Convergence to the Reynolds approximation with a double effect of roughness

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Convergence to the Reynolds approximation with a double effect of roughness

Catherine Choquet, Laurent Chupin, Marguerite Gisclon

To cite this version:

Catherine Choquet, Laurent Chupin, Marguerite Gisclon. Convergence to the Reynolds approximation with a double effect of roughness. 2010. �hal-00457146�

Convergence to the Reynolds approximation with a double effect of roughness

Catherine Choquet¹, Laurent Chupin², Marguerite Gisclon³

Key Words - Boundary conditions, upscaling, two-scale convergence, micro-fluidic, Reynolds and Stokes equations, rough boundaries, thin films.

AMS Subject Classification- 35Q30, 76A20, 76D08, 78M35, 78M40.

Abstract

We prove that the lubrication approximation is perturbed by a non-regular roughness of the boundary. We show how the flow may be accelerated using adequate rugosity profiles on the bottom. We explicit the possible effects of some abrupt changes in the profile. The limit system is mathematically justified through a variant of the notion of two-scale convergence. Finally, we present some numerical results, illustrating the limit system in the three-dimensional case.

Introduction

We study in this paper the effect of different very small domain irregularities on a thin film flow governed by the Stokes equations. There exist already some references on the subject. Roughly speaking, they may be ordered into three categories. The first one is devoted to the study of the roughness effects on the flow in a channel: the height of the channel, denoted h and depending on the horizontal componentx, is fixed and the effect of small scales of the boundary is studied. In such studies, the height of the channel is written as (see [1, 14])

h(x) =h2

x,x

ε

.

The second category is concerned with the specific study of the thin film assumption, that is when the height of channel is assumed to be small, of the form (see [9, 5, 10, 11])

h(x) =εh₁(x).

The third category combines the mechanical point of view of the latter lubrication studies with the analysis of the roughness effects. Various limit models, in special regimes, are obtained depending on the ratio between the size of the rugosities and the mean height of the domain. In [3, 4, 10, 11], the ratio is assumed to be of order one, namely

h(x) =εh₂ x,x

ε

and an asymptotic analysis is performed using an homogenization process. More recently, in [5], the authors study the case where the narrow gap is smaller than the roughness, namely

h(x) =ε h₂ x, x

ε^α

.

1Facult´e des Sciences et Techniques de Saint-J´erˆome - Universit´e P. C´ezanne - LATP UMR 6632, 13397 Marseille Cedex 20, France

Email: c.choquet@univ-cezanne.fr

2Universit´e de Lyon - INSA de Lyon - Pˆole de Math´ematiques - CNRS, UMR5208, Institut Camille Jordan - 21 av.

Jean Capelle, 69621 Villeurbanne Cedex, France Email: laurent.chupin@insa-lyon.fr

3LAMA - CNRS UMR5127 - Universit´e de Savoie, 73376 Le Bourget du Lac, France Email: gisclon@univ-savoie.fr

In all previous works, the ration between the two scales, modeled by the parameter α is such that α≤1. Moreover, the asymptotic model obtained in the caseα ≤1 is always the classical Reynolds equation. In [6], the authors consider the particular case which does not enter in the previous framework:

h(x) =ε h₂ x, x

ε²

.

They mathematically justisfy (through a variant of the notion of two-scale convergence) that an extra term modifies the standard Reynolds equation.

There exists now a huge number of available data for rough surfaces, due to the increasing efficiency of the measurements (optic or laser technics, seee.g.[7]). On the one hand, the existence of multiple scales of rough oscillations is for instance emphasized in [13, 23]. But no analytical solution is available for complex geometries. These data are also often too cumbersome for numerical simulation.

This motivates an approach using asymptotic analysis. On the other hand, the tribological literature does not provide any categoric answer for the choice of characteristic roughness parameters [15, 21].

The aim of the present paper is thus to explore the effects due to abrupt changes in the rugosity profile.

A direct application of this kind of study is the production of nano-scale electronic components which are nowadays formed by self assembly. In such a process, the solvent and the monomers are confined in a thin channel. The block copolymers create abrupt changes of the geometry (see for instance [12]).

The case of a change of speed of order one could be easily deduced from our previous work [6]. We thus focus on more abrupt changes in the rough profile. In particular, we aim to compare the speed of the change, assumed of order ε^α, α > 0, and those due to the characteristic speed of the rough oscillations, assumed of order ε², whereas ε corresponds to the caracteristic height of the channel.

We restrict ourself to the case whereα≤2 to ensure that our limit model respects the structure of the classical Reynolds approximation [20].

We thus consider a model profile of the form h(x) =εh^ε=ε

h₁(x) +ε

1−ψ x ε^α

h₂x

ε²

, 0≤α≤2.

Functionψ is introduced to mimic the change in the profile (see the Figure 1).

