Carreau law for non-newtonian fluid flow through a thin porous media

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ Esta es la versión aceptada del artículo publicado en: This is a accepted manuscript of a paper published in: The Quarterly Journal of Mechanics and Applied Mathematics (2022): 21/03/2022 DOI: https://doi.org/10.1093/qjmam/hbac004 Copyright: El acceso a la versión publicada del artículo puede requerir la suscripción de la revista. Access to the published version may require subscription. “This is a pre-copyedited, author-produced version of an article accepted for publication in [insert journal title] following peer review. The version of record Carreau law for non-newtonian fluid flow through a thin porous media, published in The Quarterly Journal of Mechanics and Applied Mathematics, vol. 75, n. 1, pp. 1-27 is available online at: https://academic.oup.com/qjmam/article/75/1/1/6551651 https://doi.org/10.1093/qjmam/hbac004.” Carreau law for non-Newtonian fluid flow through a thin porous media Mar´ıa ANGUIANO1, Matthieu BONNIVARD2, Francisco J. SU´ AREZ-GRAU3 Abstract We consider the flow of generalized Newtonian fluid through a thin porous media. The media under consideration is a bounded perforated three dimensional domain confined between two parallel plates, where the distance between the plates is described by a small parameter ". The perforation consists in an array of solid cylinders, which connect the plates in perpendicular direction, with diameter of size "and distributed periodically with period ". The flow is described by the three dimensional incompressible stationary Stokes system with a non-linear viscosity following the Carreau law. We study the limit when the thickness tends to zero and prove that the averaged velocity satisfies a non-linear two dimensional homogenized law of Carreau type. We illustrate our homogenization result by numerical simulations showing the influence of the Carreau law on the behaviour of the limit system, in the case where the flow is driven by a constant pressure gradient and for di↵erent geometries of perforations. AMS classification numbers: 76A05, 76M50, 76A20, 35B2. Keywords: Homogenization, non-Newtonian fluid, Carreau law, thin porous media. 1 Introduction Experimental observations show that the viscosity is constant for some fluids as water or honey, which are called Newtonian fluids. These fluids can be described in a very accurate way by the Navier-Stokes equations. However, many others fluids behave di↵erently, the viscosity of these fluids is no more constant, such as pastes or polymer solutions. These fluids can not be described by the Navier-Stokes equations and they are called non-Newtonian or complex fluids. The simplest idea to describe non-Newtonian fluids is to plot the viscosity measurements versus the imposed shear rate and then, to fit the obtained curve with a simple template viscosity function, adjusting some few parameters. This is the main idea of generalized Newtonian fluids models (also called quasi- Newtonian fluids models), which could be viewed as a first step inside the world of non-Newtonian fluids models (see Saramito [34, Chapter 2] for more details). In this article we consider the incompressible viscous flow of the generalized Newtonian fluid through a thin porous media, which consists in a domain of small height "and perforated by periodically distributed solid cylinders with diameter of size ". The viscosity of the fluid follows the Carreau law. This law is commonly used for fluid studied in Chemical Industry and Rheology, for instance in injection moulding of melted polymers, flow of oils, muds, etc. The incompressible generalized Newtonian fluids are characterized by the viscosity depending on the principal invariants of the symmetric stretching tensor D[u]. If uis the velocity, pthe pressure and Du the gradient velocity tensor, D[u]=(Du +Dtu)/2 denotes the symmetric stretching tensor and the stress tensor given by =pI +2⌘rD[u]. The viscosity ⌘ris constant