Igor Pa Žanin

Francisco Javier

S U Árez-Grau

Roughness-induced effects on the thermomicropolar fluid flow through a thin domain

Keywords: AMS classification numbers: 35B27, 35Q35, 76A05, 76M50 thermomicropolar fluid, thin domain, rough boundary, cooling condition, homogenization

In this paper, we study the asymptotic behavior of the thermomicropolar fluid flow through a thin channel with rough boundary. The flow is governed by the prescribed pressure drop between the channel's ends and the heat exchange through the rough wall is allowed. Depending on the limit of the ratio between channel's thickness and the wavelength of the roughness, we rigorously derive different asymptotic models clearly showing the roughness-induced effects on the average velocity and microrotation. To accomplish that, we employ the adaptation of the unfolding method to a thin-domain setting.

Introduction

The model of micropolar fluid, proposed by Eringen [START_REF] Eringen | Theory of micropolar fluids[END_REF] has been extensively studied both in the engineering and mathematical literature, due to its practical importance. Being able to take into consideration the microstructure of the fluid particles and capture the effects of its rotation, the micropolar fluid model describes the motion of numerous real fluids better than the classical Navier-Stokes equations. Liquid crystals, animal blood, muddy fluids, certain polymeric fluids or even water in models with small scales are the typical examples. The rotation of the fluid particles is mathematically described by introducing the microrotation field (along with the standard velocity and pressure fields) and, accordingly, a new governing equation coming from the conservation of angular momentum. The model of thermomicropolar fluid, introduced also by Eringen [START_REF] Eringen | Theory of thermomicrofluids[END_REF], represents an essential generalization of the micropolar fluid model acknowledging the variations of the fluid temperature as well. In such, non-isothermal, regime, the micropolar equations are being coupled with the heat conduction equation leading to a very complex system of PDEs. In particular, the 2D system describing the steady-state flow of incompressible, isotropic, thermomicropolar fluid flow between two horizontal plates in dimensionless form reads as follows (see e.g. [START_REF] Eringen | Microcontinuum field theories II: fluent media[END_REF][START_REF] Lukaszewicz | Long-time behavior of 2D micropolar fluid flows[END_REF][START_REF] Tarasinska | Global attractor for heat convection problem in a micropolar fluid[END_REF]):

                         1 P r ((u • ∇)u + ∇p) = ∆u + N 1 -N (2∇ ⊥ w + ∆u) + Ra T e 2 + f , div(u ε ) = 0, M P r (u • ∇w) = L∆w + 2N 1 -N (rot(u) -2w) + g, u • ∇T = ∆T + D∇ ⊥ w • ∇T.
(1.1)

In the above system, the velocity vector field is denoted by u, the pressure by p, w represents the microrotation and T is the temperature of the fluid. The external sources of linear and angular momentum are given by the functions f = (f 1 , f 2 ) and g, respectively. We denote by e 2 = (0, 1) ∈ R 2 the unit upward vector, whereas the positive constants appearing in (1.1) represent the following (see e.g. [START_REF] Lukaszewicz | Micropolar fluids, theory and applications[END_REF]):

• N is the coupling parameter, i.e. the relation between the Newtonian and microrotation viscosities,

• M is the relation between the moment of inertia and geometry,

• L is the couple stress parameter, i.e. the relation between the geometry and the properties of the fluid,

• D is the micropolar heat conduction parameter, i.e. the relation between the micropolar thermal conduction and the geometry,

• P r is the Prandtl number, i.e. the relation between the kinematic viscosity and the thermal diffusivity,

• Ra is the Rayleigh number, i.e. the relation between the coefficients of thermal expansion and conductivity and the geometry.

Throughout the mathematical literature, one can find many papers on the rigorous derivation of the asymptotic models describing the isothermal flow of a micropolar fluid, see e.g. [START_REF] Beneš | Rigorous derivation of the asymptotic model describing a nonsteady micropolar fluid flow through a thin pipe[END_REF], [START_REF] Bonnivard | Effects of rough boundary and nonzero boundary conditions on the lubrication process with micropolar fluid[END_REF], [START_REF] Bonnivard | A generalized Reynolds equation for micropolar flows past a ribbed surface with nonzero boundary conditions[END_REF], [START_REF] Dupuy | Asymptotic methods for micropolar fluids in a tube structure[END_REF], [START_REF] Dupuy | Asymptotic solution for a micropolar flow in a curvilinear channel[END_REF], [START_REF] Pažanin | Asymptotic behavior of micropolar fluid flow through a curved pipe[END_REF], [START_REF] Pažanin | Analysis of the thin film flow in a rough domain filled with micropolar fluid[END_REF]. Although there have been a number of recent papers concerning engineering applications of the thermomicropolar fluid model (see e.g. [START_REF] Cheng | Nonsimilar solutions for double-diffusion boundary layers on a sphere in micropolar fluids with constant wall heat and mass fluxes[END_REF], [START_REF] Md | Fluctuating Flow of Thermomicropolar Fluid Past a Vertical Surface[END_REF], [START_REF] Md | Transient Natural Convection Flow of Thermomicropolar Fluid of Micropolar Thermal Conductivity along a Nonuniformly Heated Vertical Surface[END_REF], [START_REF] Rahman | Thermo-Micropolar Fluid Flow Along a Vertical Permeable Plate With Uniform Surface Heat Flux in the Presence of Heat Generation[END_REF]), the rigorous treatments for such models are very sparse. Most recently, the system (1.1) has been studied in [START_REF] Lukaszewicz | Asymptotic analysis of the thermomicropolar fluid flow through a thin channel with cooling[END_REF] for the thermicropolar flow through a thin channel with smooth walls, namely:

Ω ε = (x 1 , x 2 ) ∈ R 2 : x 1 ∈ ω, 0 < x 2 < ε , ω = (-1/2, 1/2).
The flow is assumed to be governed by the prescribed pressure drop between channel's ends, given by q -1/2 and q 1/2 , and the heat exchange between the fluid inside the channel and the exterior medium is allowed through the upper wall by using Newton's cooling law. Using the asymptotic analysis with respect to the thickness of the channel, a higher-order asymptotic solution has been rigorously derived. In particular, assuming that f 1 and g only depends on the horizontal variable and after a dilatation in the vertical variable, it is proved that the average velocity U av = (U av 1 , U av 2 ) and the microrotation W av at the main-order term are respectively given by:

U av 1 = 1 12
1 -N P r q -1/2 -q 1/2 + P r

1/2 -1/2 f 1 (ξ) dξ , U av 2 = 0, W av = 1 12 1 L g(x 1 ), in ω. (1.2)
Moreover, the explicit expressions for the pressure approximation is obtained and for the average of the temperature as well, acknowledging the effects of fluid's microstructure through the presence of the couple stress parameter L and the micropolar heat conduction parameter D.

In great majority of the applications, the domain boundaries are not perfectly smooth, i.e. they contain some irregularities. Thus, in the present paper, we aim to generalize the results from [START_REF] Lukaszewicz | Asymptotic analysis of the thermomicropolar fluid flow through a thin channel with cooling[END_REF] to a case of a thin channel with an irregular upper wall described by

x 2 = η ε h x 1 ε ,
where η ε is the thickness of the roughness, ε is the period of the roughness and h is a positive and periodic function (see Section 2). This kind of thin rough domain has been extensively studied for the isothermal flows, see [START_REF] Bayada | Homogenization of the Stokes system in a thin film flow with rapidly varying thickness[END_REF][START_REF] Mikelić | Remark on the result on homogenization in hydrodynamical lubrication by G. Bayada and M. Chambat[END_REF] for the classical Newtonian fluid flow, [START_REF] Anguiano | Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary[END_REF] for the flow of the generalized Newtonian fluid and [START_REF] Suárez-Grau | Analysis of the roughness regime for micropolar fluids via homogenization[END_REF] for the micropolar fluid flow. In these papers, a critical size has been found between the thickness of the domain η ε and the period of the roughness ε, which is given by

λ = lim ε→0 η ε ε ∈ [0, +∞].
The critical case, λ ∈ (0, +∞), corresponds to the case in which the thickness and period of the roughness are proportional. The subcritical case, λ = 0, corresponds to a very smooth roughness, and the supercritical case, λ = +∞, corresponds to the case of a highly oscillating boundary.

As far as the authors know, the flow of a thermomicropolar fluids has not been yet considered in the above described setting. The supercritical case, due to the highly oscillating boundary, leads to the conclusion that the velocity and microrotation are zero in the roughness zone (see e.g. [START_REF] Suárez-Grau | Analysis of the roughness regime for micropolar fluids via homogenization[END_REF]) so, in the sequel, we study the asymptotic behavior of the solution in the critical and the subcritical case. By applying reduction of dimension techniques together with an adaptation of the unfolding method (see Section 3) to capture the microgeometry of the roughness, depending on the relation of ε and η ε , we rigorously derive the following expressions for the average velocity and microrotation:

U av 1 = a λ 1 -N P r q -1/2 -q 1/2 + P r 1/2 -1/2 f 1 (ξ) dξ , U av 2 = 0, W av = b λ 1 L g(x 1 ), in ω. (1.3)
where a λ , b λ ∈ R + are obtained through local problems depending on the value of λ ∈ [0, +∞) and give the roughness-induced effects on the velocity and microrotation. In the critical case λ ∈ (0, +∞), the parameters a λ , b λ are computed through local PDE problems (see Section 4, Theorem 4.3). However, in the subcritical case λ = 0, the parameters a 0 , b 0 can be explicitly computed (see Section 5, Theorem 5.3). In both cases, we obtain the same expression for the pressure as in [START_REF] Lukaszewicz | Asymptotic analysis of the thermomicropolar fluid flow through a thin channel with cooling[END_REF]. Moreover, the average of the temperature is obtained through a nonlinear problem in the critical case and is explicitly given in the subcritical case. Since the obtained findings are amenable for the numerical simulations, we believe that it could prove useful in the engineering practice as well.

Formulation of the problem and preliminaries

In this section, we first define the thin, rough 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 the domain of fixed height.