O(ε²) O(ε²)

O(ε)

O(1) O(1) O(ε^α)

Figure 1: Domain Ω^ε and the different scales The domain occupied by the fluid is then:

Ω^ε ={(x, z)∈R^d−1×R, x∈ω, 0< z < h(x)}

whereω is a domain of R^d−1. We choose d= 2 or d= 3 for the current applications. The change in the profile occurs in a subdomainw_T^ε ofω. The domainω is thus decomposed inω =ω₋^ε ∪w^ε_T ∪ω^ε₊. We assume that the functionψ belongs toL^∞(ω) and is such that ψ(x) = 0 ifx∈ω^ε₋ and ψ(x) = 1 if x ∈ ω^ε₊. The positive function h₁ is the main order part of the roughness, while function h₂ describes the oscillating part. For sake of simplicity, we assume that the fonction h^ε₂ defined by

h^ε₂(x) =h₂(x/ε²) is an ε²-periodic function. Actually, it would be sufficient for our needs to assume thath₂ is an admissible test function for the two-scale convergence (see Proposition 2.1 below). The first step of our method consists in introducing a new vertical variable. This change of variable motivates our last technical assumptions for the rough profile. We assume that ψ ∈ C²(R^d−1), h₁ ∈H²(R^d−1) andh^ε₂∈ C¹(T^d−1).

Let us present our results. We prove that the change in the rough profile gives the same type of correction at the main order than the oscillating parth₂, but only under an additional assumption linking the behavior of the roughness jumps with the one of the oscillating part of the profile. Without this additional assumption, the Reynolds approximation contains a non-explicit contribution (a kind of ”strange term coming from nowhere”, see [8]).

Theorem 0.1 Let ω be the horizontal projection of the domain of study.

(i) Let us denote by ψ⁰ ∈ L²(ω;H¹(T^d−1)) (respectively ψ⁰ ∈ L²(ω) if α < 2) the two-scale limit of the jump approximation ψ^ǫ(x) = ψ(x/ε^α) as ε → 0 and by U_b the velocity of the lower surface.

The behavior of the thin flow is approximated by the following modified Reynolds equation on the pressure p of the fluid which only depend on the horizontal variable:

div_xh³₁

12A∇_xp⁰

= div_x h₁(BU_b−Q⁰)

on ω, where the functionsA and B are defined on ω by

A= 12 N_ψ

e^Nψ/2 Z 1

0

e^−Nψt2/2dt−1

− 12

N_ψ(e^Nψ/2−1) R₁

0

R_s

0 e^{Nψ(s2−t2)/2}dtds R1

0 e^Nψs2/2ds , B = 1

N_ψ(e^Nψ/2−1) 1 R1

0 e^Nψs2/2ds, where N_ψ(x) =

Z

T^d−1

|∇_X((1−ψ⁰(x, X))h₂(X))|²dX, x ∈ ω. The ”strange” function Q⁰ is non explicit.

(ii) Assuming moreover one of the following assumption, (H1) ∇(ε²ψ^εh^ε₂) strongly two-scale converges to ∇_X(ψ⁰h₂), (H2) ψ^ε strongly two-scale converges toψ⁰,

we recover a completely explicit Reynolds-type approximation:

div_xh³₁

12A∇_xp⁰

= div_x h₁BU_b

on ω.

Remark 0.1 It is crucial to clearly separate the oscillating profile h2 and the jumps characteristic functionψ in such a study. Indeed, the limit behavior described in Theorem 0.1 clearly depends on the link between the behavior of the roughness jumps with the one of the oscillating part of the profile (see Assumptions (H1) and (H2)).

Remark 0.2 Let us describe a situation allowing the use of Assumptions (H1). Assume that ω_T^ε is a finite collection of disconnected subsets ofω with measure of order ε^α. As ε→ 0, ω_T^ε reduces to a finite discrete set of points {xⁱ_T}_i=1..N. Then

∇(ε²ψ^εh^ε₂) = XN i=1

χ_ωi,ε

+ ∇_Xh^ε₂+χ_ωi,ε T

ε^2−α∇_xψ^εh^ε₂ .

Since the Lebesgue measure does not see accidents, one easily checks with the definition of strong two-scale convergence in Proposition 2.1 that

∇(ε²ψ^εh^ε₂)→² XN

i=1

χ_ωⁱ

+∇_Xh₂. This situation is illustrated through numerical exemples in Section 4.

Remark 0.3 We recall that the usual Reynolds approximation (corresponding to h₂ = 0) reads div_xh³₁

12∇_xp⁰

= div_x h₁ 2 U_b

on ω.

The latter result is thus a perturbation of the classical Reynolds equation. The low order perturbations of the rough profile (oscillations h₂ and abrupt change in the roughness ψ) both give a perturbation of the Couette component of the Reynolds model. This is a highly desirable effect for the applications of such a lubrication model. We explicitly recover a perturbation which was detected with formal computations by [17], or more recently by [19].

Remark 0.4 The strong two-scale convergence mentioned in (ii) of Theorem 0.1 could appear as a rather technical assumption. In fact, it means that the oscillation spectrum of the sequence belongs to the integer grid (see [22] for a proof through Fourier analysis). The oscillation spectrum is then due only to the periodicity of the coefficients.

The paper is organized as follows. In the first section, we give the model and the first estimates for the velocity and the pressure. In the second section we give the definition and the properties of a convenient two-scale convergence. The third section is devoted to the rigorous study of the upscaling process in view of proving Theorem 0.1. In Section 4, we present some numerical results, illustrating the limit system in the three-dimensional lubrication context.

