Analysis of the roughness regimes for micropolar fluids via homogenization

Analysis of the roughness regimes for micropolar fluids via homogenization Francisco Javier SU´ AREZ-GRAU∗ Abstract We study the asymptotic behavior of micropolar fluid flows in a thin domain of thickness ηεwith a periodic oscillating boundary with wavelength ε. We consider the limit when εtends to zero and, depending on the limit of the ratio of ηε/ε, we prove the existence of three different regimes. In each regime, we derive a generalized Reynolds equation taking into account the microstructure of the roughness. AMS classification numbers: 76D08, 76A20, 76A05, 76M50, 35B27, 35Q35. Keywords: Homogenization; micropolar fluid flow; Reynolds equation; thin-film fluid. 1 Introduction We study in this paper the effect of small domain irregularities on thin film flows governed by the linearized 3D micropolar equations. In the case of Newtonian fluids governed by the Stokes or Navier-Stokes equations, this problem has been widely studied since Bayada and Chambat [4] provided, by means of homogenization techniques, a rigorous derivation of the classical 2D Reynolds equation div −h3 12ν∇p+b= 0 ,(1.1) where hrepresents the film thickness, pis the pressure, νis the fluid viscosity and bis a vectorial function that usually appears from the exterior forces or from the imposed velocity on a part of the boundary. In this sense, various asymptotic Reynolds-like models, in special regimes, have been obtained depending on the ratio between the size of the roughness and the thickness of the domain and the boundary conditions considered on a part of the boundary, see for example Bayada et al. [8], Benhaboucha et al. [10], Benterki et al. [11], Bresch et al. [14], Boukrouche and Ciuperca [15], Chupin and Martin [18], Letoufa et al. [24], Su´arez-Grau [33], and references therein. More precisely, a very general result was obtained in Bayada and Chambat [5, 6], see also Mikelic [29]. Assuming that the thickness of the domain is rapidly oscillating, i.e. the thickness is given by a small parameter ηεand one of the boundary is rough with small roughness of wavelength ε, it was proved that depending on the limit of the ratio ηε/ε, denoted as λ, there exist three characteristic regimes: Stokes roughness (0 < λ < +∞), Reynolds roughness (λ= 0) and high-frequency roughness (λ= +∞). In particular, it was obtained that the flow is governed by a generalized 2D Reynolds equation of the form div (−Aλ∇p+bλ)=0,(1.2) for 0 ≤λ≤+∞, where Aλand bλare macroscopic quantities known as flow factors, which take into account the microstructure of the roughness. Moreover, it holds that in the Stokes roughness regime the flow factors are calculated by solving 3D local Stokes-like problems depending on the parameter λ, while in the Reynolds ∗Departamento de Ecuaciones Diferenciales y An´alisis Num´erico. Facultad de Matem´aticas. Universidad de Sevilla. 41012-Sevilla (Spain) [email protected] 1 Francisco J. Su´arez-Grau roughness regime they are obtained by solving 2D local Reynolds-like problems, which represents a considerable simplification. In the high-frequency roughness regime, due to the highly oscillating boundary, the velocity vanishes in the oscillating zone and then, the classical Reynolds equation (1.1) is deduced in the non-oscillating zone, so there are no local problems to solve. This result has been formally generalized to the unstationary case (the rough surface is moving) in Fabricius et al. [22], and recently rigorously generalized to the case of non-Newtonian fluids governed by the 3D Navier-Stokes system with a nonlinear viscosity (power law) in Anguiano and Su´arez-Grau [2]. On the other hand, we remark that there are not many papers in the existing literature dealing with the mathematical modeling of micropolar fluid film lubrication. A generalized version of the Reynolds equation, formally obtained in a critical case when one of the non-Newtonian characteristic parameters has specific (small) order of magnitude, can be found in Singh and Sinha [32] where the authors consider a specific slider-type bearing. Later, in Bayada and Lukaszewicz [9], it was developed the rigorous derivation, obtaining the generalized version of the 2D Reynolds equation (1.1) for micropolar thin film fluids, which has the form div −h3 1−N2Φ(h, N)∇p+b= 0,(1.3) where Nis the coupling number and Φ(h, N) = 1 12 +1 4h2(1 −N2)−1 4hrN2 1−N2coth Nhp1−N2. We also refer to Dupuy et al. [19], for the case of micropolar flow in a curved channel, and to Marusic-Paloka et al. [28], for the asymptotic Brinkman-type model proposed starting from 3D micropolar equations. We remark that in previous papers, the micropolar fluid film has been considered in a simple thin domain with no roughness introduced. Recently, the roughness effects on a thin film flow have been studied as well and new mathematical models have been proposed in Boukrouche and Paoli [13], where the authors consider micropolar flow in a 2D domain assuming the roughness is of the same small order as the film thickness. Employing two-scale convergence technique, they derive the limit problem describing the macroscopic flow. Later, in Pazanin and Su´arez-Grau [31], a version of the Reynolds equation is derived in the case of a 3D domain with a particular roughness pattern, where the wavelength of the roughness is assumed to be smaller than the thickness, through a variant of the notion of two-scale convergence introduced in Bresch et al. [14]. Our goal in this paper is to give a general classification result for thin film flows of micropolar fluids with rapidly oscillating thickness in the spirit of [2, 5, 6], by considering a 3D domain with a thickness given by the parameter ηεand the wavelength of the roughness by ε. To do this, we use extension results for thin domains and an adaptation of the unfolding