The domain and some notation. Let us denote ω = (-1/2, 1/2) ⊂ R. We consider a thin domain with a rapidly oscillating thickness defined by

Ω ε = {x = (x 1 , x 2 ) ∈ R 2 : x 1 ∈ ω, 0 < x 2 < h ε (x 1 )}, (2.4)
Here, the function h ε (x 1 ) = η ε h (x 1 /ε) represents the real gap between the two surfaces. The small parameter η ε is related to the film thickness and the small parameter ε is the wavelength of the roughness. Here, we consider that η ε is of order smaller or equal than ε, i.e. we consider

η ε ≈ ε or η ε ε. (2.5)
Function h is a positive and smooth function defined for z 1 in R, Z -periodic with Z = (-1/2, 1/2) the cell of periodicity in R, and there exist h min and h max such that

0 < h min = min z1∈Z h(z 1 ), h max = max z1∈Z h(z 1 ) .
We define the boundaries of Ω ε as follows

Γ 0 = (x 1 , x 2 ) ∈ R 2 : x 1 ∈ ω, x 2 = 0 , Γ ε 1 = (x 1 , x 2 ) ∈ R 2 : x 1 ∈ ω, x 2 = h ε (x 1 ) Σ ε i = (x 1 , x 2 ) ∈ R 2 : x 1 = i, 0 < x 2 < h ε (x 1 ) , i = -1/2, 1/2.
We also define the respective rescaled sets

Ω ε = ω × (0, h(x 1 /ε)), Γ ε 1 = ω × {h(x 1 /ε)} and Σ ε i = {i} × (0, h(i/ε)), i = -1/2, 1/2.
Related to the microstructure of the periodicity of the boundary, we consider that the domain ω is divided by a mesh of size ε:

for k ∈ Z, each cell Z k ,ε = εk + εZ . We define T ε = {k ∈ Z : Z k ,ε ∩ ω = ∅}.
In this setting, there exists an exact finite number of periodic sets Z k ,ε such that k ∈ T ε . Also, we define

Z k ,ε = Z k ,ε × (0, h(z 1 )
) and Z = Z × (0, h(z 1 )), which is the reference cell in R 2 . We define the boundaries

Γ0 = Z × {0}, Γ1 = Z × {h(z 1 )}, Σi = {i} × {i} × (0, h(i)), i = -1/2, 1/2.
The quantity h max allows us to define the extended sets Ω = ω × (0, h max ) and Γ 1 = ω × {h max }.

In order to apply the unfolding method, we will use the following notation. For k ∈ Z, we define κ : R → Z by

κ(x 1 ) = k ⇐⇒ x 1 ∈ Z k ,1 . (2.6)
Remark that κ is well defined up to a set of zero measure in R (the set ∪ k∈Z ∂Y k,1 ). Moreover, for every ε > 0, we have

κ x 1 ε = k ⇐⇒ x 1 ∈ Z k ,ε .
We denote by C a generic constant which can change from line to line.

We use the following notation for the partial differential operators:

∆Φ ε = ∂ 2 Φ ε 1 ∂x 2 1 + ∂ 2 Φ ε 1 ∂x 2 2 e 1 + ∂ 2 Φ ε 2 ∂x 2 1 + ∂ 2 Φ ε 2 ∂x 2 2 e 2 , ∆ϕ ε = ∂ 2 ϕ ε ∂x 2 1 + ∂ 2 ϕ ε ∂x 2 2 , div(Φ ε ) = ∂Φ ε 1 ∂x 1 + ∂Φ ε 2 ∂x 2 , rot(Φ ε ) = ∂Φ ε 2 ∂x 1 - ∂Φ ε 1 ∂x 2 , ∇ ⊥ ϕ ε = ∂ϕ ε ∂x 2 , - ∂ϕ ε ∂x 1 ,
where

Φ ε = (Φ ε 1 , Φ ε 2
) is a vector function and ϕ ε is a scalar function defined in Ω ε .

Moreover, for Φε = ( Φε 1 , Φε 2 ) a vector function and φε a scalar function defined in Ω ε , after a dilatation in the vertical variable, we will use the following operators

∆ ηε Φε = ∂ 2 Φε 1 ∂x 2 1 + 1 η 2 ε ∂ 2 Φε 1 ∂z 2 2 e 1 + ∂ 2 Φε 2 ∂x 2 1 + 1 η 2 ε ∂ 2 Φε 2 ∂z 2 2 e 2 , ∆ ηε φε = ∂ 2 φε ∂x 2 1 + 1 η 2 ε ∂ 2 φε ∂z 2 2 , div ηε ( Φε ) = ∂ Φε 1 ∂x 1 + 1 η ε ∂ Φε 2 ∂z 2 , rot ηε ( Φε ) = ∂ Φε 2 ∂x 1 - 1 η ε ∂ Φε 1 ∂z 2 , ∇ ⊥ ηε φε = 1 η ε ∂ φε ∂z 2 , - ∂ φε ∂x 1 .
For ϕ = (ϕ 1 , ϕ 2 ) and ψ = (ψ 1 , ψ 2 ), we define ⊗ by

(ϕ ⊗ψ) ij = ϕ i ψ j , i = 1, j = 1, 2 . (2.7)
Finally, we introduce some functional spaces. L q 0 is the space of functions of L q with zero mean value. Let C ∞ # (Z) be the space of infinitely differentiable functions in R 3 that are Z -periodic. By L q # (Z) (resp. W 1,q # (Z)), 1 < q < +∞, we denote its completion in the norm L q (Z) (resp. W 1,q (Z)) and by L q 0,# (Z) the space of functions in L q # (Z) with zero mean value.

The problem. The governing equations in dimensionless form are given by

                         1 P r ((u ε • ∇)u ε + ∇p ε ) = ∆u ε + N 1 -N (2∇ ⊥ w ε + ∆u ε ) + Ra T ε e 2 + f ε in Ω ε , div(u ε ) = 0 in Ω ε , M P r (u ε • ∇w ε ) = L∆w ε + 2N 1 -N (rot(u ε ) -2w ε ) + g ε in Ω ε , u ε • ∇T ε = ∆T ε + D∇ ⊥ w ε • ∇T ε in Ω ε .
(2.8)

We complete the above system with the following boundary conditions on the bottom

u ε = 0, w ε = 0, T ε = 0 on Γ 0 , (2.9)
the following conditions on the lateral boundaries

u ε • e 2 = 0, w ε = 0, T ε = 0, p ε = 1 η 2 ε q i on Σ ε i , i = {-1/2, 1/2}, (2.10)
and the following boundary conditions on the top boundary

u ε = 0, w ε = 0, ∇T ε • n = N us(G ε -T ε ) on Γ ε 1 . (2.11)
Here, u ε = (u ε 1 , u ε 2 ) represents the velocity field, p ε the pressure, w ε the microrotation and T ε the temperature. The external body forces are given by f ε = (f ε 1 , f ε 2 ) and the external body torque by g ε .

We make the following assumptions:

-The Robin boundary condition (2.11) 3 comes from Newton's cooling law and describes the heat exchange through the upper wall between the exterior medium and the fluid inside the channel. Due to domain's microstructure, it is assumed that the exterior temperature G ε = G(x 1 /ε) with G ∈ L 2 (Z ) a given Zperiodic and bounded function depending only on the longitudinal variable.

-Following previous result [START_REF] Lukaszewicz | Asymptotic analysis of the thermomicropolar fluid flow through a thin channel with cooling[END_REF], we consider that the Nusselt number N us depends on ε, whereas all the other characteristic numbers are kept independent of ε. In fact, we compare the Nusselt number N us to the small parameter of the height η ε . Namely, we consider the following scaling of the Nusselt number

N us = η ε k, k = O(1). (2.12)
-We assume that the external source functions are independent of the variable x 2 and take the following scaling

f ε = 1 η 2 ε (f 1 (x 1 ), 0), g ε = 1 η 2 ε g(x 1 ), with f 1 , g ∈ L 2 (ω). (2.13)
Under the previous assumptions, the well-posedness of the problem (2.8)-(2.11) can be established using the methods from [START_REF] Lukaszewicz | Micropolar fluids, theory and applications[END_REF] (see also [START_REF] Kalita | Rayleigh-Bénard problem for thermomicropolar fluids[END_REF]) and prove that there exists a unique weak solution (u ε , w

ε , p ε , T ε ) ∈ H 1 (Ω ε ) 2 × H 1 0 (Ω ε ) × L 2 0 (Ω ε ) × H 1 (Ω ε ).
Our aim is to study the asymptotic behavior of u ε , w ε , p ε and T ε when ε and η ε tend to zero and identify homogenized models coupling the effects of the thickness of the domain and the roughness of the boundary. For this, we use the dilatation in the variable x 2 given by

z 2 = x 2 η ε , ( 2
                         1 P r ((ũ ε • ∇ ηε )ũ ε + ∇ ηε pε ) = ∆ ηε ũε + N 1 -N (2∇ ⊥ ηε wε + ∆ ηε ũε ) + Ra T ε e 2 + f ε in Ω ε , div ηε (ũ ε ) = 0 in Ω ε , M P r (ũ ε • ∇ ηε wε ) = L∆ ηε wε + 2N 1 -N (rot ηε (ũ ε ) -2 wε ) + g ε in Ω ε , ũε • ∇ ηε T ε = ∆ ηε T ε + D∇ ⊥ ηε wε • ∇ ηε T ε in Ω ε , (2.15)
with the boundary conditions ũε = 0, wε = 0, T ε = 0 on Γ 0 , (2.16)

ũε • e 2 = 0, wε = 0, T ε = 0, pε = 1 η 2 ε q i on Σ ε i , i = {-1/2, 1/2}, (2.17
)

ũε = 0, wε = 0, ∇ ηε T ε • n = η ε k(G ε -T ε ) on Γ ε 1 , (2.18)
The unknown functions in the above system are given by ũε

(x 1 , z 2 ) = u ε (x 1 , η ε z 2 ), pε (x 1 , z 2 ) = p ε (x 1 , η ε z 2 ), wε (x 1 , z 2 ) = w ε (x 1 , η ε z 2 ) and T ε (x 1 , z 2 ) = T ε (x 1 , η ε z 2 ) for a.e. (x 1 , z 2 ) ∈ Ω ε .
Our goal then is to describe the asymptotic behavior of this new sequences ũε , wε , pε and T ε when ε and η ε tend to zero. To do this, we establish the a priori estimates and introduce the adaptation of the unfolding method in Section 3. We obtain the limit model of the critical case (η ε ≈ ε) in Section 4 and of the sub-critical case (η ε ε) in Section 5.

A priori estimates

This section is devoted to derive the a priori estimates of the unknowns and is divided in three parts. First, we deduce the a priori estimates for velocity, microrotation and temperature and second, we derive the estimates for pressure. Finally, we introduce the adaptation of the unfolding method and derive the a priori estimates of the unfolded functions.

Estimates for velocity, microroration and temperature

To derive the desired estimates, let us recall some well-known technical results (see, e.g. [START_REF] Lukaszewicz | Asymptotic analysis of the thermomicropolar fluid flow through a thin channel with cooling[END_REF]).

