Homogenization of Bingham flow in 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 an accepted manuscript of a paper published in: Networks and Heterogeneous Media (2019): 01/12/2019 DOI: https://doi.org/10.3934/nhm.2020004 Copyright: El acceso a la versión publicada del artículo puede requerir la suscripción de la revista. Access to the published version may require subscription. Si el artículo ha sido publicado: “This article has been published in a revised form in Networks and Heterogeneous Media https://doi.org/10.3934/nhm.2020004. This version is free to download for private research and study only. Not for redistribution, re-sale or use in derivative Works”. Homogenization of Bingham Flow in thin porous media Mar´ıa ANGUIANO Departamento de An´alisis Matem´atico, Facultad de Matem´aticas Universidad de Sevilla, 41012 Sevilla, Spain Renata BUNOIU Universit´e de Lorraine, CNRS, IECL Metz, 57000, France Abstract By using dimension reduction and homogenization techniques, we study the steady flow of an incompresible viscoplastic Bingham fluid in a thin porous medium. A main feature of our study is the dependence of the yield stress of the Bingham fluid on the small parameters describing the geometry of the thin porous medium under consideration. Three di↵erent problems are obtained in the limit when the small parameter "tends to zero, following the ratio between the height "of the porous medium and the relative dimension a"of its periodically distributed pores. We conclude with the interpretation of these limit problems, which all preserve the nonlinear character of the flow. 1 Introduction In this paper we study the asymptotic behavior of the flow of a viscoplastic Bingham fluid in a thin porous medium which contains an array of bodies modelized as vertical cylindrical obstacles (the pores). We refer the reader to the very recent paper [8] and the references therein for the application of our study to problems issued from the real life applications. As a first example one can mention the flow of the volcanic lava through dense forests (see [30]). Another important application is the flow of fresh concrete spreading through networks of steel bars (see [33]). The model of thin porous medium of thickness much smaller than the distance between the pores was introduced in [34], where a stationary incompressible Navier-Stokes flow was studied. Recently, the model of thin porous medium under consideration in this paper was introduced in [20], where the flow of an incompressible viscous fluid described by the stationary Navier-Stokes equations was studied by the multiscale expansion method, which is a formal but powerful tool to analyse homogenization problems. These results were rigorously proved in [5] using an adaptation introduced in [4] of the periodic unfolding method from [16] and [17]. 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 use monotonicity arguments to pass to the limit. In [4], in particular, the flow of an incompressible stationary Stokes system with a nonlinear viscosity, being a power law, was studied. For non-stationary incompressible viscous flow in a thin porous medium see [1], where a non-stationary Stokes system is considered, and [2], where a non-stationary non-newtonian Stokes system, where the viscosity obeyed the power law, is studied. For the periodic unfolding method applied to the study of problems stated in other type of thin periodic domains we refer for instance to [23] for crane type structures and to [24], [25] for thin layers with thin beams structures, where elasticity problems are considered. In [32], the homogenization of elasticity problems in thin periodic domains of planar grids type is studied. For problems involving arrays of bodies in high-contrast materials we refer the reader to [6], Chapter 2. AMS classification numbers: 76A05, 76A20, 76M50, 35B27. Keywords: porous medium, thin