Mahdi Boukrouche
email: mahdi.boukrouche@univ-st-etienne.fr

Laetitia Paoli
email: laetitia.paoli@univ-st-etienne.fr

Fatima Ziane
email: zianefatima1@gmail.com

Ziane Achour

Micropolar Fluid Flow in a Thick Domain with Multiscale Oscillating Roughness and Friction Boundary Conditions

Keywords: Unsteady micropolar fluid flow, Tresca's friction boundary conditions, parabolic variational inequality, multi-scale convergence, reiterated homogenization 2000 MSC: 35Q35, 35K86, 76A05, 76M50, 76A20

We consider an unsteady micropolar fluid flow in a two-dimensional domain Ω ε . The velocity field is assumed to satisfy a fluid-solid friction interface condition on a part of the boundary while the micro-rotation field satisfies non-homogeneous Dirichlet boundary conditions. The problem is thus described by a non-linear coupled variational parabolic system for the fluid velocity v , the pressure p and the angular micro-rotation field Z . The thickness and the roughness of the fluid domain are described by multiple separated scales of periodic oscillations, i.e.

, and m ≥ 2. Existence, uniqueness and uniform estimates of the solution (v , p , Z ) are stated. Then we study the asymptotic behaviour of the flow as ε tends to zero by using the multiple scale convergence method for reiterated homogenization. The assumption m ≥ 2 raises several technical difficulties in the limit process and leads to non-standard divergence free conditions for the limit velocity. We derive the limit problem which is totally decoupled for the limit velocity and pressure (v 0 , p 0 ) on the one hand and the limit micro-rotation field Z 0 on the other hand. More precisely (v 0 , p 0 ) is solution of a variational elliptic inequality and Z 0 solves an elliptic partial differential equation, where the time variable appears as a parameter. Moreover we prove that v 0 , p 0 and Z 0 are uniquely determined by auxiliary well-posed problems.

Description of the problem

The problem is motivated by lubrication problems involving complex fluids satisfying the micropolar fluid theory developed by A. Eringen ([14,[START_REF] Eringen | Theory of micropolar fluids[END_REF]). Such fluids, like polymers, ferroliquid or colloidal fluids, contain suspensions of micro-structures that undergo rotations. When these particles are assumed to be rigid and isotropic, their motion is modelled by an angular micro-rotation field and the flow is described by the equilibrium of momentum, mass and moment of momentum leading to the following coupled system of partial differential equations for the fluid velocity u ε = (u ε 1 , u ε 2 ), the pressure p ε and the angular micro-rotation field ω ε :

∂u ε ∂t -(ν + ν r )∆u ε + (u ε • ∇)u ε + ∇p ε = 2ν r rot(w ε ) + f ε (1) div(u ε ) = 0 (2) ∂w ε ∂t -α∆w ε + (u ε • ∇)w ε + 4ν r w ε = 2ν r rot(u ε ) + g ε , (3)
with the initial conditions u ε (0, z) = u ε 0 (z), w ε (0, z) = w ε 0 (z) [START_REF] Boukrouche | Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary[END_REF] where f ε and g ε are external force and momentum and ν, ν r and α are given positive viscosity parameters ( [START_REF] Eringen | Simple Microfluids[END_REF][START_REF] Eringen | Theory of micropolar fluids[END_REF]).

During the last decades the study of micropolar fluid flows has become a very active research field in mathematics (see for instance the monograph [START_REF] Łukaszewicz | Micropolar Fluids. Theory and applications[END_REF] or more recent papers [START_REF] Dong | Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid flows[END_REF][START_REF] Dong | Global regularity of the 2D micropolar fluid flows with zero angular viscosity[END_REF][START_REF] Xue | Well posedness and zero microrotation viscosity limit of the 2D micropolar fluid equations[END_REF][START_REF] Chen | Global well-posedness for the micropolar fluid system in critical Besov spaces[END_REF][START_REF] Dong | Global well-posedness and large-time decay for the 2D micropolar equations[END_REF][START_REF] Jiu | On the initial-and boundary-value problem for 2D micropolar equations with only angular velocity dissipation[END_REF][START_REF] Dong | Global regularity for the 2D micropolar equations with fractional dissipation[END_REF][START_REF] Liu | Initial-boundary value problem for 2D micropolar equations without angular viscosity[END_REF][START_REF] Wang | Global regularity of the three-dimensional fractional micropolar equations[END_REF]). Motivated by lubrication problems, we consider the system (1)-(4) in the space-time domain (0, T ) × Ω ε with T > 0 and

Ω ε = z = (z 1 , z 2 ) ∈ R 2 , 0 < z 1 < L, 0 < z 2 < ε m h ε (z 1 )
where m > 1 and h ε is a smooth L-periodic function which is bounded from above and from below by two positive numbers h M and h m . The domain Ω ε can be interpreted as the crosssection of an infinite journal bearing given by two non-concentric cylinders. After a radial cut the gap between the two cylinders is described by Ω ε and, as usual in lubrication theory, its thickness is much smaller than the cylinders radii.

We decompose ∂Ω ε as ∂Ω ε = Γ 0 ∪ Γ ε 1 ∪ Γ ε L where Γ 0 = {z ∈ Ω ε : z 2 = 0}, Γ ε 1 = {z ∈ Ω ε : z 2 = ε m h ε (z 1 )} and Γ ε L is the lateral part of the boundary. Due to the original geometry of the flow domain we have u ε and w ε are L-periodic with respect to z 1 .

(

Moreover, for most lubrication devices, the inner cylinder (which corresponds to Γ 0 ) is in rotation while the outer cylinder (which corresponds to Γ ε 1 ) is at rest or rotates more slowly than the other one. Thus we will consider non-homogeneous Dirichlet boundary conditions for u ε and w ε on Γ ε 1 i.e.

u ε = U 0 e 1 = (U 0 , 0) and w ε = W 0 on (0, T ) × Γ ε 1 [START_REF] Boukrouche | Unsteady micrololar fluid flow in a thin domain with Tresca fluid-solid interface law[END_REF] where W 0 and U 0 are two given scalar functions of the time variable, while we will consider a fluid-solid friction interface condition on Γ 0 . Indeed, experimental studies have shown that some slip boundary condition of Tresca's type may occur (see [START_REF] Barnes | A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure[END_REF][START_REF] Kissi | Slip and friction of polymer melt flows[END_REF][START_REF] Pit | Friction and slip of a simple liquid at solid surface[END_REF][START_REF] Rao | The effect of the slip boundary condition on the flow of fluids in a channel[END_REF][START_REF] Pahlavan | Effect of solid properties on slip at a fluid-solid interface[END_REF][START_REF] Sochi | Slip at fluid-solid interface[END_REF] for instance). Thus we have u ε • n = 0 and w ε = 0 on (0, T ) × Γ 0 [START_REF] Chen | Global well-posedness for the micropolar fluid system in critical Besov spaces[END_REF] where n is the unit outward normal vector to ∂Ω ε . The shear stress σ ε τ remains bounded by a given positive threshold k ε and slip may occur only when |σ ε τ | = k ε with a relative slip velocity 2 in the opposite direction to σ ε τ . More precisely the stress tensor σ ε and its normal and tangential components are given by

σ ε i j = -p ε δ i j + 2(ν + ν r )d i j (u ε ), d i j (u ε ) = 1 2       ∂u ε i ∂z j + ∂u ε j ∂z i       i, j = 1, 2 and
σ ε n = (σ ε • n) • n = 2 i, j=1 σ ε i j n i n j , σ ε τ i = 2 j=1 σ ε i j n j -σ ε n n i i = 1, 2.
Then we have

|σ ε τ | ≤ k ε on (0, T ) × Γ 0 and |σ ε τ | < k ε ⇒ u ε τ = s 0 e 1 |σ ε τ | = k ε ⇒ ∃λ ≥ 0 s.t. u ε τ = s 0 e 1 -λσ ε τ on (0, T ) × Γ 0 ( 8
)
where s 0 (t)e 1 is the sliding velocity of the wall ( [START_REF] Duvaut | Les inéquations en mécanique et physique[END_REF]). This kind of nonlinear slip boundary conditions have been introduced by H. Fujita for Stokes flows in [START_REF] Fujita | Flow problems with unilateral boundary conditions[END_REF] paving the way to an abundant literature for steady or unsteady Stokes or Navier-Stokes flows ( [START_REF] Fujita | A mathematical analysis of motions of viscous incompressible fluid under leak or slip boundary conditions[END_REF][START_REF] Fujita | Analytical and numerical approaches to stationary flow problems with leak and slip boundary conditions[END_REF][START_REF] Fujita | Variational inequalities for the Stokes equation with boundary conditions of friction type[END_REF][START_REF] Saito | Regularity of solutions to the Stokes equation under a certain nonlinear boundary condition. The Navier-Stokes equations[END_REF][START_REF] Fujita | Remarks on the Stokes flows under slip and leak boundary conditions of friction type[END_REF][START_REF] Fujita | A coherent analysis of Stokes flows under boundary conditions of friction type[END_REF][START_REF] Saito | On the Stokes equations with the leak and slip boundary conditions of friction type: regularity of solutions[END_REF][START_REF] Le Roux | Steady flows of incompressible Newtonian fluids with threshold slip boundary conditions[END_REF][START_REF] Le Roux | Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions[END_REF][START_REF] Kashiwabara | On a strong solution of the non-stationary Navier-Stokes equations under slip or leak boundary conditions of friction type[END_REF][START_REF] Boukrouche | Non-isothermal Navier-Stokes system with mixed boundary conditions and friction law: uniqueness and regularity properties[END_REF]). The case of micropolar fluids has been studied in a very recent paper ( [START_REF] Boukrouche | Unsteady micrololar fluid flow in a thin domain with Tresca fluid-solid interface law[END_REF]) where the existence and uniqueness of a solution for the problem (1)-( 8) is established. Our aim is now to extend this first result and to study the asymptotic behaviour of the flow when the shape of Ω ε is described by a very oscillating function given by

z 1 → h ε (z 1 ) = h z 1 , z 1 ε , z 1 ε 2 , • • • , z 1 ε m m ≥ 2,
leading to multiple separated scales of periodic oscillations in the horizontal direction.

The paper is organized as follows. In Section 2 we introduce the mathematical framework and the variational formulation of problem (P ε ). Then we derive from [START_REF] Boukrouche | Unsteady micrololar fluid flow in a thin domain with Tresca fluid-solid interface law[END_REF] existence, uniqueness and a priori estimates for the solutions of (P ε ). In Section 3 we apply the multi-scale convergence technique for reiterated homogenization problems to get a limit triplet (v 0 , Z 0 , p 0 ). The assumption m ≥ 2 yields more technicalities in the multiple scale convergence process (see Proposition 3.3, Proposition 3.4, Proposition 3.5 and Proposition 3.7) and leads to non-standard divergence free conditions for v 0 (see Proposition 3.6). Finally in Section 4 this new type of divergence free condition plays a crucial role in the derivation of the limit problem (see Remark 4.1 and Proposition 4.1). Moreover we prove that (v 0 , p 0 ) and Z 0 solve a totally decoupled system of elliptic variational inequality and elliptic partial differential equation, where the time variable appears as a parameter (see Proposition 4.4 and Proposition 4.2). Furthermore v 0 , p 0 and Z 0 are uniquely determined through auxiliary well posed problems (see Proposition 4.3 and Theorem 4.5).