1 Model and estimates

1.1 Stokes model in lubricant context

The vector fieldU^ε= (u^ε, w^ε)∈R²×R (which describes the fluid velocity) and the pressure (given by the scalar functionp^ε) satisfy the stationary Stokes equations⁴:

−∆u^ε+∇_xp^ε= 0 in Ω^ε, (1)

−∆w^ε+∂_zp^ε= 0 in Ω^ε, (2)

div_xu^ε+∂_zw^ε= 0 in Ω^ε, (3)

U^ε=U_b^ε on ∂Ω^ε. (4)

• The viscosity of the fluid is set equal to 1 for sake of simplicity.

4In all this document, the operators (like ∆) without index denote the operators with respect all the variables. To specify the operators with respect only one variable we use subscript notations, see for instance ∆xor∂_z.

• In classical lubrication framework, the boundary conditions are the following:

U_b^ε(x, z) =





(u_b,0) x∈ω, z= 0, (εue_b(z),0) x∈∂ω,

(0,0) x∈ω, z=εh^ε,

whereu_b is a constant corresponding to the imposed velocity at the bottom of the mechanism, and ue_b is a regular fonction (for instance an affine function) connectingu_b forz= 0 and 0 for z = h^ε. Notice that for such a problem (that is for an asymptotic study ε → 0) the precise value for the velocity ue_b is not primary. In fact, the physical quantity which persists to the limit is the total flux R

∂ω

Rεh^ε

0 ue_b(z)dz, see [3].

• Finally, the pressure is normalized by Z

Ω^ε

p^ε= 0.

We introduce a new vertical variable Z defined by Z = z

h to consider the same problem in the fix domain Ω =ω×(0,1). Problem (1)-(4) now reads⁵

−∆_xu^ε+∆h

h −|∇h|² h²

Z∂_Zu^ε+ 2∇h

h ·Z∇_x∂_Zu^ε−|∇h|²

h² Z²∂_Z²u^ε− 1 h²∂²_Zu^ε +∇_xp^ε−∇h

h Z∂_Zp^ε= 0 in Ω, (5)

−∆_xw^ε+∆h

h −|∇h|² h²

Z∂_Zw^ε+ 2∇h

h ·Z∇_x∂_Zw^ε−|∇h|²

h² Z²∂_Z²w^ε− 1 h²∂_Z²w^ε +1

h∂_Zp^ε= 0 in Ω, (6) div_xu^ε−∇h

h ·Z∂_Zu^ε+ 1

h∂_Zw^ε= 0 in Ω, (7)

U^ε=U_b on ∂Ω, (8)

Observe that ∆h

h −|∇h|² h² = div

∇h h

.

1.2 Estimates and dependance with respect to the small parameter ε We begin by some estimates for the velocity.

Lemma 1.1 There exists a constant C such that the velocity components satisfy the following uni- form estimates:

k∇xU^εk_(L²_(Ω))^d ≤ C

ε, (9)

k∂_ZU^εk_(L²_(Ω))d ≤C, (10) kU^εk_(L²_(Ω))d ≤C. (11) Proof: These estimates are directly derived from the original problem after an adequate lifting of the non-homogenenous boundary conditions. We write the standard energy estimate for Stokes type system and then use the change of variables to control the derivatives. We conclude with the Poincar´e

inequality.

We now give some uniform estimates for the pressure.

5Rigorously, the functionu^ε defined in the previous Stokes system (1)-(4) is not egal to the functionu^εused in this rescale domain. However, to not introduce numerous notations, we denote always the same.

Lemma 1.2 There exists a constant C such that the pressure satisfies k∇_xp^εk_H−1(Ω)≤ C

ε², (12)

k∂_Zp^εk_H−1(Ω)≤ C

ε, (13)

kp^εk_L²_(Ω)≤ C

ε². (14)

Proof: Such estimates come from Equations (5)-(6) estimated in H⁻¹. This provides bounds on

∇_xp^ε and∂_Zp^ε. Using Poincar´e-Wirtinger inequality (with the normalization hypothesis) we get the

L²-estimates forp^ε.

Notational convention: In what follows, for any function φ ∈ H¹(ω ×T^d−1), we denote by

∇φ(x, x/ε²) =∇φ^ε=∇_xφ^ε+ε⁻²∇_Xφ^ε its horizontal gradient.

2 Definition and properties of a convenient two-scale convergence

The proof of the homogenization process will be carried out by using a variant of the two-scale convergence introduced by G. Nguetseng in [18] and developed by G. Allaire in [2]. Let us give the basic definition and properties of this concept.

Proposition 2.1 A sequence (v^ε) of functions in L²(Ω) two-scale converges to a limit v⁰(x, Z, X) belonging toL²(Ω×T^d−1),v^ε⇀ v^{2 0}, if

ε→0lim Z

Ω

v^ε(x, Z) Ψ(x, Z, x/ε²)dx dZ = Z

Ω

Z

T^d−1

v⁰(x, Z, X) Ψ(x, Z, X)dx dZ dX, for any test functionΨ(x, Z, X), X-periodic in the third variable, satisfying

ε→0lim Z

Ω

|Ψ(x, Z, x/ε²)|²dx dZ = Z

Ω

Z

T^d−1

|Ψ(x, Z, X)|²dx dZ dX.