Lemma 3.1 (Poincaré and Ladyzhenskaya inequalities). For all ϕ ∈ H 1 (Ω ε ) such that ϕ = 0 on Γ 0 , there hold the following inequalities

ϕ L 2 (Ω ε ) ≤ Cη ε ∇ϕ L 2 (Ω ε ) 2 , ϕ L 4 (Ω ε ) ≤ Cη 1 2 ε ∇ϕ L 2 (Ω ε ) 2 . (3.19)
Moreover, from the change of variables (2.14), there hold the following rescaled estimates

φ L 2 ( Ω ε ) ≤ Cη ε ∇ ηε φ L 2 ( Ω ε ) 2 , φ L 4 ( Ω ε ) ≤ Cη 3 4 ε ∇ ηε φ L 2 ( Ω ε ) 2 . (3.20)
Lemma 3.2 (Trace estimates). For all ϕ ∈ H 1 (Ω ε ) such that ϕ = 0 on Γ 0 , there hold the following trace estimates:

ϕ L 2 (Γ ε 1 ) ≤ Cη 1 2 ε ∇ϕ L 2 (Ω ε ) 2 , (3.21)
Moreover, for every case, the rescaled function satisfies the following estimate

φ L 2 ( Γ ε 1 ) ≤ Cη ε ε -1 2 ∇ ηε φ L 2 ( Ω ε ) 2 , (3.22)
Proof. Since the upper boundary Γ ε 1 is not flat, one needs to take into account the variations of the normal direction n in order to estimate the L 2 -norm of the trace of a function ϕ ∈ H 1 (Ω ε ). Intregrating on vertical lines, we obtain

Γ ε 1 |ϕ| 2 dσ = ω |ϕ(x 1 , h ε (x 1 ))| 2 1 + η ε ε 2 h x 1 ε 2 dx 1 ≤ C 1 + η 2 ε ε -2 1 2 ω |ϕ(x 1 , h ε (x 1 ))| 2 dx 1 ≤ C 1 + η 2 ε ε -2 1 2 ω hε(x1) 0 ∂ x2 ϕ(x 1 , x 2 )dx 2 2 dx 1 ≤ C 1 + η 2 ε ε -2 1 2 η ε Ω ε |∂ x2 ϕ(x 1 , x 2 )| 2 dx 1 dx 2 . Then, since η ε ε or η ε ≈ ε, it holds Γ ε 1 |ϕ| 2 dσ ≤ Cη ε Ω ε |∂ x2 ϕ(x 1 , x 2 )| 2 dx 1 dx 2 ,
which implies (3.21).

For the rescaled function, proceeding analogously, we get

Γ ε 1 | φ| 2 dσ = ω | φ(x 1 , h(x 1 /ε))| 2 1 + 1 ε 2 h x 1 ε 2 dx 1 ≤ C 1 + ε -2 1 2 ω | φ(x 1 , h(x 1 /ε))| 2 dx 1 ≤ C 1 + ε -2 1 2 ω h(x1/ε) 0 ∂ x2 φ(x 1 , x 2 )dx 2 2 dx 1 ≤ C 1 + ε -2 1 2 η 2 ε Ω ε |η -1 ε ∂ x2 φ(x 1 , x 2 )| 2 dx 1 dx 2 .
Then, we have that

Γ ε 1 | φ| 2 dσ ≤ Cη 2 ε ε -1 Ω ε |η -1 ε ∂ x2 φ(x 1 , x 2 )| 2 dx 1 dx 2 ,
which implies (3.22).

Corollary 3.3. For G ε (x 1 ) = G(x 1 /ε) with G ∈ L 2 (Z )
and Z -periodic function, we have the following estimate

G ε L 2 (Γ ε 1 ) ≤ C . (3.23)
Proof. The proof follows the line of estimates (3.21) and (3.21), just taking into account that G ∈ L 2 (Z ).

Lemma 3.4 (A priori estimates). Let (u ε , w ε , T ε ) be the solution of the problem (2.8)- (2.11). Then there hold the following estimates

u ε L 2 (Ω ε ) 2 ≤ Cη 1 2 ε , ∇u ε L 2 (Ω ε ) 2×2 ≤ Cη -1 2 ε , (3.24)
w ε L 2 (Ω ε ) ≤ Cη 1 2 ε , ∇w ε L 2 (Ω ε ) 2 ≤ Cη -1 2 ε , (3.25)
T ε L 2 (Ω ε ) ≤ Cη 5 2 ε , ∇T ε L 2 (Ω ε ) 2 ≤ Cη 3 2 ε . (3.26)
Moreover, from the change of variables (2.14), there hold the following estimates for the rescaled unknowns

ũε L 2 ( Ω ε ) 2 ≤ C, ∇ ηε ũε L 2 ( Ω ε ) 2×2 ≤ Cη -1 ε , (3.27) wε L 2 ( Ω ε ) ≤ C, ∇ ηε wε L 2 ( Ω ε ) 2 ≤ Cη -1 ε , (3.28)
T ε L 2 ( Ω ε ) ≤ Cη 2 ε , ∇ ηε T ε L 2 ( Ω ε ) 2 ≤ Cη ε . (3.29)
Proof. We divide the proof in four steps.

Step 1. First, we multiply (2.8) 3 by w ε , integrate over Ω ε to obtain

L Ω ε |∇w ε | 2 dx + 4N 1 -N Ω ε |w ε | 2 dx = - M P r Ω ε (u ε • ∇w ε )w ε dx + 2N 1 -N Ω ε rot(u ε )w ε dx + 1 η 2 ε Ω ε g w ε dx. (3.30)
For the first term on the right-hand side, since div(u ε ) = 0 and w ε = 0 on ∂Ω ε , we get

Ω ε (u ε • ∇w ε )w ε dx = 1 2 Ω ε u ε • ∇|w ε | 2 dx = - 1 2 Ω ε |w ε | 2 div(u ε ) dx = 0. (3.31)
For the rest of the terms of the right-hand side, using the Cauchy-Schwarz inequality and the Poincaré inequality (3.19), we get

Ω ε rot(u ε )w ε dx ≤ ∇u ε L 2 (Ω ε ) 2×2 w ε L 2 (Ω ε ) ≤ Cη ε ∇u ε L 2 (Ω ε ) 2×2 ∇w ε L 2 (Ω ε ) 2 , 1 η 2 ε Ω ε g w ε dx 1 dx 2 ≤ η -2 ε g L 2 (Ω ε ) w ε L 2 (Ω ε ) ≤ Cη -1 2 ε ∇w ε L 2 (Ω ε ) 2 .
(3.32)

Then, taking into account (3.31) and (3.32) in (3.30), we get

∇w ε L 2 (Ω ε ) ≤ C η ε ∇u ε L 2 (Ω ε ) + η -1 2 ε . (3.33)
Step 2. We multiply (2.8) 4 by T ε , integrate over Ω ε to obtain

Ω ε |∇T ε | 2 dx + η ε k Γ ε 1 |T ε | 2 dσ = - Ω ε (u ε • ∇)T ε T ε dx + D Ω ε (∇ ⊥ w ε • ∇)T ε T ε dx + η ε k Γ ε 1 G ε T ε dσ. (3.34)
For the first term on the right-hand side of (3.34), since

u ε = 0 on ∂Ω ε , div(u ε ) = 0 in Ω ε and T ε = 0 on Σ ε i , i = -1/2, 1/2, we get Ω ε (u ε • ∇)T ε T ε dx = 1 2 Ω ε u ε • ∇|T ε | 2 dx = - 1 2 Ω ε |T ε | 2 divu ε dx = 0. (3.35)
For the second term on the right-hand side of (3.34), we have

Ω ε ∇ ⊥ w ε • ∇T ε T ε dx = 1 2 Ω ε ∇ ⊥ w ε • ∇(T ε ) 2 dx = - 1 2 Ω ε ∇w ε × (∇(T ε ) 2 ) dx = 1 2 Ω ε w ε rot(∇(T ε ) 2 )dx - 1 2 Ω ε rot(w ε ∇(T ε ) 2 )dx = - 1 2 Ω ε n × (w ε ∇(T ε ) 2 )dx = - 1 2 Ω ε w ε ∂T ε ∂x 2 T ε n 1 - ∂T ε ∂x 1 T ε n 2 dx = 0, (3.36)
where we have used that w ε = 0 on ∂Ω ε , the identity

rot(∇(T ε ) 2 ) = rot 2 ∂T ε ∂x 1 T ε , 2 ∂T ε ∂x 2 T ε = 2 ∂ 2 T ε ∂x 2 ∂x 1 T ε + 2 ∂T ε ∂x 2 ∂T ε ∂x 1 -2 ∂ 2 T ε ∂x 1 ∂x 2 T ε -2 ∂T ε ∂x 1 ∂T ε ∂x 2 = 0.
and

Ω ε ∂w ε ∂x 2 T ε dx 1 dx 2 = - Ω ε w ε ∂T ε ∂x 2 dx 1 dx 2 .
It remains to estimate the third term of the right-hand side of (3.34). To do this, from Caychy-Schwarz's inequality, Lemma 3.2 applied to T ε and Corollary 3.3, we get

η ε k Γ ε 1 G ε T ε dσ ≤ η ε G ε L 2 (Γ ε 1 ) T ε L 2 (Γ ε 1 ) ≤ Cη 3 2 ε ∇T ε L 2 (Ω ε ) 2 . (3.37)
Then, taking into account (3.35) -(3.37) in (3.34), we have

∇T ε L 2 (Ω ε ) ≤ Cη 3 2 ε , (3.38)
which is the second estimate in (3.26). From the Poincaré inequality (3.19), we get the first estimate in (3.26).

Step 3. We multiply (2.8) 1 by u ε , integrate over Ω ε to obtain

1 1 -N Ωε |∇u ε | 2 dx = - 1 P r Ω ε (u ε • ∇)u ε u ε dx + 2N 1 -N Ω ε ∇ ⊥ w ε • u ε dx +Ra Ω ε T ε (e 2 • u ε ) dx + 1 η 2 ε Ω ε f 1 (e 1 • u ε ) dx, + 1 P r 1 η 2 ε q -1/2 Σ ε -1/2 ϕ • e 1 dx 2 - 1 P r 1 η 2 ε q 1/2 Σ ε 1/2 ϕ • e 1 dx 2 .
(3.39)

The first term on the right-han side of (3.39) satisfies

Ω ε (u ε • ∇)u ε u ε dx 1 dx 2 = 1 2 Ω ε u ε • ∇|u ε | 2 dx 1 dx 2 = - 1 2 Ω ε |u ε | 2 div u ε dx 1 dx 2 = 0. (3.40)
We estimate the rest of the terms on the right-hand side of (3.39) by using the Poincaré inequality (3.19) and using (3.33), we get

Ω ε ∇ ⊥ w ε u ε dx ≤ ∇w ε L 2 (Ω ε ) u ε L 2 (Ω ε ) ≤ Cη ε ∇w ε L 2 (Ω ε ) 2 ∇u ε L 2 (Ω ε ) 2×2 ≤ Cη 2 ε ∇u ε 2 L 2 (Ω ε ) 2×2 + η 1 2 ε ∇u ε L 2 (Ω ε ) 2×2 , Ω ε T ε (e 2 • u ε ) dx ≤ T ε L 2 (Ω ε ) u ε L 2 (Ω ε ) 2 ≤ Cη ε T ε L 2 (Ω ε ) ∇u ε L 2 (Ω ε ) 2×2 ≤ Cη 7 2 ε ∇u ε L 2 (Ω ε ) 2×2 , η -2 ε Ω ε f 1 u ε dx ≤ η -2 ε f 1 L 2 (Ω ε ) u ε L 2 (Ω ε ) 2 ≤ Cη -1 2 ε ∇u ε L 2 (Ω ε ) 2×2 , (3.41)
and

1 η 2 ε 1 P r q -1/2 Σ ε -1/2 u ε • e 1 dx 2 -q 1/2 Σ ε 1/2 u ε • e 1 dx 2 = |div((q -1/2 + (q 1/2 -q -1/2 )(x + 1/2))u ε | ≤ Cη -2 ε 1 L 2 (Ω ε ) u ε L 2 (Ω ε ) 2 ≤ Cε -1 2 ∇u ε L 2 (Ω ε ) 2×2 . ( 3
∇u ε L 2 (Ω ε ) 2×2 ≤ Cη -1 2 ε ,
which is the second estimate (3.24). By using the Poincaré inequality, we get the first estimate in (3.24). For the microrotation, from (3.33), we get the estimates of the microrotation (3.25).