Mathematical framework

We use throughout this paper the same notations as in [START_REF] Boukrouche | Unsteady micrololar fluid flow in a thin domain with Tresca fluid-solid interface law[END_REF]. We let

V ε = v ∈ C ∞ (Ω ε ) 2 : v is L-periodic in z 1 , v = 0 on Γ ε 1 , v • n = 0 on Γ 0 , 3 V ε 0 = v ∈ C ∞ (Ω ε ) 2 : v is L-periodic in z 1 , v = 0 on Γ 0 ∪ Γ ε 1 , H 1,ε = Z ∈ C ∞ (Ω ε ) : Z is L-periodic in z 1 , Z = 0 on Γ 0 ∪ Γ ε 1 , V ε = closure of V ε in H 1 (Ω ε ) 2 , V ε div = v ∈ V ε : div(v) = 0 in Ω ε , V ε 0 = closure of V ε 0 in H 1 (Ω ε ) 2 , V ε 0,div = v ∈ V ε 0 : div(v) = 0 in Ω ε , H ε = closure of V ε in L 2 (Ω ε ) 2 , H 0,ε = closure of H 1,ε in L 2 (Ω ε ), H 1,ε = closure of H 1,ε in H 1 (Ω ε ), L 2 0 (Ω ε ) = q ∈ L 2 (Ω ε ) : Ω ε q(z)dz = 0 .
These functional spaces are endowed with the inner products and norms defined by

[v, Θ] = (v, ϕ) + (Z, ψ) with the norm [v] = [v, v] for all v = (v, Z) ∈ H ε × H 0,ε , Θ = (ϕ, ψ) ∈ H ε × H 0,ε and [[v, Θ]] = (∇v, ∇ϕ) + (∇Z, ∇ψ) with the norm [[v]] = [[v, v]] for all v = (v, Z) ∈ V ε × H 1,ε , Θ = (ϕ, ψ) ∈ V ε × H 1,ε where (•, •) denotes the inner product of L 2 (Ω ε ) d with d = 1, 2 or 4.
In order to deal with homogeneous boundary conditions on Γ ε 1 we introduce new unknown velocity and micro-rotation fields

v ε (t, z) = u ε (t, z) -U ε (t, z 2 )e 1 in (0, T ) × Ω ε Z ε (t, z) = w ε (t, z) -W ε (t, z 2 ) in (0, T ) × Ω ε
where U ε and W ε are given by

U ε (t, z 2 ) = s 0 (t) + U 0 (t) -s 0 (t) U ε (z 2 ) = s 0 (t) + U 0 (t) -s 0 (t) U z 2 ε m W ε (t, z 2 ) = W 0 (t)W ε (z 2 ) = W 0 (t)W z 2 ε m with functions U ∈ C ∞ (R) and W ∈ C ∞ (R) such that U(0) = U (0) = 0 = W(0), U(X) = W(X) = 1 ∀X ∈ [h m , h M ]. Thus U ε (t, 0) = s 0 (t), W ε (t, 0) = 0 U ε t, ε m h ε (z 1 ) = U 0 (t), W ε (t, ε m h ε (z 1 )) = W 0 (t)
for all (t, z 1 ) ∈ (0, T ) × (0, L) and U ε and W ε are two extensions of U 0 and W 0 . Moreover

∂U ε ∂z 2 (t, 0) = 0 which yields σ(u ε , p ε ) = σ(v ε , p ε ) on (0, T ) × Γ 0 . Moreover rot v ε = ∂v ε 2 ∂z 1 - ∂v ε 1 ∂z 2 = rot u ε - ∂U ε ∂z 2 , rot Z ε = ∂Z ε ∂z 2 , - ∂Z ε ∂z 1 = rot w ε - ∂W ε ∂z 2 , 0 .
We assume that k ε ∈ L ∞ 0, T ; L ∞ + (Γ 0 ) and we denote as j ε the Tresca's functional i.e.

j ε (ϕ) = T 0 Γ 0 k ε |ϕ| dz 1 dt ∀ϕ ∈ L 2 0, T ; L 2 (Γ 0 ) 2 .
The variational formulation of problem ( 1)-( 8) is given by

Problem (P ε ) : Find (v ε , Z ε , p ε ) such that v ε = (v ε , Z ε ) ∈ C [0, T ]; H ε ∩ L 2 (0, T ; V ε div ) × C [0, T ]; H 0,ε ∩ L 2 (0, T ; H 1,ε ) , p ε ∈ H -1 0, T ; L 2 0 (Ω ε ) and ∂ ∂t [v ε , Θ ε ], θ D (0,T ),D(0,T ) + T 0 a(v ε , Θ ε )θ dt + T 0 B(v ε , v ε , Θ ε )θ dt + T 0 R(v ε , Θ ε )θ dt + j ε (ϕ ε θ + v ε ) -j ε (v ε ) ≥ T 0 p ε , div(ϕ ε ) θ dt + T 0 F ε (v ε ), Θ ε θ dt ∀θ ∈ D(0, T ), ∀Θ ε = (ϕ ε , ψ ε ) ∈ V ε × H 1,ε (9)
with the initial condition

v ε (0) = (v ε 0 , Z ε 0 ) = (u ε 0 -U ε (0, •)e 1 , w ε 0 -W ε (0, •)) (10)
where

(F ε (v ε ), Θ) = -a(ξ ε , Θ) -B(ξ ε , v ε , Θ) -B(v ε , ξ ε , Θ) -R(ξ ε , Θ) - ∂ξ ε ∂t , Θ + [ f ε , Θ], ξ ε = (U ε e 1 , W ε ), f ε = ( f ε , g ε ) and a(v, Θ) = (ν + ν r )(∇v, ∇ϕ) + α(∇Z, ∇ψ) R(v, Θ) = -2ν r (rot Z, ϕ) -2ν r (rot v, ψ) + 4ν r (Z, ψ) B(v, u, Θ) = b(v, u, ϕ) + b 1 (v, w, ψ) = 2 i, j=1 Ω ε v i ∂u j ∂z i ϕ j dz + 2 i=1 Ω ε v i ∂w ∂z i ψ dz for all v = (v, Z), u = (u, w) and Θ = (ϕ, ψ) in V ε × H 1,ε
Let us observe that we identify ϕ ε θ + v ε and v ε with their trace on Γ 0 in the definition of

j ε (ϕ ε θ + v ε ) and j ε (v ε ). Furthermore by choosing Θ ε = ±(ϕ ε , ψ ε ) ∈ V ε 0 × H 1,ε we get j ε (±ϕ ε θ + v ε ) -j ε (v ε ) = T 0 Γ 0 k ε | ± ϕ ε θ + v ε | -|v ε | dz 1 dt = 0,
and the variational inequality (9) reduces to

∂ ∂t [v ε , Θ ε ], θ D (0,T ),D(0,T ) + T 0 a(v ε , Θ ε )θ dt + T 0 B(v ε , v ε , Θ ε )θ dt + T 0 R(v ε , Θ ε )θ dt = T 0 p ε , div(ϕ ε ) θ dt + T 0 F ε (v ε ), Θ ε θ dt. (11)
With Theorem 1 in [START_REF] Boukrouche | Unsteady micrololar fluid flow in a thin domain with Tresca fluid-solid interface law[END_REF] we obtain immediately an existence and uniqueness result for Problem (P ε ).

Theorem 2.1. Let ε > 0 and assume that (s 0 , U 0 , W 0 ) ∈ (H 1 (0, T )) 3 , ( f ε , g ε ) ∈ L 2 (0, T )×Ω ε 3 , k ε ∈ L ∞ 0, T ; L ∞ + (Γ 0 ) and (v ε 0 , Z ε 0 ) ∈ H ε × H 0,ε . Then Problem (P ε ) admits an unique solution. Moreover ∂v ε ∂t ∈ L 2 0, T ; (V ε div × H 1,ε ) .
Remark 2.1. It follows that the inertia term in ( 9) can be rewritten as

∂ ∂t [v ε , Θ ε ], θ D (0,T ),D(0,T ) = T 0 ∂v ε ∂t , θΘ ε (V ε div ×H 1,ε ) ,V ε div ×H 1,ε dt for all Θ ε = θΘ ε ∈ D(0, T )⊗(V ε div × H 1,ε ). By density of D(0, T )⊗(V ε div × H 1,ε ) into L 2 (0, T ; V ε div × H 1,ε ) we may consider any test-function Θ ε ∈ L 2 (0, T ; V ε div × H 1,ε ) in (9) and we get T 0 ∂v ε ∂t , Θ ε (V ε div ×H 1,ε ) ,V ε div ×H 1,ε dt + T 0 a(v ε , Θ ε ) dt + T 0 B(v ε , v ε , Θ ε ) dt + T 0 R(v ε , Θ ε ) dt + j ε (ϕ ε + v ε ) -j ε (v ε ) ≥ T 0 F ε (v ε ), Θ ε dt ( 12
)
for all Θ ε = (ϕ ε , ψ ε ) ∈ L 2 (0, T ; V ε div × H 1,ε ).
We consider in this paper a thick domain with a multiscale oscillating roughness described by

h ε (z 1 ) = h z 1 , z 1 ε , z 1 ε 2 , • • • , z 1 ε m
with m > 1 and h a positive C ∞ -function on R m+1 . We assume that L ε k ∈ N for all k ∈ {1, . . . , m} and h : (z 1 , η 11 , . . . , η 1m ) → h(z 1 , η 11 , . . . , η 1m ) is L-periodic with respect to z 1 and 1-periodic with respect to η 11 , . . ., η 1m . Hence h ε is L-periodic in z 1 and we have to deal with multiple rapid scales z 1 ε , . . ., z 1 ε m in the horizontal direction. In order to study the asymptotic behaviour as ε tends to zero we introduce the following scaling

y 1 = z 1 , y 2 = z 2 ε m h ε (z 1 )
which transforms Ω ε into Ω = (0, L) × (0, 1) and we denote

Γ 1 = [0, L] × {1}, Γ L = {0, 1} × [0, 1].
We may now consider the unknowns of the problem as functions of t and the new space variables (y 1 , y 2 ) and we will still denote them as v ε , p ε and Z ε i.e.

v ε (t, z) = v ε (t, z 1 , z 2 ) = v ε (t, y 1 , ε m h ε (y 1 )y 2 ) := v ε (t, y 1 , y 2 ) = v ε (t, y) p ε (t, z) := p ε (t, y), Z ε (t, z) := Z ε (t, y)
and similarly

f ε (t, z) := f ε (t, y), g ε (t, z) := g ε (t, y).
With the chain rule we get

∂ ∂z 2 = 1 ε m h ε ∂ ∂y 2 , ∂ ∂z 1 = ∂ ∂y 1 ∂y 1 ∂z 1 + ∂ ∂y 2 ∂y 2 ∂z 1 = ∂ ∂y 1 + - y 2 h ε ∂h ε ∂y 1 ∂ ∂y 2
and for the sake of notational simplicity we define the following differential operator b ε • ∇

∂ ∂z 1 = 1, - y 2 h ε ∂h ε ∂y 1            ∂ ∂y 1 ∂ ∂y 2            := b ε • ∇
Then we obtain the following a priori estimates:

Proposition 2.2. Let (s 0 , U 0 , W 0 ) ∈ H 1 (0, T ) 3 and assume that ε 2m ( f ε , g ε ), ε m k ε and ε m (v ε 0 , Z ε 0 ) are uniformly bounded with respect to ε in L 2 0, T ; L 2 (Ω) 3 , L ∞ 0, T ; L ∞ + (Γ 0 ) and L 2 (Ω) 3
respectively. Then there exists a constant C, which does not depend on ε, such that the following uniform estimates are satisfied

(ε m b ε • ∇v ε i ) L 2 (0,T ;L 2 (Ω)) ≤ C, (ε m b ε • ∇Z ε ) L 2 (0,T ;L 2 (Ω)) ≤ C, ( 13
)
∂v ε i ∂y 2 L 2 (0,T ;L 2 (Ω)) ≤ C, ∂Z ε ∂y 2 L 2 (0,T ;L 2 (Ω)) ≤ C, (14)
∂v ε i ∂y 1 L 2 (0,T ;L 2 (Ω)) ≤ C ε m , ∂Z ε ∂y 1 L 2 (0,T ;L 2 (Ω)) ≤ C ε m , (15)
v ε i L 2 (0,T ;L 2 (Ω)) ≤ C, Z ε L 2 (0,T ;L 2 (Ω)) ≤ C, (16)
ε 2m p ε H -1 (0,T ;L 2 (Ω)) ≤ C, (17)
for i = 1, 2 and v ε 1 L 2 (0,T ;L 2 (Γ 0 )) ≤ C, (18)
where v ε 1 is identified to its trace on Γ 0 .

Proof. We simply replace ε by ε m in Proposition 1 [START_REF] Boukrouche | Unsteady micrololar fluid flow in a thin domain with Tresca fluid-solid interface law[END_REF] (see also Proposition 3.2 and Proposition 3.3 in [START_REF] Boukrouche | Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary[END_REF]).

Multiscale convergence

Since the boundary of the fluid domain Ω ε oscillates periodically on several separated scales, the most convenient tool to study the asymptotic behaviour as ε tends to zero is the multiple scale convergence method, introduced by G. Allaire and M. Briane in [START_REF] Allaire | Multiscale convergence and reiterated homogenisation[END_REF] as a generalization of the two-scale convergence method ( [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF][START_REF] Allaire | Homogenization and two-scale convergence[END_REF][START_REF] Łukkassen | Two-scale convergence[END_REF]) for elliptic reiterated homogenization problems. Of course we need to adapt the definition and main compactness result of [START_REF] Allaire | Multiscale convergence and reiterated homogenisation[END_REF] to our time-dependent setting.

Let O be an open bounded subset of R N with N ≥ 1, n be a positive integer and ε 1 > . . . > ε n be n positive functions of ε > 0 which converge to zero as ε does. The different scales are assumed to be separated i.e.

lim ε→0 ε k+1 ε k = 0 ∀k ∈ {1, . . . , n -1}.
Let Y = [0, 1] N . We denote as C # (Y n ) the space of continuous functions which are Y-periodic with respect to η k for all k ∈ {1, . . . , n} and H 1 # (Y) the space of functions of H 1 loc (R N ) which are Y-periodic.