Such a functionΨis called an admissible test function for the two-scale convergence. Note that Z is only a parameter for this definition.

(i) From each bounded sequence(v^ε)inL²(Ω)one can extract a subsequence which two-scale converges to some limitv⁰ ∈L²(Ω×T^d−1). The weak L²-limit of v^ε is v(x, Z) =R

T^d−1v⁰(x, Z, X)dX.

(ii) Let(v^ε) be a bounded sequence inL²(0,1;H¹(ω))which converges weakly tov in L²(0,1;H¹(ω)).

Thenv^ε⇀ v² and there exists a function v² ∈L²(Ω;H¹(T^d−1)) such that, up to a subsequence,

∇v^ε⇀² ∇v(x) +∇_Xv²(x, y).

(iii) Let (v^ε) be a bounded sequence in L²(Ω) such that (ε²∇_xv^ε) is bounded in (L²(Ω))^d−1. Then, there exists a function v⁰ ∈L²(Ω;H¹(T^d−1))such that, up to a subsequence,

v^ε⇀ v^{2 0} and ε²∇v^ε⇀∇² _Xv⁰(x, Z, X).

• Due to our assumptions for h₁ and h₂, functionsh₁(x), h₂(x/ε²), ∂_xih₁(x), ∂_xi(ε²h₂(x/ε²)) =

∂_Xih₂(x/ε²) can obviously be considered as admissible test functions for the two-scale conver- gence.

• We also note that the function v₂ appearing in part (ii) of the previous proposition is the rigorous counterpart of the third term of the formal anisotropic expansion associated with the present setting:

v^ε= ˜v⁰(x, Z, x

ε²) +ε˜v¹(x, Z, x

ε²) +ε²v˜²(x, Z, x ε²) +. . .

• An consequence of the above definition of the two-scale convergence is the following.

Lemma 2.1 Let(v^ε)be a bounded sequence inL²(Ω)such that(ε^η∇v^ε)is bounded inL², withη <2.

Then the two-scale limitv⁰∈L²(Ω;H¹(T^d−1)) of v^ε is such that

∇_Xv⁰ = 0.

Proof: By (iii) of Proposition 2.1, ε²∇v^ε⇀∇² _Xv⁰. Since (ε^η∇v^ε) is bounded in L², it two-scale converges to some limitη⁰ and (ε²∇v^ε =ε^2−ηε^η∇v^ε) two-scale converges to 0. Thus ∇_Xv⁰ = 0.

Note that the above lemma is a key result to treat the anisotropy of the rough profile.

3 Rigorous derivation of the limit model

3.1 Convergence results

We infer from the estimates derived in Subsection 1.2 the following result.

Lemma 3.1 There exist limit functions

p⁰ ∈L²(Ω;L²(T^d−1)), u⁰∈L²(Ω;H¹(T^d−1)) and w⁰∈L²(Ω;H¹(T^d−1)) such that

ε²p^ε⇀ p^{2 0}, u^ε⇀ u^{2 0}, ∇_Xu⁰= 0, w^ε⇀ w^{2 0} and ∇_Xw⁰ = 0.

Proof: In view of Lemma 1.1, we claim the existence of limits (up to a subsequence). Moreover,

using Lemma 2.1, we assert that∇_Xu⁰ = 0 and∇_Xw⁰ = 0.

Before passing to the limit in the equations, we state some auxiliary results. We begin by the pressure function.

Lemma 3.2 The two-scale limit pressure is such that

∇_Xp⁰ = 0 and ∂_Zp⁰= 0.

Proof: We multiply Equation (5) by an admissible test function in the formε⁴φ(x, Z, x/ε²) and we integrate by parts. Unsing in particular ∆h

h −|∇h|² h² = div

∇h h

, we get Z

Ω

∇_xu^ε·(ε⁴∇_xφ^ε+ε²∇_Xφ^ε)dx dZ

− Z

Ω

ε⁴(∇_xh₁+ε⁻¹(1−ψ^ε)∇_Xh^ε₂−ε^1−αh^ε₂∇_xψ^ε)· 1

h₁+ε(1−ψ^ε)h^ε₂∇_xu^ε∂_Z(Zφ^ε)dx dZ

− Z

Ω

ε⁴(∇_xh₁+ε⁻¹(1−ψ^ε)∇_Xh^ε₂−ε^1−αh^ε₂∇_xψ^ε)· 1

h₁+ε(1−ψ^ε)h^ε₂Z∂_Zu^ε(∇_xφ^ε+ 1

ε²∇_Xφ^ε)dx dZ +

Z

Ω

ε⁴|∇_xh₁+ε⁻¹(1−ψ^ε)∇_Xh^ε₂−ε^1−αh^ε₂∇_xψ^ε|²

|h₁+ε(1−ψ^ε)h^ε₂|² ∂Zu^ε·∂Z(Z²φ^ε)dx dZ +

Z

Ω

ε⁴ 1

ε²|h₁+ε(1−ψ^ε)h^ε₂|²∂Zu^ε·∂Zφ^εdx dZ − Z

Ω

ε²p^ε(ε²divxφ^ε+ divXφ^ε)dx dZ +

Z

Ω

ε²(∇_xh₁+ε⁻¹(1−ψ^ε)∇_Xh^ε₂−ε^1−αh^ε₂∇_xψ^ε)· 1

h₁+ε(1−ψ^ε)h^ε₂ε²p^ε∂_Z(Zφ^ε)dx dZ = 0.