domain, Bingham fluid 1 Mar´ıa Anguiano and Renata Bunoiu If ⇧is a three-dimensional domain with smooth boundary @⇧and f=(f1,f 2,f 3) are external given forces defined on ⇧, then the velocity u=(u1,u 2,u 3) of a fluid and its pressure psatisfy the equations of motion  3 X j=1 @xi((p, u))ij =fiin ⇧,1i3,(1) completed with the fluid’s incompressibility condition div u=P3 i=1 @xiui= 0 in ⇧, and the no-slip boundary condition u= 0 on the boundary @⇧. What distinguishes di↵erent fluids is the expression of the stress tensor . Newtonian fluids are the most encountered ones in real life and as typical examples one can mention the water and the air. For a newtonian fluid, the entries of the stress tensor (p, u) are given by ((p, u))ij =pij +2µ(D(u))ij,1i, j 3 (2) where ij is the Kronecker symbol, the real positive µis the viscosity of the fluid and the entries of the strain tensor are (D(u))ij =(@xjui+@xiuj)/2.If fbelongs to (L2(⇧))3and the space Vis defined by V={v2 (H1 0(⇧))3|div v=0},then uand psatisfying (1) with (2) are such that (see for instance [22]): (Stokes) There is a unique u2Vand a unique (up to an additive real constant) p2L2(⇧)such that (if <·,·>is the dual pairing between (H1(⇧))3and (H1 0(⇧))3) a(u, v)=l(v)<rp, v >, 8v2(H1 0(⇧))3,(3) with a(u, v)=2µZ⇧ D(u): D(v)dx and l(v)=Z⇧ f·vdx. A fluid whose stress is not defined by relation (2) is called a non-newtonian fluid. There are several classes of non-newtonian fluids, as the power law, Carreau, Cross, Bingham fluids. It is on the study of the last type of fluid that we are interested in this paper. We refer to [18] for a review on non-newtonian fluids. For a Bingham fluid, the nonlinear stress tensor is defined by (see [19]) ((p, u))ij =pij +2µ(D(u))ij +p2g(D(u))ij |D(u)|,(4) where |D(u)|2=D(u): D(u)6= 0 and the positive number grepresents the yield stress of the fluid. If g= 0, then (4) becomes (2). Viscoplastic Bingham fluids are quite often encountered in real life. As examples one can mention volcanic lava, fresh concrete, the drilling mud, oils, clays and some paintings. For ugand pgsatisfying (1) with (4), according to [19], one has the following result: (Bingham) There is a unique ug2Vand a (non-unique) pg2L2(⇧)/Rsuch that a(ug,vug)+j(v)j(ug)l(vug)<rpg,vug>, 8v2(H1 0(⇧))3.(5) Here a, l, < ·,·>are as before and j(v)=p2gZ⇧|D(v)|dx, 8v2(H1 0(⇧))3. If the yield stress of the Bingham fluid is of the form g("), with "2]0,1[ and such that g(")tendstozero when "tends to zero, then, according to [[19], Chapter 6, Th´eor`eme 5.1.], the following result holds When "tends to zero, one has for the solution u"of problem (5) corresponding to g(")the following convergence u"*uweakly in V, where uis the solution of problem (3). This means that, in a fixed domain, the nonlinear character of the Bingham flow is lost in the limit (when the yield stress tends to zero), as it is expected. A natural question that arises is the following: If the yield stress g(") 2 Mar´ıa Anguiano and Renata Bunoiu is as before and, moreover, the domain ⇧itself depends on the small parameter ", what happens when "tends to zero? The answer is that, in the limit, the nonlinear character of the flow may be preserved. For instance, if ⇧"is a classical rigid porous medium, it was proven in [29] with the asymptotic expansion method that, in a range of parameters, the nonlinear character of the Bingham flow is preserved in the homogenized problem, which is a nonlinear Darcy equation. The convergence corresponding to the above mentioned result was proven in [10] with the two-scale convergence method and then recovered in [12] with the periodic unfolding method. The case of a doubly periodic rigid porous medium was studied in [11], where a more involved nonlinear Darcy equation is derived. Another class of domains for which the nonlinear character of the flow may be preserved in the limit is those of thin