Definition 3.1. A sequence (u ε ) ε>0 of L 2 (0, T ) × O (resp. H -1 0, T ; L 2 (O) ) multi-scale con- verges to u 0 ∈ L 2 0, T ; L 2 (O × Y n ) (resp. u 0 ∈ H -1 0, T ; L 2 (O × Y n ) ) if and only if lim ε→0 T 0 O u ε (t, y)ϕ y, y ε 1 , . . . , y ε n θ(t) dydt = T 0 O×Y n u 0 (t, y, η 1 , . . . , η n )ϕ(y, η 1 , . . . , η n )θ(t) dη n . . . dη 1 dydt for all θ ∈ D(0, T ), for all ϕ ∈ C O; C # (Y n ) .
In such a case we will denote w ε →→ n+1 w 0 .

Then we have the following compactness result:

Theorem 3.2. Let (u ε ) ε>0 be a bounded sequence in L 2 (0, T ) × O (resp. in H -1 0, T ; L 2 (O) ).

Then, possibly extracting a subsequence still denoted (u ε ) ε>0 , there exists u 0 ∈ L 2 0, T ;

L 2 (O × Y n ) (resp. u 0 ∈ H -1 0, T ; L 2 (O × Y n ) ) such that u ε →→ n+1 u 0 .
The proof is a straighforward adaptation of Theorem 4.2 in [START_REF] Boukrouche | Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary[END_REF] where the case n = 1 is considered (see also Theorem 2.3 in [START_REF] Allaire | Multiscale convergence and reiterated homogenisation[END_REF]).

In the rest of the paper we let n = m, N = 2, O = Ω = (0, L) × (0, 1) and ε k = ε k for all k ∈ {1, . . . , m}. We denote η k = (η 1k , η 2k ) ∈ Y for all k ∈ {1, . . . , m}. As a first step we obtain the following (m+1)-scale convergence result for the velocity and micro-rotation:

Proposition 3.3. Under the assumptions of Proposition 2.2, there exist v 0 = (v 0 1 , v 0 2 ) and Z 0 depending on (t, y, η 11 , . . . , η 1m ) such that v 0 i , Z 0 ∈ L 2 0, T ; L 2 Ω × (0, 1) m-1 ; H 1 # (0, 1) for i = 1, 2,
and, possibly extracting subsequences still denoted (v ε i ) ε>0 and (Z ε ) ε>0 , the following convergences hold:

v ε i →→ m+1 v 0 i , ε m ∂v ε i ∂y 1 →→ m+1 ∂v 0 i ∂η 1m for i = 1, 2 (19)
and

Z ε →→ m+1 Z 0 , ε m ∂Z ε ∂y 1 →→ m+1 ∂Z 0 ∂η 1m . ( 20
)
Moreover

∂v 0 i ∂y 2 , ∂Z 0 ∂y 2 ∈ L 2 0, T ; L 2 (Ω × (0, 1) m ) for i = 1, 2. Remark 3.1. Let us emphasize that v 0 = (v 0 1 , v 0
2 ) and Z 0 do not depend on η 2k for k ∈ {1, . . . , m}.

Proof. From Proposition 2.2, the sequences (v ε i ) ε>0 ,

∂v ε i ∂y 2 ε>0 and ε m ∂v ε i ∂y 1 ε>0
are bounded in

L 2 ((0, T ) × Ω).
It follows that, possibly extracting subsequences, they (m + 1)-scale converge to v 0 i , ξ 0 i and

ξ 1 i respectively, with v 0 i , ξ 0 i , ξ 1 i ∈ L 2 0, T ; L 2 (Ω × Y m ) . Hence we have lim ε→0 T 0 Ω v ε i (t, y)ϕ(y, y ε , ..., y ε m )θ(t) dydt = T 0 Ω Y m v 0 i (t, y, η 1 , ..., η m )ϕ(y, η 1 , ..., η m )θ(t) dη m ...dη 1 dydt (21) lim ε→0 T 0 Ω ∂v ε i ∂y 2 (t, y)ϕ(y, y ε , ..., y ε m )θ(t) dydt = T 0 Ω Y m ξ 0 i (t, y, η 1 , ..., η m )ϕ(y, η 1 , ..., η m )θ(t) dη m ...dη 1 dydt (22)
and

lim ε→0 T 0 Ω ε m ∂v ε i ∂y 1 (t, y)ϕ(y, y ε , ..., y ε m )θ(t) dydt = T 0 Ω Y m ξ 1 i (t, y, η 1 , ..., η m )ϕ(y, η 1 , ..., η m )θ(t) dη m ...dη 1 dydt (23)
for all θ ∈ D(0, T ) and ϕ ∈ C Ω; C # (Y m ) . From ( 21) and ( 23) we obtain

lim ε→0 T 0 Ω ε m ∂v ε i ∂y 1 (t, y)ϕ(y, y ε , ..., y ε m )θ(t) dydt = -lim ε→0 T 0 Ω v ε i (t, y) ε m ∂ϕ ∂y 1 (y, y ε , ..., y ε m ) + m k=1 ε m-k ∂ϕ ∂η 1k (y, y ε , ..., y ε m ) θ(t) dydt = - T 0 Ω Y m v 0 i (t, y, η 1 , ..., η m ) ∂ϕ ∂η 1m (y, η 1 , ..., η m )θ(t) dη m ...dη 1 dydt = T 0 Ω Y m ξ 1 i (t, y, η 1 , ..., η m )ϕ(y, η 1 , ..., η m )θ(t) dη m ...dη 1 dydt (24)
for all θ ∈ D(0, T ) and

ϕ ∈ D Ω; C ∞ # (Y m ) which yields ξ 1 i = ∂v 0 i ∂η 1m ∈ L 2 (0, T ; L 2 (Ω × Y m )) and (19) holds.
Next we observe that

T 0 Ω ∂v ε i ∂y 2 (t, y)ϕ(y, y ε , ..., y ε m )θ(t) dydt = - T 0 Ω v ε i (t, y) ∂ϕ ∂y 2 (y, y ε , ..., y ε m ) + m k=1 1 ε k ∂ϕ ∂η 2k (y, y ε , ..., y ε m ) θ(t) dydt for all θ ∈ D(0, T ) and ϕ ∈ D Ω; C ∞ # (Y m
) . Then we multiply both sides of this equality by ε m and we pass to the limit as ε tends to zero. Recalling that

∂v ε i ∂y 2 ε>0 is bounded in L 2 (0, T ) × Ω , we infer from (21) that T 0 Ω Y m v 0 i (t, y, η 1 , ..., η m ) ∂ϕ ∂η 2m (y, η 1 , ..., η m )θ(t) dη m ...dη 1 dydt = 0
and so v 0 does not depend on η 2m . Now let us assume that v 0 does not depend on η 2m , η 2(m-1) , ..., η 2 for some in {2, ..., m} and let us choose a test-function

ϕ ∈ D Ω; C ∞ # (Y -1 × [0, 1] m-+1 ) (i.e. ϕ does not depend on η 2m , η 2(m-1) , ..., η 2 ). We get T 0 Ω ∂v ε i ∂y 2 (t, y)ϕ y, y ε , ..., y ε -1 , y 1 ε , ..., y 1 ε m θ(t) dydt = - T 0 Ω v ε i (t, y) ∂ϕ ∂y 2 y, y ε , ..., y ε -1 , y 1 ε , ..., y 1 ε m θ(t) dydt - T 0 Ω v ε i (t, y)         -1 k=1 1 ε k ∂ϕ ∂η 2k y, y ε , ..., y ε -1 , y 1 ε , ..., y 1 ε m         θ(t) dydt.
We multiply both sides of this equation by ε -1 , and we pass to the limit as ε tends to zero. We obtain

T 0 Ω Y -1 ×[0,1] m-+1 v 0 i (t, y, η 1 , ..., η -1 , η 1 , ..., η 1m ) × ∂ϕ ∂η 2( -1) θ dη 1m ...dη 1 dη -1 ...dη 1 dydt = 0.
Hence v 0 does not depend on η 2( -1) and by induction we may conclude that v 0 does not depend on η 2k for any k ∈ {1, ..., m}.

Finally let ϕ ∈ D Ω; C ∞ # ([0, 1] m ) (i.e.
ϕ does not depend on η 21 , . . . , η 2m ) in [START_REF] Jiu | On the initial-and boundary-value problem for 2D micropolar equations with only angular velocity dissipation[END_REF]. At the limit we get Reminding that ξ 0 i ∈ L 2 0, T ; L 2 (Ω × Y m ) we infer that

lim ε→0 T 0 Ω ∂v ε i ∂y 2 (t, y)ϕ(y, y 1 ε , ..., y 1 ε m )θ(t) dydt = -lim ε→0 T 0 Ω v ε i (t, y) ∂ϕ ∂y 2 (y, y 1 ε , ..., y 1 ε m )θ(t) dydt = T 0 Ω Y m ξ 0 i (t, y, η 1 , ..., η m )ϕ(y, η 11 , ..., η 1m )θ(t) dη m ...dη 1 dydt = - T 0 Ω [0,1] m v 0 i (t,
∂v 0 i ∂y 2 ∈ L 2 0, T ; L 2 (Ω × (0, 1) m ) .
Then with ϕ ∈ D Ω × (0, 1) m-1 in ( 24) we obtain

0 = T 0 Ω [0,1] m ∂v 0 i ∂η 1m (t,
= T 0 Ω [0,1] m-1 v 0 i (t, y, η 11 , ..., η 1(m-1) , 1)ϕ(y, η 11 , ..., η 1(m-1) )θ(t) dη 1(m-1) ...dη 11 dydt - T 0 Ω [0,1] m-1 v 0 i (t, y, η 11 , ..., η 1(m-1) , 0)ϕ(y, η 11 , ..., η 1(m-1) )θ(t) dη 1(m-1) ...dη 11 dydt and thus v 0 i ∈ L 2 (0, T ; L 2 (Ω × (0, 1) m-1 ; H 1 # (0, 1))).
We may perform the same kind of computations for Z ε , which allows us to conclude.

v k i , Z k ∈ L 2 (0, T ) × (0, L) × (0, 1) m ; L 2 (0, 1) × (0, 1) k-1 ; H 1 # (0, 1)
for i = 1, 2 and k ∈ {1, ..., m} and, possibly extracting subsequences still denoted (v ε i ) ε>0 and (Z ε ) ε>0 , we have the following convergences

∂v ε i ∂y 2 →→ m+1 ∂v 0 i ∂y 2 + m k=1 ∂v k i ∂η 2k , ∂Z ε ∂y 2 →→ m+1 ∂Z 0 ∂y 2 + m k=1 ∂Z k i ∂η 2k .
Proof. The result relies on the same kind of arguments as in the proof of Theorem 2.6 in [START_REF] Allaire | Multiscale convergence and reiterated homogenisation[END_REF] for well-separated scales. Let us introduce the following subspaces H k , k ∈ {1, . . . , m} of functions ϕ = ϕ(y 2 , η 21 , . . . , η 2m ) given by

H m = ϕ ∈ L 2 (0, 1); L 2 # (0, 1) m : 1 0 ϕ ∂ψ ∂η 2m dη 2m = 0 ∀ψ = ψ(η 2m ) ∈ H 1 # (0, 1)
and for all k ∈ {1, . . . , m -1}

H k =          ϕ ∈ L 2 (0, 1); L 2 # (0, 1) m : [0,1] m-k+1 ϕ ∂ψ ∂η 2k dη 2k dη 2(k+1) • • • dη 2m = 0 ∀ψ = ψ(η 2k ) ∈ H 1 # (0, 1)          . Then we let H = m k=1 H k .
Now let us choose a test-function ϕ in [START_REF] Jiu | On the initial-and boundary-value problem for 2D micropolar equations with only angular velocity dissipation[END_REF] given by ϕ = ϕ 1 ϕ 2 with

ϕ 1 = ϕ 1 (y 1 , η 11 , . . . , η 1m ) ∈ D (0, L); C ∞ # [0, 1] m and ϕ 2 = ϕ 2 (y 2 , η 21 , . . . , η 2m ) ∈ D (0, 1); C ∞ # [0, 1] m ∩ H.
We define ϕ ε 1 and ϕ ε 2 by

ϕ ε 1 (y 1 ) = ϕ 1 y 1 , y 1 ε , • • • , y 1 ε m , ϕ ε 2 (y 2 ) = ϕ 2 y 2 , y 2 ε , • • • , y 2 ε m
for all (y 1 , y 2 ) ∈ Ω. For all θ ∈ D(0, T ) we obtain