We bear in mind that the termε^1−α is of lower order thanε⁻¹ if 0≤α <2 and of the same order if α= 2. Passing to the limitε→0, we obtain

ε→0lim Z

Ω

ε²p^εdiv_Xφ^εdx dZ = Z

Ω

Z

T^d−1

p⁰div_Xφ dx dZ dX = 0.

Thus∇_Xp⁰ = 0 and the two-scale limit and the weak L² limit of (ε²p^ε) coincide.

Now let φ∈L²(ω;H₀¹(0,1)). We write R

Ω∂Z(ε²p^ε)φ dx dZ =εhε∂Zp^ε, φi_H−1×H¹ →0

=− Z

Ω

ε²p^ε∂Zφ dx dZ → − Z

Ω

p⁰∂Zφ dx dZ.

It follows that ∂_Zp⁰= 0. This ends the proof of the lemma.

We now introduce auxiliary limit functions for sake of clearness in the computations. We define, γ⁰ ∈L²(Ω;H¹(T^d−1)) andξ⁰ ∈L²(Ω;H¹(T^d−1)) such that

(1−ψ^ε)u^ε⇀ γ^{2 0}, ε²∇_x((1−ψ^ε)u^ε)⇀² ∇_Xγ⁰, (1−ψ^ε)²u^ε⇀ ξ^{2 0}, ε²∇x((1−ψ^ε)²u^ε)⇀² ∇Xξ⁰.

Of course, if 0≤α <2, we have ∇_Xγ⁰ = 0 and ∇_Xξ⁰ = 0. We now study the two-scale limit of the vertical velocity component.

Lemma 3.3 The vertical velocity component is such that w^ε⇀² 0.

Proof: We already mentioned that ∇_Xw⁰ = 0 (see lemma 3.1). We now prove that ∂_Zw⁰ = 0. We multiply the divergence Equation (7) byεφ(x, Z) whereφ∈H₀¹(Ω). Integrating by parts, we obtain

− Z

Ω

εu^ε· ∇_xφ+ Z

Ω

ε

h₁+ε(1−ψ^ε)h^ε₂∇_xh₁·u^ε∂_Z(Zφ) +

Z

Ω

1

h₁+ε(1−ψ^ε)h^ε₂(1−ψ^ε)∇Xh^ε₂·u^ε∂Z(Zφ)− Z

Ω

ε^2−α

h₁+ε(1−ψ^ε)h^ε₂h^ε₂∇xψ^ε·u^ε∂Z(Zφ)

− Z

Ω

1

h₁+ε(1−ψ^ε)h^ε₂w^ε∂Zφ= 0. (15)

Thanks to the relation

ε→0lim Z

Ω

1

h1+ε(1−ψ^ε)h^ε₂∇_Xh^ε₂·(1−ψ^ε)u^ε∂_Z(Zφ) = Z

Ω

1 h1

Z

T^d−1

∇_Xh₂γ⁰dX

∂_Z(Zφ), which is justified by (1−ψ^ε)u^ε⇀ γ^{2 0} and to the relation

−lim

ε→0

Z

Ω

ε^2−α

h₁+ε(1−ψ^ε)h^ε₂h^ε₂∇_xψ^ε·u^ε∂_Z(Zφ)

=−lim

ε→0

Z

Ω

1

h₁+ε(1−ψ^ε)h^ε₂h^ε₂ε²∇_x(ψ(x/ε^α)u^ε)∂_Z(Zφ) + lim

ε→0

Z

Ω

1

h₁+ε(1−ψ^ε)h^ε₂h^ε₂ψ^εε²div_xu^ε∂_Z(Zφ)

= lim

ε→0

Z

Ω

1

h₁+ε(1−ψ^ε)h^ε₂h^ε₂ε²∇_x((1−ψ(x/ε^α))u^ε)∂_Z(Zφ)

= Z

Ω

1

h₁∂_Z(Zφ) Z

T^d−1

h₂∇_Xγ⁰dX, which is justified byε²∇_x((1−ψ^ε)u^ε)⇀² ∇_Xγ⁰, we compute

ε→0lim Z

Ω

1

h₁+ε(1−ψ^ε)h^ε₂∇Xh^ε₂·(1−ψ^ε)u^ε∂Z(Zφ)− Z

Ω

ε^2−α

h₁+ε(1−ψ^ε)h^ε₂h^ε₂∇xψ^ε·u^ε∂Z(Zφ)

= Z

Ω

1

h₁∂_Z(Zφ) Z

∇_X(h₂γ⁰)dX

= 0 because of the periodicity of h₂γ⁰. We thus infer from (15) that

ε→0lim Z

Ω

1

h₁+ε(1−ψ^ε)h^ε₂w^ε∂_Zφ= Z

Ω

1

h₁w⁰∂_Zφ= 0.