Step 4. Finally, the estimates of the rescaled unknown are obtained by applying to estimates of the unknowns the change of variables (2.14).

The extension of (ũ ε , wε , T ε ) to the whole domain. The sequence of solutions (ũ ε , wε , T ε ) is defined in a varying set Ω ε , but not defined in a fixed domain independent of ε. In order to pass to the limit if ε tends to zero, convergences in fixed Sobolev spaces (defined in Ω) are used, which requires first that (ũ ε , wε , T ε ) be extended to the whole domain Ω. We extend each unknown in the following:

-From the boundary conditions satisfied by ũε and wε , we extend them by zero in Ω \ Ω ε and denote the extensions by Ũε and W ε , respectively.

-For the temperature T ε , we use the extension operator described in [23, Lemma 2.3] called P ε which allows us to extend functions from H 1 ( Ω ε ), which are zero on the lateral boundaries, to H 1 (Ω). Moreover, this extension satisfies

P ε ( φ) L 2 (Ω) ≤ C φ L 2 ( Ω ε ) , ∂ x1 P ε ( φ) L 2 (Ω) ≤ C ∂ x1 φ L 2 ( Ω ε ) + 1 ε ∂ z2 φ L 2 ( Ω ε ) , ∂ z2 P ε ( φ) L 2 (Ω) ≤ C ∂ z2 φ L 2 ( Ω ε ) , (3.43)
for every function φ ∈ H 1 ( Ω ε ). Thus, we denote by θε the extension of T ε , i.e. θε = P ε ( T ε ).

We have the following result.

Lemma 3.5 (Estimates of extended functions). The extended functions ( Ũ ε , W ε , θε ) of (ũ ε , wε , T ε ) satisfy the following estimates

Ũε L 2 (Ω) 2 ≤ C, ∇ ηε Ũε L 2 (Ω) 2×2 ≤ Cη -1 ε , (3.44) W ε L 2 (Ω) ≤ C, ∇ ηε W ε L 2 (Ω) 2 ≤ Cη -1 ε , (3.45) θε L 2 (Ω) ≤ Cη 2 ε , ∇ ηε θε L 2 (Ω) 2 ≤ Cη ε . ( 3

Estimates for pressure

Let us first give a more accurate estimate for pressure p ε . For this, we need to recall a version of the decomposition result for p ε whose proof can be found in [START_REF] Casado-Díaz | A decomposition result for the pressure of a fluid in a thin domain and extensions to elasticity problems[END_REF]Corollary 3.4] (see also [START_REF] Bonnivard | A generalized Reynolds equation for micropolar flows past a ribbed surface with nonzero boundary conditions[END_REF][START_REF] Casado-Díaz | Asymptotic behavior of the Navier-Stokes system in a thin domain with Navier condition on a slightly rough boundary[END_REF]).

Proposition 3.6. The following decomposition for p ε ∈ L 2 0 (Ω ε ) holds

p ε = p ε 0 + p ε 1 , (3.47)
where p ε 0 ∈ H 1 (ω), which is independent of x 2 , and p ε 1 ∈ L 2 (Ω ε ). Moreover, the following estimates hold

η 3 2 ε p ε 0 H 1 (ω) + p ε 1 L 2 (Ω ε ) ≤ C ∇p ε H -1 (Ω ε ) 2 ,
that is

p ε 0 H 1 (ω) ≤ Cη -3 2 ε ∇p ε H -1 (Ω ε ) 2 , p ε 1 L 2 (Ω ε ) ≤ C ∇p ε H -1 (Ω ε ) 2 . (3.48)
We denote by pε 1 the rescaled function associated with p ε 1 defined by pε

1 (x 1 , z 2 ) = (x 1 , η ε z 2 ) for a.e. (x 1 , z 2 ) ∈ Ω ε .
As consequence, we have the following result.

Corollary 3.7. The pressures p ε 0 , p ε 1 and pε 1 satisfy the following estimates

p ε 0 H 1 (ω) ≤ Cη -2 ε , p ε 1 L 2 (Ω ε ) ≤ Cη -1 2 ε , pε 1 L 2 ( Ω ε ) ≤ Cη -1 ε . (3.49)
Proof. Thank to (3.48), we just need to obtain the estimate for ∇p ε given by

∇p ε H -1 (Ω ε ) 2 ≤ Cη -1 2 ε , (3.50)
to derive (3.49). To do this, we consider ϕ ∈ H 1 0 (Ω ε ), and taking into account the variational formulation (3.62), we get

∇p ε , ϕ = - P r 1 -N Ωε ∇u ε : ∇ϕ dx - Ω ε (u ε • ∇)u ε ϕ dx + 2N P r 1 -N Ω ε ∇ ⊥ w ε • ϕ dx + P r Ra Ω ε T ε (e 2 • ϕ) dx + P r η 2 ε Ω ε f 1 (e 1 • ϕ) dx. (3.51)
Estimating the terms on the right-hand side of (3.51) using Lemmas 3.1 and 3.4, we get

P r 1 -N Ωε ∇u ε : ∇ϕ dx ≤ C ∇u ε L 2 (Ω ε ) 2×2 ∇ϕ L 2 (Ω ε ) 2×2 ≤ Cη -1 2 ε ϕ H 1 0 (Ω ε ) 2 , Ω ε (u ε • ∇)u ε ϕ dx ≤ u ε L 4 (Ω ε ) 2 ∇u ε L 2 (Ω ε ) 2×2 ϕ L 4 (Ω ε ) 2 ≤ Cη ε ∇u ε 2 L 2 (Ω ε ) 2×2 ∇ϕ L 2 (Ω ε ) 2×2 ≤ ϕ H 1 0 (Ω ε ) 2 , 2N P r 1 -N Ω ε ∇ ⊥ w ε ϕ dx ≤ C ∇w ε L 2 (Ω ε ) 2 ϕ L 2 (Ω ε ) 2 ≤ η 1 2 ε ϕ H 1 0 (Ω ε ) 2 , P r Ra Ω ε T ε (e 2 • ϕ) dx ≤ C T ε L 2 (Ω ε ) ϕ L 2 (Ω ε ) 2 ≤ Cη 7 2 ε ϕ H 1 0 (Ω ε ) 2 , P r η 2 ε Ω ε f 1 (e 1 • ϕ) dx ≤ Cη -2 ε f 1 L 2 (Ω ε ) ϕ L 2 (Ω ε ) 2 ≤ Cη -1 2 ε ϕ H 1 0 (Ω ε ) 2 ,
which together with (3.51) gives

∇p ε , ϕ ≤ Cη -1 2 ε ϕ H 1 0 (Ω ε ) 2 , ∀ ϕ ∈ H 1 0 (Ω ε ) 2 .
This gives the desired estimate (3.50), which finishes the proof.

Adaptation of the unfolding method

The change of variables (2.14) does not provide the information we need about the behavior of rescaled unknown in the microstructure associated to Ω ε . To solve this difficulty, we use an adaptation of the unfolding method (see [START_REF] Cioranescu | Periodic unfolding and homogenization[END_REF][START_REF] Cioranescu | The periodic unfolding method in homogenization[END_REF] for more details) introduced to this context in [START_REF] Anguiano | Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary[END_REF] (see also [START_REF] Pažanin | Homogenization of the Darcy-Lapwood-Brinkman flow through a thin domain with highly oscillating boundaries[END_REF][START_REF] Suárez-Grau | Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary[END_REF][START_REF] Suárez-Grau | Effective boundary condition for a quasi-newtonian fluid at a slightly rough boundary starting from a Navier condition[END_REF][START_REF] Suárez-Grau | Analysis of the roughness regime for micropolar fluids via homogenization[END_REF]).

Let us recall that this adaptation of the unfolding method divides the domain Ω ε in cubes of lateral length ε and vertical length h(z 1 ). Thus, given the unknowns ũε , wε , T ε , p ε 0 and pε 1 , we define

ûε (x 1 , z) = ũε εκ x 1 ε + εz 1 , z 2 a.e. (x 1 , z) ∈ ω × Z, (3.52) ŵε (x 1 , z) = wε εκ x 1 ε + εz 1 , z 2 a.e. (x, z) ∈ ω × Z, (3.53)
T ε (x 1 , z) = T ε εκ x 1 ε + εz 1 , z 2 a.e. (x 1 , z) ∈ ω × Z, (3.54)
pε 0 (x 1 , z 1 ) = p ε 0 εκ x 1 ε + εz 1 a.e. (x 1 , z 1 ) ∈ ω × Z . (3.55) pε 1 (x 1 , z) = pε 1 εκ x 1 ε + εz 1 , z 2 a.e. (x 1 , z) ∈ ω × Z. (3.56)
where the function κ is defined by (2.6).

Remark 3.8. For k ∈ T ε , the restriction of (û ε , ŵε , T ε , pε 1 ) to Z k ,ε × Z does not depend on x 1 , while as a function of z it is obtained from (ũ ε , wε , θε , pε 1 ) by using the change of variables

z 1 = x 1 -εk ε , which transform Z k ,ε into Z.
Analogously, the restriction of pε 0 to Z k ,ε × Z does not depend on x 1 , while as a function of z 1 it is obtained from p ε 0 by using the previous change of variables.

We are now in position to obtain estimates for the unfolded unknowns (û ε , ŵε , T ε , pε 0 , pε 1 ).