domains. The case of a domain ⇧"which is thin in one direction was addressed in [14] and [15]. We refer to [13] for the asymptotic analysis of a Bingham fluid in a thin T-like shaped domain. In all these cases, a lower-dimensional Bingham-like law was exhibited in the limit. This law was already encountered in engineering (see [31]), but no rigurous mathematical justification was previously known. A first mathematical result combining both periodic and thin domains for the Bingham flow was announced in [3]. For the shallow flow of a viscoplastic fluid we refer the reader to [21], [9], [26], [27] and [28]. For a homogenized non-newtonian viscoelastic model we refer to [7]. The paper is organized as follows. In Section 2. we state the problem: we define in (6) the thin porous medium ⌦"(see also Figure 1), of height "and relative dimension a"of its periodically distributed pores. In ⌦"we consider the flow of a viscoplastic Bingham fluid with velocity u"and pressure p"verifying the nonlinear variational inequality (9). In Section 3. we give some aprioriestimates for the velocity and for the pressure obtained after the change of variables (10) and verifying (12), and then for the velocity and for the pressure defined in (21). In Section 4. by passing to the limit "!0, we prove the main convergence results of our paper, stated in Theorems 4.2, 4.4 and 4.6, respectively. Up to our knowledge, problems (36), (57) and (78) are new in the mathematical literature. We conclude in Section 5. with the interpretation of these limit problems, which all three preserve the nonlinear character of the flow; both e↵ects of a nonlinear Darcy equation and a lower dimensional Bingham-like law appear. The paper ends with a list of References. 2 Statement of the problem A 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 Y0 s(Y0 fSY0 s=Y0 and Y0 fTY0 s=?). More precisely, we assume that !is a smooth, bounded, connected set in R2, and that Y0 sis an open connected subset of Y0with a smooth boundary @Y0 s, such that Y0 sis strictly included in Y0. The microscale of a porous medium is a small positive number a". The domain !is covered by a regular mesh of size a": for k02Z2, each cell Y0 k0,a"=a"k0+a"Y0is divided in a fluid part Y0 fk0,a"and a solid part Y0 sk0,a", i.e. is similar to the unit cell Y0rescaled to size a".WedefineY=Y0⇥(0,1) ⇢R3, which is divided in a fluid part Yfand a solid part Ys, and consequently Yk0,a"=Y0 k0,a"⇥(0,1) ⇢R3, which is also divided in a fluid part Yfk0,a"and a solid part Ysk0,a". We denote by ⌧(Y0 sk0,a") the set of all translated images of Y0 sk0,a".Theset⌧(Y0 sk0,a") represents the solids in R2. The fluid part of the bottom !"⇢R2of the porous medium is defined by !"=!\Sk02K"Y0 sk0,a",where K"={k02Z2:Y0 k0,a"\!6=;}. The whole fluid part ⌦"⇢R3in the thin porous medium is defined by ⌦"={(x1,x 2,x 3)2!"⇥R:0<x 3<"}.(6) We make the assumption that the solids ⌧(Y0 sk0,a") do not intersect the boundary @!.WedefineY" sk0,a"= Y0 sk0,a"⇥(0,"). Denote by S"the set of the solids contained in ⌦".Then,S"is a finite union of solids, i.e. S"=Sk02K"Y" sk0,a". We define e ⌦"=!"⇥(0,1),⌦=!⇥(0,1),and Q"=!⇥(0,").We observe that e ⌦"=⌦\Sk02K"Ysk0,a",and we define T"=Sk02K"Ysk0,a"as the set of the solids contained in e ⌦". 3 Mar´ıa Anguiano and Renata Bunoiu Figure 1: View of the domain ⌦" We denote by : the full contraction of two matrices; for A=(ai,j)1i,j3and B=(bi,j )1i,j3,wehave A:B=P3 i,j=1 aijbij. In order to apply the unfolding method, we will need the following notation. For k02Z2,wedefine:R2! Z2by (x0)=k0() x02Y0 k0,1.(7) Remark that is well defined up to a set of zero measure in R2(the set [k02Z2@Y0 k0,1). Moreover, for every a">0, we have ✓x0 a"◆=k0() x02Y0 k0,a". We denote by Ca generic positive constant which can change from line to line. 