for a Newtonian fluid but dependent on the shear rate, i.e. ⌘r=⌘r(D[u]), for viscous non-Newtonian fluids. The deviatoric stress tensor ⌧, i.e. the part of the total stress 1Departamento de An´alisis Matem´atico. Facultad de Matem´aticas. Universidad de Sevilla, 41012 Sevilla (Spain) [email protected] 2Sorbonne Paris Cit´e, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS. Universit´e Paris Diderot, 75205 Paris (France) bonniv[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 tensor that is zero at equilibrium, is then a nonlinear function of the shear rate D[u], i.e. ⌧=⌘r(D[u])D[u] (see Barnes et al. [17], Bird et al. [18] and Mikeli´c [32] for more details). The most widely used laws in engineering practice are the power law or Ostwald-de Waele model (Ostwald, 1925; de Waele, 1923) and the Carreau law (Carreau, 1972). It is observed that the power law correctly describes the behavior of polymers at high shear rates. It o↵ers the advantage of allowing 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 r2(1,2) (see Agassant et al. [2], p. 49), whereas for real fluids it tends to some constant value ⌘0called the zero-shear-rate viscosity. Thus, as a generalization we consider the Carreau law, which is an alternative generalized Newtonian model that enables the description of the plateaus in viscosity that are expected when the shear rate is very small or very large. The empiricism for the viscosity ⌘rused in the Carreau law is defined by ⌘r(D[u]) = (⌘0⌘1)(1 + |D[u]|2)r 21+⌘1,1<r<2,⌘ 0>⌘ 1>0,>0,(1) where ⌘1is the high-shear-rate limit of the viscosity, the parameter is a time constant and r1isadimen- sionless constant describing the slope in the power law region. The matrix norm |·|is defined by |⇠|2=Tr(⇠⇠t) with ⇠2R3. We recall that in this case the viscosity is decreasing with the shear rate and the fluid is said shear thinning or pseudoplastic (see Saramito [34, Chapter 2]). Homogenization applied to porous media is a mathematical method that allows to upscale the fundamental equations from continuum physics, being valid at the microscopic level. The homogenization theory of heterogeneous media studies the e↵ects of the micro-structure (i.e. of the pore structure) upon solutions of PDEs of the continuum mechanics, so allow to derive rigorously equations describing filtration of a generalized Newtonian fluid (see Mikeli´c [32] for more details). To find the homogenized law describing the generalized Newtonian fluid flow through a porous media ⌦"⇢R3, which is a domain with fixed height and periodically perforated by obstacles of size ", Bourgeat and Mikeli´c in [20] (see also Bourgeat et al. [21], G¨otz and Parhusip [27]) used the homogenization technique called the two-scale convergence. The domain without perforations is the bounded smooth domain ⌦⇢R3which is made of two parts, the fluid part ⌦"and the solid part ⌦\⌦". Moreover, assuming that the Reynolds number is proportional to "and the flow is sufficiently slow to neglect inertial e↵ects, then the following stationary Stokes system with a non-linear viscosity following the Carreau law (1), with 1 <r<2, was considered 8 > > > < > > > : "div (⌘r(D[u"])D[u"]) + rp"=fin ⌦", div u"=0 in⌦ ", u"= 0 on @⌦". (2) When "tends to zero, two types of averaged momentum equations were rigorously derived depending on the value of (i.e. the Reynolds number) connecting the velocity and the pressure gradient: -If 6= 1, the homogenized law is the classical 3D Darcy’s law for Newtonian fluids V(x)=K µ(f(x)rxp(x)) in ⌦,divxV(x)=0 in⌦,V(x)·n= 0 on @⌦, 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 µis equal to ⌘0if <1 and equal to ⌘1if >1. -If = 1, 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 @⌦, where U:R3!R3is a permeability function, 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. 2 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau On the other hand, in [19] Boughanim and Tapi´ero considered the polymer flow through a thin slab ⌦"= !