Lemma 3.9. There exists a constant C > 0 independent of ε, such that ûε , ŵε , T ε , pε 0 and pε 1 defined by (3.52)-(3.56) respectively satisfy

ûε L 2 (ω×Z) 2 ≤ C, ∂ z1 ûε L 2 (ω×Z) 2 ≤ Cεη -1 ε , ∂ z2 ûε L 2 (ω×Z) 2 ≤ C, (3.57) ŵε L 2 (ω×Z) ≤ C, ∂ z1 ŵε L 2 (ω×Z) ≤ Cεη -1 ε , ∂ z2 ŵε L 2 (ω×Z) ≤ C, (3.58)
T ε L 2 (ω×Z) ≤ Cη 2 ε , ∂ z1 T ε L 2 (ω×Z) ≤ Cεη ε , ∂ z2 T ε L 2 (ω×Z) ≤ Cη 2 ε , (3.59) pε 0 L 2 (ω×Z ) ≤ Cη -2 ε , ∂ z1 pε 0 L 2 (ω×Z ) ≤ Cεη -2 ε , pε 1 L 2 (ω×Z) ≤ Cη -1 ε . (3.60)
Proof. From the proof of Lemma 4.9 in [START_REF] Anguiano | Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary[END_REF] in the case p = 2, we have the following properties concerning the estimates of a function φε and its respective unfolding function φε : Weak variational formulation. We give the equivalent weak variational formulation of system (2.8)-(2.11) and the rescaled system (2.15)-(2.18), which will be useful in next sections in order to obtain the limit system taking into account the effects of the rough boundary.

φε L 2 (ω×Z) 2 = φε L 2 ( Ω ε ) 2 , ∂ z1 φε L 2 (ω×Z) 2×1 = ε ∂ x1 φε L 2 ( Ω ε ) 2 , ∂ z2 φε L 2 (ω×Z) 2 = ∂ z2 φε L 2 ( Ω ε ) 2 . ( 3
Taking into account the decomposition of the pressure and

∇p ε , ϕ = Ω ε ∂ x1 p ε 0 (x 1 ) ϕ 1 dx 1 dz 2 - Ω ε pε 1 div(ϕ) dx 1 dz 2 , ∀ ϕ ∈ H 1 0 (Ω ε ) 2 ,
then, the weak variational formulation for system (2.8)-(2.11) is the following

1 1 -N Ωε ∇u ε • ∇ϕ dx + 1 P r Ω ε ∂ x1 p ε 0 (x 1 ) ϕ 1 dx 1 dx 2 - 1 P r Ω ε p ε 1 div(ϕ) dx = - 1 P r Ω ε (u ε • ∇)u ε ϕ dx + 2N 1 -N Ω ε ∇ ⊥ w ε • ϕ dx + Ra Ω ε T ε (e 2 • ϕ) dx + 1 η 2 ε Ω ε f 1 (e 1 • ϕ) dx (3.62) L Ω ε ∇w ε • ∇ψ dx + 4N 1 -N Ω ε w ε ψ dx = - M P r Ω ε (u ε • ∇w ε )ψ dx + 2N 1 -N Ω ε rot(u ε )ψ dx + 1 η 2 ε Ω ε g ψ dx, (3.63
)

Ω ε ∇T ε • ∇φ dx + η ε k Γ ε 1 T ε φ dσ = - Ω ε (u ε • ∇)T ε φ dx + D Ω ε ∇ ⊥ w ε • ∇T ε φ dx + η ε k Γ ε 1 G ε φ dσ . (3.64) for every ϕ ∈ H 1 0 (Ω ε ) 2 , ψ ∈ H 1 0 (Ω ε ) and φ ∈ H 1 (Ω ε ) such that φ = 0 on ∂Ω ε \ Γ ε 1 .
The equivalent weak variational formulation for the rescaled system (2.15)-(2.18) reads as follows

1 1 -N Ωε ∇ ηε ũε • ∇ ηε φ dx 1 dz 2 + 1 P r Ω ε ∂ x1 p ε 0 (x 1 ) φ1 dx 1 dz 2 - 1 P r Ω ε pε 1 div ηε φ dx 1 dz 2 = - 1 P r Ω ε (ũ ε • ∇ ηε )ũ ε φ dx 1 dz 2 + 2N 1 -N Ω ε ∇ ⊥ ηε wε • φ dx 1 dz 2 +Ra Ω ε T ε (e 2 • φ) dx 1 dz 2 + 1 η 2 ε Ω ε f 1 (e 1 • φ) dx 1 dz 2 , (3.65)
L

Ω ε ∇ ηε wε • ∇ ηε ψ dx 1 dz 2 + 4N 1 -N Ω ε wε ψ dx 1 dz 2 = - M P r Ω ε (ũ ε • ∇ ηε wε ) ψ dx 1 dz 2 + 2N 1 -N Ω ε rot ηε (ũ ε ) ψ dxdz 2 + 1 η 2 ε Ω ε g ψ dx 1 dz 2 , (3.66
)

Ω ε ∇ ηε T ε • ∇ ηε φ dx 1 dz 2 + k Γ ε 1 T ε φ dσ = - Ω ε (ũ ε • ∇ ηε ) T ε φ dx 1 dz 2 + D Ω ε ∇ ⊥ ηε wε • ∇ ηε T ε φ dx 1 dz 2 + k Γ ε 1 G ε φ dσ. (3.67) for every φ ∈ H 1 0 ( Ω ε ) 2 , ψ ∈ H 1 0 ( Ω ε ) and φ ∈ H 1 ( Ω ε ) such that φ = 0 on ∂ Ω ε \ Γ ε 1 , and φ(x 1 , z 2 ) = ϕ(x 1 , η ε z 2 ), ψ(x 1 , z 2 ) = ψ(x 1 , η ε z 2 ) and φ(x 1 , z 2 ) = φ(x 1 , η ε z 2 ) for a.e. (x 1 , z 2 ) ∈ Ω ε .
Next, according previous estimates of the unfolding functions, we consider as test functions in (3.65)-(3.67) the following ones

ϕ ε (x 1 , z 2 ) = η 2 ε ϕ(x 1 , x 1 /ε, z 2 ) with ϕ(x 1 , z) ∈ D(ω; C ∞ # (Z) 2 ), ψ ε (x 1 , z 2 ) = η 2 ε ψ(x 1 , x 1 /ε, z 2 ) with ψ(x 1 , z) ∈ D(ω; C ∞ # (Z)), φ ε (x 1 , z 2 ) = φ(x 1 , x 1 /ε, z 2 ) with φ(x 1 , z) ∈ D(ω; C ∞ # (Z)
). Taking into account this, the formulation (3.65) reads

1 1 -N Ωε η 2 ε ∇ ηε ũε • ∇ ηε ϕ ε dx 1 dz 2 + 1 P r Ω ε η 2 ε ∂ x1 p ε 0 (x 1 ) ϕ ε 1 dx 1 dz 2 - 1 P r Ω ε η 2 ε pε 1 div ηε ϕ ε dx 1 dz 2 = - 1 P r Ω ε η 2 ε (ũ ε • ∇ ηε )ũ ε ϕ ε dx 1 dz 2 + 2N 1 -N Ω ε η 2 ε ∇ ⊥ ηε wε ϕ ε dx 1 dz 2 +Ra Ω ε η 2 ε T ε (e 2 • ϕ ε ) dx 1 dz 2 + Ω ε f 1 (e 1 • ϕ ε ) dx 1 dz 2 , (3.68)
the formulation (3.66) reads

L Ω ε η 2 ε ∇ ηε wε • ∇ ηε ψ ε dx 1 dz 2 + 4N 1 -N Ω ε η 2 ε wε ψ ε dx 1 dz 2 = - M P r Ω ε η 2 ε (ũ ε • ∇ ηε wε )ψ ε dx 1 dz 2 + 2N 1 -N Ω ε η 2 ε rot ηε (ũ ε )ψ ε dx 1 dz 2 + Ω ε g ψ ε dx 1 dz 2 , (3.69)
and the formulation (3.67) reads reads

Ω ε ∇ ηε T ε • ∇ ηε φ ε dx 1 dz 2 + k Γ ε 1 T ε φ ε dσ = - Ω ε (ũ ε • ∇ ηε ) T ε φ ε dx 1 dz 2 + D Ω ε ∇ ⊥ ηε wε • ∇ ηε T ε φ ε dx 1 dz 2 + k Γ ε 1 G ε φ ε dσ. (3.70)
By the unfolding change of variables (see [START_REF] Anguiano | Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary[END_REF][START_REF] Suárez-Grau | Analysis of the roughness regime for micropolar fluids via homogenization[END_REF] for more details), we get

1 P r η 2 ε Ω ε (ũ ε • ∇ ηε )ũ ε ϕ ε dx 1 dz 2 = - η 2 ε P r Ω ε ũε ⊗ũ ε ∂ x1 ϕ dx 1 dz 2 + η ε P r Ω ε ∂ z2 ũε 2 ũε • ϕ dx 1 dz 2 + Ω ε ũε 2 ∂ z2 ũε • ϕ dx 1 dz 2 = - η 2 ε ε -1 P r ω×Z ûε ⊗û ε • ∂ z1 ϕ dx 1 dz + η ε P r ω×Z ∂ z2 ûε 2 ûε • ϕ dx 1 dz + ω×Z ûε 2 ∂ z2 ûε • ϕ dx 1 dz + O ε ,
where the operation ⊗ is defined by (2.7) and O ε tends to zero. With this and by applying the unfolding change of variables to the rest of the terms in (3.68), we get

1 1 -N ω×Z η 2 ε ε -2 ∂ z1 ûε • ∂ z1 ϕ dx 1 dz + 1 1 -N ω×Z ∂ z2 ûε • ∂ z2 ϕ dx 1 dz + 1 P r ω×Z η 2 ε ε -1 ∂ z1 pε 0 ϕ 1 dx 1 dz - 1 P r ω×Z η 2 ε ε -1 pε 1 ∂ z1 ϕ 1 dx 1 dz - 1 P r ω×Z η ε pε 1 ∂ z2 ϕ 2 dx 1 dz = η 2 ε ε -1 P r ω×Z ûε ⊗û ε • ∂ z1 ϕ dx 1 dz - η ε P r ω×Z ∂ z2 ûε 2 ûε • ϕ dx 1 dz + ω×Z η 2 ε ûε 2 ∂ z2 ûε ϕ dx 1 dz + 2N 1 -N ω×Z ∂ z2 ŵε ϕ 1 dx 1 dz - 2N 1 -N ω×Z η 2 ε ε -1 ∂ z1 ŵε ϕ 2 dx 1 dz +Ra ω×Z η 2 ε T ε (e 2 • ϕ) dx 1 dz + ω×Z f 1 (e 1 • ϕ) dx 1 dz + O ε , (3.71)
with O ε devoted to tend to zero.

Analogously, applying the unfolding change of variables to the equation (3.69), we get

L ω×Z η 2 ε ε -2 ∂ z1 ŵε ∂ z1 ψ dx 1 dz + L ω×Z ∂ z2 ŵε ∂ z2 ψ dx 1 dz + 4N 1 -N ω×Z η 2 ε ŵε ψ dx 1 dz = - M P r ω×Z η 2 ε ε -1 ûε 1 ∂ z1 ŵε ψ dx 1 dz - M P r ω×Z η ε ûε 2 ∂ z2 ŵε ψ dx 1 dz + 2N 1 -N ω×Z η 2 ε ε -1 ∂ z1 ûε 2 ψ dx 1 dz - 2N 1 -N ω×Z η ε ∂ z2 ûε 1 ψ dx 1 dz + ω×Z g ψ dx 1 dz + O ε , (3.72)
with O ε devoted to tend to zero.