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. In ⌦"we consider the stationary flow of an incompressible Bingham fluid. As already seen in the Introduction, following Duvaut and Lions [19], the problem is formulated in terms of a variational inequality. For a vectorial function v=(v0,v 3), we define (D(v))i,j =1 2@xjvi+@xivj,1i, j 3,|D(v)|2=D(v):D(v). We introduce the following spaces V(⌦")={v2(H1 0(⌦"))3|div v=0in⌦ "},H(⌦")={v2(L2(⌦"))3|div v=0in⌦ ",v·n= 0 on @⌦"}. For u, v 2(H1 0(⌦"))3,weintroduce a(u, v)=2µZ⌦" D(u):D(v)dx, j(v)=p2g(")Z⌦"|D(v)|dx, (u, v)⌦"=Z⌦" u·vdx, where the yield stress g(") will be made precise in Section 3.1. Let f2(L2(⌦))3be given such that f=(f0,0). Let f"2(L2(⌦"))3be defined by f"(x)=f(x0,x 3/"),a.e. x2⌦". The model of the flow is described by the following variational inequality: Find u"2V(⌦") such that a(u",vu")+j(v)j(u")(f",vu")⌦",8v2V(⌦").(8) 4 Mar´ıa Anguiano and Renata Bunoiu From Duvaut and Lions [19], we know that there exists a unique u"2V(⌦") solution of problem (8). Moreover, from Bourgeat and Mikeli´c [10], we know that if p"is the pressure of the fluid in ⌦", then problem (8) is equivalent to the following one: Find u"2V(⌦") and p"2L2 0(⌦") such that a(u",vu")+j(v)j(u")(f",vu")⌦"+(p",div (vu"))⌦",8v2(H1 0(⌦"))3.(9) Problem (9) admits a unique solution u"2V(⌦") and a (non) unique solution p"2L2 0(⌦"), where L2 0(⌦") denotes the space of functions belonging to L2(⌦") and of mean value zero. Our aim is to study the asymptotic behavior of u"and p"when "tends to zero. For this purpose, we first use the dilatation of the domain ⌦"in the variable x3, namely y3=x3 ",(10) in order to have the functions defined in an open set with fixed height, denoted e ⌦". Namely, we define ˜u"2(H1 0(e ⌦"))3,˜p"2L2 0(e ⌦")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 @y3=(0,0,@ @y3)t. Moreover, associated to the change of variables (10), we introduce the operators: D",D",div " and r", defined by (D"v)i,j =@xjvifor i=1,2,3,j=1,2,(D"v)i,3=1 "@y3vifor i=1,2,3, D"[v]=1 2D"v+Dt "v,|D"[v]|2=D"[v]:D"[v],div"v=div x0v0+1 "@y3v3,r"w=(rx0w, 1 "@y3w)t. We introduce the following spaces V(e ⌦")={˜v2(H1 0(e ⌦"))3|div"˜v=0in e ⌦"},H(e ⌦")={˜v2(L2(e ⌦"))3|div"˜v=0in e ⌦",˜v·n= 0 on @e ⌦"}. For ˜u, ˜v2V(e ⌦"), we introduce a"(˜u, ˜v)=2µZe ⌦" D"[˜u]:D"[˜v]dx0dy3,j "(˜v)=p2g(")Ze ⌦"|D"[˜v]|dx0dy3,(˜u, ˜v)e ⌦"=Ze ⌦" ˜u·˜vdx0dy3. Using the transformation (10), the variational inequality (8) can be rewritten as: Find ˜u"2V(e ⌦") such that a"(˜u",˜v˜u")+j"(˜v)j"(˜u")(f, ˜v˜u")e ⌦",8˜v2V(e ⌦"),(11) and (9) can be rewritten as: Find ˜u"2V(e ⌦") and ˜p"2L2 0(e ⌦") such that a"(˜u",˜v˜u")+j"(˜v)j"(˜u")(f, ˜v˜u")e ⌦"+(˜p",div"(˜v˜u"))e ⌦",8˜v2(H1 0(e ⌦"))3.(12) Our goal now is to describe the asymptotic behavior of this new sequence (˜u",˜p"). 5 Mar´ıa Anguiano and Renata Bunoiu 3 A Priori Estimates We start by obtaining some aprioriestimates for ˜u". Lemma 3.1. There exists a constant Cindependent of ", such that if ˜u"2(H1 0(e ⌦"))3is the solution of problem (11), one has i) if a"⇡",witha"/"!,0<<+1,ora"⌧", then k˜u"k(L2( e ⌦"))3C µa2 ",kD"[˜u"]k(L2( e ⌦"))3⇥3C µa",kD"˜u"k(L2( e ⌦"))3⇥3C µa",(13) ii) if a"", then k˜u"k(L2( e ⌦"))3C µ"2,kD"[˜u"]k(L2( e ⌦"))3⇥3C µ",kD"˜u"k(L2( e ⌦"))3⇥3C µ".(14) Proof. Setting successively ˜v=2˜u"and ˜v= 0 in (11), we have 2µZe ⌦" D"[˜u"]:D"[˜u"]dx0dy3+p2g(")Ze ⌦"|D"[˜u"]|dx0dy3=Ze ⌦" f·˜u"dx0dy3.(15) Using Cauchy-Schwarz’s inequality and the assumption of f, we obtain that Ze ⌦" f·˜u"dx0dy3Ck˜u"k(L2( e ⌦"))3, and taking into account that Re ⌦"|D"[˜u"]|dx0dy30,by (15), we have kD"[˜u"]k2 (L2( e ⌦"))3⇥3C µk˜u"k(L2( e ⌦"))3. For the cases a"⇡"or a"⌧", taking into account Remark 4.3(i) in [4], we obtain the second estimate in (13), and, consequently, from classical Korn’s inequality we obtain the last estimate in (13). Now, from the second estimate in (13) and Remark 4.3(i) in [4], we deduce the first estimate in (13). For the case a"", proceeding similarly with Remark 4.3(ii) in [4], we obtain the desired result. 