⇥(0,")⇢R3,where"is the small thickness of the slab. Starting from the Stokes system (2), with external body force of the form f=(f0,0) such that f0=(f1,f 2), by using dimension reduction and homogenization techniques, they studied the limit when the thickness tends to zero. According to the value of , they proved the following: -If 6= 1, 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 @!, where V0=(V1,V 2), x0=(x1,x 2). As previously, the viscosity µis equal to ⌘0if <1 and equal to ⌘1if >1. -If = 1, 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. In this paper, we consider a thin porous media ⌦"=!"⇥(0,")⇢R3of small height "which is perforated by periodically distributed solid cylinders of diameter of size ". Here, the bottom of the domain without perforations !⇢R2is made of two parts, the fluid part !"and the solid part !\!". This model of thin porous media has been recently introduced by Fabricius et al. [25], where the flow of an incompressible viscous fluid described by the stationary Navier-Stokes equations has been studied by the multiscale expansion method, which is a formal tool to analyse homogenization problems. These results have been rigorously proved in [13] using an adaptation, introduced in [10], of the periodic unfolding method from Cioranescu et al. [23, 24]. This adaptation consists of a combination of the unfolding method with a rescaling in the height variable, in order to work with a domain of fixed height and to pass to the limit. In particular, the generalized Newtonian fluids obeying the power law in the thin porous media ⌦"have been studied rigorously in Anguiano and Su´arez-Grau [10] where we have obtained a 2D Darcy’s law when the domain thickness tends to zero (see also [15] for the extension to the case of a thin porous media with an array of cylinders with small diameter). Also, the Bingham plastic behavior in the thin porous media ⌦"has been studied in [8, 9]. For other studies concerning thin porous media, we refer to Anguiano [3, 4, 5, 6, 7], Anguiano and Su´arez-Grau [11, 12, 14], Jouybari and T. S. Lundstr¨om [29], Prat and Aga¨esse [33], Su´arez-Grau [35], Yeghiazarian et al. [36] and Zhengan and Hongxing [37]. However, as far as we know, in the previous literature there is no study for the homogenization of three-dimensional incompressible stationary Stokes system with a non-linear viscosity following the Carreau law in a thin porous media, as we consider in this article. Therefore, taking into account the previous results, we consider the Stokes system (2) in the thin porous media ⌦"assuming = 1. This choice is the most challenging one and answers to the question addressed in the paper, namely it preserves the nonlinear character of the flow in the limit. After the homogenization process, we obtain the following lower-dimensional homogenization law 8 < : V0(x0)=U(f0(x0)rx0p(x0)) ,V 3(x0)=0in!, divx0V0(x0)=0 in!, V 0(x0)·n= 0 on @!, where the permeability function U:R2!R2is defined through the solutions of 3D local Stokes problems with non-linear viscosity following the Carreau law and posed in a reference cell, see Theorem 2.1. 3 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau In order to illustrate the influence of the non-linear character of the Carreau law on the behaviour of the twodimensional limit system, we conclude the paper with a numerical study of a flow of a generalized Newtonian fluid, driven by a constant pressure gradient, in a thin porous medium confined between two parallel plates. Using di↵erent geometries for the inclusion T, we show that, depending on the choice of parameters and rin the viscosity law (1), the amplitude of the mean filtration velocity associated to a given pressure gradient can be dramatically increased in comparison with the Newtonian case r= 2. 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 result (Theorem 2.1). The proof of the main result is provided in Section 3. Section 4 is dedicated to the numerical simulations of the homogenized model, in the case of Carreau fluid driven by a constant pressure gradient. We finish the paper with a list of references. 