T 0 Ω ∂v ε i ∂y 2 (t, y)ϕ ε 1 (y 1 )ϕ ε 2 (y 2 ) θ(t) dydt = - T 0 Ω v ε i (t, y)ϕ ε 1 (y 1 )
∂ϕ ε 2 ∂y 2 (y 2 ) θ(t) dydt = - T 0 Ω v ε i (t, y)ϕ 1 y 1 , y 1 ε , • • • , y 1 ε m ∂ϕ 2 ∂y 2 y 2 , y 2 ε , • • • , y 2 ε m θ(t) dydt - m k=1 T 0 Ω v ε i (t, y)ϕ 1 y 1 , y 1 ε , • • • , y 1 ε m ε -k ∂ϕ 2 ∂η 2k y 2 , y 2 ε , • • • , y 2 ε m θ(t) dydt = - T 0 Ω v ε i (t, y)ϕ 1 y 1 , y 1 ε , • • • , y 1 ε m ∂ϕ 2 ∂y 2 y 2 , y 2 ε , • • • , y 2 ε m θ(t) dydt - m k=1 1 0 ε -k ∂ϕ 2 ∂η 2k y 2 , y 2 ε , • • • , y 2 ε m T 0 L 0 v ε i (t, y)ϕ 1 y 1 , y 1 ε , • • • , y 1 ε m θ(t) dy 1 dt dy 2 .
By applying Corollary 3.4 in [START_REF] Allaire | Multiscale convergence and reiterated homogenisation[END_REF] with N = 1, Ω = (0, 1), n = m and ε k = ε k for all k ∈ {1, . . . , m}, the assumption of well-separated scales is satisfied (see Definition 3.1 in [START_REF] Allaire | Multiscale convergence and reiterated homogenisation[END_REF]) and we infer that

1 ε k+1 ∂ϕ 2 ∂η 2k y 2 , y 2 ε , • • • , y 2 ε m is bounded in H 1 (0, 1) independently of ε.
Thus we may pass to the limit in the previous equality and we get Moreover with Lemma 3.7 (i) in [START_REF] Allaire | Multiscale convergence and reiterated homogenisation[END_REF] we know that D (0, 1);

T 0 Ω Y m ξ 0 i (t, y, η)ϕ(y, η)θ(t) dη m . . . dη 1 dydt = - T 0 Ω Y m v 0 i (t,
C ∞ # [0, 1] m ∩ H is dense into H. It follows that T 0 Ω Y m       ξ 0 i (t, y, η) - ∂v 0 i ∂y 2 (t, y, η 11 , . . . , η 1m )       ϕ(t, y, η) dη m . . . dη 1 dydt = 0 for all ϕ ∈ L 2 (0, T ) × (0, L) × (0, 1) m ; H = m k=1 L 2 (0, T ) × (0, L) × (0, 1) m ; H k . Hence ξ 0 i (t, y, η) - ∂v 0 i ∂y 2 ∈ m k=1 L 2 (0, T ) × (0, L) × (0, 1) m ; H k ⊥ .
With Lemma 3.7 (ii) in [START_REF] Allaire | Multiscale convergence and reiterated homogenisation[END_REF] we obtain that m k=1 L 2 (0,

T )×(0, L)×(0, 1) m ; H k ⊥ = m k=1 L 2 (0, T )× (0, L) × (0, 1) m ; H ⊥ k with H ⊥ k = ∇ η 2k q(y 2 , η 21 , • • • , η 2k ); q ∈ L 2 (0, 1) × (0, 1) k-1 ; H 1 # (0, 1)
for all k ∈ {1, • • • , m} and the conclusion follows.

The same result holds for Z ε .

Proposition 3.5. Under the assumptions of Proposition 2.2 the (m + 1)-scale limits v 0 and Z 0 satisfy the following boundary conditions:

v 0 2 = 0, Z 0 = 0 on (0, T ) × (Γ 0 ∪ Γ 1 ) × (0, 1) m v 0 1 = 0 on (0, T ) × Γ 1 × (0, 1) m and we have v ε 1 →→ m+1 v 0 1 on (0, T ) × Γ 0 × (0, 1) m .
Proof. With estimates [START_REF] Fujita | Analytical and numerical approaches to stationary flow problems with leak and slip boundary conditions[END_REF] we know that the trace of

v ε 1 on Γ 0 is uniformly bounded in L 2 0, T ; L 2 (Γ 0 ) = L 2 (0, T ) × (0, L) so it admits a (m + 1)-scale limit ξ ∈ L 2 0, T ; L 2 (0, L) × (0, 1) m . Let θ ∈ D(0, T ) and ϕ ∈ C ∞ Ω; C ∞ # [0, 1] m
. By using the boundary conditions ( 6)-( 7) we obtain

T 0 Ω ∂v ε 1 ∂y 2 (t, y)ϕ(y, y 1 ε , ..., y 1 ε m )θ(t) dydt = - T 0 Ω v ε 1 (t, y) ∂ϕ ∂y 2 (y, y 1 ε , ..., y 1 ε m )θ(t) dydt - T 0 L 0 v ε 1 (t, y 1 , 0)ϕ(y 1 , 0, y 1 ε , ..., y 1 ε m )θ(t) dy 1 dt
and passing to the (m + 1)-scale limit as ε tends to zero, we get

T 0 Ω Y m        ∂v 0 1 ∂y 2 (t,
       ×ϕ(y, η 11 , . . . , η 1m )θ dη m . . . dη 1 dydt = - T 0 Ω [0,1] m v 0 1 (t,
Recalling that v k 1 is 1-periodic with respect to η 2k and the test-function ϕ does not depend on η 2k for all k ∈ {1, . . . , m}, Green's formula implies Then we choose ϕ(y, η By density of D(0,

T 0 Ω Y m ∂v k 1 ∂η 2k (t,
+ T 0 L 0 [0,1] m v 0 1 (t,
1 ) = ϕ(y 2 ) ϕ(y 1 , η 1 ) with ϕ ∈ C ∞ [0, L]; C ∞ # [0, 1] m and ϕ ∈ C ∞ [0, 1] such that ϕ(1) = 0 and ϕ(0) = 1 first, then ϕ(1) = 1 and ϕ(0) = 0. We obtain T 0 L 0 [0,1] m ξ(t, y 1 , η 11 , . . . , η 1m ) -v 0 1 (t,
T ) ⊗ C ∞ [0, L]; C ∞ # [0, 1] m into L 2 0, T ; L 2 (0, L) × (0, 1) m we infer that ξ = v 0 1 on (0, T ) × Γ 0 × (0, 1) m and v 0 1 = 0 on (0, T ) × Γ 1 × (0, 1) m .
Starting now from

T 0 Ω ∂v ε 2 ∂y 2 (t, y)ϕ(y, y 1 ε , ..., y 1 ε m )θ(t) dydt = - T 0 Ω v ε 2 (t, y) ∂ϕ ∂y 2 (y, y 1 ε , ..., y 1 ε m )θ(t) dydt and T 0 Ω ∂Z ε ∂y 2 (t, y)ϕ(y, y 1 ε , ..., y 1 ε m )θ(t) dydt = - T 0 Ω Z ε (t, y) ∂ϕ ∂y 2 (y, y 1 ε , ..., y 1 ε m )θ(t) dydt
we obtain similarly that v 0 2 = 0 and Z 0 = 0 on (0, T ) × (Γ 0 ∪ Γ 1 ) × (0, 1) m .

Proposition 3.6. Under the assumptions of Proposition 2.2 the (m + 1)-scale limit v 0 satisfies the following divergence free conditions:

h ∂v 0 1 ∂η 1m -y 2 ∂h ∂η 1m ∂v 0 1 ∂y 2 + ∂v 0 2 ∂y 2 = 0 in (0, T ) × Ω × (0, 1) m (26) ∂ ∂η 1m 1 0 hv 0 1 dy 2 = 0 in D (0, T ) × (0, L) × (0, 1) m (27)
∂ ∂η 1(m-1) [0,1] 2 hv 0 1 dη 1m dy 2 = 0 in D (0, T ) × (0, L) × (0, 1) m-1 (28)
and

∂ ∂η 1k [0,1] m-k+1 hv 0 1 dη 1(k+1) . . . dη 1m dy 2 = 0 in D (0, T ) × (0, L) × (0, 1) k (29)
for all k ∈ {1, . . . , m -2}.

Proof. Let us prove first [START_REF] Liu | Initial-boundary value problem for 2D micropolar equations without angular viscosity[END_REF]. For any ϕ ∈ D Ω × (0, 1) m we define ϕ ε and ϕ ε by

ϕ ε (y) = ϕ y 1 , y 2 , y 1 ε , • • • , y 1 ε m for all y = (y 1 , y 2 ) ∈ Ω and ϕ ε (z) = ϕ ε z 1 , z 2 ε m h ε (z 1 ) for all z = (z 1 , z 2 ) ∈ Ω ε . From (2) we get 0 = T 0 Ω ε ∂v ε 1 ∂z 1 (t, z) + ∂v ε 2 ∂z 2 (t, z) ϕ ε (z)θ(t) dzdt = - T 0 Ω ε v ε 1 (t, z) ∂ϕ ε ∂z 1 (z) + v ε 2 (t, z) ∂ϕ ε ∂z 2 (z) θ(t) dzdt = - T 0 Ω v ε 1 (t, y)(b ε • ∇ϕ ε )(y) + v ε 2 (t, y) 1 ε m h ε (y 1 ) ∂ϕ ε ∂y 2 (y) ε m h ε (y 1 )θ(t) dydt. (30)
With the definition of the operator b ε • ∇ we have

ε m (b ε • ∇ϕ ε )(y)h ε (y 1 ) = ε m h y 1 , y 1 ε , ..., y 1 ε m ∂ϕ ∂y 1 y, y 1 ε , ..., y 1 ε m + m k=1 ε m-k h ∂ϕ ∂η 1k y, y 1 ε , ..., y 1 ε m -y 2 ε m ∂h ∂y 1 y 1 , y 1 ε , ..., y 1 ε m + m k=1 ε m-k ∂h ∂η 1k y 1 , y 1 ε , ..., y 1 ε m ∂ϕ ∂y 2 y, y 1 ε , ..., y 1 ε m .
Passing to the (m + 1)-scale limit as ε tends to 0 in [START_REF] Pahlavan | Effect of solid properties on slip at a fluid-solid interface[END_REF] we obtain for all ϕ ∈ D Ω × (0, 1) m and θ ∈ D(0, T ) and the conclusion follows from the density of

0 = - T 0 Ω [0,1] m v 0 1 (t,
D(0, T ) ⊗ D Ω × (0, 1) m into L 2 0, T ; L 2 Ω × (0, 1) m .
Let us assume now that ϕ ∈ D (0, L)×(0, 1) m (i.e. ϕ does not depend on y 2 ) and θ ∈ D(0, T ). Then we have

T 0 Ω v ε 1        ε m h ∂ϕ ∂y 1 + m k=1 ε m-k h ∂ϕ ∂η 1k        θ dydt = 0 ( 31
)
and at the limit we get Let us assume now that ϕ ∈ D (0, L) × (0, 1) m-1 (i.e. ϕ does not depend on y 2 nor on η 1m ) and θ ∈ D(0, T ). Then, by dividing (31) by ε we get

T 0 Ω [0,1] m v 0 1 h ∂ϕ ∂η 1m θ dη 1m . . .
T 0 Ω v ε 1         ε m-1 h ∂ϕ ∂y 1 + m-1 k=1 ε m-1-k h ∂ϕ ∂η 1k         θ dydt = 0
and at the limit we obtain

T 0 Ω [0,1] m v 0 1 h ∂ϕ ∂η 1(m-1)
θ dη 1m dη 11 dydt

= T 0 L 0 [0,1] m-1 [0,1] 2 hv 0 1 dη 1m dy 2 ∂ϕ ∂η 1(m-1)
θ dη 1(m-1) . . . dη 11 dy 1 dt = 0 which proves [START_REF] Łukkassen | Two-scale convergence[END_REF]. Then by choosing ϕ ∈ D (0, L) × (0, 1) k (i.e. ϕ does not depend on y 2 nor on η 1m , . . . , η 1(k+1) ), we obtain by induction with similar computations that

∂ ∂η 1k [0,1] m-k+1
hv 0 1 dη 1(k+1) . . . dη 1m dy 2 = 0 in D (0, T ) × (0, L) × (0, 1) k for all k ∈ {1, . . . , m -1}.