3.1 The extension of (˜u",˜p")to the whole domain ⌦ We extend the velocity ˜u"by zero to the ⌦\e ⌦"and denote the extension by the same symbol. Obviously, estimates (13)-(14) remain valid and the extension is divergence free too. We study in the sequel the following cases for the value of the yield stress g("): i) if a"⇡",witha"/"!,0<<+1, or a"⌧", then g(")=ga ", ii) if a"", then g(")=g", where gis a positive number. These choices are the most challenging ones and they answer to the question adressed in the paper, namely they all preserve in the limit the nonlinear character of the flow. In order to extend the pressure to the whole domain ⌦, the mapping R"(defined in Lemma 4.5 in [4] as R" 2) allows us to extend the pressure p"to Q"by introducing F"in (H1(Q"))3: hF",wiQ"=hrp",R "wi⌦",for any w2(H1 0(Q"))3.(16) 6 Mar´ıa Anguiano and Renata Bunoiu Setting succesively v=u"+R"wand v=u"R"win (9) we get the inequality |hF",wiQ"||a(u",R "w)|+|(f",R "w)⌦"|+j(R"w).(17) Moreover, if div w=0thenhF",wiQ"=0,and the DeRham Theorem gives the existence of P"in L2 0(Q")with F"=rP". Using the change of variables (10), we get for any ˜w2(H1 0(⌦))3where ˜w(x0,y 3)=w(x0,"y 3), Dr"˜ P",˜wE⌦=Z⌦ ˜ P"div"˜wdx 0dy3="1ZQ" P"div wdx="1hrP",wiQ". Then, using the identification (16) of F"and the inequality (17), |Dr"˜ P",˜wE⌦|"1(|a(u",R "w)|+|(f",R "w)⌦"|+j(R"w)) . and applying the change of variables (10), |Dr"˜ P",˜wE⌦||a"(˜u",˜ R"˜w)|+|(f, ˜ R"˜w)e ⌦"|+j"(˜ R"˜w),(18) where ˜ R"˜w=R"wfor any ˜w2(H1 0(⌦))3. Now, we estimate the right-hand side of (18) using the estimates given in Lemma 4.6 in [4]. Lemma 3.2. There exists a constant Cindependent of ", such that the extension ˜ P"2L2 0(⌦)of the pressure ˜p" satisfies   ˜ P"  L2 0(⌦)C. (19) Proof. Let us estimate r"˜ P"in the cases a"⇡"or a"⌧". We estimate the right-hand side of (18). Using Cauchy-Schwarz’s inequality and from the second estimate in (13) we have |a"(˜u",˜ R"˜w)|2µkD"[˜u"]k(L2( e ⌦"))3⇥3  D"˜ R"˜w  (L2( e ⌦"))3⇥3Ca"  D"˜ R"˜w  (L2( e ⌦"))3⇥3. Using the assumption made on the function f, we obtain |(f, ˜ R"˜w)e ⌦"|C  ˜ R"˜w  (L2( e ⌦"))3, and by Cauchy-Schwarz’s inequality and taking into account that |e ⌦"||⌦|, we obtain j"(˜ R"˜w)Ca "  D"˜ R"˜w  (L2( e ⌦"))3⇥3. Then, from (18), we deduce Dr"˜ P",˜wE⌦Ca"  D"˜ R"˜w  (L2( e ⌦"))3⇥3+C  ˜ R"˜w  (L2( e ⌦"))3. Taking into account the third point in Lemma 4.6 in [4], we have Dr"˜ P",˜wE⌦Ca"✓1 a"k˜wk(L2(⌦))3+kD"˜wk(L2(⌦))3⇥3◆+C⇣k˜wk(L2(⌦))3+a"kD"˜wk(L2(⌦))3⇥3⌘. If a"⇡"we take into account that a"⌧1, and if a"⌧"we take into account that a"/"⌧1 and a"⌧1, and we see that there exists a positive constant Csuch that Dr"˜ P",˜wE⌦Ck˜wk(H1 0(⌦))3,8˜w2(H1 0(⌦))3, 7 Mar´ıa Anguiano and Renata Bunoiu and consequently   r"˜ P"  (H1(⌦))3C. It follows that (see for instance Girault and Raviart [22], Chapter I, Corollary 2.1) there exists a representative of ˜ P"2L2 0(⌦) such that   ˜ P"  L2 0(⌦)C  r˜ P"  (H1(⌦))3C  r"˜ P"  (H1(⌦))3C. Finally, let us estimate r"˜ P"in the case a"". Similarly to the previous case, we estimate the right side of (18) by using Cauchy-Schwarz’s inequality and from the second estimate in (14), and we have Dr"˜ P",˜wE⌦C"  D"˜ R"˜w  (L2( e ⌦"))3⇥3+C  ˜ R"˜w  (L2( e ⌦"))3. Taking into account the proof in Lemma 4.5 in [4], the change of variables (10) and that a"", we can deduce Dr"˜ P",˜wE⌦C"✓1 "k˜wk(L2(⌦))3+1 "kDx0˜wk(L2(⌦))3⇥2+1 "k@y3˜wk(L2(⌦))3◆ +C⇣k˜wk(L2(⌦))3+a"kDx0˜wk(L2(⌦))3⇥2+k@y3˜wk(L2(⌦))3⌘, and using that a"⌧1, we see that there exists a positive constant Csuch that Dr"˜ P",˜wE⌦Ck˜wk(H1 0(⌦))3,8˜w2(H1 0(⌦))3, and reasing as the previous case, we have the estimate (19). According to these extensions, problem (12) can be written as: 2µZ⌦ D"[˜u"]:D"[˜v˜u"]dx0dy3+p2g(")Z⌦|D"[˜v]|dx0dy3p2g(")Z⌦|D"[˜u"]|dx0dy3(20) Z⌦ f·(˜v˜u")dx0dy3+Z⌦ ˜ P"div"(˜v˜u")dx0dy3, for every ˜vthat is the extension by zero to the whole ⌦of a function in (H1 0(e ⌦"))3. 