method (see Cioranescu et. al [16, 17]) developed in [2]. As a result, we deduce that the three characteristic regimes fluids are still valid for micropolar fluids, and moreover, we derive a generalized version of the Reynolds equation of the form (1.2) depending on λ. Also, the flow factors are calculated in a different way depending on the regime. More precisely, in the Stokes roughness regime (0 < λ < +∞) the flow factors are calculated by solving 3D local micropolar Stokes-like problems depending on the parameter λ, while in the Reynolds roughness regime (λ= 0) they are obtained by solving 2D local micropolar Reynolds-like problems. Finally, in the high-frequency roughness regime (λ= +∞) due to the highly oscillating boundary, the classical micropolar Reynolds equation (1.3) is deduced in the non-oscillating zone, and there are no local problems to solve. The paper is organized as follows. In Section 2 we introduce the domain and some useful notation, and we state the problem. In Section 3, we give some a priori estimates for the velocity, microrotation and pressure, and we introduce the extension results and the version of the unfolding method necessary to pass to the limit depending on each regime. The Stokes roughness regime is considered in Section 4, the Reynolds roughness regime in Section 5, and the high-frequency roughness regime in Section 6. The corresponding main concergence results are stated in Theorems 4.3, 5.3 and 6.2, respectively. The paper ends with a conclusion section, an appendix, where we recall the computation of the coefficients of the classical micropolar Reynolds equation (1.3), and with a list of references. 2 Francisco J. Su´arez-Grau 2 Statement of the problem In this section, we first define the thin domain and some sets necessary to study the asymptotic behavior of the solutions. Next, we introduce the problem considered in the thin domain and also, the rescaled problem posed in a domain of fixed height. We finish this section giving the equivalent weak variational formulation for both problems. The domain. A thin domain with a rapidly oscillating thickness is defined by a domain ωand an associated microstructure given by a function hε(x0) = ηεh(x0/ε) that models the roughness of the upper surface and depends on two small positive parameters ηεand ε, representing the thickness of the domain and the wavelength of the roughness, respectively. More precisely, we assume that ωis an open, smooth, bounded and connected set of R2, and his a positive and smooth function, defined for y0in R2,Y0-periodic with Y0= (−1/2,1/2)2the cell of periodicity in R2, and there exist hmin and hmax such that 0< hmin = min y0∈Y0h(y0), hmax = max y0∈Y0h(y0). We remark that along this paper, the points x∈R3will be decomposed as x= (x0, x3) with x0∈R2,x3∈R. We also use the notation x0to denote a generic vector of R2. Thus, we define the thin domain Ωε⊂R3by Ωε=(x0, x3)∈R2×R:x0∈ω, 0< x3< hε(x0), and the oscillating part of the boundary by Σε=ω× {hε(x0)}. We also define the respective rescaled sets e Ωε=ω×(0, h(x0/ε)) and e Σε=ω× {h(x0/ε)}. Related to the microstructure of the periodicity of the boundary, we consider that the domain ωis covered by a rectangular mesh of size ε: for k0∈Z2, each cell Y0 k0,ε =εk0+εY 0, and for simplicity, we assume that there exists an exact finite number of periodic sets Y0 k0,ε. We define Tε={k0∈Z2:Y0 k0,ε ∩ω6=∅}. Also, we define Yk0,ε =Y0 k0,ε ×(0, h(y0)) and Y=Y0×(0, h(y0)), which is the reference cell in R3. Two quantities hmin and hmax allow us to define: – The extended sets Qε=ω×(0, ηεhmax), Ω = ω×(0, hmax) and Σ = ω× {hmax}. – The extended cube e Qk0,ε =Y0 k0,ε ×(0, hmax) for k0∈Z2. – The restricted sets Ω+ ε=ω×(ηεhmin, hε(x0)), e Ω+ ε=ω×(hmin, h(x0/ε)), Ω+=ω×(hmin, hmax) and Ω−=ω×(0, hmin). – The extended and restricted basic cells Π = Y0×(0, hmax), Π+=Y0×(hmin, hmax) and Π−=Y0×(0, hmin). In order to apply the unfolding method, we will use the following notation. For k0∈Z2, we define κ:R2→Z2 by κ(x0) = k0⇐⇒ x0∈Y0 k0,1.(2.4) Remark that κis well defined up to a set of zero measure in R2(the set ∪k0∈Z2∂Y 0 k0,1). Moreover, for every ε > 0, we have κx0 ε=k0⇐⇒ x0∈Y0 k0,ε . We denote by Oεa generic real sequence which tends to zero with εand can change from line to line. We denote by Ca generic constant which can change from line to line. To finish, let C∞ #(Y) be the space of infinitely differentiable 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. 3 Francisco J. Su´arez-Grau The problem. When the distance between two surfaces becomes very small, the experimental results from the tribology literature (see e.g. [23, 26, 27]) suggest that the fluid’s internal structure should be taken into account as well. Among various non-Newtonian models, the model of micropolar fluid (proposed by Eringen [21] in 60’s) turns out to be the most appropriate since it acknowledges the effects of the local structure and micromotions of the fluid elements. Physically, micropolar fluids consist in a large number of small spherical particles uniformly dispersed in a viscous medium. Assuming that the particles are rigid and ignoring their deformations, the related mathematical model expresses the balance of momentum, mass and angular momentum. A new unknown function called microrotation (i.e. the angular velocity field of rotation of particles) is added to the usual velocity and pressure fields. Consequently, Navier-Stokes equations become coupled with a new vector equation coming from the conservation of angular momentum with four microrotation viscosities introduced (see [25] for more details). Being able to describe numerous real fluids better than the classical (Newtonian) model, micropolar fluid models have been extensively studied in recent years (see e.g. [12, 13, 19, 31]). Taking into account the application we want to model (lubrication with micropolar fluid), it is reasonable to assume a small Reynolds number and omit the inertial terms in momentum equations of the micropolar system. Also, it has been observed that the magnitude of the viscosity coefficients appearing in the micropolar equations may influence the effective flow. Thus, it is reasonable to work with the system written in a non-dimensional form (see e.g. [9] for more details). Thus, we consider the stationary flow of an incompressible micropolar fluid in Ωεwhich is governed by the following linearized micropolar system formulated in a non-dimensional form        −div(Duε) + ∇pε= 2N2rot wε+fεin Ωε, div uε= 0 in Ωε, −RMdiv(Dwε)+4N2wε= 2N2rot uε+gεin Ωε, (2.5) with homogeneous boundary conditions (it does not alter the generality of the problem under consideration), uε=wε= 0 on ∂Ωε.