It follows that ∂Zw⁰ = 0. We conclude using the boundary conditions.

Remark 3.1 We recall that the aim of this work is to get a perturbation of the Reynolds approxi- mation. And the previous result is characteristic of such a lubrication approximation: it states that we can neglect the vertical component of the velocity.

3.2 Divergence equation Lemma 3.4 We have the relation

h₁div_xu⁰− ∇_xh₁·Z ∂_Zu⁰+∂_Z(w¹+L) = 0, where w¹ = (1/d−1)R

T^d−1w˜₁dX, with w˜₁ ∈L²(Ω;H¹(T^d−1)) is defined by the following two-scale

convergence (

εw^ε⇀² 0,

∇(εw^ε)⇀² ∇_Xw˜₁, while∂_ZL is defined by the following weak convergence





∂_ZL= lim

ε→0∂_Z

Zε(1−ψ^ε)h^ε₂ h1

divu^ε

in D^′(Ω), L_|Z=0 =L_|Z=1 = 0.

Remark 3.2 The term w˜¹ is the rigorous counterpart of the first order term of a formal anisotropic expansion ofw^ε:

w^ε= ˜w⁰(x, Z, x/ε²) +εw˜¹(x, Z, x/ε²) +ε²w˜²(x, Z, x/ε²) +· · ·

Proof: We multiply the divergence Equation (7) by a test function φ(x, Z)∈ D(Ω). We obtain:

− Z

Ω

u^ε· ∇_xφ+ Z

Ω

1

h₁+ε(1−ψ^ε)h^ε₂∇_xh₁·u^ε∂_Z(Zφ)dx dZ +

Z

Ω

1

h₁+ε(1−ψ^ε)h^ε₂

(1−ψ^ε)

ε ∇_Xh^ε₂·u^ε∂_Z(Zφ)dx dZ

− Z

Ω

1

h₁+ε(1−ψ^ε)h^ε₂ h^ε₂

ε^α−1∇xψ^ε·u^ε∂Z(Zφ)dx dZ− Z

Ω

1

h₁+ε(1−ψ^ε)h^ε₂ 1

εw^ε∂Zφ dx dZ= 0.

It follows that

Z

Ω

u⁰· ∇_xφ+ Z

Ω

1

h₁∇_xh₁·Z∂_Zu⁰φ dx dZ + lim

ε→0

Z

Ω

1 ε

(1−ψ^ε)∇Xh^ε₂−ε^2−αh^ε₂∇xψ^ε

h₁+ε(1−ψ^ε)h^ε₂ ·Z∂_Zu^εφ dx dZ+ Z

Ω

1 ε

1

h₁+ε(1−ψ^ε)h^ε₂w^ε∂_Zφ dx dZ

= 0.(16) We note that 1/(h₁+ε(1−ψ^ε)h^ε₂) is a computable perturbation of 1/h₁ for our convergences needs.

Indeed, we have

1

h1+ε(1−ψ^ε)h^ε₂ − 1 h1

= −ε(1−ψ^ε)h^ε₂ h1(h1+ε(1−ψ^ε)h^ε₂). On the one hand, using (1−ψ^ε)²u^ε⇀ ξ^{2 0} andε²∇_x(1−ψ^ε)²u^ε⇀² ∇_Xξ⁰, we write

ε→0lim Z

Ω

1 ε

−ε(1−ψ^ε)h^ε₂

h₁(h₁+ε(1−ψ^ε)h^ε₂) (1−ψ^ε)∇_Xh^ε₂−ε^2−αh^ε₂∇_xψ^ε

·Z∂_Zu^εφ

=− Z

Ω

1 h²₁

Z

h₂∇_Xh₂Z ∂_Zξ⁰φ dX

−lim

ε→0

Z

Ω

h^ε₂²

h₁(h₁+ε(1−ψ^ε)h^ε₂) ε²

2∇_x

(1−ψ(x/ε^α))²∂_Zu^ε Z φ + lim

ε→0

Z

Ω

h^ε₂²

h1(h1+ε(1−ψ^ε)h^ε₂) ε²

2(1−ψ(x/ε^α))²∇_x(∂_Zu^ε)Z φ

=− Z

Ω

1 h²₁

Z

h2∇Xh2Z ∂Zξ⁰φ dX + lim

ε→0

Z

Ω

h^ε₂²

h₁(h₁+ε(1−ψ^ε)h^ε₂) ε²

2∇_x

(1−ψ(x/ε^α))²u^ε

∂_Z(Z φ)

−lim

ε→0

Z

Ω

h^ε₂²

h₁(h₁+ε(1−ψ^ε)h^ε₂) ε²

2(1−ψ(x/ε^α))²∇_x(u^ε)∂_Z(Z φ)

=− Z

Ω

1 h²₁

Z

h₂∇_Xh₂Z ∂_Zξ⁰φ dX + Z

Ω

1 2h²₁

Z

h²₂∇_X(ξ⁰)∂_Z(Z φ)dX

=− Z

Ω

1 h²₁

Z

h₂∇_Xh₂∂_Zξ⁰Z φ dX− Z

Ω

1 2h²₁

Z

h²₂∇_X(∂_Zξ⁰)Z φ dX

=− Z

Ω

1 2h²₁

Z

∇X h²₂∂Zξ⁰

Z φ dX

= 0

because of the periodicity of h₂ξ⁰.