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 Y0=(1/2,1/2)2, which is made of two complementary parts: the fluid part Y0 f, and the solid part T0(Y0 fST0=Y0and Y0 fTT0=;). More precisely, we assume that !is a smooth, bounded, connected set in R2, and that T0is an open connected subset of Y0with a smooth boundary @T0, such that ¯ T0is strictly included in Y0. The microscale of a porous medium is a small positive number ". The domain !is covered by a regular mesh of square of size ": for k02Z2, each cell Y0 k0,"="k0+"Y0is divided in a fluid part Y0 fk0,"and a solid part T0 k0,", i.e. is similar to the unit cell Y0rescaled to size ".WedefineY=Y0⇥(0,1) ⇢R3, which is divided in a fluid part Yf=Y0 f⇥(0,1) and a solid part T=T0⇥(0,1), and consequently Yk0,"=Y0 k0,"⇥(0,1) ⇢R3, which is also divided in a fluid part Yfk0,"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,") represents the obstacles in R2. 1 1 Yf T Y0 f 1 1 T0 Figure 1: View of the 3D reference cells Y(left) and the 2D reference cell Y0(right). " " T0 k0,"⇥(0,") Y0 fk0,"⇥(0,") " " Y0 fk0," T0 k0," Figure 2: View of the 3D reference cells Yk0,"(left) and the 2D reference cell Y0 k0,"(right). The fluid part of the bottom !"⇢R2of a porous medium is defined by !"=!\Sk02K" ¯ T0 k0,",where K"={k02Z2:Y0 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<"}.(3) 4 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau " " S" " Figure 3: View of the thin porous media ⌦"(left) and domain without perforations Q"(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,"). We define e ⌦"=!"⇥(0,1),⌦=!⇥(0,1),Q "=!⇥(0,").(4) 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 along the paper. The points x2R3will 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 #(Y) be the space of infinitely di↵erentiable functions in R3that are Y0-periodic. By L2 #(Y) (resp. H1 #(Y)) we denote its completion in the norm L2(Y)(resp. H1(Y)) and by L2 0,#(Y) the space of functions in L2 #(Y) with zero mean value. Statement of the problem. Let us consider the following stationary Stokes system with the non-linear viscosity following the Carreau law (1) in ⌦", with a Dirichlet boundary condition on the exterior boundary @Q" and the cylinders @S",8 > > > < > > > : "div (⌘r(D[u"])D[u"]) + rp"=fin ⌦", div u"=0 in⌦ ", u"= 0 on @Q"[@S", (5) where the second member fis of the form f(x)=(f0(x0),0) with f02L1(!)2.(6) We remark that the assumptions of neglecting the vertical component of the exterior force and the independence of the vertical variable are usual when dealing with fluids in through thin domains (see [19] for more details). The classical theory from Lions [30] gives the existence of a unique solution (u",p ")2H1 0(⌦")3⇥L2 0(⌦"), where L2 0is the space of functions of L2with zero mean value. Our goal is to study the asymptotic behavior of u"and p"when "tends to zero. For this purpose, we use the dilatation in the variable x3as follows y3=x3 ",(7) in order to have the functions defined in the open set with fixed height e ⌦". Namely, we define ˜u"and ˜p"by ˜u"(x0,y 3)=u"(x0,"y 3),˜p"(x0,y 3)=p"(x0,"y 3), a.e. (x0,y 3)2e ⌦". Let us introduce some notation which will be useful in the following. For a vectorial function v=(v0,v 3) and a scalar function w, we will denote Dx0[v]=1 2(Dx0v+Dt x0v) and @y3[v]=1 2(@y3v+@t y3v), where we denote 5 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau @y3=(0,0,@ @y3)t. Moreover, associated to the change of variables (7), we introduce the operators: D",D",div " and r",by D"[v]=1 2D"v+Dt "v, (D"v)i,j =@xjvifor i=1,2,3,j=1,2,(D"v)i,3="1@y3vifor i=1,2,3, div"v=div x0v0+"1@y3v3,r"w=(rx0w,"1@y3w)t. Using the transformation (7), system (5) can be rewritten as 8 > > > < > > > : "div"(⌘r(D"[˜u"])D"[˜u"]) + r"˜p"=fin e ⌦", div"˜u"=0in e ⌦", ˜u"= 0 on @⌦[@T". (8) Our goal then is to describe the asymptotic behavior of this new sequence (˜u",˜p"). The sequences of solutions (˜u",˜p")2H1 0(e ⌦")3⇥L2 0(e ⌦") is not defined in a fixed domain