Proposition 3.7. Under the assumptions of Proposition 2.2, there exists p 0 ∈ H -1 0, T ; L 2 (Ω × Y m ) such that, possibly extracting a subsequence still denoted (p ε ) ε>0 , we have

ε 2m p ε →→ m+1 p 0 . ( 32
)
Moreover, p 0 depends only on t and y 1 and we have p 0 ∈ H -1 (0, T ; H 1 # (0, L)) and ∂p 0 ∂y 1 ∈ L 2 (0, T )× (0, L) with

(z) = ϕ ε z 1 , z 2 ε m h ε (z 1 )
for all z ∈ Ω ε . We choose Θ ε = (0, ϕ ε ), 0) in [START_REF] Dong | Global regularity of the 2D micropolar fluid flows with zero angular viscosity[END_REF]. For any θ ∈ D(0, T ) we obtain

T 0 Ω ε m p ε ∂ϕ ε ∂y 2 θ dydt = - T 0 Ω ε 2m v ε 2 ϕ ε h ε θ dydt - T 0 Ω ε 2m f ε 2 ϕ ε h ε θ dydt +(ν + ν r ) T 0 Ω (ε m b ε • ∇v ε 2 )(ε m b ε • ∇ϕ ε )h ε + 1 h ε ∂v ε 2 ∂y 2 ∂ϕ ε ∂y 2 θ dydt + T 0 Ω ε m v ε 1 (ε m b ε • ∇v ε 2 )ϕ ε h ε + ε m v ε 2 ∂v ε 2 ∂y 2 ϕ ε θ dydt +2ν r T 0 Ω ε m (ε m b ε • ∇Z ε )ϕ ε h ε θ dydt + T 0 Ω ε m U ε (t, y)(ε m b ε • ∇v ε 2 )ϕ ε h ε θ dydt with U ε (t, y) = s 0 (t) + (U 0 (t) -s 0 (t))U(h ε (y 1 )y 2 ) ∀(t, y 1 , y 2 ) ∈ [0, T ] × Ω
where we recall that h ε (y 1 ) = h y 1 , y 1 ε , ..., y 1 ε m for all y 1 ∈ [0, L]. By using the classical interpolation inequality

u L 3 (Ω) ≤ u 1 2 L 2 (Ω) u 1 2 L 6 (Ω) ∀u ∈ L 2 (Ω) ∩ L 6 (Ω) we get ε m T 0 Ω p ε ∂ϕ ε ∂y 2 θ dydt ≤ ε 2m h M v ε 2 ϕ ε θ + h M ε 2m f ε 2 ϕ ε θ +(ν + ν r )h M (ε m b ε • ∇v ε 2 ) (ε m b ε • ∇ϕ ε )θ + (ν + ν r ) h m ∂v ε 2 ∂y 2 ∂ϕ ε ∂y 2 θ +ε m C 1 2 v ε 2 1 2 v ε 2 1 2 L 2 (0,T ;H 1 (Ω)) ∂v ε 2 ∂y 2 ϕ ε θ L ∞ (0,T ;L 6 (Ω)) +ε m h M C 1/2 v ε 1 1 2 v ε 1 1 2 L 2 (0,T ;H 1 (Ω)) ε m b ε • ∇v ε 2 ϕ ε θ L ∞ (0,T ;L 6 (Ω)) +ε m h M s 0 L ∞ (0,T ) + U 0 -s 0 L ∞ (0,T ) U L ∞ (0,h M ) (ε m b ε • ∇v ε 2 ) ϕ ε θ +2ε m ν r h M (ε m b ε • ∇Z ε ) ϕ ε θ (34)
where C denotes the norm of the canonical injection of L 6 (Ω) into H 1 (Ω) and • denotes the canonical norm of L 2 0, T ; L 2 (Ω) .

Then with Proposition 2.2 we infer that there exists a constant C, independent of ε, such that

ε m T 0 Ω p ε ∂ϕ ε ∂y 2 θ dydt = T 0 Ω ε m p ε (t, y) ∂ϕ ∂y 2 y, y ε , ..., y ε m + m k=1 1 ε k ∂ϕ ∂η 2k y, y ε , ..., y ε m θ(t) dydt ≤ Cε m ϕ ε θ + ε m ϕ ε θ +C ε m 2 ϕ ε θ L ∞ (0,T ;L 6 (Ω)) + ϕ ε θ + (ε m b ε • ∇ϕ ε )θ + ∂ϕ ε ∂y 2 θ . (35)
Since ϕ ∈ D Ω; C ∞ # (Y m ) the terms ϕ ε θ L ∞ (0,T ;L 6 (Ω)) , ϕ ε θ and ϕ ε θ are uniformly bounded with respect to ε. In order to estimate the two last terms we recall that

ε m b ε • ∇ϕ ε = ε m ∂ϕ ε ∂y 1 -ε m y 2 h ε (y 1 ) ∂h ε ∂y 1 ∂ϕ ε ∂y 2 = ε m ∂ϕ ∂y 1 + m k=1 ε m-k ∂ϕ ∂η 1k -ε m y 2 h ε ∂h ∂y 1 + m k=1
+ m k=1 1 ε k ∂ϕ ∂η 2k y, y ε , ..., y ε m θ(t) dydt ≤ O(ε) + C θ L 2 (0,T ) ∂ϕ ∂η 2m L 2 (Ω) . (36)
By multiplying this inequality by ε m we get at the limit 36) by ε -1 we get at the limit

T 0 Ω Y -1 ×[0,1] m-+1 p 0 (t, y, η 1 , ..., η -1 , η 1 , ..., η 1m ) × ∂ϕ ∂η 2( -1)
(y, η 1 , ..., η -1 , η 1 , ..., η 1m )θ(t) dη 1m ...dη 1 dη -1 ...dη 1 dydt = 0 and thus p 0 does not depend on η 2( -1) . Hence p 0 does not depend on η 2k for all k ∈ {1, ..., m}.

Finally with ϕ ∈ D Ω; C ∞ # [0, 1] m (i.e. ϕ does not depend on η 2k for all k ∈ {1, . . . m}) and we may pass to the (m + 1)-scale limit as ε tends to zero in [START_REF] Wang | Global regularity of the three-dimensional fractional micropolar equations[END_REF] and we get Now we consider again ϕ ∈ D Ω; C ∞ # [0, 1] m ) and we define ϕ ε and ϕ ε as previously. Then we choose Θ ε = (ϕ ε , 0), 0 in [START_REF] Dong | Global regularity of the 2D micropolar fluid flows with zero angular viscosity[END_REF]. For all θ ∈ D(0, T ) we obtain

T 0 Ω [0,1] m p 0 (t,
ε 2m T 0 Ω p ε ∂ϕ ε ∂y 1 - y 2 h ε ∂h ε ∂y 1 ∂ϕ ε ∂y 2 h ε θ dydt = -ε 2m T 0 Ω v ε 1 ϕ ε h ε θ dydt +(ν + ν r ) T 0 Ω h ε (ε m b ε • ∇v ε 1 )(ε m b ε • ∇ϕ ε ) + 1 h ε ∂v ε 1 ∂y 2 ∂ϕ ε ∂y 2 θ dydt + T 0 Ω ε m v ε 1 (ε m b ε • ∇v ε 1 )ϕ ε h ε + ε m v ε 2 ∂v ε 1 ∂y 2 ϕ ε θ dydt +(ν + ν r ) T 0 Ω 1 h ε ∂ U ε ∂y 2 ∂ϕ ε ∂y 2 θ dydt -2ν r T 0 Ω ε m ∂Z ε ∂y 2 ϕ ε θ dydt + T 0 Ω ε m U ε (ε m b ε • ∇v ε 1 )ϕ ε h ε θ dydt - T 0 Ω ε m v ε 1 (ε m b ε • ∇ϕ ε ) U ε h ε θ dydt - T 0 Ω ε m v ε 2 ∂ϕ ε ∂y 2 U ε θ dydt -2ν r T 0 Ω ε m ∂ W ε ∂y 2 ϕ ε θ dydt - T 0 Ω ε 2m f ε 1 ϕ ε - ∂ U ε ∂t ϕ ε h ε θ dydt where W ε (t, y) = W 0 (t)W h ε (y 1 )y 2 for all (t, y 1 , y 2 ) ∈ [0, T ] × Ω, which yields T 0 Ω ε 2m p ε ∂ϕ ∂y 1 y, y 1 ε , ..., y 1 ε m h y 1 , y 1 ε , ..., y 1 ε m θ dydt + T 0 Ω ε 2m p ε m k=1 ε -k ∂ϕ ∂η 1k y, y 1 ε , ..., y 1 ε m h y 1 , y 1 ε , ..., y 1 ε m θ dydt - T 0 Ω ε 2m p ε y 2 ∂h ∂y 1 y 1 , y 1 ε , ..., y 1 ε m ∂ϕ ∂y 2 y, y 1 ε , ..., y 1 ε m θ dydt - T 0 Ω ε 2m p ε y 2 m k=1 ε -k ∂h ∂η 1k y 1 , y 1 ε , ..., y 1 ε m ∂ϕ ∂y 2 y, y 1 ε , ..., y 1 ε m θ dydt ≤ O(1). ( 37
)
By multiplying this estimate by ε m and passing to the (m + 1)-scale limit as ε tends to zero, we get It follows that p 0 does not depend on η 1m . Now again by induction, we assume that p 0 does not depend on η 1m , η 1(m-1) , ..., η 1 for in {2, ..., m} and we choose ϕ ∈ D Ω; C ∞ # [0, 1] -1 (i.e. ϕ does not depend on η 1m , η 1(m-1) , ..., η 1 ). By multiplying (37) by ε -1 , then taking the (m+1)-scale limit as ε tends to zero, we may conclude that p 0 does not depend on η 1( -1) . Hence p 0 depends only on t and y 1 .

T 0 Ω [0,1] m p 0 ∂ϕ ∂η 1m h -y 2 ∂h
Now let ϕ ∈ C ∞ # ([0, L]
) and θ ∈ D(0, T ). We define ϕ ε by

ϕ ε (y) = ϕ(y 1 ) h ε (y 1 ) y 2 (1 -y 2 )e 1 + ε m y 2 2 (1 -y 2 )
∂h ε ∂y 1 (y 1 )e 2 for all (y 1 , y 2 ) ∈ Ω.

We obtain that ϕ ε is L-periodic in y 1 and ϕ ε = 0 on Γ 0 ∪ Γ 1 . With the previous estimates we obtain

T 0 Ω ε 2m p ε (b ε • ∇ϕ ε 1 ) + 1 ε m h ε ∂ϕ ε 2 ∂y 2 h ε θ dydt ≤ T 0 Ω ε 2m p ε (b ε • ∇ϕ ε 1 )h ε θ dydt + T 0 Ω ε m p ε ∂ϕ ε 2 ∂y 2 θ dydt ≤ C ϕ ε 1 θ + (ε m b ε • ∇ϕ ε 1 )θ + ∂y 2 h ε θ(t) dydt ≤ O(ε m 2 ) + C ϕθ
where C and C is two constants independent of ε. With (38) we get

T 0 Ω ε 2m p ε (b ε • ∇ϕ ε 1 ) + 1 ε m h ε ∂ϕ ε 2 ∂y 2 h ε θ(t) dydt = T 0 Ω ε 2m p ε (t, y)y 2 (1 -y 2 ) ∂ϕ ∂y 1 (y 1 )θ(t) dydt ≤ O(ε m 2 ) + C ϕθ L 2 ((0,T )×(0,L))
and at the limit

T 0 Ω p 0 y 2 (1 -y 2 ) ∂ϕ ∂y 1 (y 1 )θ(t) dydt = 1 6 T 0 L 0 p 0 ∂ϕ ∂y 1 (y 1 )θ(t) dydt ≤ C ϕθ L 2 ((0,T )×(0,L)) .
Hence ∂p 0 ∂y 1 ∈ L 2 (0, T ) × (0, L) and p 0 ∈ H -1 (0, T ; H 1 # (0, L)).

Finally recalling that

T 0 Ω ε 2m p ε (t, y)h ε (y 1 )θ(t) dydt = 0
for all θ ∈ D(0, T ), we obtain

T 0 L 0 [0,1] m
p 0 (t, y 1 )h(y 1 , η 11 , . . . , η 1m )θ(t) dη 1m . . . dη 11 dydt = 0 for all θ ∈ D(0, T ) which concludes the proof.

