f1 0.2 0.4 0.6 0.8 1 1 2 3 ·102 f1 0.2 0.4 0.6 0.8 1 0.5 1 1.5 ·102 f1 V1 r=1.7 r=2 r=2.3 r=2.6 0.2 0.4 0.6 0.8 1 0.5 1 1.5 ·102 f1 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 ·102 f1 Figure 7: Case = 1 (Carreau law). Component V1of the mean filtration velocity V0plotted against f1,withf0=(f1,0), for r2{1.7,2,2.3,2.6}, in the case of elliptic inclusions E1(first line), E2(second line), E3(third line) and E4(fourth line). From left to right: =1,= 10,= 100. 37 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 =1,r=1.7 3 4 5 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 = 100,r=1.7 3 4 5 6 7 8 ·102 0 0.2 0.4 0.6 0.811.2 0 0.5 1 f1 f2 =1,r=2 3 4 5 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 = 100,r=2 3 4 5 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 =1,r=2.3 3 4 5 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 = 100,r=2.3 2 2.5 3 3.5 4 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 =1,r=2.6 3 4 5 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 = 100,r=2.6 1.5 2 2.5 3 ·102 Figure 8: Case = 1 (Carreau law). Representation of ˜ V0when f0is a unit vector of the form (f1,f 2) = (cos ✓,sin ✓)with✓2[0,⇡/2], in the case of an elliptic inclusion E2. Each vector ˜ V0is represented by a vector of length 0.2, localized at point (f1,f 2) and colored according to |˜ V0|.Theleft column corresponds to = 1 and the right one to = 100. Each line from top to bottom corresponds respectively to r=1.7, r= 2, r=2.3 and r=2.6. 38 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau 4.3 Power law (r>2,>1) In order to allow for comparisons, we perform similar simulations for the power law regime (r>2, >1) as we did for the Carreau regime (= 1), taking 2{1,10,100}and r2{2.3,2.6}, to separately test the influence of the amplitude of f0, and of the angle formed by f0and the x1axis. In each case, we provide the Newtonian behaviour corresponding to r= 2, as a reference model. 4.3.1 Influence of the amplitude of f0 We have plotted in Fig. 9 the horizontal component of ˜ V0as a function of f1,whenf0takes the form f0=(f1,0). Contrary to what we can observe in Fig. 7, for a given value of f1, the order between the computed value of ˜ V1for rin {2,2.3,2.6}depends on the choice of : the behaviour of the e↵ective system is no longer monotonous with respect to r. In particular, for = 100, the di↵erent curves intersect for a specific value of f1, which seems to be unique and depends on the shape of the obstacle. For instance, the f1component of this intersection point is between 0.3 and 0.4 for E1and close to 0.5 for E4. Moreover, when f1exceeds this value, we recover a behaviour that is very similar to what appeared in Fig. 7: for r2{2.3,2.6},V1is an increasing concave function of f1, and V1is smaller for r=2.6 than for r=2.3. We may interpret these features as follows. When |f0|is small and ⌘rfollows the power law ⌘r(Dz[w⇠0]) = |Dz[w⇠0]|r2, the deformation rate tensor Dz[wf0] associated to the solution wf0of system (2.12) (with ⇠0=f0) also has a small amplitude. Thus, as mentioned in the Introduction, the power law is not well-suited to capture the behaviour of a quasi-Newtonian fluid in such regime. On the opposite, for high values of parameter such as = 100 in our simulations, and f1large enough, the qualitative behaviour of the nonlinear 2DDarcy law associated with the Carreau law or with the power law become very similar, since the Carreau law behaves as a power law for large values of the deformation rate. 4.3.2 Influence of the orientation of f0 Finally, we have represented in Fig.10 the vector ˜ V0computed for di↵erent orientations of the imposed pressure gradient f0, similarly as in Fig.8. We can observe for = 100 and r2{2.3,2.6}a very similar behaviour of the e↵ective systems (2.10) and (2.13), which seems to confirm the above interpretation. The di↵erences between both systems appear for = 1 and concern only the amplitude of ˜ V0,which increases as rincreases in the case of the power law, while it was not a↵ected by variations of rin the case of Carreau law and for this particular value of . 39 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau 0.2 0.4 0.6 0.8 1 0 2·102 4·102 6·102 8·102 0.1 f1 V1 =1 r=2 r=2.3 r=2.6 0.2 0.4 0.6 0.8 1 2 4 6 ·102 f1 = 10 0.2 0.4 0.6 0.8 1 2 4 6 ·102 f1 = 100 0.2 0.4 0.6 0.8 1 0 2·102 4·102 6·102 8·102 0.1 f1 V1 r=2 r=2.3 r=2.6 0.2 0.4 0.6 0.8 1 2 4 6 ·102 f1 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6·102 f1 0.2 0.4 0.6 0.8 1 0 1 2 3 4·102 f1 V1 r=2 r=2.3 r=2.6 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 2.5 ·102 f1 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 ·102 f1 0.2 0.4 0.6 0.8 1 0 1 2 3·102 f1 V1 =1 r=2 r=2.3 r=2.6 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2·102 f1 = 10 r=2 r=2.3 r=2.6 0.2 0.4 0.6 0.8 1 0.5 1 1.5 ·102 f1 = 100 r=2 r=2.3 r=2.6 Figure 9: Case >1, r>1 (power law). Component ˜ V1of the mean filtration velocity ˜ V0plotted against f1,withf0=(f1,0), for r2{2,2.3,2.6}and elliptic inclusions E1(first line), E2(second line), E3(third line) and E4(fourth line). From left to right: =1,= 10,= 100. 40 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau 0 0.2 0.4 0.6 0.811.2 0 0.5 1 f1 f2 =1,r=2 3 4 5 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 = 100,r=2 3 4 5 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 =1,r=2.3 3 4 5 6 7 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 = 100,r=2.3 2 2.5 3 3.5 4 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 =1,r=2.6 4 5 6 7 8 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 = 100,r=2.6 2 2.5 3 3.5 ·102 Figure 10: Case >1, r>2 (power law). Representation of ˜ V0when f0=(f1,f 2) takes the form (f1,f 2) = (cos ✓,sin ✓)with✓2[0,⇡/2], in the case of the obstacle shape E2. As in Fig. 8, each vector ˜ V0is represented by a vector of length 0.2, localized at point (f1,f 2) and colored according to |˜ V0|. The left column corresponds to = 1, the right one to = 100, and each line from top to bottom corresponds respectively to r= 2, r=2.3 and r=2.6. 41