(2.6) In system (2.5)-(2.6), the velocity uε, the pressure pεand the microrotation wεare unknown. Dimensionless (non- Newtonian) parameter N2characterizes the coupling between the equations for the velocity and microrotation and it is of order O(1), in fact N2lies between zero and one. The second dimensionless parameter, denoted by RMis, in fact, related to the characteristic length of the microrotation effects and is compared with the small parameter ηεby assuming that RM=O(η2 ε), namely RM=η2 εRcwith Rc=O(1) .(2.7) This case is the situation that is commonly introduced to study the micropolar fluid because the third equation of (2.5) shows a strong coupling between velocity and microrotation in the limit (see [7, 9]). Under assumptions that fε, gε∈L2(Ωε)3, it is well known that problem (2.5)-(2.6) has a unique weak solution (uε, wε, pε)∈H1 0(Ωε)3×H1 0(Ωε)3×L2 0(Ωε) (see [25]), where the space L2 0is the space of functions of L2with null integral. Our aim is to study the asymptotic behavior of uε,wεand pεwhen εand ηεtend to zero and identify homogenized models coupling the effects of the thickness of the domain and the roughness of the boundary. For this purpose, as usual when we deal with thin domains, we use the dilatation in the variable x3given by y3=x3 ηε ,(2.8) in order to have the functions defined in the open set with fixed height e Ωεwith oscillating boundary e Σε. Namely, we define ˜uε,˜wε∈H1 0(e Ωε)3and ˜pε∈L2 0(e Ωε) by ˜uε(x0, y3) = uε(x0, ηεy3),˜wε(x0, y3) = wε(x0, ηεy3), ˜pε(x0, y3) = pε(x0, ηεy3),a.e. (x0, y3)∈e Ωε.(2.9) 4 Francisco J. Su´arez-Grau Let us introduce some notation which will be useful in the following. For a vectorial function v= (v0, v3) and a scalar function w, we introduce the operators Dηε,∇ηεand rotηεby (Dηεv)ij =∂xjvifor i= 1,2,3, j = 1,2,(Dηε)i,3=1 ηε∂y3vifor i= 1,2,3, ∇ηεw= (∇x0w, 1 ηε∂y3w)t,divηεv= divx0v0+1 ηε∂y3v3, rotηεv=rotx0v3+1 ηεroty3v0,Rotx0v0t , where, denoting (v0)⊥= (−v2, v1)t, we define rotx0v3= (∂x2v3,−∂x1v3)t,roty3v0=∂y3(v0)⊥, Rotx0v0=∂x1v2−∂x2v1.(2.10) Using the transformation (2.8), the rescaled system (2.5)-(2.6) can be rewritten as        −divηε(Dηε˜uε) + ∇ηε˜pε= 2N2rotηε˜wε+˜ fεin e Ωε, divηε˜uε= 0 in e Ωε, −η2 εRcdivηε(Dηε˜wε)+4N2˜wε= 2N2rotηε˜uε+ ˜gεin e Ωε, (2.11) with homogeneous boundary conditions ˜uε= ˜wε= 0 on ∂e Ωε,(2.12) where ˜ fεand ˜gεare defined similarly as in (2.9). Our goal then is to describe the asymptotic behavior of this new sequences ˜uε, ˜wεand ˜pεwhen εand ηεtend to zero. To do this, we need to obtain appropriate estimates, so it is usual to consider for fεand gεthe following estimates kfεkL2(Ωε)3≤Cη 1 2 ε,kgεkL2(Ωε)3≤Cη 3 2 ε.(2.13) For example, assuming f, g ∈L2(Ω), we can consider as external forces satisfying (2.13) the following ones fε(x) = fx0,x3 ηε, gε(x) = ηεgx0,x3 ηεa.e. x∈Ωε. We point out that due to the thickness of the domain, it is usual to assume that the vertical components of the external forces can be neglected and, moreover the forces can be considered independent of the vertical variable. Thus, for sake of simplicity, assuming f0, g0∈L2(ω)2, along the paper we will consider the following assumptions: (i) If ηε≈ε, with ηε/ε →λ, 0 <λ<+∞, or ηεε, then fε(x)=(f0(x0),0), gε= (ηεg0(x0),0),a.e. x∈Ωε.(2.14) (ii) If ηεε, then fε(x)=(f0(x0),0), gε= (εg0(x0),0),a.e. x∈Ωε.(2.15) We observe that in this case ˜ fε=fεand ˜gε=gεand that in (i) the external forces satisfy (2.13). However, in the case (ii), due to the high oscillations of the boundary, to obtain appropriate estimates it is necessary to assume that gεsatisfies a more precise estimate, that is kgεkL2(Ωε)3≤Cεη 1 2 ε(see proof of Lemma 3.3 for more details). 5 Francisco J. Su´arez-Grau Weak variational formulations. We finish this section by giving the equivalent weak variational formulation of system (2.5)-(2.6) and the rescaled system (2.11)-(2.12), which will be useful in next sections. For problem (2.5)-(2.6), the weak variational formulation is to find uε, wε∈H1 0(Ωε)3and pε∈L2 0(Ωε) such that                              ZΩε Duε:Dϕ dx −ZΩε pεdiv ϕ dx = 2N2ZΩε rot wε·ϕ dx +ZΩε fε·ϕ dx, η2 εRcZΩε Dwε:Dψ dx + 4N2ZΩε wε·ψ dx = 2N2ZΩε rot uε·ψ dx +ZΩε gε·ψ dx , (2.16) for every ϕ, ψ ∈H1 0(Ωε)3, and the equivalent weak variational formulation for the rescaled system (2.11)-(2.12) is to find ˜uε,˜wε∈H1 0(e Ωε)3and ˜pε∈L2 0(e Ωε) such that                              Ze Ωε Dηε˜uε:Dηε˜ϕ dx0dy3−Ze Ωε ˜pεdivηε˜ϕ dx0dy3 = 2N2Ze Ωε rotηε˜wε·˜ϕ dx0dy3+Ze Ωε ˜ fε·˜ϕ dx0dy3, η2 εRc Ze Ωε Dηε˜wε:Dηε˜ ψ dx0dy3+ 4N2Ze Ωε ˜wε·˜ ψ dx0dy3 = 2N2Ze Ωε rotηε˜uε·˜ ψ dx0dy3+Ze Ωε ˜gε·˜ ψ dx0dy3, (2.17) for every ˜ϕ, ˜ ψ∈H1 0(e Ωε)3. 3A priori estimates First, we recall the Poincar´e inequality in a domain with thickness ηε(see [30]). Lemma 3.1. For every v∈H1 0(Ωε)3, the following inequality holds kvkL2(Ωε)3≤c2ηεkDvkL2(Ωε)3×3,(3.18) where c2>0is independent of v,εand ηε. Next, we give the following results relating the derivative and the rotational. Lemma 3.2. For v∈H1 0(Ωε)3, the following inequality holds krot vkL2(Ωε)3≤ kDvkL2(Ωε)3×3,(3.19) and, if moreover div v= 0 in Ωε, then it holds krot vkL2(Ωε)3=kDvkL2(Ωε)3×3.