3.2 Adaptation of the Unfolding Method The change of variable (10) does not provide the information we need about the behavior of ˜u"in the microstructure associated to e ⌦". To solve this difficulty, we use an adaptation introduced in [4] of the unfolding method from [16] and [17]. Let us recall this adaptation of the unfolding method in which we divide the domain ⌦in cubes of lateral length a"and vertical length 1. For this purpose, given (˜u",˜ P")2(H1 0(⌦))3⇥L2 0(⌦), we define (ˆu",ˆ P")by ˆu"(x0,y)=˜u"✓a"✓x0 a"◆+a"y0,y 3◆,ˆ P"(x0,y)= ˜ P"✓a"✓x0 a"◆+a"y0,y 3◆,a.e. (x0,y)2!⇥Y, (21) where the function is defined in (7). Remark 3.3. For k02K ", the restriction of (ˆu",ˆ P")to Y0 k0,a"⇥Ydoes not depend on x0, whereas as a function of yit is obtained from (˜u",˜ P")by using the change of variables y0=x0a"k0 a" ,which transforms Yk0,a"into Y. 8 Mar´ıa Anguiano and Renata Bunoiu In the left-hand side, we only give the details of convergence for the first nonlinear term, the most challenging one. p2ga "Z!⇥YDx0[˜v]+ 1 a" Dy0[˜v]+1 "@y3[˜v]dx0dy p2gZ!⇥Y|Dy0[˜v]|dx0dy p2gZ!⇥Ya"Dx0[˜v]+Dy0[˜v]+a" "@y3[˜v]Dy0[˜v]dx0dy p2gZ!⇥Y|a"Dx0[˜v]|dx0dy +p2gZ!⇥Ya" "@y3[˜v]dx0dy !0,as "!0. Using (48) the two first terms in the right hand side of (59) tend to the following limit Z!⇥Y f0·(˜v0ˆu0)dx0dy. We consider now the terms which involve the pressure. Taking into account the convergence of the pressure (48) the first term of the pressure tends to the following limit R!⇥Yˆ Pdivx0˜v0dx0dy, and using (50) and taking into account that ˆ Pdoes not depend on y, we have (45). Finally, using that divy0˜v0= 0, we have 1 a"Z!⇥Y ˆ P"divy0˜v0dx0dy =0.(60) It is straightforward to obtain that ˆu3= 0 and therefore we get (57). 4.3 Supercritical case a""(=+1) We obtain some compactness results about the behavior of the sequences (˜u",˜ P") and (ˆu",ˆ P") satisfying the a priori estimates given in Lemmas 3.1-ii) and 3.4-ii), respectively. Lemma 4.5 (Supercritical case).For a subsequence of "still denote by ", there exist ˜u2H1(0,1; L2(!)3), where ˜u3=0and ˜u=0on y3={0,1},ˆu2H1(0,1; L2 ](!⇥Y0)3)(“]” denotes Y0-periodicity), with ˆu=0in !⇥Ys, ˆu=0on y3={0,1}such that RYˆu(x0,y)dy =R1 0˜u(x0,y 3)dy3with RYˆu3dy =0and ˆu3independent of y3,and ˆ P2L2 0(!⇥Y), independent of y, such that ˜u" "2*(˜u0,0) in H1(0,1; L2(!)3),(61) ˆu" "2*ˆuin H1(0,1; L2(!⇥Y0)3),ˆ P"*ˆ Pin L2 0(!⇥Y),(62) divx0✓Z1 0 ˜u0(x0,y 3)dy3◆=0in !,✓Z1 0 ˜u0(x0,y 3)dy3◆·n=0on @!,(63) divy0ˆu0=0in !⇥Y, divx0✓ZY ˆu0(x0,y)dy◆=0in !,✓ZY ˆu0(x0,y)dy◆·n=0on @!.(64) Proof. See Lemmas 5.2, 5.3 and 5.4 in [4] for the proof of (61)-(64). Here, we prove that ˆ Pdoes not depend on the microscopic variable y. To do this, we choose as test function ˜v(x0,y)2D(!;C1 ](Y)3)with˜v(x0,y)=02!⇥Ys (thus, ˜v(x0,x 0/a",y 3)2(H1 0(e ⌦"))3). In order to prove that ˆ Pdoes not depend on y3,weset"˜v(x0,x 0/a",y 3)in (20) (we recall that g(")=g"))and using that div"˜u"= 0, we have 2µ"Z⌦ D"[˜u"]:✓Dx0[˜v]+ 1 a" Dy0[˜v]+1 "@y3[˜v]◆dx0dy32µZ⌦|D"[˜u"]|2dx0dy3(65) +p2g"2Z⌦Dx0[˜v]+ 1 a" Dy0[˜v]+1 "@y3[˜v]dx0dy3p2g"Z⌦|D"[˜u"]|dx0dy3 "Z⌦ f0·˜v0dx0dy3Z⌦ f0·˜u0 "dx0dy3+"Z⌦ ˜ P"divx0˜v0dx0dy3+" a"Z⌦ ˜ P"divy0˜v0dx0dy3+Z⌦ ˜ P"@y3˜v3dx0dy3. 15 Mar´ıa Anguiano and Renata Bunoiu Applying the change of variables given in Remark 3.3 to relation (65) and taking into account (27)-(29), we obtain 2µ"Z!⇥Y✓1 a" Dy0[ˆu"]+1 "@y3[ˆu"]◆:✓1 a" Dy0[˜v]+1 "@y3[˜v]◆dx0dy (66) +p2g"2Z!⇥YDx0[˜v]+ 1 a" Dy0[˜v]+1 "@y3[˜v]dx0dy p2g"Z!⇥Y 1 a" Dy0[ˆu"]+1 "@y3[ˆu"]dx0dy +O" "Z!⇥Y f0·˜v0dx0dy Z!⇥Y f0·ˆu0 "dx0dy +"Z!⇥Y ˆ P"divx0˜v0dx0dy +" a"Z!⇥Y ˆ P"divy0˜v0dx0dy +Z!⇥Y ˆ P"@y3˜v3dx0dy +O". According with (62) and using that a"", one has for the first term in relation (66) 2µ"Z!⇥Y✓" a" 1 "2Dy0[ˆu"]+ 1 "2@y3[ˆu"]◆:✓" a" Dy0[˜v]+@y3[˜v]◆dx0dy !0,as "!0.(67) We pass to the limit in the first nonlinear term and we have p2g"Z!⇥Y"Dx0[˜v]+ " a" Dy0[˜v]+@y3[˜v]dx0dy !0,as "!0.