independent of "but rather in a varying set e ⌦". In order to pass the limit if "tends to zero, convergences in fixed Sobolev spaces (defined in ⌦) are used which requires first that (˜u",˜p") be extended to the whole domain ⌦. Then, an extension (˜u",˜ P")2H1 0(⌦)3⇥L2 0(⌦) is defined on ⌦and coincides with (˜u",˜p") on e ⌦"(we will use the same notation, ˜u", for the velocity in e ⌦"and its continuation in ⌦). Our main result referred to the asymptotic behavior of the solution of (8) is given by the following theorem. Theorem 2.1 (Main Theorem).There exist ˜u2H1 0(0,1; L2(!)3)with ˜u3=0and ˜ P2L2 0(!), such that the extension (˜u",˜ P")of the solution of (8) satisfies the following convergences "1˜u"*˜uweakly in H1(0,1; L2(!)3),˜ P"!˜ Pstrongly in L2(⌦). Moreover, defining V(x0)=R1 0˜u(x0,y 3)dy3,itholdsthat(V, ˜ P)2L2(!)3⇥(L2 0(!)\H1(!)) is the unique solution of the lower-dimensional homogenization law 8 > < > : V0(x0)=U⇣f0(x0)rx0˜ P(x0)⌘,V 3(x0)=0 in !, divx0V0(x0)=0 in !, V 0(x0)·n=0 on @!,(9) where the permeability function U:R2!R2is defined by U(⇠0)=ZYf w0 ⇠0(y)dy, 8⇠02R2,(10) with w⇠0, for every ⇠02R2, the unique solution of the local Stokes system with the non-linear viscosity following the Carreau law (1), 8 > > > > > > < > > > > > > : divy(⌘r(Dy[w⇠0]) Dy[w⇠0]) + ry⇡⇠0=⇠0in Yf, divyw⇠0=0 in Yf, w⇠0=0 on @T, (w⇠0,⇡ ⇠0)2H1 #(Yf)3⇥L2 0,#(Yf). (11) Remark 2.2. According to [21, Lemma 2], the permeability function Uis coercive and strictly monotone. 3 Proof of the main result In this section we provide the proof of the main result (Theorem 2.1). To to this, first we establish some apriori estimates of the solution of (8) and we define the extension of the solution. Second, we introduce the version of the unfolding method depending on ". Next, a compactness result, which is the main key when we will pass to the limit later, is addressed and finally, the proof of the Theorem 2.1 is given. 6 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau 3.1 A priori estimates. In this subsection, we establish sharp aprioriestimates of the dilated solution in e ⌦". To do this, we first need the Poincar´e and Korn inequalities in e ⌦", which can be found in [10]. Lemma 3.1 (Remark 4.3-(i) in [10]).There exists a positive constant C, independent of ", such that k˜vkL2( e ⌦")3C"kD"˜vkL2( e ⌦")3⇥3,8˜v2H1 0(e ⌦")3(Poincar´e’s inequality),(12) kD"˜vkL2( e ⌦")3⇥3CkD"[˜v]kL2( e ⌦")3⇥3,8˜v2H1 0(e ⌦")3(Korn’s inequality).(13) We give aprioriestimates for velocity ˜u"in e ⌦". Lemma 3.2. There exists a positive constant C, independent of ", such that k˜u"kL2( e ⌦")3C",kD"˜u"kL2( e ⌦")3⇥3C, kD"[˜u"]kL2( e ⌦")3⇥3C. (14) Proof. Multiplying (8) by ˜u"and integrating over e ⌦", we get "(⌘0⌘1)Ze ⌦"1+|D"[˜u"]|2r 21|D"[˜u"]|2dx0dy3+"⌘1Ze ⌦"|D"[˜u"]|2dx0dy3=Ze ⌦" f0·˜u0 "dx0dy3. Taking into account that ⌘0>⌘ 1, and >0, we have "(⌘0⌘1)Ze ⌦"1+|D"[˜u"]|2r 21|D"[˜u"]|2dx0dy30, and then, from Cauchy-Schwarz’s inequality and the assumption on f0given in (6), we get "⌘1kD"[˜u"]k2 L2( e ⌦")3⇥3Ck˜u"kL2( e ⌦")3. Applying Poincar´e’s inequality (12) and Korn’s inequality (13) in the right-hand side, we have kD"[˜u"]k2 L2( e ⌦")3⇥3Ck|D"[˜u"]kL2( e ⌦")3⇥3, which gives (14)3. Finally, applying again (12) and (13), we obtain (14)1and (14)2. Remark 3.3. We extend the velocity ˜u"by zero in ⌦\e ⌦"(this is compatible with the homogeneous boundary condition on @⌦[@T"), and denote the extension by same symbol. Obviously, estimates given in Lemma 3.2 remain valid and the extension ˜u"is divergence free too. In order to extend the pressure ˜p"to the whole domain ⌦and obtain aprioriestimates, we recall a result in which is concerned with the extension of the pressure p"to the whole domain Q". Thus, we first use a restriction operator R"from H1 0(Q")3into H1 0(⌦")3, which is introduced in [10] as R" 2, next we extend the gradient of the pressure by duality in H1(Q")3and finall, by means of the dilatation, we extend ˜p"to ⌦. Lemma 3.4 (Lemma 4.5-(i) in [10]).There exists a (restriction) operator R"acting from H1 0(Q")3into H1 0(⌦")3 such that 1. R"v=v,ifv2H1 0(⌦")3(elements of H1 0(⌦")are extended by 0to Q"). 