(3.20) Proof. By using relation −∆v= rot (rot v)− ∇ divϕ, it can be proved (see [20]) that ZΩε |Dv|2dx =ZΩε |rot v|2dx +ZΩε |div v|2dx, ∀v∈H1 0(Ωε)3. Then, (3.19) easily holds, and (3.20) is a consequence of the free divergence condition.  6 Francisco J. Su´arez-Grau We start by obtaining some a priori estimates for uε,wε, ˜uεand ˜wε. Lemma 3.3. There exists a constant Cindependent of ε, such that the solution (uε, wε)of problem (2.5)-(2.6) and the corresponding rescaled solution (˜uε,˜wε)of the problem (2.11)-(2.12) satisfy kuεkL2(Ωε)3≤Cη 5 2 ε,kDuεkL2(Ωε)3×3≤Cη 3 2 ε,(3.21) kwεkL2(Ωε)3≤Cη 3 2 ε,kDwεkL2(Ωε)3×3≤Cη 1 2 ε,(3.22) k˜uεkL2( e Ωε)3≤Cη2 ε,kDηε˜uεkL2( e Ωε)3×3≤Cηε,(3.23) k˜wεkL2( e Ωε)3≤Cηε,kDηε˜wεkL2( e Ωε)3×3≤C . (3.24) Moreover, in the case ηεε, defining the restriction functions u+ ε:= uε|Ω+ ε,w+ ε:= wε|Ω+ ε,˜u+ ε:= ˜uε|e Ω+ εand ˜w+ ε:= ˜wε|e Ω+ ε, we also have the following estimates ku+ εkL2(Ω+ ε)3≤η 1 2 εε2,kDu+ εkL2(Ω+ ε)3×3≤η 1 2 εε, (3.25) kw+ εkL2(Ω+ ε)3≤η−3 2 εε3,kDw+ εkL2(Ω+ ε)3×3≤η−3 2 εε2,(3.26) k˜u+ εkL2( e Ω+ ε)3≤ε2,kDηε˜u+ εkL2( e Ω+ ε)3×3≤ε, (3.27) k˜w+ εkL2( e Ω+ ε)3≤η−2 εε3,kDηε˜w+ εkL2( e Ω+ ε)3×3≤η−2 εε2.(3.28) Proof. For every cases, taking ϕ=uεas test function in the first equation of (2.16), taking into account RΩεrot wε·uεdx =RΩεrot uε·wεdx, applying Cauchy-Schwarz’s inequality and from (2.14), (2.15), (3.18) and (3.20), we have kDuεk2 L2(Ωε)3×3= 2N2ZΩε rot wε·uεdx +ZΩε fε·uεdx = 2N2ZΩε wε·rot uεdx +ZΩε f0(x0)·u0 εdx ≤2N2kwεkL2(Ωε)3kDuεkL2(Ωε)3×3+η 3 2 εc2kf0kL2(ω)2kDuεkL2(Ωε)3×3, (3.29) which implies η−3 2 εkDuεkL2(Ωε)3×3≤η−3 2 ε2N2kwεkL2(Ωε)3+c2kf0kL2(ω)2.(3.30) In the cases ηε≈εand ηεε, taking ψ=wεas test function in the second equation of (2.16), applying Cauchy-Schwarz’s inequality and taking into account (2.14), we have η2 εRckDwεk2 L2(Ωε)3×3+ 4N2kwεk2 L2(Ωε)3 = 2N2ZΩε rot uε·wεdx +ηεZΩε g0(x0)·w0 εdx ≤2N2kwεkL2(Ωε)3kDuεkL2(Ωε)3×3+η 3 2 εkg0kL2(ω)2kwεkL2(Ωε)3, (3.31) which implies η−3 2 ε2N2kwεkL2(Ωε)3≤η−3 2 εN2kDuεkL2(Ωε)3×3+1 2kg0kL2(ω)2.(3.32) In the case ηεε, proceeding as above by taking into account (2.15), and using that in this case εη 1 2 εkg0kL2(ω)2kwεkL2(Ωε)3≤η 3 2 εkg0kL2(ω)2kwεkL2(Ωε)3, 7 Francisco J. Su´arez-Grau then estimate (3.32) also holds. Then, from (3.30) and (3.32), we conclude for every cases that η−3 2 εkDuεkL2(Ωε)3×3≤c2 1−N2kf0kL2(ω)2+1 2(1 −N2)kg0kL2(ω)2, which gives the second estimate in (3.21). This together with (3.18) gives the first one. Moreover, by means of the dilatation (2.8), we get (3.23). To get the second estimate in (3.22), we use RΩεrot uε·wεdx =RΩεrot wε·uεdx in (3.31), (3.18) and (3.19), and proceeding as above we obtain in every cases η2 εRckDwεk2 L2(Ωε)3×3+ 4N2kwεk2 L2(Ωε)3 ≤2N2kuεkL2(Ωε)3kDwεkL2(Ωε)3×3+η 5 2 εkg0kL2(ω)2kDwεkL2(Ωε)3×3, (3.33) which, by using the estimate of uεgiven in (3.21), provides η2 εRckDwεkL2(Ωε)3×3≤2N2η 5 2 εC+η 5 2 εc2kg0kL2(ω)2. This implies (3.22), and by means of the dilatation, we get (3.24). Finally, in the case ηεε, estimates (3.25)-(3.26) in Ω+ εare obtained similarly as above by using the following Poincar´e’s inequality in Ω+ ε, kvkL2(Ω+ ε)3≤CεkDvkL2(Ω+ ε)3×3,∀v∈H1 0(Ω+ ε)3.(3.34) This estimate is obtained by using the fact that in the case ηεε, in Ω+ εwe can find the boundary with homogeneous boundary condition at distance εintegrating along the horizontal variable x0. Thus, taking u+ εas test function in the first equation of (2.16) and using (3.34), we get kDu+ εk2 L2(Ω+ ε)3×3 ≤2N2kw+ εkL2(Ω+ ε)3kDu+ εkL2(Ω+ ε)3×3+εη 1 2 εc2kf0kL2(ω)2kDu+ εkL2(Ω+ ε)3×3, and then ε−1η−1 2 εkDu+ εkL2(Ω+ ε)3×3≤ε−1η−1 2 ε2N2kw+ εkL2(Ω+ ε)3+c2kf0kL2(ω)2.(3.35) Next, we obtain η2 εRckDw+ εk2 L2(Ω+ ε)3×3+ 4N2kw+ εk2 L2(Ω+ ε)3 ≤2N2kw+ εkL2(Ω+ ε)3kDu+ εkL2(Ω+ ε)3×3+εη 1 2 εkg0kL2(ω)2kw+ εkL2(Ω+ ε)3, and then ε−1η−1 2 ε2N2kw+ εkL2(Ω+ ε)3≤ε−1η−1 2 εN2kDu+ εkL2(Ω+ ε)3×3+1 2kg0kL2(ω)2. From the above estimates, we get the second estimate in (3.25) and by (3.34), the first one. By means of the dilatation we deduce (3.27). Finally, by applying (3.34), we have η2 εRckDw+ εk2 L2(Ω+ ε)3×3+ 4N2kw+ εk2 L2(Ω+ ε)3 ≤2N2ku+ εkL2(Ω+ ε)3kDw+ εkL2(Ω+ ε)3×3+ε2η 1 2 εkg0kL2(ω)2kDw+ εkL2(Ω+ ε)3×3, which, by using the estimate of u+ εgiven in (3.25), provides the second estimate in (3.26), and then the first one. Moreover, by means of the dilatation we deduce (3.28) which ends the proof.  8 Francisco J. Su´arez-Grau 3.1 The extension of (˜uε,˜wε,˜pε)to the whole domain Ω The sequence of solutions (˜uε,˜wε,˜pε)∈H1 0(e Ωε)3×H1 0(e Ωε)3×L2 0(e Ωε) is not defined in a fixed domain independent of εbut rather in a varying set e Ωε. In order to pass to the limit if εtends to zero, convergences in fixed Sobolev spaces (defined in Ω) are used, which requires first that (˜uε,˜wε,˜pε) be extended to the whole domain Ω. Therefore, we extend ˜uεand ˜wεby zero in Ω \e Ωε(this is compatible with the homogeneous boundary condition on ∂e Ωε), and denote the extensions by the same symbol. Obviously, estimates (3.21)-(3.24) remain valid and the extension ˜uεis divergence free too. Extending the pressure is a much more difficult task. A continuation of the pressure for a flow in a porous media was introduced in [34]. This construction applies to periodic holes in a domain Ωεwhen each hole is strictly contained into the periodic cell. In this context, we can not use directly this result because the “holes” are along the boundary Σεof Ωεand moreover, the scale of the vertical direction is smaller than the scales of the horizontal directions. This fact will induce several limitations in the results obtained by using the method, especially in view of the convergence for the pressure. In this sense, for the case of Newtonian fluids, an operator Rεgeneralizing the results of [34] to this context (extending the pressure from Ωεto Qε) was introduced in [6, 29], and later extended to the case of non-Newtonian (power law) fluids [2] by defining an extension operator Rε p, for every flow index p > 1. Then, in order to extend the pressure to the whole domain Ω, the mapping Rε(defined in [2, Lemma 4.6] as Rε 2) allows us to extend the pressure pεfrom Ωεto Qεby introducing Fεin H−1(Qε)3as follows (brackets are for duality products between H−1and H1 0) hFε, ϕiQε=h∇pε, Rε(ϕ)iΩεfor any ϕ∈H1 0(Qε)3.