On the other hand, we definew^aux by

ε→0lim Z

Ω

(1−ψ^ε)h^ε₂

h1(h1+ε(1−ψ^ε)h^ε₂)w^ε∂_Zφ= Z

Ω

Z

T^d−1

w^auxdX

∂_Zφ=− Z

Ω

∂_ZZ

T^d−1

w^auxdX

φ, (17)

which means that (1−ψ^ε)h^ε₂w^ε/h²₁ ⇀R

T^d−1w^auxdX weakly inL²(Ω). Then Relation (16) reads:

Z

Ω

u⁰· ∇_xφ+ Z

Ω

1

h₁∇_xh₁·Z∂_Zu⁰φ+ Z

Ω

∂_ZZ

T^d−1

w^auxdX

φ=−A−B (18)

where

A= lim

ε→0

Z

Ω

1 ε

1

h₁ (1−ψ^ε)∇Xh^ε₂−ε^2−αh^ε₂∇xψ^ε

·Z∂Zu^εφ, B= lim

ε→0

Z

Ω

1 ε

1

h₁w^ε∂_Zφ.

We write

A= lim

ε→0

Z

Ω

1

h₁∇_x(ε(1−ψ^ε)h^ε₂)·(Z∂_Zu^ε)φ

= lim

ε→0

Z

Ω

ε(1−ψ^ε)h^ε₂ h1

div_xu^ε∂_Z(Zφ)− Z

Ω

ε(1−ψ^ε)h^ε₂

h²₁ ∇_xh₁·u^ε∂_Z(Zφ) +

Z

Ω

ε(1−ψ^ε)h^ε₂

h₁ u^ε· ∇_x(∂_Z(Zφ))

= lim

ε→0

Z

Ω

ε(1−ψ^ε)h^ε₂

h₁ div_xu^ε∂_Z(Zφ)

= lim

ε→0

Z

Ω

ε(1−ψ^ε)h^ε₂

h₁ div_xu^εZ∂_Zφ+ lim

ε→0

Z

Ω

ε(1−ψ^ε)h^ε₂

h₁ div_xu^εφ

=−lim

ε→0

Z

Ω

∂_Z

Zε(1−ψ^ε)h^ε₂ h1

div_xu^ε

φ+ lim

ε→0

Z

Ω

ε(1−ψ^ε)h^ε₂ h1

div_xu^εφ. (19)

The first term in the right hand side will give the conservative contribution∂_ZLto the final divergence equation. Let us compute the last term. We multiply the divergence Equation (7) by the test function

ε(1−ψ^ε)h^ε₂φ(x, Z). We get

ε→0lim Z

Ω

(1−ψ^ε)h^ε₂

h₁+ε(1−ψ^ε)h^ε₂w^ε∂Zφ

= lim

ε→0

Z

Ω

εdiv_xu^ε(1−ψ^ε)h^ε₂φ+ lim

ε→0

Z

Ω

ε(1−ψ^ε)h^ε₂

h₁+ε(1−ψ^ε)h^ε₂∇_xh₁·u^ε∂_Z(Zφ) + lim

ε→0

Z

Ω

h^ε₂

h₁+ε(1−ψ^ε)h^ε₂∇_Xh^ε₂·(1−ψ^ε)²u^ε∂_Z(Zφ)

−lim

ε→0

Z

Ω

ε^2−αh^ε₂²

h₁+ε(1−ψ^ε)h^ε₂∇_xψ^ε·(1−ψ^ε)u^ε∂_Z(Zφ)

= lim

ε→0

Z

Ω

εdivxu^ε(1−ψ^ε)h^ε₂φ+ Z

Ω

1 2h₁

Z

∇X(h²₂)·ξ⁰∂Z(Zφ)dX + lim

ε→0

Z

Ω

h^ε₂²

h₁+ε(1−ψ^ε)h^ε₂ ε²∇_x1

2(1−ψ^ε)²u^ε

∂_Z(Zφ)

−lim

ε→0

Z

Ω

h^ε₂² h₁+ε(1−ψ^ε)h^ε₂

ε²

2(1−ψ^ε)²div_xu^ε∂_Z(Zφ)

= lim

ε→0

Z

Ω

εdiv_xu^ε(1−ψ^ε)h^ε₂φ+ Z

Ω

1 2h1

Z

∇_X(h²₂)·ξ⁰∂_Z(Zφ)dX+ Z

Ω

1 2h1

Z

h²₂∇_Xξ⁰∂_Z(Zφ)dX

= lim

ε→0

Z

Ω

εdiv_xu^ε(1−ψ^ε)h^ε₂φ+1 2

Z

Ω

1 h1

Z

∇_X(h²₂ξ⁰)∂_Z(Zφ)dX

= lim

ε→0

Z

Ω

εdiv_xu^ε(1−ψ^ε)h^ε₂φ because of the periodicity ofh₂ξ⁰.