2. divR"v=0in ⌦",ifdiv v=0on Q". 3. For every v2H1 0(Q")3, there exists a positive constant C, independent of vand ", such that kR"vkL2(⌦")3+"kDR"vkL2(⌦")3⇥3CkvkL2(Q")3+"kDvkL2(Q")3⇥3.(15) 7 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau Using the restriction operator R"given in Lemma 3.4, we introduce F"in H1(Q")3in the following way hF",viH1(Q")3,H1 0(Q")3=hrp",R "viH1(⌦")3,H1 0(⌦")3,for any v2H1 0(Q")3,(16) and calculate the right hand side of (16) by using the variational formulation of problem (5), which gives hF",viH1(Q")3,H1 0(Q")3="(⌘0⌘1)Z⌦" (1 + |D[u"]|2)r 21D[u"]:DR"vdx "⌘1Z⌦" D[u"]:DR"vdx+Z⌦" f0·(R"v)0dx . (17) Using Lemma 3.2 for fixed ", we see that it is a bounded functional on H1 0(Q") (see the proof of Lemma 3.5 below), and in fact F"2H1(Q")3. Moreover, div v= 0 implies hF",vi=0,and the DeRham theorem gives the existence of P"in L2 0(Q")withF"=rP". Next, we get for every ˜v2H1 0(⌦)3where ˜v(x0,y 3)=v(x0,"y 3), using the change of variables (7), that hr"˜ P",˜viH1(⌦)3,H1 0(⌦)3=Z⌦ ˜ P"div"˜vdx 0dy3="1ZQ" P"div vdx="1hrP",viH1(Q")3,H1 0(Q")3. Using the identification (17) of F",wehave hr"˜ P",˜viH1(⌦)3,H1 0(⌦)3="1✓"(⌘0⌘1)Z⌦" (1 + |D[u"]|2)r 21D[u"]:DR"vdx "⌘1Z⌦" D[u"]:DR"vdx+Z⌦" f0·(R"v)0dx◆, and applying the change of variables (7), we obtain hr"˜ P",˜viH1(⌦)3,H1 0(⌦)3="(⌘0⌘1)Ze ⌦" (1 + |D"[˜u"]|2)r 21D"[˜u"]:D"˜ R"˜vdx 0dy3 "⌘1Ze ⌦" D"[˜u"]:D"˜ R"˜vdx+Ze ⌦" f0(x0)·(˜ R"˜v)0dx0dy3, (18) where ˜ R"˜v=R"vfor any ˜v2H1 0(⌦)3. Finally, we estimate the right-hand side of (18) and give the estimate of the extended pressure ˜ P". Lemma 3.5. There exists a positive constant Cindependent of ", such that k˜ P"kL2(⌦)C, kr"˜ P"kH1(⌦)3C. (19) Proof. Applying the dilatation in (15), we have that ˜ R"˜vsatisfies the following estimate k˜ R"˜vkL2( e ⌦")3+"kD"˜ R"˜vkL2( e ⌦")3⇥3Ck˜vkL2(⌦)3+"kD"˜vkL2(⌦)3⇥3,(20) and since "⌧1, we have k˜ R"˜vkL2( e ⌦")3Ck˜vkH1 0(⌦)3,kD"˜ R"˜vkL2( e ⌦")3⇥3C "k˜vkH1 0(⌦)3.(21) 8 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau Throughout this numerical section, we assume that the flow is driven by a constant pressure gradient in the horizontal direction, which amounts to considering a constant external force f=(f0,0) with f02R2.Such assumption is realistic in applications like enhanced oil recovery, where the flow, frequently described by the Carreau law in the engineering literature, is mainly driven by the pressure di↵erence between the injection point and the well [18, Chapter 4]. Since f0is constant, one gets that ˜ P⌘0 and V0is also constant, given by V0=RYfw0 f0(y)dy where w0 f0is the solution to system (11) with ⇠0=f0. In order to compute the solution to system (11), we rely on a mixed formulation of the problem and solve it by a finite element method, using FreeFem++ software [28]. Since the system (11) is nonlinear (for 1 <r<2), we solve the corresponding mixed formulation by a fixed point algorithm (see for instance [34, Section 2.8]). We consider the Taylor-Hood approximation for the velocity-pressure pair, i.e. P2elements for the velocity field and P1elements for the pressure. It is well known that this choice is compatible with the Babuˇska-Brezzi condition [22, 26]. The three-dimensional mesh of the cell Yfis obtained by constrained Delaunay tetrahedralization. The results that we present in the next subsections are obtained using approximately 8000 tetrahedra in the mesh of Yf. We propose to explore numerically the influence of the amplitude of the constant pressure gradient f0, and of its orientation, on the mean filtration velocity V0. In our simulations, we will use di↵erent shapes for the inclusion T, that are presented in the next paragraph. 