Finally, from (3.70), we deduce

ε -2 ω×Z ∂ z1 T ε ∂ z1 φ dx 1 dz + η -2 ε ω×Z ∂ z2 T ε ∂ z2 φ dx 1 dz = -η ε ω×Z ûε • ∇ ηε,ε T ε φ dx 1 dz +D ω×Z ∇ ⊥ ηε,ε ŵε • ∇ ηε,ε (η -2 ε T ε )φ dx 1 dz + k ω× Γ1 G φ dx 1 dσ + O ε . (3.73)
where we use the operators

∇ ηε,ε = (η ε ε -1 ∂ z1 , ∂ z2 ) and ∇ ⊥ ηε,ε = (∂ z2 , -η ε ε -1 ∂ z1 ) and O ε is devoted to tend to zero.
Here, we have used (3.22) and (3.29), which gives

k Γ ε 1 T ε φ ε dσ ≤ C T ε L 2 ( Γε 1 ) ≤ Cη ε ε -1 2 ∇ ηε T ε L 2 ( Ω ε ) 2 ≤ Cη 2 ε ε -1 2 → 0,
and by the unfolding change of variables with respect to x 1 , the periodicity of h(z 1 ) and G(z 1 ), it holds

k Γ ε 1 G ε φ ε dσ = k Γ ε 1 G(x 1 /ε)φ ε dσ = ω× Γ1 G(z 1 )φ dx 1 dσ + O ε .

Homogenized model in the critical case

It corresponds to the critical case when the thickness of the domain is proportional to the wavelength of the roughness, with λ the proportionality constant, that is

η ε ≈ ε, with η ε /ε → λ, 0 < λ < +∞.
Let us introduce some notation which will be useful along this section. For a vectorial function v = (v 1 , v 2 ) and a scalar w, we introduce the operators ∇ λ , ∆ λ , div λ and by

(∇ λ v) i,1 = λ∂ z1 v i , (∇ λ v) i,2 = ∂ z2 v i for i = 1, 2, ∆ λ v = λ 2 ∂ 2 z1 v + ∂ 2 z2 v, ∇ λ w = (λ∂ z1 w, ∂ z2 w) t , div λ v = λ∂ z1 v 1 + ∂ z2 v 2 , ∇ ⊥ λ w ε = (∂ z2 w, -λ∂ z1 w) t
Next, we give some compactness results about the behavior of the sequences ( Ũε , W ε , θε , p ε 0 , pε 1 ) and the related unfolding functions (û ε , ŵε , T ε , pε 0 , pε 1 ) satisfying the a priori estimates given in Lemma 3.5, Corollary 3.7 and Lemma 3.9 respectively. Lemma 4.1. For a subsequence of ε still denote by ε, we have the following convergence results:

(i) (Velocity) There exist Ũ = ( Ũ1 , Ũ2 ) ∈ H 1 (0, h max ; L 2 (ω) 2 ), with Ũ = 0 on z 2 = {0, h max } and Ũ2 = 0, such that Ũε Ũ in H 1 (0, h max ; L 2 (ω) 2 ), (4.74)
∂ x1 hmax 0 Ũ1 (x 1 , z 2 ) dz 2 = 0 in ω, (4.75)
and û = (û 1 , û2 ) ∈ L 2 (ω; H 1 # (Z)) 2 , with û = 0 on z 2 = {0, h(z 1 )} such that it hold Z û(x 1 , z)dz = hmax 0 Ũ(x 1 , z 2 ) dz 2 with Z û2 (x 1 , z) dz = 0, and moreover

ûε û in L 2 (ω; H 1 (Z) 2 ), (4.76)
div λ û = 0 in ω × Z, (4.77)
∂ x1 Z û1 (x 1 , z) dz = 0 in ω . (4.78) (ii) (Microrotation) There exist W ∈ H 1 (0, h max ; L 2 (ω)), with W = 0 on z 2 = {0, h max }, such that W ε W in H 1 (0, h max ; L 2 (ω)), (4.79
)

and ŵ ∈ L 2 (ω; H 1 # (Z)), with ŵ = 0 on z 2 = {0, h(z 1 )} such that Z ŵ(x 1 , z)dz = hmax 0 W (x 1 , z 2 ) dz 2 ,

and moreover

ŵε ŵ in L 2 (ω; H 1 (Z)). (4.80) (iii) (Temperature) There exist θ ∈ H 1 (0, h max ; L 2 (ω)), with θ = 0 on z 2 = {0}, such that η -2 ε θε θ in H 1 (0, h max ; L 2 (ω)), (4.81)
and T ∈ L 2 (ω; H 1 # (Z)), with T = 0 on z 2 = {0}, such that Z T (x 1 , z)dz = hmax 0 θ(x 1 , z 2 ) dz 2 , and moreover

η -2 ε T ε T in L 2 (ω; H 1 (Z)). (4.82) (iv) (Pressure) There exist three functions p ∈ L 2 0 (ω) ∩ H 1 (ω), independent of z 2 with p(i) = q i , i = -1/2, 1/2, p0 ∈ L 2 (ω; H 1 # (Z )) and p1 ∈ L 2 (ω; L 2 # (Z)) such that η 2 ε p ε 0 p in H 1 (ω), (4.83) η 2 ε ε -1 ∂ z1 pε 0 ∂ z1 p + ∂ z1 p0 in L 2 (ω; L 2 (Z )), η ε pε 1 p1 in L 2 (ω; L 2 (Z)). (4.84)
Proof. We will only give some remarks and, for more details, we refer the reader to Lemmas 5.2-i) and 5.4-i) in [START_REF] Anguiano | Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary[END_REF].

We start with the extension Ũ ε . Estimates (3.44) imply the existence of Ũ ∈ H 1 (0, h max ; L 2 (ω) 2 ) such that convergence (4.74) holds, and the continuity of the trace applications from the space of Ũ such that Ũ L 2 and ∂ z2 Ũ L 2 are bounded to L 2 (Γ 1 ) and to L 2 (Γ 0 ) implies Ũ = 0 on Γ 1 and Γ 0 . Next, from the free divergence condition div ηε ( Ũε ) = 0, it can be deduced that Ũ2 is independent of z 2 , which together with the boundary conditions satisfied by Ũ2 on z 2 = {0, h max } implies that Ũ2 = 0. Finally, from the free divergence condition and the convergence (4.74) of Ũε , it is straightforward the corresponding free divergence condition in a thin domain given in (4.75).

Concerning ûε , estimates given in (3.57) imply the existence of û ∈ L 2 (ω; H 1 (Z) 3 ) such that convergence (4.76) holds. It can be proved the Z -periodicity of û, and applying the unfolding change of variables to the free divergence condition div ηε ũε = 0, passing to the limit, we get divergence condition (4.77). Finally, it can be proved that Z û(x 1 , z) dz = hmax 0 Ũ(x 1 , z 2 ) dz 2 which together with Ũ2 = 0 implies hmax 0 Ũ2 (x 1 , z 2 ) dz 2 = 0, and together with property (4.75) implies the divergence condition given in (4.78).

We continue proving (ii). From estimates (3.45), convergence (4.79) and that W = 0 on z 2 = {0, h max } are obtained straighfordward. The proofs of the convergence of W ε and identity Z ŵ dz = hmax 0 W dz 2 are similar to the ones of ûε .

We continue with (iii). The proof is similar to (ii), but we have to take into account estimates (3.46) and that the dirichlet boundary condition for temperature is imposed on the bottom and not on the top. We finish the proof with (iv). From estimates of p ε 0 and pε 0 , and the classical compactness result for the unfolding method for a bounded sequence in H 1 , we get convergences for (4.83) and (4.84) 1 . Estimate for pε 1 implies convergence (4.84) 2 . From the boundary conditions of pε on x 1 = {-1/2, 1/2}, the decomposition of the pressure and the convergences of pε 0 and pε 1 , it holds the boundary conditions for p. Finally, since pε has mean value zero, from the decomposition of the pressure, we have

0 = Ω ε η 2 ε pε dx 1 dz 2 = ω h(x 1 /ε)η 2 ε p ε 0 dx 1 + Ω ε η 2 ε pε 1 dx 1 dz 2 .
Taking into account that h is x 1 -periodic, the convergence of η 2 ε p ε 0 to p and that

Ω ε η 2 ε pε 1 dx 1 dz 2 ≤ Cη ε → 0, we get Z h dz 1 ω p dx 1 = 0,
and so that p has null mean value in ω.

Using previous convergences, in the following theorem we give the two-pressured homogenized system satisfied by (û, ŵ, P , T ).

Theorem 4.2 (Limit unfolded problems). In the case η ε ≈ ε, with η ε /ε → λ, 0 < λ < +∞, then the functions û, ŵ, T and p given in Lemma 4.1 satisfy

• (û, p) ∈ L 2 (ω; H 1 # (Z) 2 ) × (L 2 0 (ω) ∩ H 1 (ω)) is the unique solution of the two-pressure homogenized Stokes problem                                          - 1 1 -N ∆ λ û + 1 P r ∇ λ q = f 1 (x 1 ) - 1 P r ∂ x1 p(x 1 ) e 1 in ω × Z, div λ û = 0 in ω × Z, û = 0 on ω × ( Γ0 ∪ Γ1 ), ∂ x1 Z û1 (x 1 , z) dz = 0 in ω, Z û2 dz = 0 in ω, p(i) = q i i = -1/2, 1/2, q(x 1 , z) ∈ L 2 (ω; L 2 # (Z)).
(4.85)

• ŵ ∈ L 2 (ω; H 1 # (Z)) is the unique solution of the Laplace problem -L∆ λ ŵ = g(x 1 ) in ω × Z, ŵ = 0 on ω × ( Γ0 ∪ Γ1 ), (4.86)
• T ∈ L 2 (ω; H 1 # (Z)) is the unique solution of the nonlinear problem        -∆ λ T -D∇ ⊥ λ ŵ • ∇ λ T = 0 in ω × Z, T = 0 on z 2 = ω × Γ0 , ∇ λ T • n = k G(z 1 ) on ω × Γ1 .
(4.87)

Proof. We only have to prove (4.85) 1 , (4.86) 1 and (4.87) 1,3 . The rest follows from Lemma 4.1. We divide the proof in three steps, where we are going to pass to the limit in variational formulation (3.71)-(3.73), taking into account that η ε /ε → λ, 0 < λ < +∞.