(68) In order to pass the limit in the second nonlinear term, we taking into account that p2g"Z!⇥Y 1 a" Dy0[ˆu"]+1 "@y3[ˆu"]dx0dy =p2g"2Z!⇥Y " a" 1 "2Dy0[ˆu"]+ 1 "2@y3[ˆu"]dx0dy, and using (62), with a"", and the fact that the function E(')=|'|is proper convex continuous, we can deduce that lim inf "!0 p2g"Z!⇥Y 1 a" Dy0[ˆu"]+1 "@y3[ˆu"]dx0dy 0.(69) Moreover, using (62) the two first terms in the right hand side of (66) can be written by "Z!⇥Y f0·˜v0dx0dy "2Z!⇥Y f0·ˆu0 " "2dx0dy !0,as "!0.(70) We consider now the terms which involve the pressure. Taking into account the convergence of the pressure (62) and a"", passing to the limit when "tends to zero, we have Z!⇥Y ˆ P@ y3˜v3dx0dy. (71) Therefore, taking into account (67)-(71), when we pass to the limit in (66) when "tends to zero, we have 0R!⇥Yˆ P@ y3˜v3dx0dy. Now, if we choose as test function "˜v(x0,x 0/a",y 3) in (20) and we argue similarly, we can deduce that ˆ Pdoes not depend on y3. Now, in order to prove that ˆ Pdoes not depend on y0,weseta"˜v=a"(˜v0(x0,x 0/a",y 3),0) in (20) and using that div"˜u"= 0, we have 2µa"Z⌦ D"[˜u"]:✓Dx0[˜v]+ 1 a" Dy0[˜v]+1 "@y3[˜v]◆dx0dy32µZ⌦|D"[˜u"]|2dx0dy3(72) +p2g"a"Z⌦Dx0[˜v]+ 1 a" Dy0[˜v]+1 "@y3[˜v]dx0dy3p2g"Z⌦|D"[˜u"]|dx0dy3 a"Z⌦ f0·˜v0dx0dy3Z⌦ f0·˜u0 "dx0dy3+a"Z⌦ ˜ P"divx0˜v0dx0dy3+Z⌦ ˜ P"divy0˜v0dx0dy3. 16 Mar´ıa Anguiano and Renata Bunoiu Applying the change of variables given in Remark 3.3 to relation (72) and taking into account (27)-(29), we obtain 2µa"Z!⇥Y✓1 a" Dy0[ˆu"]+1 "@y3[ˆu"]◆:✓1 a" Dy0[˜v]+1 "@y3[˜v]◆dx0dy (73) +p2g"a"Z!⇥YDx0[˜v]+ 1 a" Dy0[˜v]+1 "@y3[˜v]dx0dy p2g"Z!⇥Y 1 a" Dy0[ˆu"]+1 "@y3[ˆu"]dx0dy +O" a"Z!⇥Y f0·˜v0dx0dy Z!⇥Y f0·ˆu0 "dx0dy +a"Z!⇥Y ˆ P"divx0˜v0dx0dy +Z!⇥Y ˆ P"divy0˜v0dx0dy. According with (62) and using that a"", the first term in relation (73) can be written by the following way 2µa"Z!⇥Y✓" a" 1 "2Dy0[ˆu"]+ 1 "2@y3[ˆu"]◆:✓" a" Dy0[˜v]+@y3[˜v]◆dx0dy !0,as "!0.(74) In order to pass to the limit in the first nonlinear term, we have p2ga"Z!⇥Y"Dx0[˜v]+ " a" Dy0[˜v]+@y3[˜v]dx0dy !0,as "!0.(75) Moreover, using (62) the two first terms in the right hand side of (73) can be written by a"Z!⇥Y f0·˜v0dx0dy "2Z!⇥Y f0·ˆu0 " "2dx0dy !0,as "!0.(76) We consider now the terms which involve the pressure. Taking into account the convergence of the pressure (62), passing to the limit when "tends to zero, we have Z!⇥Y ˆ Pdivy0˜v0dx0dy. (77) Therefore, taking into account (69) and (74)-(77), when we pass to the limit in (73) when "tends to zero, we have 0 R!⇥Yˆ Pdivy0˜v0dx0dy. Now, if we choose as test function a"˜v=a"(˜v0(x0,x 0/a",y 3),0) in (20) and we argue similarly, we can deduce that ˆ Pdoes not depend on y0,so ˆ Pdoes not depend on y. Theorem 4.6 (Supercritical case).If a"", then (ˆu"/"2,ˆ P")converges to (ˆu, ˆ P)in H1(0,1; L2(!⇥Y0)3)⇥ L2 0(!⇥Y), which satisfies the following variational equality 2µZ!⇥Y @y3[ˆu0]:(@y3[˜v0]@y3[ˆu0]) dx0dy +p2gZ!⇥Y|@y3[˜v0]|dx0dy p2gZ!⇥Y|@y3[ˆu0]|dx0dy Z!⇥Y f0·(˜v0ˆu0)dx0dy Z!⇥Yrx0ˆ P(˜v0ˆu0)dx0dy, (78) for every ˜v2H1(0,1; L2(!⇥Y0)3)such that ˜v(x0,y)=0in !⇥Ys,divy0˜v0=0in !⇥Y, ✓ZY ˜v0(x0,y)dy◆·n=0on @!. Proof. We choose a test function ˜v(x0,y)2D(!;C1 ](Y)3)with˜v(x0,y)=02!⇥Ys(thus, ˜v(x0,x 0/a",y 3)2 (H1 0(e ⌦"))3). We first multiply (20) by "2and we use that div"˜u"= 0. Then, we take a test function "2˜v(x0,x 0/a",y 3), with ˜v3independent of y3and with ˜v(x0,y)=0in!⇥Ysand satisfying the incompressibility conditions (64), that is, divy0˜v0=0in!⇥Yand RY˜v0(x0,y)dy·n= 0 on @!, and we have 2µZ⌦ D"[˜u"]:✓Dx0[˜v]+ 1 a" Dy0[˜v]+1 "@y3[˜v]◆dx0dy32µ1 "2Z⌦|D"[˜u"]|2dx0dy3(79) +p2g"Z⌦Dx0[˜v]+ 1 a" Dy0[˜v]+1 "@y3[˜v]dx0dy3p2g1 "Z⌦|D"[˜u"]|dx0dy3 Z⌦ f0·˜v0dx0dy31 "2Z⌦ f0·˜u0 "dx0dy3+Z⌦ ˜ P"divx0˜v0dx0dy3+1 a"Z⌦ ˜ P"divy0˜v0dx0dy3. 17 Mar´ıa Anguiano and Renata Bunoiu Applying the change of variables given in Remark 3.3 to relation (79), arguing as in the critical case, we obtain 2µZ!