4.1 Geometry of the inclusion T In the numerical simulations of system (11), we have chosen to consider two types of shapes for the horizontal projection T0of the solid inclusion T=T0⇥(0,1): ellipses and rectangles. •Ellipses are built using two possible values for each semi-axis: 0.1 and 0.3. This leads to 4 di↵erent geometries: two disks of respective radius 0.1 and 0.3, and the ellipse of semi-major axis 0.3 and semiminor axis 0.1, parallel to the xor the yaxis (see Fig. 4). These shapes will be numbered E1,E 2,E 3and E4in the rest of this section. •Similarly, we construct 4 rectangular shapes by selecting the length of opposite sides in {0.15,0.35}(see Fig. 5). These shapes are numbered R1,R 2,R 3and R4. These shapes are commonly used in the literature on flows in porous media, and o↵er a variety of geometric features, such as size, regularity and isotropy, that favor comparisons. 4.2 Influence of the amplitude of the imposed pressure gradient f0 In order to test the impact of the variations of |f0|, we have computed the mean filtration velocity V0associated to a pressure gradient f0directed by e1, in the form f0=(f1,0) with f12(0,1). In that case, V0is also directed by e1, and reads V0=(V1,0). The results that we have obtained are plotted in Fig. 6 (in the case of elliptic inclusions E1to E4) and in Fig. 7 (in the case of rectangular inclusions R1to R4). We can observe that the amplitude of the mean filtration velocity, which is simply equal to |V0|=V1in this setting, increases as ris diminished and is augmented. For = 1, V1remains very close to the reference value (Newtonian case r= 2). On the opposite, increasing up to = 100 a↵ects strongly the behaviour of the fluid as a function of r. For instance, fixing = 100 and f1= 1, and taking r=1.3 drastically increases the velocity V1with respect to the computed value for r= 2. Comparing E1to E4and R1to R4, we observe that, as expected, the filtration velocity is smaller for large volume inclusions than for small volume ones. Our results also confirm the intuition that aligning the obstacle in the sense of the flow (as E2and R2), or perpendicularly to it (as E3and R3), a↵ects the filtration velocity: V1 is much smaller for E3(resp. R3) than for E2(resp. R2). Let us notice that this e↵ect is much more pronounced for rectangles that for ellipses, which is probably linked with the di↵erence of regularity of the boundaries of the corresponding shapes. 15 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau Shape E1Shape E2 Shape E3Shape E4 Figure 4: Representation of 2Dreference cells Y0, in the case of elliptic shapes for T0inclusions (in grey), surrounded by Y0 f(in blue). From left to right and top to bottom: disk or radius 0.1, ellipse of semi-major axis 0.3 and semi-minor axis 0.1, oriented along the xdirection and the ydirection resp., and disk of radius 0.3. These shapes are numbered E1,E 2,E 3and E4. 16 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau Shape R1Shape R2 Shape R3Shape R4 Figure 5: Representation of 2Dreference cells Y0, in the case of rectangular shapes for T0inclusions (in grey), surrounded by Y0 f(in blue). From left to right and top to bottom: square or side 0.15, rectangle of length 0.35 and width 0.15, oriented along the xdirection and the ydirection resp., and square of side 0.35. These shapes are numbered R1,R 2,R 3and R4. 17 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8·102 f1 V1 =1 r=1.3 r=1.5 r=1.7 r=2 0 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 = 10 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 f1 = 100 0 0.2 0.4 0.6 0.8 1 0 2 4 6·102 f1 V1 r=1.3 r=1.5 r=1.7 r=2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 ·102 f1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 f1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 ·102 f1 V1 r=1.3 r=1.5 r=1.7 r=2 0 0.2 0.4 0.6 0.8 1 0 1 2 3·102 f1 0 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 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 ·102 f1 V1 r=1.3 r=1.5 r=1.7 r=2 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2·102 f1 0 0.2 0.4 0.6 0.8 1 0 2 4 6·102 f1 Figure 6: Component V1of the mean filtration velocity V0V0plotted against f1,withf0=(f1,0), for r2 {1.3,1.5,1.7,2}, 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. 