(3.36) We calcule the right hand side of (3.36) by using the first equation of (2.16) and we have hFε, ϕiQε=−ZΩε Duε:DRε(ϕ)dx + 2N2ZΩε rot wε·Rε(ϕ)dx +ZΩε f0(x0)·Rε(ϕ)0dx . (3.37) Using Lemma 3.3 for fixed ε, we see that it is a bounded functional on H1 0(Qε) (see the proof of Lemma 3.4 below) and in fact Fε∈H−1(Qε)3. Moreover, divϕ= 0 implies hFε, ϕiQε= 0, and the DeRham theorem gives the existence of Pεin L2 0(Qε) with Fε=∇Pε. Defining the rescaled extended pressure ˜ Pε∈L2 0(Ω) by ˜ Pε(x0, y3) = Pε(x0, ηεy3),a.e. (x0, y3)∈Ω, we get for any ˜ϕ∈H1 0(Ω)3where ˜ϕ(x0, y3) = ϕ(x0, ηεy3) that h∇ηε˜ Pε,˜ϕiΩ=−ZΩ ˜ Pεdivηε˜ϕ dx0dy3 =−η−1 εZQε Pεdiv ϕ dx =η−1 εh∇Pε, ϕiQε. Then, using the identification (3.37) of Fε, we get h∇ηε˜ Pε,˜ϕiΩ=η−1 ε−ZΩε Duε:DRε(ϕ)dx + 2N2ZΩε rot wε·Rε(ϕ)dx +ZΩε f0(x0)·Rε(ϕ)0dx, 9 Francisco J. Su´arez-Grau Theorem 4.2. In the case ηε≈ε, with ηε/ε →λ,0< λ < +∞, then the sequence (η−2 εˆuε, η−1 εˆwε)converges weakly to (ˆu, ˆw)in L2(ω;H1(Y)3)×L2(ω;H1(Y)3)and ˆ Pεconverges strongly to ˜ Pin L2(Ω), where (ˆu, ˆw, ˜ P)∈ L2(ω;H1 #(Y)3)×L2(ω;H1 #(Y)3)×(L2 0(ω)∩H1(ω)), with RYˆu3dy =RYˆw3dy = 0, is the unique solution of the following homogenized system                                        −∆λˆu+∇λˆq= 2N2rotλˆw+f0(x0)− ∇x0˜ P(x0)in ω×Y, divλˆu= 0 in ω×Y, −Rc∆λˆw+ 4N2ˆw= 2N2rotλˆu+g0(x0)in ω×Y, ˆu= 0 on y3={0, h(y0)}, divx0ZY ˆu0(x0, y)dy= 0 in ω, ZY ˆu0(x0, y)dy·n= 0 on ∂ω, ˆq(x0, y)∈L2(ω;L2 0,#(Y)). (4.63) Proof. From Lemma 4.1, conditions (4.63)2,4,5,6hold. To prove that (ˆu, ˆw, ˜ P) satisfies the momentum equations given in (4.63), we consider ϕ∈ D(ω;C∞ #(Y)3) with divλϕ= 0 in ω×Yand divx0(RYϕ0dy) = 0 in ω, and we choose ϕε= (λ(ε/ηε)ϕ0, ϕ3) in (3.54). Taking into account that thanks to divλϕ= 0 in ω×Y, we have that 1 ηεZω×Π ˆ Pε(λdivy0ϕ0+∂y3ϕ3)dx0dy = 0 . Thus, passing to the limit using the convergences (4.56) and (4.59), and taking into account that λ(ε/ηε)→1, we obtain Zω×Y Dλˆu:Dyˆϕ dx0dy −Zω×Π ˜ Pdivx0ϕ0dx0dy = 2N2Zω×Y (λroty0ˆw3·ϕ0+ roty3ˆw0·ϕ0+λroty0ˆw0ϕ3)dx0dy +Zω×Y f0·ϕ0dx0dy . (4.64) Since ˜ Pdoes not depend on yand divx0RYϕ0dy = 0 in ω, we have that Zω×Y ˜ Pdivx0ϕ0dx0dy =Zω ˜ Pdivx0ZY ϕ0dydx0= 0, so we get Zω×Y Dλˆu:Dyϕ dx0dy = 2N2Zω×Y rotλˆw·ϕ dx0dy +Zω×Y f0·ϕ0dx0dy . (4.65) Next, for every ψ∈ D(ω;C∞ #(Y)3), we choose ψε=η−1 εψin (3.55). Then, passing to the limit using convergences (4.56) and (4.59), we get Zω×Y Dλˆw:Dyψ dx0dy + 4N2Zω×Y ˆw·ψ dx0dy = 2N2Zω×Y rotλˆu·ψ dx0dy +Zω×Y g0·ψ0dx0dy . (4.66) By density (4.65) holds for every function ϕin the Hilbert space Vdefined by V=     ϕ(x0, y)∈L2(ω;H1 #(Y)3),such that divλϕ(x0, y) = 0 in ω×Y, divx0ZY ϕ0(x0, y)dy= 0 in ω, ZY ϕ0(x0, y)dy·n= 0 on ∂ω     , 16 Francisco J. Su´arez-Grau and (4.66) in L2(ω;H1 #(Y)3). From [25, Part III, Theorem 2.4.2], the variational formulation (4.65)-(4.66) admits a unique solution (ˆu, ˆw) in V×L2(ω;H1 #(Y)3). Following [1], the orthogonal of Vwith respect to the usual scalar product in L2(ω×Y) is made of gradients of the form ∇x0q(x0) + ∇λˆq(x0, y), with q(x0)∈L2(ω)/Rand ˆq(x0, y)∈L2(ω;L2 #(Y)/R). Therefore, by integration by parts, the variational formulations (4.65)-(4.66) are equivalent to the homogenized system (4.63). It remains to prove that the pressure q(x0), arising as a Lagrange multiplier of the incompressibility constraint divx0(RYˆu0(x0, y)dy) = 0, is the same as the limit of the pressure ˆ Pε. This can be easily done by considering in equation (3.54) a test function only with divλequal to zero, obtain the variational formulation (4.64) and indentifying limits. Since 2N2rotλˆw+f0∈L2(ω×Y)3and Yis smooth enough, we deduce that ˜ P∈H1(ω). Finally, since from [25, Part III, Lemma 2.4.1] we have that (4.63) admits a unique solution, and then the complete sequence (η2 εˆuε, η−1 εˆwε,ˆ Pε) converges to the unique solution (ˆu(x0, y),ˆw(x0, y),˜ P(x0)).  Let us define the local problems which are useful to eliminate the variable yof the previous homogenized problem and then obtain a Reynolds equation for the pressure ˜ P. For every i, k = 1,2 and 0 <λ<+∞, we consider the following 3D local micropolar problems                    −∆λui,k +∇λπi,k −2N2rotλwi,k =eiδ1kin Y, divλui,k = 0 in Y, −Rc∆λwi,k + 4N2wi,k −2N2rotλui,k =eiδ2kin Y, ui,k =wi,k = 0 on y3={0, h(y0)}, ui,k(y), wi,k(y), πi,k(y)Y0−periodic. (4.67) It is known (see [25, Part III, Lemma 2.5.1]) that there exist a unique solution (ui,k, wi,k, πi,k)∈H1 #(Y)3× H1 #(Y)3×L2 0(Y) of problem (4.67), and moreover πi,k ∈H1(Y). We give the main result concerning the homogenized flow. Theorem 4.3. Let (ˆu, ˆw, ˜ P)∈L2(ω;H1 #(Y)3)×L2(ω;H1 #(Y)3)×(L2 0(ω)∩H1(ω)) be the unique weak solution of problem (4.63). Then, the extensions (η−2 ε˜uε, η−1 ε˜wε)and ˜ Pεof the solution of problem (2.11)-(2.12) converge weakly to (˜u, ˜w)in H1(ω, hmax;L2(ω)3)×H1(ω, hmax;L2(ω)3)and strongly to ˜ Pin L2(Ω) respectively, with ˜u3= ˜w3= 0. Moreover, defining e U(x0) = Rhmax 0˜u(x0, y3)dy3and f W(x0) = Rhmax 0˜w(x0, y3)dy3, it holds e U0(x0) = K(1) λf0(x0)− ∇x0˜ P(x0)+K(2) λg(x0),e U3(x0)=0 in ω, f W0(x0) = L(1) λf0(x0)− ∇x0˜ P(x0)+L(2) λg(x0),f W3(x0)=0 in ω, (4.68) where K(k) λ,L(k) λ∈R2×2,k= 1,2, are matrices with coefficients K(k) λij =ZY ui,k j(y)dy, L(k) λij =ZY wi,k j(y)dy, i, j = 1,2, with ui,k,wi,k,i, k = 1,2, the solutions of the local micropolar problems defined in (4.67). Here, ˜ P∈H1(ω)∩L2 0(ω)is the unique solution of the Reynolds problem      divx0−Aλ∇x0˜ P(x0) + bλ(x0)= 0 in ω, −Aλ∇x0˜ P(x0) + bλ(x0)·n= 0 on ∂ω, (4.69) where the flow factors are given by Aλ=K(1) λand bλ(x0) = K(1) λf0(x0) + K(2) λg0(x0). 