The limit problem

In order to identify the limit problem for (v 0 , Z 0 , p 0 ) it is convenient to introduce the following differential operator

b • ∇ = 1, - y 2 h(y 1 , η 1 ) ∂h ∂η 1m (y 1 , η 1 )            ∂ ∂η 1m ∂ ∂y 2            , η 1 = de f (η 11 , ..., η 1m ) ∈ (0, 1) m .
Indeed with the previous convergence results (see Proposition 3.3 and Proposition 3.4) we have

ε m b ε • ∇v ε i = ε m ∂v ε i ∂y 1 (t, y) - y 2 h ε (y 1 ) ε m ∂h ∂y 1 (y 1 , y 1 ε , • • • , y 1 ε m ) + m k=1 ε m-k ∂h ∂η 1k (y 1 , y 1 ε , • • • , y 1 ε m ) ∂v ε i ∂y 2 (t, y) →→ m+1 ∂v 0 i ∂η 1m (t, y, η 1 ) - y 2 h(y 1 , η 1 ) ∂h ∂η 1m (y 1 , η 1 ) ∂v 0 i ∂y 2 (t, y, η 1 ) + m k=1 ∂v k i ∂η 2k (t, y 1 , η 11 , . . . , η 1m , y 2 , η 21 , . . . , η 2k ) = (b • ∇v 0 i )(t, y, η 1 ) - y 2 h(y, η 1 ) ∂h ∂η 1m (y, η 1 ) m k=1 ∂v k i ∂η 2k (t,
ε m b ε • ∇Z ε →→ m+1 (b • ∇Z 0 )(t, y, η 1 ) - y 2 h(y, η 1 ) ∂h ∂η 1m (y, η 1 ) m k=1 ∂Z k ∂η 2k (t, y 1 , η 11 , . . . η 1m , y 2 , η 21 , ..., η 2k ).
Hence the appropriate framework to study the limit problem is given by the following functional spaces:

V =        ϕ ∈ C ∞ Ω; C ∞ # ([0, 1] m ) 2 : L-periodic in y 1 , ϕ = 0 on Γ 1 × (0, 1) m , ϕ • n = 0 on Γ 0 × (0, 1) m        , V div =                              ϕ ∈ V : h ∂ϕ 1 ∂η 1m -y 2 ∂h ∂η 1m ∂ϕ 1 ∂y 2 + ∂ϕ 2 ∂y 2 = 0 in Ω × (0, 1) m , ∂ ∂η 1m 1 0 hϕ 1 dy 2 = 0 in D (0, L) × (0, 1) m , ∂ ∂η 1k [0,1] m-k+1 hϕ 1 dη 1(k+1) . . . dη 1m dy 2 = 0 in D (0, L) × (0, 1) k , for all k ∈ {1, . . . , m -1}                              , H 1 = ψ ∈ C ∞ Ω; C ∞ # ([0, 1] m ) : L-periodic in y 1 , ψ = 0 in (Γ 0 ∪ Γ 1 ) × (0, 1) m ,
and

V 0 = H 1 2 , V 0,div = V 0 ∩ V div . We let F = ϕ ∈ L 2 (0, 1) × (0, 1) m-1 ; H 1 # (0, 1) : ∂ϕ ∂y 2 ∈ L 2 (0, 1) × (0, 1) m
and we define V 0 , V div , V 0,div and H 1 0,# as for all (y 1 , η 1 ) = (y 1 , η 11 , . . . , η 1m )

V 0 = closure of V 0 in L 2 (0, L; F 2 ), V div = closure of V div in L 2 (0, L; F 2 ), V 0,div = closure of V 0,div in L 2 (0, L; F 2 ), H 1 0,# = closure of H 1 in L 2 (0, L; F ). Remark 4.1. For any ϕ ∈ V div we may define Φ 1 ∈ C ∞ [0, L]; C ∞ # ([0, 1] m ) by Φ 1 (y 1 , η 1 ) = Φ 1 (
∈ [0, L] × [0, 1] m . Since ϕ ∈ V div , we have ∂Φ 1 ∂η 1m = 0 in D (0, L) × (0, 1) m i.e. (0,L)×(0,1) m ∂Φ 1 ∂η 1m ψ dη 11 . . . dη 1m dy 1 = - (0,L)×(0,1) m Φ 1 ∂ψ ∂η 1m dη 11 . . . dη 1m dy 1 = 0 for all ψ ∈ D (0, L) × (0, 1) m . Observing that ∂Φ 1 ∂η 1m ∈ C ∞ [0, L]; C ∞ # ([0, 1] m ) we have by density (0,L)×(0,1) m ∂Φ 1 ∂η 1m ψ dη 11 . . . dη 1m dy 1 = 0 ∀ψ ∈ L 2 (0, L) × (0, 1) m which yields that ∂Φ 1 ∂η 1m = 0 in L 2 (0, L) × (0, 1) m and thus ∂Φ 1 ∂η 1m = 0 a.e. in [0, L] × [0, 1] m .
By continuity we may conclude that ∂Φ 1 ∂η 1m = 0 for all (y 1 , η 11 , . . . , η 1m ) ∈ [0, L]×[0, 1] m which implies that Φ 1 does not depend on η 1m and we have

Φ 1 = Φ 1 (y 1 , η 11 , . . . , η 1(m-1) ) ∈ C ∞ [0, L]; C ∞ # ([0, 1] m-1
) .

Then we have also

∂ ∂η 1(m-1) 1 0 1 0 h(y 1 , η 1 )ϕ 1 (y 1 , y 2 , η 1 ) dη 1m dy 2 = 0 in D (0, L)×(0, 1) m-1 i.e. (0,L)×(0,1) m-1 ∂ ∂η 1(m-1) 1 0 1 0 h(y 1 , η 1 )ϕ 1 (y 1 , y 2 , η 1 ) dy 2 Φ 1 =Φ 1 (y 1 ,η 11 ,...,η 1(m-1) ) dη 1m ψ dη 11 . . . dη 1(m-1) dy 1 = (0,L)×(0,1) m-1 ∂ ∂η 1(m-1) 1 0 Φ 1 (y 1 , η 11 , . . . η 1(m-1) )dη 1m =Φ 1 (y 1 ,η 11 ,...,η 1(m-1) ) ψ dη 11 . . . dη 1(m-1) dy 1 = - (0,L)×(0,1) m-1 Φ 1 (y 1 , η 11 , . . . , η 1(m-1) ) ∂ψ ∂η 1(m-1) dη 11 . . . dη 1(m-1) dy 1 = 0 for all ψ ∈ D (0, L) × (0, 1) m-1 . Owing that ∂Φ 1 ∂η 1(m-1) ∈ C ∞ [0, L]; C ∞ # ([0, 1] m-1 ) we infer that ∂Φ 1 ∂η 1(m-1)
= 0 for all (y 1 , η 11 , . . . , η

1(m-1) ) ∈ [0, L] × [0, 1] m-1 which implies that Φ 1 does not depend on η 1(m-1) i.e. Φ 1 = Φ 1 (y 1 , η 11 , . . . , η 1(m-2) ) ∈ C ∞ [0, L]; C ∞ # ([0, 1] m-2 )
. With an immediate induction argument we get finally that

Φ 1 = Φ 1 (y 1 ) ∈ C ∞ [0, L] .
Then we may obtain a first relation between the limit velocity v 0 , the limit micro-rotation Z 0 and the limit pressure p 0 .

Proposition 4.1. Let s 0 , U 0 , W 0 belong to H 1 (0, T ) and ε m (v ε 0 , Z ε 0 ) be bounded in (L 2 (Ω)) 3 . Let us assume that there exist f ∈ C [0, T ]; C Ω; C # ([0, 1] m ) 2 , g ∈ C [0, T ]; C Ω; C # ([0, 1] m ) and k ∈ C [0, T ]; C [0, L]; C # ([0, 1] m )
, such that k takes its values in R * + , f , g and k are L-periodic in y 1 and

ε 2m f ε (t, y) = f (t, y, y 1 ε , ..., y 1 ε m ), ε 2m g ε (t, y) = g(t, y, y 1 ε , ..., y 1 ε m ), εk ε (t, y 1 ) = k(t, y 1 , y 1 ε , ..., y 1 ε m )
for all (t, y) ∈ [0, T ] × Ω. Then, v 0 , Z 0 and p 0 satisfy for all Θ = (ϕ, ψ) ∈ V 0,div × H 1 and for all θ ∈ D(0, T ), where U(t, y 1 , y 2 , η 1 ) = s 0 (t) + U 0 (t)s 0 (t) U(h(y 1 , η 1 )y 2 ),

(ν + ν r ) T 0 Ω×(0,1) m 2 i=1 h(b • ∇v 0 i )(b • ∇ϕ i ) + 1 h ∂v 0 i ∂y 2 ∂ϕ i ∂y 2 θ dη 1m . . . dη 11 dydt + T 0 Ω×(0,1) m α h(b • ∇Z 0 )(b • ∇ψ) + 1 h ∂Z 0 ∂y 2 ∂ψ ∂y 2 + hϕ 1 ∂p 0 ∂y 1 θ dη 1m . . . dη 11 dydt = -(ν + ν r ) T 0 Ω×(0,1) m h(b • ∇U)(b • ∇ϕ 1 ) + 1
W(t, y 1 , y 2 , η 1 ) = W 0 (t)W(h(y 1 , η 1 )y 2 )
for all (t, y 1 , y 2 , η 1 )

∈ [0, T ] × Ω × [0, 1] m . Proof. Let θ ∈ D(0, T ), Θ = (ϕ, ψ) ∈ V 0 × H 1 .
We define ϕ ε and ψ ε as follows

ϕ ε (y) = ϕ(y 1 , y 2 , y 1 ε , • • • , y 1 ε m ) + ε m y 2 ∂h ∂y 1 (y 1 , y 1 ε , • • • , y 1 ε m )ϕ 1 (y 1 , y 2 , y 1 ε , • • • , y 1 ε m )e 2 + y 2 m-1 k=1 ε m-k ∂h ∂η 1k (y 1 , y 1 ε , • • • , y 1 ε m )ϕ 1 (y 1 , y 2 , y 1 ε , • • • , y 1 ε m )e 2
and

ψ ε (y) = ψ(y 1 , y 2 , y 1 ε , .., y 1 ε m )
for all y = (y 1 , y 2 ) ∈ Ω. We have Then we let Θ ε = (ϕ ε , ψ ε ) with

ε m b ε • ∇ϕ ε 1 = ε m ∂ϕ ε 1 ∂y 1 - ε m y 2 h ε ∂h ε ∂y 1 ∂ϕ ε 1 ∂y 2 = ε m ∂ϕ 1 ∂y 1 (y 1 , y 2 , y 1 ε , • • • , y 1 ε m ) + m k=1 ε m-k ∂ϕ 1 ∂η 1k (y 1 , y 2 , y 1 ε , • • • , y 1 ε m ) - ε m y 2 h(y 1 , y 1 ε , • • • , y 1 ε m )        ∂h ∂y 1 (y 1 , y 1 ε , • • • , y 1 ε m ) + m k=1 ε -k ∂h ∂η 1k (y 1 , y 1 ε , • • • , y 1 ε m )        × ∂ϕ 1 ∂y 2 (y 1 , y 2 , y 1 ε , • • • , y 1 ε m ) →→ m+1 ∂ϕ 1 ∂η 1m (
ϕ ε (z 1 , z 2 ) = ϕ ε z 1 , z 2 ε m h ε (z 1 ) , ψ ε (z 1 , z 2 ) = ψ ε z 1 , z 2 ε m h ε (z 1 )
for all z = (z 1 , z 2 ) ∈ Ω ε . Reminding that Θ = (ϕ, ψ) ∈ V 0 × H 1 , we obtain that Θ ε ∈ V ε 0 × H 1,ε and with ( 11)

ε m ∂ ∂t [v ε , Θ ε ], θ D (0,T ),D(0,T ) + ε m T 0 a(v ε , Θ ε )θ dt + ε m T 0 B(v ε , v ε , Θ ε )θ dt +ε m T 0 R(v ε , Θ ε )θ dt = ε m T 0 p ε , div(ϕ ε ) θ dt + ε m T 0 F ε (v ε ), Θ ε θ dt.
We pass now to the limit as ε tends to zero with the same kind of computations as in Theorem 5.1 in [START_REF] Boukrouche | Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary[END_REF] (see also Proposition 5 in [START_REF] Boukrouche | Unsteady micrololar fluid flow in a thin domain with Tresca fluid-solid interface law[END_REF]). With Proposition 2.2 we have the following estimates

ε m ∂ ∂t [v ε , Θ ε ], θ D (0,T ),D(0,T ) = -ε 2m T 0 Ω         2 i=1 v ε i ϕ ε i + Z ε ψ ε         h ε θ (t)dydt = O(ε 2m ) ε m T 0 B(v ε , v ε , Θ ε )θdt = ε m T 0 Ω 2 i=1 v ε 1 (ε m b ε • ∇v ε i )ϕ ε i + v ε 2 h ε ∂v ε i ∂y 2 ϕ ε i h ε θ dydt +ε m T 0 Ω v ε 1 (ε m b ε • ∇Z ε )ψ ε + v ε 2 h ε ∂Z ε ∂y 2 ψ ε h ε θ dydt = O(ε m 2 )
and

ε m T 0 R(v ε , Θ ε )θ dt = -2ν r ε m T 0 Ω 1 h ε ∂Z ε ∂y 2 ϕ ε 1 -(ε m b ε • ∇Z ε )ϕ ε 2 h ε θ dydt -2ν r ε m T 0 Ω (ε m b ε • ∇v ε 2 ) - 1 h ε ∂v ε 1 ∂y 2 ψ ε h ε θ dydt + 4ν r ε 2m T 0 Ω Z ε ψ ε h ε θ dydt = O(ε m ).
With the (m + 1)-scale convergence results of Proposition 3.3 and Proposition 3.4 we get