⇥Y✓1 a" Dy0[ˆu"]+1 "@y3[ˆu"]◆:✓1 a" Dy0[˜v]+1 "@y3[˜v]◆dx0dy (80) 2µ1 "2Z!⇥Y 1 a" Dy0[ˆu"]+1 "@y3[ˆu"] 2 dx0dy +p2g"Z!⇥YDx0[˜v]+ 1 a" Dy0[˜v]+1 "@y3[˜v]dx0dy p2g1 "Z!⇥Y 1 a" Dy0[ˆu"]+1 "@y3[ˆu"]dx0dy +O" Z!⇥Y f0·˜v0dx0dy 1 "2Z!⇥Y f0·ˆu0 "dx0dy +Z!⇥Y ˆ P"divx0˜v0dx0dy +1 a"Z!⇥Y ˆ P"divy0˜v0dx0dy +O". According with (62), the first term in relation (80) can be written by the following way 2µZ!⇥Y✓" a" 1 "2Dy0[ˆu"]+ 1 "2@y3[ˆu"]◆:✓" a" Dy0[˜v]+@y3[˜v]◆dx0dy, and, taking into account that a"", this term tends to the following limit 2µZ!⇥Y @y3[ˆu0]:@y3[˜v0]dx0dy. (81) The second term in relation (80) writes 2µZ!⇥Y✓" a" 1 "2Dy0[ˆu"]+ 1 "2@y3[ˆu"]◆:✓" a" 1 "2Dy0[ˆu"]+ 1 "2@y3[ˆu"]◆dx0dy, and, taking into account that the function B(')=|'|is proper convex continuous and a"", we get that the lim inf"!0of this second is greater or equal than 2µZ!⇥Y @y3[ˆu0]:@y3[ˆu0]dx0dy. (82) In order to pass to the limit in the first nonlinear term, using that a"",wehave p2g"Z!⇥YDx0[˜v]+ 1 a" Dy0[˜v]+1 "@y3[˜v]dx0dy p2gZ!⇥Y|@y3[˜v0]|dx0dy p2gZ!⇥Y"Dx0[˜v]+ " a" Dy0[˜v]+@y3[˜v]@y3[˜v]dx0dy p2gZ!⇥Y|"Dx0[˜v]|dx0dy +p2g" a"Z!⇥Y|Dy0[˜v]|dx0dy !0,as "!0. Now, in order to pass the limit in the second nonlinear term, taking into account that p2g1 "Z!⇥Y 1 a" Dy0[ˆu"]+1 "@y3[ˆu"]dx0dy =p2gZ!⇥Y " a" 1 "2Dy0[ˆu"]+ 1 "2@y3[ˆu"]dx0dy, and using (62) and the fact that the function E(')=|'|is proper convex continuous and a"", we can deduce that lim inf "!0 p2g1 "Z!⇥Y 1 a" Dy0[ˆu"]+1 "@y3[ˆu"]dx0dy p2gZ!⇥Y|@y3[ˆu]|dx0dy. (83) Moreover, using (62) the two first terms in the right hand side of (80) tend to the following limit Z!⇥Y f0·(˜v0ˆu0)dx0dy. (84) We consider now the terms which involve the pressure. Taking into account the convergence of the pressure (62) the first term of the pressure tends to the following limit R!⇥Yˆ Pdivx0˜v0dx0dy, and using (64) and taking into account that ˆ Pdoes not depend on y, we have (45). Finally using that divy0˜v0= 0, we have (60). Therefore, taking into account (45), (60) and (81)-(84), we get (78). 18 Mar´ıa Anguiano and Renata Bunoiu 5 Conclusions By using dimension reduction and homogenization techniques, we studied the limiting behavior of the velocity and of the pressure for a nonlinear viscoplastic Bingham flow with small yield stress, in a thin porous medium of small height "and for which the relative dimension of the pores is a". Three cases are studied following the value of =lim "!0a"/"and, at the limit, they all preserve the nonlinear character of the flow. More precisely, according to [29], each of the limit problems (36), (57) and (78), is written as a nonlinear Darcy equation: 8 > < > : ˜ U0(x0)=K⇣f0(x0)rx0ˆ P(x0)⌘in !, divx0˜ U0(x0)=0 in!, ˜ U0(x0)·n= 0 on @!. (85) The velocity of filtration ˜ U(x0)=⇣˜ U0(x0),˜ U3(x0)⌘is defined by ˜ U(x0)=Z Y ˆu(x0,y)dy = 1 Z 0 0 @Z Y0 ˆu(x0,y0,y 3)dy01 Ady3= 1 Z 0 ˜u(x0,y 3)dy3. We remark that in all three cases, the vertical component ˜ U3of the velocity of filtration equals zero and this result is in accordance with the previous mathematical studies of the flow in this thin porous medium, for newtonian fluids (Stokes and Navier-Stokes equations) and for power law fluids (see [20], [1], [2], [4], [5]). Moreover, despite the fact that the limit pressure is not unique, the velocity of filtration is uniquely determined (see Section 4.3 in [29]). In (85), the function K:R2! R2is nonlinear and its expression can not be made explicit for the Bingham flow (see [29]). Nevertheless, in each case, for a given ⇠2R2, one has K(⇠)=R Y ⇠ (y)dy, with ⇠ solution of a local problem stated in the cell Y.If0<<+1, the local problem is a 3-D Bingham problem. If = 0, the local problem is a 2-D Bingham problem (defined for each y32]0,1[), while if =+1 the 1-D local problem (defined for each y02Y0) corresponds to a lower-dimensional Bingham-like law (see [15]). 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