Step 1. To prove (4.85) 1 , we consider (3.71) with ϕ replaced by φε = (λ

(ε/η ε )ϕ 1 , ϕ 2 ) with ϕ = (ϕ 1 , ϕ 2 ) ∈ D(ω; C ∞ # (Z) 2
). This gives the following variational formulation:

1 1 -N ω×Z η ε ε -1 λ∂ z1 ûε 1 ∂ z1 ϕ 1 dx 1 dz + 1 1 -N ω×Z η 2 ε ε -2 ∂ z1 ûε 2 ∂ z1 ϕ 2 dx 1 dz + 1 1 -N ω×Z λ(ε/η ε )∂ z2 ûε 1 ∂ z2 ϕ 1 dx 1 dz + 1 1 -N ω×Z ∂ z2 ûε 2 ∂ z2 ϕ 2 dx 1 dz + 1 P r λ(ε/η ε ) ω×Z η 2 ε ε -1 ∂ z1 pε 0 ∂ z1 ϕ 1 dx 1 dz - 1 P r ω×Z λη ε pε 1 ∂ z1 ϕ 1 dx 1 dz - 1 P r ω×Z η ε pε 1 ∂ z2 ϕ 2 dx 1 dz = η 2 ε ε -1 P r ω×Z ûε ⊗û ε ∂ z1 φε dx 1 dz - η ε P r ω×Z ∂ z2 ûε 2 ûε • φε dx 1 dz + ω×Z ûε 2 ∂ z2 ûε φε dx 1 dz + 2N 1 -N ω×Z η ε λ(ε/η ε )∂ z2 ŵε ϕ 1 dx 1 dz - 2N 1 -N ω×Z η 2 ε ε -1 ∂ z1 ŵε ϕ 2 dx 1 dz +Ra ω×Z η 2 ε T ε ϕ 2 dx 1 dz + ω×Z λ(ε/η ε )f 1 ϕ 1 dx 1 dz + O ε , (4.88)
where O ε is devoted to tends to zero when ε → 0. Below, let us pass to the limit when ε tends to zero in each term of (4.88):

• For the first fourth terms in the left-hand side of (4.88), taking into account convergence (4.76) and that that η ε /ε → λ and λ(ε/η ε ) → 1, we get that

1 1 -N ω×Z η ε ε -1 λ∂ z1 ûε 1 ∂ z1 ϕ 1 dx 1 dz + 1 1 -N ω×Z η 2 ε ε -2 ∂ z1 ûε 2 ∂ z1 ϕ 2 dx 1 dz + 1 1 -N ω×Z λ(ε/η ε )∂ z2 ûε 1 ∂ z2 ϕ 1 dx 1 dz + 1 1 -N ω×Z ∂ z2 ûε 2 ∂ z2 ϕ 2 dx 1 dz converges to 1 1 -N ω×Z λ 2 ∂ z1 ûε • ∂ z1 ϕ dx 1 dz + 1 1 -N ω×Z ∂ z2 ûε • ∂ z2 ϕ dx 1 dz = 1 1 -N ω×Z ∇ λ û • ∇ λ ϕ dx 1 dz.
• For the fifth to eighth terms in the left hand side of (4.88), taking into account that λ(ε/η ε ) → 1 and the convergences (4.83) and (4.84), we have the following convergences

1 P r λ(ε/η ε ) ω×Z η 2 ε ε -1 ∂ z1 pε 0 ϕ 1 dx 1 dz → 1 P r ω×Z (∂ x1 p + ∂ z1 p0 ) ϕ 1 dx 1 dz, - 1 P r ω×Z λη ε pε 1 ∂ z1 ϕ 1 dx 1 dz - 1 P r ω×Z ηε p ε 1 ∂ z2 ϕ 2 dx 1 dz → - 1 P r ω×Z p1 div λ ϕ dx 1 dz.
• For the first three terms in the right-hand side of (4.88), by taking into account the estimates (3.57), we get

η 2 ε ε -1 P r ω×Z ûε ⊗û ε ∂ z1 φε dx 1 dz ≤ Cη 2 ε ε -1 ûε 2 L 2 (ω×Z) 2 ∂ z1 ϕ L ∞ (ω×Z) 2 ≤ Cη 2 ε ε -1 → 0, and
η ε P r ω×Z ∂ z2 ûε 2 ûε • φε dx 1 dz + ω×Z ûε 2 ∂ z2 ûε φε dx 1 dz ≤ η ε ûε L 2 (ω×Z) 2 ∂ z2 ûε L 2 (ω×Z) 2 ϕ L ∞ (ω×Z) 2 ≤ Cη ε → 0.
Then, we deduce that the convective terms satisfy

η 2 ε ε -1 P r ω×Z ûε ⊗û ε ∂ z1 φε dx 1 dz - η ε P r ω×Z ∂ z2 ûε 2 ûε • φε dx 1 dz + ω×Z ûε 2 ∂ z2 ûε φε dx 1 dz → 0.
• For the fourth and fifth terms in the right-hand side of (4.88), by taking into account convergence (4.80), we have

2N 1 -N ω×Z η ε λ(ε/η ε )∂ z2 ŵε ϕ 1 dx 1 dz - 2N 1 -N ω×Z η 2 ε ε -1 ∂ z1 ŵε ϕ 2 dx 1 dz → 0.
• For the sixth term in the right-hand side of (4.88), by taking into account convergence (4.82), we get

Ra ω×Z η 2 ε T ε ϕ 2 dx 1 dz → 0.
• For the last term in the right-hand side of (4.88), by taking into account that λ(ε/η ε ) → 1, we get

ω×Z λ(ε/η ε )f 1 ϕ 1 dx 1 dz → ω×Z f 1 ϕ 1 dx 1 dz.
Therefore, by previous convergences, and taking q := p0 /λ + p1 ∈ L 2 (ω; L 2 # (Z)), we deduce that the limit variational formulation is given by the following one

1 1 -N ω×Z ∇ λ û • ∇ λ ϕ dx 1 dz + 1 P r ω×Z ∂ x1 p (e 1 • ϕ) dx 1 dz - 1 P r ω×Z q div λ ϕ dx 1 dz = ω×Z f 1 (e 1 • ϕ) dx 1 dz. (4.89)
By density, (4.89) holds for every function ϕ ∈ L 2 (ω;

H 1 # (Z) 2
) and is equivalent to the system (4.85) 1 . We also remark that (4.89) admits a unique solution, and then the convergence is for the complete sequences of the unknowns.

Step 2. Next, we prove (4.86) 1 . Let us pass to the limit when ε tends to zero in each term of the variational formulation (3.72):

• For the first two terms in the left-hand side of (3.72), by using convergence (4.80) and η ε /ε → λ, we get

L ω×Z η 2 ε ε -2 ∂ z1 ŵε ∂ z1 ψ dx 1 dz + L ω×Z ∂ z2 ŵε ∂ z2 ψ dx 1 dz → L ω×Z ∇ λ ŵ • ∇ λ ψ dx 1 dz.
• For the third term in the left-hand side of (3.72), by using convergence (4.80), we have

4N 1 -N ω×Z η 2 ε ŵε ψ dx 1 dz → 0.
• For the first two terms in the right-hand side of (3.72), by using estimates (3.57) and (3.58), we get

- M P r ω×Z η 2 ε ε -1 ûε 1 ∂ z1 ŵε ψ dx 1 dz ≤ Cη 2 ε ε -1 ûε L 2 (ω×Z) 2 ∂ z1 ŵε L 2 (ω×Z) ≤ Cη ε , - M P r ω×Z η ε ûε 2 ∂ z2 ŵε ψ dx 1 dz ≤ Cη ε ûε L 2 (ω×Z) 2 ∂ z2 ŵε L 2 (ω×Z) ≤ Cη ε .
Thus, we get

- M P r ω×Z η 2 ε ε -1 ûε 1 ∂ z1 ŵε ψ dx 1 dz - M P r ω×Z η ε ûε 2 ∂ z2 ŵε ψ dx 1 dz → 0.
• For the third and fourth terms in the right-hand side of (3.72), by using estimates (3.57) and (3.58), we get

2N 1 -N ω×Z η 2 ε ε -1 ∂ z1 ûε 2 ψ dx 1 dz ≤ Cη 2 ε ε -1 ∂ z1 ûε L 2 (ω×Z) 2 ≤ Cη ε , - 2N 1 -N ω×Z η ε ∂ z2 ûε 1 ψ dx 1 dz ≤ Cη ε ∂ z2 ûε L 2 (ω×Z) 2 ≤ Cη ε . Thus, we have 2N 1 -N ω×Z η 2 ε ε -1 ∂ z1 ûε 2 ψ dx 1 dz - 2N 1 -N ω×Z η ε ∂ z2 ûε 1 ψ dx 1 dz → 0.
Then, from the above convergences, we get that the limit variational formulation for ŵ is given by

L ω×Z ∇ λ ŵ • ∇ λ ψ dx 1 dz = ω×Z g ψ dx 1 dz. (4.90)
By density, (5.117) holds for every function ψ in L 2 (ω; H 1 # (Z)) and is equivalent to problem (4.86). We remark that (5.117) admits a unique solution, and then the complete sequence ŵε converges to the unique solution ŵ(x 1 , z).

Before passing to the next step, we need to prove that ∇ ηε,ε ŵε converges strongly to ∇ λ ŵ in L 2 (ω × Z) 2 . To do this, we take ŵε as test function in (3.72) and ŵ in (5.117). Then, it is easy to prove that lim

ε→0 ω×Z |∇ ηε,ε ŵε | 2 dx 1 dz = 1 L ω×Z g ŵ dx 1 dz = ω×Z |∇ λ ŵ| 2 dx 1 dz
This together with the weak convergence of ∇ ηε,ε ŵε to ∇ λ ŵ in L 2 (ω×Z) 2 , it gives the desired strong convergence.

Step 3. To prove (4.87) 1,4 , we take into account that the variational formulation (3.73) can be written as follows

η 2 ε ε -2 ω×Z η -2 ε ∂ z1 T ε ∂ z1 φ dx 1 dz + η -2 ε ω×Z ∂ z2 T ε ∂ z2 φ dx 1 dz = -η ε ω×Z ûε • ∇ ηε,ε T ε φ dx 1 dz +D ω×Z ∇ ⊥ ηε,ε ŵε • ∇ ηε,ε (η -2 ε T ε )φ dx 1 dz + k ω× Γ1 G φ dx 1 dσ + O ε , (4.91)
where O ε tends to zero. Below, we pass to the limit in every terms:

• For the first two terms in the left-hand side of (5.118), by using convergence (4.82), we get

η 2 ε ε -2 ω×Z η -2 ε ∂ z1 T ε ∂ z1 φ dx 1 dz + η -2 ε ω×Z ∂ z2 T ε ∂ z2 φ dx 1 dz → ω×Z ∇ λ T • ∇ λ φ dx 1 dz.
• For the first term in the right-hand side of (5.118), by using estimates (3.57) and (3.59), we get

-η ε ω×Z ûε • ∇ ηε,ε T ε φ dx 1 dz ≤ Cη ε ûε L 2 (ω×Z) 2 ∇ ηε,ε T ε L 2 (ω×Z) ≤ Cη 3 ε , so we have -η ε ω×Z ûε • ∇ ηε,ε T ε φ dx 1 dz → 0.
• For the second term in the right-hand side of (5.118), by using convergences (4.82) and the strong convergence of ∇ ⊥ ηε,ε ŵε to ∇ ⊥ λ ŵ, we get

D ω×Z ∇ ⊥ ηε,ε ŵε • ∇ ηε,ε (η -2 ε T ε )φ dx 1 dz → D ω×Z ∇ ⊥ λ ŵ • ∇ λ T φ dx 1 dz.
Then, using previous convergences, we get that the limit variational formulation for T is given by

ω×Z ∇ λ T • ∇ λ φ dx 1 dz = D ω×Z ∇ ⊥ λ ŵ • ∇ λ T φ dx 1 dz + k ω× Γ1 G(z 1 )φ dx 1 dσ. (4.92)
By density, (4.92) holds for every function φ in L 2 (ω; H 1 # (Y )) and is equivalent to problem (4.87). We remark that (4.92) admits a unique solution, and then the complete sequence T ε converges to the unique solution T (x 1 , z).