ε m T 0 a(v ε , Θ ε )θ dt = (ν + ν r ) T 0 Ω 2 i=1 h ε (ε m b ε • ∇v ε i )(ε m b ε • ∇ϕ ε i ) + 1 h ε ∂v ε i ∂y 2 ∂ϕ ε i ∂y 2 θdydt +α T 0 Ω h ε (ε m b ε • ∇Z ε )(ε m b ε • ∇ψ ε ) + 1 h ε ∂Z ε ∂y 2 ∂ψ ε ∂y 2 θdydt → (ν + ν r ) T 0 Ω×Y m 2 i=1 h(b • ∇v 0 i )(b • ∇ϕ i ) + 1 h ∂v 0 i ∂y 2 ∂ϕ i ∂y 2 θ dηdydt +α T 0 Ω×Y m h(b • ∇Z 0 )(b • ∇ψ) + 1 h ∂Z 0 ∂y 2 ∂ψ ∂y 2 θ dηdydt +(ν + ν r ) T 0 Ω×Y m 2 i=1 -y 2 ∂h ∂η 1m m k=1 ∂v k i ∂η 2k (b • ∇ϕ i ) + 1 h m k=1 ∂v k i ∂η 2k ∂ϕ i ∂y 2 θ dηdydt +α T 0 Ω×Y m -y 2 ∂h ∂η 1m m k=1 ∂Z k ∂η 2k (b • ∇ψ) + 1 h m k=1 ∂Z k ∂η 2k ∂ψ ∂y 2 θ dηdydt.
Since ϕ, ψ and h do not depend on η 2k and v k and Z k are η 2k -periodic for all k ∈ {1, ..., m}, we infer that the two last integral terms are equal to 0 and thus

ε m T 0 a(v ε , Θ ε )θ dt → T 0 a(v 0 , Θ)θ dt with v 0 = (v 0 , Z 0 ), where a(v, Θ) = (ν + ν r ) Ω×(0,1) m 2 i=1 h(b • ∇v i )(b • ∇ϕ i ) + 1
for all v = (v, Z) ∈ L 2 (0, L; F 2 ) × L 2 (0, L; F ) and Θ = (ϕ, ψ) ∈ L 2 (0, L; F 2 ) × L 2 (0, L; F ). Let us recall that ξ ε = (U ε e 1 , W ε ) with U ε (t, z) = s 0 (t) + U 0 (t) -s 0 (t) U(h ε (y 1 )y 2 ) W ε (t, z) = W 0 (t)W(h ε (y 1 )y 2 )
for all (t, z) ∈ [0, T ] × Ω ε , where U and W belong to C ∞ (R). Thus

U ε (t, z) = U(t, y 1 , y 2 , y 1 ε , ..., y 1 ε m ), W ε (t, z) = W(t, y 1 , y 2 , y 1 ε , ..., y 1 ε m ) and U and W belong to C [0, T ]; C 1 Ω; C 1 # ([0, 1] m ) . It follows that ε m T 0 dξ ε dt , Θ ε θ dt = O(ε 2m ), ε m T 0 R(ξ ε , Θ ε )θ dt = O(ε m ) ε m T 0 B(ξ ε , v ε , Θ ε )θ dt = O(ε m ), ε m T 0 B(v ε , ξ ε , Θ ε )θ dt = O(ε m ) and ε m T 0 a(ξ ε , Θ ε )θ dt → T 0 a(ξ, Θ)θ dt with ξ = (Ue 1 , W). Finally ε 2m T 0 Ω f ε ϕ ε θh ε dydt → T 0 Ω [0,1] m f ϕθh dη 1m . . . dη 11 dydt, ε 2m T 0 Ω g ε ψ ε θh ε dydt → T 0 Ω [0,1] m gψθh dη 1m . . . dη 11 dydt. 0 Ω ε m p ε         ε m ∂h ∂y 1 ϕ 1 + m-1 k=1 ε m-k ∂h ∂η 1k ϕ 1 + y 2 m-1 k=1 ε m-k ∂h ∂η 1k ∂ϕ 1 ∂y 2         θ dydt + T 0 Ω ε m p ε h ∂ϕ 1 ∂η 1m -y 2 ∂h ∂η 1m ∂ϕ 1 ∂y 2 + ∂ϕ 2 ∂y 2 θ dydt.
Let us assume now that ϕ ∈ V 0,div . Then

ε m T 0 Ω ε p ε div(ϕ ε )θ dzdt = T 0 Ω ε 2m p ε ∂(hϕ 1 ) ∂y 1 θ dydt + m-1 k=1 T 0 Ω ε 2m p ε ε -k ∂(hϕ 1 ) ∂η 1k θ dydt. ( 40
) Since h ∈ C ∞ [0, L]; C ∞ # ([0, 1] m ) and ϕ 1 ∈ C ∞ Ω; C ∞ # ([0, 1] m ) , we may define ψ k (y 1 , τ, η 1 ) = τ 0 ∂(hϕ 1 ) ∂η 1k dy 2 for all (y 1 , τ, η 1 ) ∈ Ω × [0, 1] m . Hence ψ k belongs to C ∞ Ω; C ∞ # [0, 1] m , is L-periodic with respect to y 1 and ψ k = 0 on Γ 0 × (0, 1) m . Moreover since ϕ ∈ V 0,div we have Φ 1 = 1 0 hϕ 1 dy 2 ∈ C ∞ ([0, L]). Thus ∂ ∂η 1k 1 0 hϕ 1 dy 2 = 1 0 ∂(hϕ 1 ) ∂η 1k dy 2 ≡ 0. It follows that ψ k = 0 on Γ 1 × (0, 1) m and ψ k ∈ H 1 . Let ψ ε k (y) = ψ k (y, y 1 ε , • • • , y 1 ε m ) for all y = (y 1 , y 2 ) ∈ Ω. Then T 0 Ω ε 2m p ε ε -k ∂(hϕ 1 ) ∂η 1k θ dydt = ε m-k T 0 Ω ε m p ε ∂ψ ε k ∂y 2 θ dydt.
With the same computations as in Proposition 3.7 (see [START_REF] Sochi | Slip at fluid-solid interface[END_REF]), by choosing

ϕ ε = ψ ε k we obtain T 0 Ω ε m p ε ∂ψ ε k ∂y 2 θ dydt = O(1) which implies that lim ε→0 ε m-k T 0 Ω ε m p ε ∂ψ ε k ∂y 2 θ dydt = 0 for all k ∈ {1, . . . , m -1}.
Hence we may pass to the limit as ε tends to zero in (40) and we get lim

ε→0 ε 2m T 0 Ω ε p ε div(ϕ ε )θ dzdt = T 0 Ω×Y m p 0 ∂(hϕ 1 ) ∂y 1 θ dηdydt = - T 0 Ω×(0,1) m ∂p 0 ∂y 1 hϕ 1 θ dη 1m . . . dη 11 dydt
which allows us to conclude.

By density equation (39) still holds for all Θ = (ϕ, ψ) ∈ V 0,div × H 1 0,# but it does not provide a variational formulation for the limit triplet (v 0 , Z 0 , p 0 ) since v 0 and the test-function ϕ do not belong to the same functional space (indeed we do not expect v 0 to vanish on (0, T )×Γ 0 ×(0, 1) m ). Nevertheless we observe that, by choosing ϕ ≡ 0, we obtain a totally decoupled problem for Z 0 given by Find Z 0 ∈ L 2 (0, T ; H 1 0,# ) such that

In this problem the time variable appears only as a parameter. For the sake of notational simplicity, we define for all y 1 ∈ [0, L], the bilinear symetric form a y 1 by for all (w, ϕ) ∈ F × F and

a y 1 (w, ϕ) = (0,1) m+1 h(y 1 , η 1 )(b • ∇w)(b • ∇ϕ) + 1 h(y 1 , η
H 1 y 1 = ψ ∈ C ∞ [0, 1]; C ∞ # ([0, 1] m ) : ψ(0, •) = ψ(1, •) = 0 in (0, 1) m , H 1 0,y 1 ,# = closure of H 1 y 1 in F .
By a straighforward adaptation of Proposition 5.2 in [START_REF] Boukrouche | Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary[END_REF] we infer that a y 1 is continuous on F 2 and uniformly coercive with respect to y 1 on H

for all ψ ∈ H 1 # (0, L), such that L 0 Q(w ) [0,1] m h(y 1 , η 1 ) dη 1m . . . dη 11 dy 1 = 0.
The corresponding mapping Q given by:

Q : V div -→ H 1 # (0, L) |R w -→ Q(w )
belongs to L c (V div , H 1 # (0, L) |R ) and p 1 (t, .) = Q(w (t, .)) for a.a. t in [0, T ].

We are now able to identify the limit problem for (v 0 , p 0 ). for all ϕ ∈ L 2 (0, T, V div ).

Proof. Let θ ∈ D(0, T ), Θ = (ϕ, ψ) ∈ V div × H 1 . We define ϕ ε and ψ ε , then ϕ ε and ψ ε as in Proposition 4.1 and we let Θ ε = (ϕ ε , ψ ε ) in ( 9). Then we choose Θ ε = (-v ε , 0) in ( 12) and we add the two inequalities. Observing that B v ε , v ε , (v ε , 0) = 0, j ε (0) = 0 and

j ε (ϕ ε θ + v ε ) -j ε (v ε ) = T 0 L 0 k ε (|ϕ ε θ + v ε | -|v ε |) dy 1 dt ≤ j ε (ϕ ε θ)
+ ε m 2 v ε (T ) 2 L 2 (Ω ε ) - ε m 2 v ε (0) 2 L 2 (Ω ε ) .
The compatibility assumption (42) on u ε 0 allows us to choose the extension U ε given by U ε (t, z) = s 0 (t) + u 0 (t)s 0 (t) U z 2 ε m ∀(t, z) ∈ [0, T ] × Ω ε and it follows that v ε (0) = v ε 0 = 0. Then using the same computations as in the proof of Theorem 4.1 we may pass to the (m + 1)-scales to pass to the limit in all terms except the non-linear terms ε m T 0 a v ε , (v ε , 0) dt and ε m j ε (v ε ). For the latter we can use a straighforward adaptation of lemma 4 in [START_REF] Boukrouche | Unsteady micrololar fluid flow in a thin domain with Tresca fluid-solid interface law[END_REF] For the former we apply classical results in multi-scale convergence techniques (see Proposition 1.6 in [START_REF] Allaire | Homogenization and two-scale convergence[END_REF] and its obvious generalization to (m + 1)-scale convergence) which yields lim inf ε m which satisfies the non-homogeneous Dirichlet boundary condition (6) on Γ ε 1 and coincide with the sliding velocity of the wall Γ 0 at t = 0. We may weaken this condition by assuming only that (ε m u ε 0 ) ε>0 converges to zero in L 2 (Ω ε ) 2

.

Finally we can state the main result of this paper. Proof. By introducing the decomposition of v 0 into (43) we obtain the variational inequality (44) for w . Then existence and uniqueness of the solution of (44) is a direct consequence of the uniform coercivity property of a y 1 with respect to y 1 ∈ [0, L] on H 1 0,y 1 # and the linearity and continuity properties of Q.

y 1 ,y 1 , 1 ,

 111 0, η 11 , . . . , η 1m ) × ϕ(y 1 , η 11 , . . . , η 1m )θ dη 1m . . . dη 11 dy 1 dtdt = 0 and η 11 , . . . , η 1m ) ϕ(y 1 , η 11 , . . . , η 1m )θ(t) dη 1m . . . dη 11 dy 1 dt = 0.

dη 11 dydt = 0 which

 0 θ dη 1m . . . dη 11 dy 1 dt = 0 which proves[START_REF] Łukaszewicz | Micropolar Fluids. Theory and applications[END_REF].

  1 , η 11 , . . . , η 1m ) dη 1m . . . dη 11 dy 1 = 0 in D (0, T ). (33) Proof. The convergence property (32) follows from the estimate (17) and Theorem 3.2. Let ϕ ∈ D Ω; C ∞ # (Y m ) . We denote ϕ ε (y) = ϕ y, y ε , ..., y ε m for all y ∈ Ω and ϕ ε

  η)θ(t) dηdydt = 0 which implies that p 0 does not depend on η 2m . Then we proceed by induction. Indeed let us assume that p 0 does not depend on η 2m , η 2(m-1) , • • • , η 2 for in {2, • • • , m}. Then by choosing a test-function ϕ ∈ D Ω; C ∞ # Y -1 × [0, 1] m-+1 (i.e. ϕ does not depend on η 2m , η 2(m-1) , • • • , η 2 ) and multiplying (

2 θ

 2 dη 1m . . . dη 11 dydt + T 0 Ω×(0,1) m gψθh dη 1m . . . dη 11 dydt ∀ψ ∈ H 1 0,# , ∀θ ∈ D(0, T ).

Proposition 4 . 4 .