Bearing in mind (17), we infer from the latter relation that:

ε→0lim Z

Ω

(1−ψ^ε)h^ε₂

h₁(h₁+ε(1−ψ^ε)h^ε₂)w^ε∂_Zφ= lim

ε→0

Z

Ω

h^ε₂

h₁ε(1−ψ^ε)φdiv_xu^ε

=− Z

Ω

∂_ZZ

T^d−1

w^auxdX

φ (20)

because of the definition ofw^aux. Let us now compute B = lim

ε→0

Z

Ω

1 ε

1

h₁w^ε∂_Zφ. We write Z

Ω

1 ε

1 h1

w^ε∂_Zφ= Z

Ω

1

ε²(εw^ε 1 h1

∂_Zφ)

= 1

d−1 Z

Ω

div(I(x/ε²))(εw^ε 1 h1

∂_Zφ)

=− 1 d−1

Z

Ω

I(x/ε²)· ∇(εw^ε 1 h₁∂_Zφ) whereI is the 1-periodic function such thatI(x) =x ifx∈T^d−1. Then

B=− 1 d−1

Z

Ω

Z

T^d−1

X· ∇_Xw˜₁dX 1 h₁ ∂_Zφ

= 1

d−1 Z

Ω

Z

T^d−1

˜

w₁dX 1 h₁ ∂_Zφ

= Z

Ω

w¹

h₁ ∂_Zφ=− Z

Ω

∂Zw¹ h₁ φ

(21)

wherew¹ = _d−1¹ R

T^d−1w˜₁dX, the function ˜w₁∈L²(Ω;H¹(T^d−1)) being defined by ( εw^ε⇀² 0,

∇(εw^ε)⇀² ∇_Xw˜₁.

Using (18)-(21), we obtain the announced result.

The conservative form of the limit divergence equation obtained in the previous lemma is sufficient to get an explicit Reynolds approximation of the flow, even if functionL is not explicitly computed (see [6]). By the way, we end this section by computingL under an additional assumption.

Lemma 3.5 Assuming moreover one of the following assumption, (H1)∇(ψ^εh^ε₂) strongly two-scale converges to ∇_X(ψ⁰h₂),

(H2)ψ^ε strongly two-scale converges to ψ⁰, (H3)α <2,

then L= 0 and the limit divergence equation reads

h1divxu⁰− ∇xh1·Z∂Zu⁰+∂Zw¹ = 0.

Proof: From the assumptions (H1) or (H2), it follows that h2(X)γ⁰(x, X) =h2(X) 1−ψ⁰(x, X)

u0(x, Z), h₂(X)ξ⁰(x, X) =h₂(X) 1−ψ⁰(x, X)2

u₀(x, Z).

The two scale limit ofZ ε(1−ψ^ε)h^ε₂divu^εish₁L=Z(1−ψ⁰)h₂div_Xu¹ whereu¹∈L²(Ω;H¹(T^d−1)) is the anisotropic two-scale limit defined by

∇(εu^ε)⇀² ∇_Xu¹.

Let us compute div_Xu¹. To this aim, we multiply the divergence Equation (7) by εφ(x, Z, x/ε²) whereφis an admissible test function for the two scale convergence. Integrating by parts, we obtain

Z

Ω

εdivu^εφ^ε− Z

Ω

(1−ψ^ε)∇_Xh^ε₂−ε^2−αh^ε₂∇_xψ^ε

·Z∂_Zu^ε φ^ε h₁+ε(1−ψ^ε)h^ε₂

− Z

Ω

ε∇_xh₁·Z∂_Zu^ε φ^ε

h₁+ε(1−ψ^ε)h^ε₂ + Z

Ω

∂_Zw^ε φ^ε

h₁+ε(1−ψ^ε)h^ε₂ = 0.

We recall that ε∂_Zu^ε→² 0 (a strong two-scale convergence due to the fact that ∂_Zu^ε is bounded in (L²(Ω))^d) and w^ε⇀² 0 (see Lemma 3.3). We also have

ε→0lim Z

Ω

(1−ψ^ε)∇_Xh^ε₂−ε^2−αh^ε₂∇_xψ^ε

·Z∂_Zu^ε φ^ε

h₁+ε(1−ψ^ε)h^ε₂ = Z

Ω

Z

T^d−1

1

h₁ div_X(h₂Z∂_Zγ⁰)φ dX, because

(1−ψ^ε)∇_Xh^ε₂−ε^2−αh^ε₂∇_xψ^ε

u^ε=ε²div_x (1−ψ^ε)h^ε₂u^ε

−ε²(1−ψ^ε)h^ε₂div_xu^ε

⇀2 div_X h₂γ⁰ . We thus conclude that

div_Xu¹ = 1

h₁div_X(h₂·Z∂_Zγ⁰). (22)