17 Francisco J. Su´arez-Grau Proof. We eliminate the microscopic variable yin the effective problem (4.63). To do that, we consider the following identification ˆu0(x0, y) = 2 X i=1 hfi(x0)−∂xi˜ P(x0)(ui,1)0(y) + gi(x0)(ui,2)0(y)i, ˆw0(x0, y) = 2 X i=1 hfi(x0)−∂xi˜ P(x0)(wi,1)0(y) + gi(x0)(wi,2)0(y)i, ˆq(x0, y) = 2 X i=1 hfi(x0)−∂xi˜ P(x0)πi,1(y) + gi(x0)πi,2(y)i. From to the identities for the velocity RYˆu0(x0, y)dy =Rhmax 0˜u0(x0, y3)dy3and RYˆu3dy = 0, and for the microrotation RYˆw0(x0, y)dy =Rhmax 0˜w0(x0, y3)dy3and RYˆw3dy = 0 given in Lemma 4.1, we deduce that e Uand f W are given by (4.68). Finally, the divergence condition with respect to the variable x0given in (4.63) together with the expression of e U0(x0) gives (4.69).  5 Reynolds roughness regime (λ= 0) It corresponds to the case when the wavelength of the roughness is much greater than the film thickness, i.e. ηεεwhich is equivalent to λ= 0. Next, we give some compactness results about the behavior of the extended sequences (˜uε,˜wε,˜ Pε) and the unfolding functions (ˆuε,ˆwε,ˆ Pε) satisfying the a priori estimates given in Lemmas 3.3 and 3.4, and Lemma 3.6, respectively. Lemma 5.1. For a subsequence of εstill denoted by ε, we have that (i) (Velocity) there exist ˜u∈H1(0, hmax;L2(ω)3), with ˜u= 0 on y3={0, hmax}and ˜u3= 0, and ˆu∈ H1(0, h(y0); L2 #(ω×Y0)3), with ˆu= 0 on y3={0, h(y0)}and ˆu3independent of y3, such that RYˆu(x0, y)dy = Rhmax 0˜u(x0, y3)dy3with RYˆu3dy = 0, and moreover η−2 ε˜uε*(˜u0,0) in H1(0, hmax;L2(ω)3), η−2 εˆuε*ˆuin H1(0, h(y0); L2(ω×Y0)3),(5.70) divx0 Zhmax 0 ˜u0(x0, y3)dy3!= 0 in ω, Zhmax 0 ˜u0(x0, y3)dy3!·n= 0 in ∂ω , (5.71) divy0ˆu0= 0 in ω×Y, divx0ZY ˆu0(x0, y)dy= 0 in ω, ZY ˆu0(x0, y)dy·n= 0 in ∂ω , (5.72) (ii) (Microrotation) there exist ˜w∈H1(0, hmax;L2(ω)3), with ˜w= 0 on y3={0, hmax}and ˜w3= 0, and ˆw∈H1(0, h(y0); L2 #(ω×Y0)3), with ˆw= 0 on y3={0, h(y0)}, such that RYˆw(x0, y)dy =Rhmax 0˜w(x0, y3)dy3with RYˆw3dy = 0, and moreover η−1 ε˜wε*( ˜w, 0) in H1(0, hmax;L2(ω)3), η−1 εˆwε*ˆwin H1(0, h(y0); L2(ω×Y0)3),(5.73) 18 Francisco J. Su´arez-Grau (iii) (Pressure) there exists ˜ P∈L2 0(Ω) independent of y3, such that ˜ Pε→˜ Pin L2(Ω),ˆ Pε→˜ Pin L2(ω×Π).(5.74) Proof. The proof of (i) is similar to the critical case, but we have to take into account that applying the change of variables (3.48) to the divergence condition divηε˜uε, multiplying by η−1 εand passing to the limit, we prove that ˆu3is independent of y3. Thus, the divergence condition on y0given in (5.72) straightforward. For more details, we refer the reader to the proof of Lemmas 5.2-i) and 5.4-ii) in [2]. The proofs of (ii) and (iii) are similar to the critical case, so we omit it.  Next, we give the homogenized system satisfied by (ˆu, ˆw, ˜ P). Theorem 5.2. In the case ηεε, then the sequence (η−2 εˆuε, η−1 εˆwε)converges weakly to (ˆu, ˆw)in H1(0, h(y0); L2(ω× Y0)3)×H1(0, h(y0); L2(ω×Y0)3)and ˆ Pεconverges strongly to ˜ Pin L2(Ω), where (ˆu, ˆw, ˜ P)in H1(0, h(y0); L2 #(ω× Y0)3)×H1(0, h(y0); L2 #(ω×Y0)3)×(L2 0(ω)∩H1(ω)) with RYˆu3dy = 0,ˆu3independent of y3and ˆw3= 0, is the unique solution of the following homogenized system                                        −∂2 y3ˆu0+∇y0ˆq= 2N2roty3ˆw0+f0(x0)− ∇x0˜ P(x0)in ω×Y, divy0ˆu0= 0 in ω×Y, −Rc∂2 y3ˆw0+ 4N2ˆw0= 2N2roty3ˆu0+g0(x0)in ω×Y, ˆu0= 0 on y3={0, h(y0)}, divx0ZY ˆu0(x0, y)dy= 0 in ω, ZY ˆu0(x0, y)dy·n= 0 on ∂ω, ˆu(x0, y),ˆw(x0, y),ˆq(x0, y0)Y0−periodic. (5.75) Proof. From Lemma 5.1, conditions (5.75)2,4,5,6hold. To prove that (ˆu, ˆw, ˜ P) satisfies the momentum equations given in (5.75), we consider ϕ∈ D(ω;C∞ #(Y)3) with ϕ3independent of y3, divy0ϕ0= 0 in ω×Yand divx0RYϕ0dy = 0 in ω, and we choose ϕε= (ϕ0, ϕ3) in (3.54). Taking into account that divy0ϕ0= 0 in ω×Yand ϕ3is independent of y3, we have that 1 ηεZω×Π ˆ Pεdivy0ϕ0dx0dy = 0 and 1 ηεZω×Π ˆ Pε∂y3ϕ3dx0dy = 0 . Also, from Cauchy-Schwarz’s inequality, the second estimate in (3.49), the first estimate in (3.50) and ηε/ε →0, we have that  1 ε2Zω×Y Dy0ˆuε:Dy0ϕ dx0dy3≤C ε2kDy0ˆuεkL2(ω×Y)3×2≤Cηε ε→0, and  1 εZω×Y roty0ˆwε,3·ϕ0dx0dy= 1 εZω×Y ˆwε,3·roty0ϕ0dx0dy ≤C εkwεkL2(ω×Y)3≤Cηε ε→0. Thus, passing to the limit using the convergences (5.70), (5.73) and (5.74), we obtain Zω×Y ∂y3ˆu0·∂y3ϕ0dx0dy −Zω×Y ˜ Pdivx0ϕ0dx0dy = 2N2Zω×Y roty3ˆw0·ϕ0dx0dy +Zω×Y f0·ϕ0dx0dy . 19 Francisco J. Su´arez-Grau Since ˆ Pdoes not depend on yand divx0RYϕ0dy = 0 in ω, we have that Zω×Y ˜ Pdivx0ϕ0dx0dy =Zω ˜ Pdivx0ZY ϕ0dydx0= 0, so we get Zω×Y ∂y3ˆu0·∂y3ϕ0dx0dy = 2N2Zω×Y roty3ˆw0·ϕ0dx0dy +Zω×Y f0·ϕ0dx0dy . (5.76) Next, for every ψ∈ D(ω;C∞ #(Y)3), we choose ψε=η−1 εψin (3.55). Then, proceeding similarly as above and passing to the limit using previous convergences, we get RcZω×Y ∂y3ˆw0·∂y3ψ0dx0dy + 4N2Zω×Y ˆw0·ψ0dx0dy = 2N2Zω×Y roty3ˆu0·ψ0dx0dy +Zω×Y g0·ψ0dx0dy. (5.77) Finally, we can prove ˆw3= 0. For this, we take in (3.55) the test function ψε= (0, η−1 εψ3), and passing to the limit as above, we get RcZω×Y ∂y3ˆw3:∂y3ψ3dx0dy + 4N2Zω×Y ˆw3·ψ3dx0dy = 0, which is equivalent to the equation −Rc∂2 y3ˆw3+2N2ˆw3= 0. This together with the boundary conditions ˆw3= 0 on y3={0, h(y0)}implies that ˆw3= 0. By density, and reasoning as in the proof of Theorem 4.63, problem (5.76)-(5.77) is equivalent to the homogenized system (5.75) (observe that the condition divy0ϕ0= 0 implies that ˆqdoes not depend on y3). Since ∂y3ˆu0+ 2N2roty3ˆw0+f0∈L2(ω×Y), it can be easily proved that ∇x0˜ P∈L2(ω)2and so ˜ P∈H1(ω) and also that system (5.75) has a unique solution (see for example Proposition 3.3 and 3.5 in [30]).  Let us define the local problems which are useful to eliminate the variable yof the previous homogenized problem and then obtain a Reynolds equation for ˜ P. We define Φ and Ψ by Φ(h(y0), N, Rc) = 1 12 +Rc 4h2(y0)(1 −N2) −1 4h(y0)rN2Rc 1−N2coth  Nh(y0)s1−N2 Rc , (5.78) Ψ(h(y0), N, Rc) = tanh Nh(y0)q1−N2 Rc 1−N h(y0)q1−N2 Rctanh Nh(y0)q1−N2 Rc,(5.79) and for every i, k = 1,2, we consider the following local Reynolds problems −divy0h3(y0) 1−N2Φ(h(y0), N, Rc)∇y0πi,k(y0) + eiδ1k= 0 in Y0.