 44 Let (s 0 , U 0 , W 0 ) ∈ H 1 (0, T ) 3 and let us assume that there exist ( f, g)∈ C [0, T ]; C Ω; C # ([0, 1] m ) 3 and k ∈ C [0, T ]; C [0, L]; C # ([0, 1] m ), such that k takes its values in R * + , f , g and k are L-periodic in y 1 andε 2m f ε (t, y) = f (t, y, y 1 ε , ..., y 1 ε m ), ε 2m g ε (t, y) = g(t, y, for all (t, y) ∈ [0, T ] × Ω.Let us assume moreover that ε m Z ε 0 is bounded in L 2 (Ω) and u ε 0 satisfies the following compatibility condition: there existsU ∈ C ∞ (R) such that U(0) = U (0) = 0, U(X) = 1 for all X ∈ [h m , h M ],andu ε 0 = s 0 (0)e 1 + U 0 (0)s 0 (0) U z 2 ε m e 1 in Ω ε . (42)Then (v 0 , p 0 ) satisfies the following variational inequality T 1 , η 1 )ϕ 1 (t, y, η 1 ) dη 1m . . . dη 11 dy 2 dy 1 dt + , y 1 , η 1 ) |ϕ(t, y 1 , 0, η 1 ) + v 0 (t, y 1 , 0, η 1 )| -|v 0 (t, y 1 , 0, η 1 )| dη 1m . . . dη 11 dy 1 dt m+1 f (t, y, η 1 )h(y 1 , η 1 )ϕ(t, y, η 1 ) dη 1m . . . dη 11 dy 2 dy 1 dt (43)

0 B 0 R 0 F

 000 , Θ ε ]θ dt + ε m T 0 a(v ε , Θ ε )θ dt + ε m T (v ε , v ε , Θ ε )θ dt +ε m T (v ε , Θ ε )θ dt -ε m T 0 R v ε , (v ε , 0) dt + ε m j ε (ϕ ε θ) ≥ ε m j ε (v ε ) + ε m T 0 p ε , div(ϕ ε ) θ dt + ε m T 0 a v ε , (v ε , 0) dt +ε m T 0 (F(v ε ), Θ ε )θ dt -ε m T (v ε ), (v ε , 0) dt

  y 1 , η 1 )|v 0 (t, y 1 , 0, η 1 )| dη 1m . . . dη 11 dy 1 dt.

a 0 a 0 L 0 a y 1 (Remark 4 . 4 .

 000144 (v ε , 0), (v ε , 0) dt = T (v 0 , 0), (v 0 , 0) dt = T v 0 , v 0 ) dy 1 dtand the conclusion follows. The compatibility condition (42) means that the initial velocity u e 0 is a smooth function of z 2

Theorem 4 . 5 . 1 v ε →→ m+1 v 0 = ∂p 0 ∂y 1 w 1 y 1 + ∂p 1 ∂y 1 w 1 y 1 + 1 + z 2 t,y 1 where p 1 = 0 L 0 a y 1 0 L 0 a y 1 (

 45111211001001 Under the assumptions of Theorem 4.4, we haveε 2m p ε →→ m+1 p 0 = p 0 + p (U 0s 0 )w 2 y 1 + w 3 t,y 1 + w Z ε →→ m+1 Z 0 = W 0 z 1 y Q(w ) and w ∈ L 2 (0, T ; V div )is the unique solution of the following variational inequality: + w | -|w |) dη 1m . . . dη 11 dy 1 dt + T (ϕ) ≥ 0 (44) for all ϕ ∈ L 2 (0, T ; V div ) with b(w, ϕ) = T (w, ϕ) dy 1 dt + m+1 hϕ dη 1m . . . dη 11 dy 2 dy 1 dt + T Ue 1 , ϕ) dy 1 dti )h dη 1m . . . dη 11 dy 2 dy 1 dt for all (w, ϕ) ∈ L 2 (0, T ; V div ) 2 .

  y, η 11 , ..., η 1m )ϕ(y, η 11 , ..., η 1(m-1) )θ(t) dη 1m ...dη 11 dydt

  Proposition 3.4. Under the assumptions of Proposition 2.2, there exist v k i and Z k , depending on (t, y 1 , η 11 , . . . , η 1m , y 2 , η 21 , . . . , η 2k ) such that

  y, η 11 , . . . , η 1m ) ∂ϕ ∂y 2 (y, η 11 , . . . , η 1m )θ dη 1m . . . dη 11 dydt , η 11 , . . . , η 1m )ϕ(y 1 , 0, η 11 , . . . , η 1m )θ dη 1m . . . dη 11 dy 1 dt.

	T	L
	-		ξ(t, y 1
	0	0	[0,1] m

  y 1 , η 11 , . . . η 1m , y 2 , η 21 , . . . , η 2k )ϕ(y, η 11 , . . . , η 1m )θ(t) dηdydt = 0 for all k ∈ {1, ..., m}. Thus (25) reduces to y, η 11 , . . . , η 1m )ϕ(y, η 11 , . . . , η 1m )θ(t) dη 1m . . . dη 11 dydt (t, y, η 11 , . . . , η 1m )ϕ(y 1 , 0, η 11 , . . . , η 1m )θ(t) dη 1m . . . dη 11 dy 1 dt

	T (t, = -0 Ω [0,1] m ∂v 0 1 ∂y 2 T 0 Ω [0,1] m v 0 1 (t, y, η 11 , . . . , η 1m )	∂ϕ ∂y 2	(y, η 11 , . . . , η 1m )θ(t) dη 1m . . . dη 11 dydt
		T	L
	-
		0	0	[0,1] m
	and using again Green's formula
	T	L
	0	0	[0,1]

ξm ξ(t, y 1 , η 11 , . . . , η 1m )v 0 1 (t,

y 1 , 0, η 11 , . . . , η 1m ) ×ϕ(y 1 , 0, η 11 , . . . , η 1m )θ dη 1m . . . dη 11 dy 1 dt

  y 1 , 1, η 11 , . . . , η 1m )ϕ(y 1 , 1, η 11 , . . . , η 1m )θ dη 1m . . . dη 11 dy 1 dt = 0.

  y, η 11 , ..., η 1m )h(y 1 , η 11 , ..., η 1m ) 11 , ..., η 1m )θ dη 1m . . . dη 11 dydt 11 , ..., η 1m )θ dη 1m . . . dη 11 dydt

	× (y, η + ∂ϕ ∂η 1m T 0 Ω [0,1] m y 2 v 0 1 ∂h ∂η 1m (y 1 , η 11 , ..., η 1m ) (y, η -∂ϕ ∂y 2 T 0 Ω [0,1] m v 0 2 (t, y, η 11 , ..., η 1m ) ∂ϕ ∂y 2 (y, η 11 , ..., η 1m )θ dη 1m . . . dη 11 dydt
	=	0	T	Ω [0,1] m	h	∂v 0 1 ∂η 1m	-y 2	∂h ∂η 1m	∂v 0 1 ∂y 2	+	∂v 0 2 ∂y 2	ϕθ dη 1m . . . dη 11 dydt

  y 1 , y 2 , η 11 , ..., η 1m ) ∂ϕ ∂y 2 (y 1 , y 2 , , η 11 , ..., η 1m )θ(t) dη 1m ...dη 11 dydt = 0 which allows us to conclude that p 0 does not depend on y 2 .

  11 , . . . , η 1m ) dη 1m . . . dη 11 dydt = 0.Reminding that h is a C ∞ -function from R m+1 to R * + which is L-periodic in y 1 and 1-periodic in η 11 , ..., η 1m , for any φ ∈ D Ω; C ∞ 11 , . . . , η 1m ) dη 1m . . . dη 11 dydt = 0.

	∂η 1m θ dη 1m . . . dη 11 dydt ∂ϕ ∂y 2 θ dη 1m . . . dη 11 dydt = 0 ϕ θ dη 1m . . . dη 11 dydt ∂ϕ ∂y 2 p 0 (t, y 1 , η 11 , . . . , η 1m ) and since p 0 does not depend on y 2 T 0 Ω [0,1] m p 0 ∂ϕ ∂η 1m h -y 2 ∂h ∂η 1m = T 0 Ω [0,1] m p 0 ∂ϕ ∂η 1m h + ∂h ∂η 1m = T 0 Ω [0,1] m ∂(hϕ) ∂η 1m h ∈ D Ω; C ∞ # [0, 1] m and thus (y, η # [0, 1] m we may define ϕ = φ T 0 Ω [0,1] m p 0 (t, y 1 , η 1 ) ∂φ ∂η 1m (y, η

  y 1 , y 2 , η 11 , . . . , η 1m )

		-	y 2 h(y 1 , η 11 , . . . , η 1m )	∂h ∂η 1m	(y 1 , η 11 , . . . , η 1m )	∂ϕ 1 ∂y 2	(y 1 , y 2 , η 11 , . . . , η 1m )
		= b • ∇ϕ 1
	and similarly
	ε m b ε • ∇ϕ ε 2	→→ m+1 ∂ϕ 2 ∂η 1m	(y 1 , y 2 , η 11 , . . . , η 1m )
			-	y 2 h(y 1 , η 11 , . . . , η 1m )	∂h ∂η 1m	(y 1 , η 11 , . . . , η 1m )	∂ϕ 2 ∂y 2	(y 1 , y 2 , η 11 , . . . , η 1m )
			= b • ∇ϕ 2
	and
	∂ϕ ε 2 ∂y 2	→→ m+1 ∂ϕ 2 ∂y 2	(y 1 , y 2 , η 11 , . . . , η 1m ),	∂ϕ ε 1 ∂y 2	→→ m+1 ∂ϕ 1 ∂y 2	(y 1 , y 2 , η 11 , . . . , η 1m ).

  , y 2 , η 1 )h(y 1 , η 1 ) dη 1m . . . dη 11 dy 2 dy 1 , η 1 ) dη 1m . . . dη 11 dy 1 = 0 ∀a.a. t ∈ (0, T ), i = 0, 1.Proof. The proof is a straighforward adaptation of Proposition 7 in[START_REF] Boukrouche | Unsteady micrololar fluid flow in a thin domain with Tresca fluid-solid interface law[END_REF].

	1 h(y 1 Remark 4.3. For all w ∈ V div , we may define Q(w ) as the unique solution in H 1 L 0 ∂p 1 ∂y 1 ∂ψ ∂y 1 a y 1 (w 1 y 1 , w 1 y 1 ) dy 1 = L 0 ∂ψ ∂y 1 (0,1) m+1 w 1 (t, y 1 for all ψ ∈ H 1 # (0, L), such that L 0 p i (t, y 1 ) [0,1] m # (0, L) |R of the Reynolds problem 0,y 1 ,# . Then (41) can be rewritten as and L 0 ∂Q(w ) ∂y 1 ∂ψ a y 1 (w 1 y 1 , w 1 y 1 ) dy 1 ∂y 1 = L 0 ∂ψ ∂y 1 (0,1) m+1 w (y

1 , y 2 , η 1 )h(y 1 , η 1 ) dη 1m . . . dη 11 dy 2 dy 1

Remark 4.5. It follows that the limit triplet (v 0 , Z 0 , p 0 ) is uniquely defined and the whole sequence (v ε , p ε , Z ε ) ε>0(m + 1)-scale converge to its limit.

Find Z 0 ∈ L 2 (0, T ; H 1 0,# ) such that α L 0 a y 1 (Z 0 (t), ψ)dy 1 = -α L 0 a y 1 (W(t), ψ) dy 1 + L 0 (0,1) m+1 g(t, y 1 , y 2 , η 1 )h(y 1 , η 1 )ψdη 1m . . . dη 11 dy 2 dy 1 for all ψ ∈ H 1 0,# and for almost all t ∈ (0, T ). We obtain Proposition 4.2. Under the assumptions of Proposition 4.1, the (m+1)-scale limit Z 0 is uniquely given by

where z 1 y 1 ∈ H 1 0,y 1 ,# and z 2 t,y 1 ∈ H 1 0,y 1 ,# are the unique solutions of the following auxiliary problems

Proof. The proof is a straighforward adaptation of the proof of Proposition 5.2 in [START_REF] Boukrouche | Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary[END_REF] (see also Proposition 6 in [START_REF] Boukrouche | Unsteady micrololar fluid flow in a thin domain with Tresca fluid-solid interface law[END_REF]).

Similarly we may introduce auxiliary problems for the limit velocity. More precisely, for all y 1 ∈ [0, L] we let

and we define

By using the properties of a y 1 we infer immediately that a y 1 is continuous and uniformly coercive with respect to y 1 on V y 1 ,div . Hence the following auxiliary problems (L1)

admit an unique solution for all (t, y 1 )

Moreover the continuity properties of h, U and f imply that y 1 → w i y 1 belong to C # ([0, L]; V 0y 1 ,div ) (for i = 1, 2) and (t, y 1 ) → w 3 t,y 1 belong to C [0, T ]; C # ([0, L]; V 0y 1 ,div ) . Thus v 0 can be decomposed as follows

∀a.a. (t, y, η 1 ) ∈ (0, T ) × Ω × (0, 1) m with w ∈ L 2 (0, T ; V div ). With the definition of w 1 y 1 , w 2 y 1 , w 3 t,y 1 and (39) we obtain that L 0 a y 1 w (t, y 1 , •, •), ϕ(y 1 , •) dy 1 = 0 ∀ϕ ∈ V 0,div , ∀a.a. t ∈ (0, T ) and we introduce

Remark 4.2. The subspace V div can be interpreted as the orthogonal of V 0,div with respect to the inner product defined by L 0 a y 1 (•, •) dy 1 in V div .

We obtain the following result for the limit pressure p 0 : Proposition 4.3. Under the assumptions of Proposition 4.1, the limit pressure p 0 can be decomposed as p 0 + p 1 where p 0 (t, •) and p 1 (t, •) are given as the unique solutions in H