18 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 ·102 f1 V1 =1 r=1.3 r=1.5 r=1.7 r=2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 ·102 f1 = 10 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 f1 = 100 0 0.2 0.4 0.6 0.8 1 0 1 2 3 ·102 f1 V1 r=1.3 r=1.5 r=1.7 r=2 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 ·102 f1 0 0.2 0.4 0.6 0.8 1 0 5·102 0.1 0.15 0.2 f1 0 0.2 0.4 0.6 0.8 1 0 2 4 6 ·103 f1 V1 r=1.3 r=1.5 r=1.7 r=2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 ·103 f1 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 ·102 f1 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 ·103 f1 V1 r=1.3 r=1.5 r=1.7 r=2 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 ·103 f1 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 ·103 f1 Figure 7: Component V1of the mean filtration velocity V0plotted against f1,withf0=(f1,0), for r2 {1.3,1.5,1.7,2}, in the case of rectangular inclusions R1(first line), R2(second line), R3(third line) and R4 (fourth line). From left to right: =1,= 10,= 100. 19 Mar´ıa Anguiano, Matthieu Bonnivard and Francisco J. Su´arez-Grau 4.3 Dependency on the orientation of the pressure gradient We complete our previous results on the amplitude of f0by testing the influence of its orientation, on both the orientation and amplitude of V0. To this aim, we consider a family of pressure gradients f0= (cos ✓,sin ✓) and the anisotropic shapes of inclusions E2and R2. By symmetry, we can restrict the angular parameter ✓to ✓2[0,⇡/2]. The results that we obtain are represented in Fig. 8 (for ellipse E2) and in Fig. 9 (for rectangle R2). As observed in the previous paragraph, for = 1, the value of rvalue does not seem to have a strong impact on the behaviour of the model. For both shapes E2and R2, the value and orientation of V0remain essentially the same as in the reference Newtonian case r= 2. On the other hand, for = 100, the influence of rbecomes clear. For example, in case of inclusion R2, the maximal amplitude of the velocity (achieved for ✓= 0) is close to 0.2withr=1.3, while it was merely equal to 0.03 for r= 2. A comparable change of order of magnitude can be observed for E2. As regards the orientation of the vector V0, we may observe a slight di↵erence of orientation in case = 100, between the reference case r= 2 and the other cases r2{1.3,1.5,1.7}, concerning mainly the values of angle ✓ close to ⇡/2. However, from a global perspective, it appears that modifiying and rdoes not have a definite influence on the response of the system to a rotation of f0. 20 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=1.3 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=1.3 0.2 0.3 0.4 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 =1,r=1.5 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=1.5 6·102 8·102 0.1 0.12 0.14 0 0.2 0.4 0.6 0.811.2 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 0.811.2 0 0.5 1 f1 f2 = 100,r=2 3 4 5 ·102 Figure 8: 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|. The left column corresponds to = 1 and the right one to = 100. Each line from top to bottom corresponds respectively to r=1.3, r=1.5, r=1.7 and r= 2. 21 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=1.3 1 1.5 2 2.5 3 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 = 100,r=1.3 5·102 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 =1,r=1.5 1 1.5 2 2.5 3 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 = 100,r=1.5 2 4 6 8 ·102 0 0.2 0.4 0.6 0.811.2 0 0.5 1 f1 f2 =1,r=1.7 1 1.5 2 2.5 3 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 = 100,r=1.7 1 2 3 4 5 ·102 0 0.2 0.4 0.6 0.811.2 0 0.5 1 f1 f2 =1,r=2 1 1.5 2 2.5 3 ·102 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 f1 f2 = 100,r=2 1 1.5 2 2.5 3 ·102 Figure 9: Representation of V0when f0is a unit vector of the form (f1,f 2) = (cos ✓,sin ✓)with✓2[0,⇡/2], in the case of a rectangular inclusion R2. 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 and the right one to = 100. 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