(5.80) It is known that from the positivity of function Φ, problem (5.80) has a unique solution for πi,k ∈H1 #(Y0) (see [7] for more details). Next, we give the main result of this section. 20 Francisco J. Su´arez-Grau Theorem 5.3. Let (ˆu, ˆw, ˜ P)∈L2(ω;H1 #(Y)3)×L2(ω;H1 #(Y)3)×(L2 0(ω)∩H1(ω)) be the unique weak solution of problem (5.75). Then, the extensions (η−2 ε˜uε, η−1 ε˜wε)and ˜ Pεof the solution of problem (2.11)-(2.12) converge weakly to (˜u, ˜w)in H1(0, hmax;L2(ω)3)×H1(0, hmax;L2(ω)3)and strongly to ˜ Pin L2(Ω) respectively, with ˜u3= ˜w3= 0. Moreover, defining e U(x0) = Rhmax 0˜u(x0, y3)dy3and f W(x0) = Rhmax 0˜w(x0, y3)dy3, it holds e U0(x0) = K(1) 0f0(x0)− ∇x0˜ P(x0)+K(2) 0g0(x0),e U3(x0) = 0 in ω, f W0(x0) = L(2) 0g0(x0),f W3(x0) = 0 in ω, (5.81) where the matrices K(k) 0,k= 1,2, and L(2) 0are matrices with coefficients K(k) 0ij =1 1−N2ZY0 h3(y0)Φ(h(y0), N, Rc)∂yiπj,k(y0) + δij δ1kdy0, L(2) 0ij =−1 4N3rRc 1−N2ZY0 Ψ(h(y0), N)dy0δij , (5.82) for i, j = 1,2, with Φand Ψgiven by (5.78) and (5.79), respectively, and πi,k ∈H1 #(Y0),i, k = 1,2, the unique solutions of the cell problems (5.80). Here, ˜ P∈H1(ω)∩L2 0(ω)is the unique solution of problem      divx0−A0∇x0˜ P(x0) + b0(x0)= 0 in ω, −A0∇x0˜ P(x0) + b0(x0)·n= 0 on ∂ω, (5.83) where the flow factors are given by A0=K(1) 0and b0(x0) = K(1) 0f0(x0) + K(2) 0g0(x0). Proof. We proceed as in in the proof of Theorem 4.3 in order to obtain (5.81). Thus, expressions for e Uand f W can be obtained by defining ˆu0(x0, y) = 2 X i=1 h∂xi˜ P(x0)−fi(x0)ui,1(y)−gi(x0)ui,2(y)i, ˆw0(x0, y) = 2 X i=1 h∂xi˜ P(x0)−fi(x0)wi,1(y)−gi(x0)wi,2(y)i,(5.84) ˆq(x0, y) = 2 X i=1 h∂xi˜ P(x0)−fi(x0)πi,1(y0)−gi(x0)πi,2(y0)i, where (ui,k, wi,k)∈H1 #(Y)2×H1 #(Y)2,i, k = 1,2, are the unique solutions of                    −∂2 y3ui,k +∇y0πi,k −2N2roty3wi,k =−eiδ1kin Y, divy0ui,k = 0 in Y, −Rc∂2 y3wi,k + 4N2wi,k −2N2roty3ui,k =−eiδ2kin Y, ui,k =wi,k = 0 on y3={0, h(y0)}, ui,k(y), wi,k(y), πi,k(y0)Y0−periodic . (5.85) 21 Francisco J. Su´arez-Grau Then, thanks to the identities RYˆu0(x0, y)dy =Rhmax 0˜u0(x0, y3)dy3,RYˆu3dy = 0, RYˆw0(x0, y)dy =Rhmax 0˜w0(x0, y3)dy3 and ˆw3= 0 given in Lemma 5.1, it holds e U0(x0) = ZY ˆu0(x0, y)dy =−K(1) 0∇x0˜ P(x0)−f0(x0)+K(2) 0g0(x0), e U3(x0) = ZY ˆu3(x0, y0)dy = 0 in ω, f W0(x0) = ZY ˆw0(x0, y)dy =−L(1) 0∇x0˜ P(x0)−f0(x0)+L(2) 0g0(x0), f W3(x0) = ZY ˆw3(x0, y)dy = 0 in ω, (5.86) where K(k) 0,L(k) 0,k= 1,2, are matrices defined by their coefficients K(k) 0ij =−ZY ui,k j(y)dy, L(k) 0ij =−ZY wi,k j(y)dy , i, j = 1,2.(5.87) Then, by the divergence condition in the variable x0given in (5.75), we get the generalized Reynolds equation (5.83). However, we observe that (5.85) can be viewed as a system of ordinary differential equations with constant coefficients, with respect to the variable y3and unkowns functions y37→ ui,k 1(y0, y3), wi,k 2(y0, y3), ui,k 2(y0, y3), wi,k 1(y0, y3), where y0is a parameter, y0∈Y0. Thus, we can give explicit expressions for ui,k and wi,k. The procedure to obtain a solution to the previous system is given in the Appendix (see also in [7, 9]). Thus, considering ¯u0=ui,k, ¯w0=wi,k,¯ f0=−eiδik and ¯g0=−eiδ2kin (7.114)-(7.115), we obtain that ui,k, wi,k are given in terms of πi,k by the expressions ui,k(y) = 1 2(1−N2)hy2 3−h(y0)y3 +h(y0)N2 ksinh(ky3)−(cosh(ky3)−1) coth kh(y0) 2i∇y0πi,k(y0) + eiδ1k +h(y0) N2h2N2 ksinh(ky3)−2y3A+2N2 k(cosh(ky3)−1)B−y3i(eiδ2k)⊥, (5.88) wi,k(y) = 1 4(1−N2)h2y3 +h(y0)cosh(ky3)−1−sinh (ky3) coth kh(y0) 2i∇y0πi,k(y0) + eiδ1k⊥ −h(y0) 2N2hcosh(ky3)A+ sinh(ky3)Bieiδ2k, where k=q4N2(1−N2) Rcand A,Bare given by A(y0) = sinh(kh(y0)) −2h(y0) sinh(kh(y0))+ 4N2 k(cosh(kh(y0))−1) , B(y0) = −(cosh(kh(y0)−1) −2h(y0) sinh(kh(y0))+ 4N2 k(cosh(kh(y0))−1) . (5.89) Taking into account that from (7.116) it holds Zh(y0) 0 ui,k(y0, y3)dy3=−h3(y0) 1−N2Φ(h(y0), N, Rc)∇y0πi,k +eiδ1k, Zh(y0) 0 wi,k(y0, y3)dy3=−1 4N3rRc 1−N2Ψ(h(y0), N, Rc)eiδ2k, (5.90) 22 Francisco J. Su´arez-Grau with Φ and Ψ given by (5.78) and (5.79), respectively, we get that πi,k satisfies the generalized Reynolds cell problem (5.80). Using the expressions of ui,k and wi,k together with (5.86), (5.87) and (5.90), we easily get (5.81). Observe that, from the second equation in (5.90) with k= 2, we have L(1) 0= 0, which ends the proof.  6 High-frequency roughness regime (λ= +∞) It corresponds to the case when the wavelength of the roughness is much smaller than the film thickness, i.e. ηεεwhich is equivalent to λ= +∞. Next, we give some compactness results about the behavior of the extended sequence (˜uε,˜wε,˜ Pε) and the unfolding functions (ˆuε,ˆwε,ˆ Pε) satisfying the a priori estimates given in Lemmas 3.3 and 3.4, and Lemma 3.6, respectively. Lemma 6.1. For a subsequence of εstill denoted by ε, we have that (i) (Velocity) there exists ˜u∈H1(0, hmax;L2(ω)3), with ˜u= 0 on y3={0, hmax}and ˜u3= 0, such that η−2 ε˜uε*(˜u0,0) in H1(0, hmax;L2(ω)3), η−2 ε˜uε*0in H1(hmin, hmax;L2(ω)3),(6.91) η−2 εˆuε*(˜u0,0) in H1(0, hmin;L2(ω)3),(6.92) divx0 Zhmin 0 ˜u0(x0, y3)dy3!= 0 in ω, Zhmin 0 ˜u0(x0, y3)dy3!·n= 0 in ∂ω , (6.93) (ii) (Microrotation) there exists ˜w∈H1(0, hmax;L2(ω)3), with ˜w= 0 on y3={0, hmax}and ˜w3= 0, such that η−1 ε˜wε*( ˜w0,0) in H1(0, hmax;L2(ω)3), η−1 ε˜wε*0in H1(hmin, hmax;L2(ω)3),(6.94) η−1 εˆwε*( ˜w0,0) in H1(0, hmin;L2(ω)3),(6.95) (iii) (Pressure) there exists a function ˜ P∈L2 0(Ω) independent of y3, such that ˜ Pε→˜ Pin L2(Ω),ˆ Pε→˜ Pin L2(ω×Π) .(6.96) Proof. We start proving (i). We will only give some remarks and for more details, we refer to the reader to Lemmas 5.2-ii) and 5.4-ii) in [2]. As previous cases, we can prove that there exists ˜u∈H1(0, hmax;L2(ω)3) such that η−2 ε˜uεconverges weakly to ˜uin H1(0, hmax;L2(ω)3). On the other hand, from estimate (3.27), ε/ηε→0 and taking into account that η−2 ε= ( ε ηε)2ε−2, then second convergence in (6.91) holds and so ˜u= 0 in Ω+. Then, reasoning as previous cases, we can prove that ˜u3= 0, ˜v0= 0 on y3={0, hmin}and also, the divergence condition (6.93). From estimates (3.49), we deduce that there exists ˆu∈H1(0, h(y0); L2(ω×Y0)3) such that ˆuε*ˆuin H1(0, h(y0); L2(ω×Y0)3).(6.97) Since ε−1η−1 εDyˆuεis bounded in L2(ω×Y)3, we observe that η−2 εDyˆuεis also bounded, and tends to zero. This together with (6.97) implies η−2 εDy0ˆuεconverges weakly to zero in H1(0, h(y0); L2(ω×Y0)3×2), and so ˆudoes not depend on y0. 23 Francisco J. Su´arez-Grau Proceeding as previous cases, but taking ϕ∈C1 c(Ω+), we can prove that Zω×Π+ ˆu(x0, y)ϕ(x0, y3)dy =ZΩ+ ˜u(x0, y3)ϕ(x0, y3)dx0dy3, and taking into account that ˜u= 0 on Ω+, we deduce that ˆu= 0 in ω×Π+. Then, we can prove that Rω×Π−ˆu(x0, y)ϕ(x0, y3)dy =RΩ−˜u(x0, y3)ϕ(x0, y3)dx0dy3holds and, since ˆudoes not depend on y0, we have that ˆu= (˜u0,0). For the proof of (ii) for microrotation, we can proceed as for the velocity. By considering estimate (3.24), we prove the existence of the weak limit ˜w∈H1(0, h(y0); L2(ω×Y0)3) of the sequence η−1 ε˜wε, and taking into account estimate (3.28), ε/ηε→0 and that η−1 ε= (η2 εε3)( ε ηε)3, we prove the second convergence in (6.94). Moreover, as in the case of the velocity, it can be proved that ˜w= 0 on y3={0, hmin}. To prove that ˜w3= 0, we argue as in the critical case, by taking a test function ψε= (0,0, η−1 εψ3) in (3.53), passing to the limit and considering the previous boundary conditions. For the proof of (6.95), we proceed as the case of the velocity by taking into account estimates (3.50). Finally, to prove (iii), we proceed as in the critical case. First we prove weak convergence of the extended pressure ˜ Pεto a function ˜ Pin L2 0(Ω) and next, we prove that ˜ Pindependent of y3. Finally, we prove strong convergence of the pressure, but in this case we have to take into account te behavior of ˜uεand ˜wεon the oscillating part. Thus, we consider σε∈H1 0(Ω)3such that σε* σ in H1 0(Ω)3. Denoting ˜σε= (σ0 ε, εσε,3) and ˜σ= (σ0,0), we have ˜σε*˜σin H1 0(Ω)3.(6.98) Then, <∇x0,y3˜ Pε, σε>Ω+−<∇x0˜ P, ˜σ >Ω+ ≤<∇x0,y3˜ Pε− ∇x0˜ P, ˜σ >Ω++<∇x0,y3˜ Pε, σε−˜σ >Ω+. On the one hand, using the weak convergence of the pressure, we have <∇x0,y3˜ Pε− ∇x0˜ P, ˜σ >Ω+=ZΩ+˜ Pε−˜ Pdivx0σ0dx→0,as ε→0. On the other hand, proceeding as in Lemma 3.4, we have that <∇x0,y3˜ Pε, σε−˜σ >Ω+=<∇ηε˜ Pε,˜σε−˜σ)>Ω+ ≤Ck˜σε−˜σkL2(Ω)3+εkDx0,y3(σε−˜σ)kL2(Ω)3×3, which tends to zero because of the convergence of the sequence σεand the Rellich theorem. Then, reasoning similarly as above by considering in Ω−, we deduce that <∇x0,y3˜ Pε, σε>Ω−−<∇x0˜ P, σ >Ω−→0, which together with previous convergence, implies the convergence of ∇x0,y3˜ Pεto ∇x0˜ Pstrongly in H−1(Ω)3. This together with the Ne˘cas inequality (3.44) implies the first convergence in (6.96). Finally, we remark that the strong convergence of sequence ˆ Pεto ˜ Pis a consequence of the strong convergence of ˜ Pεto ˜ P(see [17, Proposition 2.9]).  As seen in the previous compactness result, the microstructure of e Ωεwill not be involved in the homogenized system and thus, we will obtain a Reynolds equation satisfied by ˜ Pin the non-oscillating part of the domain, that is Ω−. 24 Francisco J. Su´arez-Grau Theorem 6.2. In the case ηεε, then the extensions (η−2 ε˜uε, η−1 ε˜wε)and ˜ Pεof the solution of problem (2.11)- (2.12) converge weakly to H1(0, hmin;L2(ω)3)×H1(0, hmin;L2(ω)3)and strongly to ˜ Pin L2(Ω) respectively, with ˜u3= ˜w3= 0, where ˜u0and ˜w0are given by the following expressions in terms of the pressure ˜ Pin Ω−, ˜u0(x0, y3) = y2 3 2(1 −N2)+1 4(1 −N2)2N2 ksinh(ky3)−2y3 −hmin 2(1 −N2) N2 k(cosh(khmin)−1) coth khmin 2∇x0˜ P(x0)−f0(x0) ¯w0(x0, y3) = hy3 2(1 −N2)+hmin 4(1 −N2)cosh(ky3)−1 (6.99) −coth khmin 2sinh(ky3)i∇x0˜ P(x0)−f0(x0)⊥ , with k=q4N2(1−N2) Rc. Moreover, defining e U(x0) = Rhmin 0˜u(x0, y3)dy3and f W(x0) = Rhmin 0˜w(x0, y3)dy3, it holds e U0(x0) = hmin 1−N2Φ(hmin, N, Rc)f0(x0)− ∇x0˜ P(x0),e U3(x0) = 0 in ω, f W0(x0) = 0,f W3(x0) = 0 in ω, (6.100) where Φis given by (5.78), and ˜ P∈H1(ω)×L2 0(ω)is the unique solution of the Reynolds problem      divx0−A∞∇x0˜ P(x0) + b∞(x0)= 0 in ω , −A∞∇x0˜ P(x0) + b∞(x0)·n= 0 on ∂ω . (6.101) Here, the flow factors are given by A∞=hmin 1−N2Φ(hmin, N, Rc)and b∞(x0) = hmin 1−N2Φ(hmin, N, Rc)f0(x0). Proof. From Lemma 6.1, we observe that at main order, the microstructure does not appear because the high oscillation of the boundary. Thus, we choose in the first equation of the variational formulation (3.53), extended to Ω, the following test function ϕε(x0, y3)=(ϕ0(x0, y3),0) ∈ D(Ω−)3satisfying the divergence condition divx0Rhmin 0ϕ0(x0, y3)dy3= 0 in ω. Passing to the limit by using convergences (6.91), (6.94) and (6.96), we get ZΩ− ∂y3˜u0·∂y3ϕ0dx0dy3+ZΩ− ˜ Pdivx0ϕ0dx0dy3 = 2N2ZΩ− roty3˜w0·ϕ0dx0dy3+ZΩ− f0(x0)·ϕ0. Since ˜ Pdoes not depend on y3and the divergence condition on the variable x0satisfied by ϕ0, we have that ZΩ− ˜ Pdivx0ϕ0dx0dy3=Zω ˜ Pdivx0 Zhmin 0 ϕ0dy3!dx0= 0, and so ZΩ− ∂y3˜u0·∂y3ϕ0dx0dy3= 2N2ZΩ− roty3˜w0·ϕ0dx0dy3+ZΩ− f0(x0)·ϕ0. Next, we choose in the second equation of of the variational formulation (3.53), extended to Ω, the following test function ψε(x0, y3)=(η−1 εψ0(x0, y3),0) ∈ D(Ω−)3and taking into account that ε/ηε→0, we pass to the limit and we get RcZΩ− ∂y3˜w0·∂y3ψ0dx0dy3+ 4N2 ZΩ− ˜w0·ψ0dx0dy3= 2N2 ZΩ− roty3˜u0·ϕ0dx0dy3. 25 Francisco J. Su´arez-Grau [33] Su´arez-Grau, F.J.: Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary. Nonlinear Analysis 117, 99-123 (2015). [34] Tartar, L.: Incompressible fluid flow in a porous medium convergence of the homogenization process. In: Appendix to Lecture Notes in Physics, 127. Springer-Velag, Berlin (1980). 32