Finally, we give the main result concerning the homogenized flow.

U av (x 1 ) = hmax 0 Ũ(x 1 , z 2 ) dz 2 , W av (x 1 ) = hmax 0 W (x 1 , z 2 ) dz 2 , T av (x 1 ) = hmax 0 θ(x 1 , z 2 ) dz 2 .
We have the following:

• The average velocity is given by

U av 1 = a λ 1 -N P r q -1/2 -q 1/2 + P r 1/2 -1/2 f 1 (ξ) dξ , U av 2 (x 1 ) = 0 in ω, (4.93)
where a λ ∈ R is given by

a λ = Z |∇ λ u bl (z)| 2 dz, with (u bl , π bl ) ∈ H 1 # (Z) 2 × L 2 # (Z) the solution of the local Stokes problem                  -∆ λ u bl + ∇ λ π bl = e 1 in Z, div λ u bl = 0 in Z, u bl = 0 on z 2 = {0, h(z 1 )}, Z u bl 2 (z)dz = 0. (4.94)
• The pressure p is given by p(x 1 ) = q -1/2 -q -1/2 -q 1/2 + P r

1/2 -1/2 f 1 (ξ) dξ x 1 + 1 2 + P r x1 -1/2 f (ξ) dξ in ω. (4.95)
• The average microrotation is given by • The average temperature is given by

W av (x 1 ) = b λ 1 L g(x 1 ) in ω, (4.96
T av (x 1 ) = Z T bl (x 1 , z) dz in ω, (4.98)
with T bl ∈ L 2 (ω; H 1 # (Z)) the unique solution of the nonlinear local problem

         -∆ λ T bl - D L g(x 1 )(∇ ⊥ λ w bl • ∇ λ )T bl = 0 in ω × Z, T bl = 0 on ω × Γ0 , ∇ λ T bl • n = k G(z 1 ) on ω × Γ1 .
Proof. First, we proceed to eliminate the microscopic variable z in the effective linear problems (4.85) and (4.86).

To do that, we consider the following identification

û(x 1 , z) = (1 -N ) f 1 (x 1 ) -1 P r ∂ x1 P (x 1 ) u bl (z), q(x 1 , z) = P r f 1 (x 1 ) -1 P r ∂ x1 P (x 1 ) π bl (z), ŵ(x 1 , z) = g(x 1 ) L w bl (z),
where (u bl , π bl ) and w bl satisfies (4.94) and (4.97), respectively.

From the identities for velocity Z û1 (x 1 , z) dz = hmax 0 Ũ1 (x 1 , z 2 ) dz 2 and Z û2 dz = 0, and for the microrotation

Z ŵ(x 1 , z) dz = hmax 0 W (x 1 , z 2 ) dz 2
given in Lemma 4.1, by linearity we deduce that U av is given by

U av 1 = a λ (1 -N ) f 1 (x 1 ) - 1 P r ∂ x1 p(x 1 ) U av 2 = 0, in ω,
and W av is given by (4.96).

Next, the divergence condition with respect to the variable x 1 given in (4.75) together with the expression of

U av 1 gives that U av 1 = a λ (1 -N ) f 1 (x 1 ) -1 P r ∂ x1 p(x 1 ) = C 1 , C 1 ∈ R. (4.99)
Then, integrating with respecto to x 1 , and taking into account that p(-1/2) = q -1/2 , it holds

p(x 1 ) = q -1/2 - P r a λ (1 -N ) C 1 x 1 + 1 2 + P r x1 -1/2 f (ξ) dξ.
Finally, since p(1/2) = q 1/2 , we deduce

C 1 = a λ (1 -N ) P r q -1/2 -q 1/2 + P r 1/2 -1/2 f 1 (ξ) dξ .
This implies (4.95). Then, by using the expression of p, we deduce that the average velocity U av 1 is given by (4.93).

Finally, the formula for T av follows from (4.93) 3 and the identity Z T (x 1 , z) dz = hmax 0 θ(x 1 , z 2 ) dz 2 by renaming T bl ≡ T .

Homogenized model in the subcritical case

It corresponds to the case when the wavelength of the roughness is much greater than the film thickness, i.e., η ε ε, which is equivalent to λ = 0.

We start by giving some compactness results about the behavior of the extended sequences ( Ũε , W ε , θε , p ε 0 , pε 1 ) and the related unfolding functions (û ε , ŵε , T ε , pε 0 , pε 1 ) satisfying the a priori estimates given in Lemmas 3.5 and Lemma 3.9 respectively. Lemma 5.1. For a subsequence of ε still denote by ε, we have the following convergence results: (i) (Velocity) There exist Ũ = ( Ũ1 , Ũ2 ) ∈ H 1 (0, h max ; L 2 (ω) 2 ), with Ũ = 0 on z 2 = {0, h max } and Ũ2 = 0, such that (5.104)

Ũε Ũ in H 1 (0, h max ; L 2 (ω) 2 ), ( 5
(ii) (Microrotation) There exist W ∈ H 1 (0, h max ; L 2 (ω)), with W = 0 on z 2 = {0, h max }, such that W ε W in H 1 (0, h max ; L 2 (ω)), (5.105
)

and ŵ ∈ H 1 (0, h(z 1 ); L 2 # (ω × Z )), with ŵ = 0 on z 2 = {0, h(z 1 )} such that it hold Z ŵ(x 1 , z)dz = hmax 0 W (x 1 , z 2 ) dz 2 ,

and moreover

ŵε ŵ in H 1 (0, h(z 1 ); L 2 (ω × Z )).
(5.106)

(iii) (Temperature) There exist θ ∈ H 1 (0, h max ; L 2 (ω)), with θ = 0 on z 2 = {0}, such that η -2 ε θε θ in H 1 (0, h max ; L 2 (ω)), (5.107)
and T ∈ H 1 (0, h(z 1 ); L 2 # (ω × Z )), with T = 0 on z 2 = {0}, such that Z T (x 1 , z)dz = hmax 0 θ(x 1 , z 2 ) dz 2 , and moreover

η -2 ε T ε
T in H 1 (0, h(z 1 ); L 2 (ω × Z )).

(5.108) (iv) (Pressure) There exist three functions p ∈ L 2 0 (ω) ∩ H 1 (ω), independent of z 2 with with p(i

) = q i , i = -1/2, 1/2, p0 ∈ L 2 (ω; H 1 # (Z )) and p1 ∈ L 2 (ω; L 2 # (Z)) such that η 2 ε p ε 0 p in H 1 (ω), (5.109)
η 2 ε ε -1 ∂ z1 pε 0 ∂ z1 p + ∂ z1 p0 in L 2 (ω; L 2 (Z )), η ε pε 1 p1 in L 2 (ω; L 2 (Z)).
(5.110)

Proof. Proof The proof of (i) is similar to the critical case, but we have to take into account that applying the unfolded change of variables to the divergence condition div ηε (ũ ε ) = 0 and multiplying by η ε , we get

η ε ε ∂ z1 ûε 1 + ∂ z2 ûε 2 = 0.
(5.111)

Passing to the limit, since η ε ε, we get ∂ z2 û2 = 0, which means that û2 is independent of z 2 . Due to the boundary conditions on the top and bottom, it holds that û2 = 0. Now, multiplying (5.111) by εη -1 ε ϕ with ϕ independent of z 2 and integrating by parts, we get ω×Z h(z1)

0 ûε 1 dz 2 ∂ z1 ϕ dx 1 dz 1 = 0.
Passing to the limit and integrating by parts, we get (5.103). For more details, we refer the reader to the proof of Lemmas 5.2-i) and 5.4-ii) in [START_REF] Anguiano | Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary[END_REF] (see also [START_REF] Suárez-Grau | Analysis of the roughness regime for micropolar fluids via homogenization[END_REF]). The proofs of (ii), (iii) and (iv) are similar to the critical case, so we omit it.

Using previous convergences, in the following theorem we give the two-pressured homogenized system satisfied by (û, ŵ, P , T ).

Theorem 5.2 (Limit unfolded problems). In the case η ε ε, then the functions û, ŵ, T and p given in Lemma 5.1 satisfy

• (û, p) ∈ H 1 (0, h(z 1 ); L 2 # (ω × Z )) × (L 2 0 (ω) ∩ H 1 (ω))
with û2 = 0 is the unique solution of the two-pressure homogenized reduced Stokes problem

                                   - 1 1 -N ∂ 2 z2 û1 + 1 P r ∂ z1 p0 = f 1 (x 1 ) - 1 P r ∂ x1 p(x 1 ) in ω × Z, ∂ z1 h(z1) 0 û1 dz 2 = 0 in ω × Z , û1 = 0 on ω × ( Γ0 ∪ Γ1 ), ∂ x1 Z û1 (x 1 , z) dz = 0 in ω, p(i) = q i i = -1/2, 1/2, p0 ∈ L 2 # (ω × Z ).
(5.112)

• ŵ ∈ L 2 (ω; H 1 # (Z)) is the unique solution of the Laplace problem -L∂ 2 z2 ŵ = g(x 1 ) in ω × Z, ŵ = 0 on ω × ( Γ0 ∪ Γ1 ),
(5.113)

• T ∈ L 2 (ω; H 1 # (Z))
is the unique solution of the nonlinear problem

       ∂ 2 z2 T = 0 in ω × Z, T = 0 on z 2 = ω × Γ0 , ∂ z2 T = k G(z 1 ) on ω × Γ1 .
(5.114)

Proof. We divide the proof in three steps.

Step 1. To prove (5.112) 1 , we consider in (3.71) where ϕ(x , z) ∈ D(ω; C ∞ # (Z) 2 ) with ϕ 2 = 0 in ω × Z. This gives the following variational formulation:

1 1 -N ω×Z η 2 ε ε -2 ∂ z1 ûε 1 ∂ z1 ϕ 1 dx 1 dz + 1 1 -N ω×Z ∂ z2 ûε 1 ∂ z2 ϕ 1 dx 1 dz + 1 P r ω×Z η 2 ε ε -1 ∂ z1 pε 0 ϕ 1 dx 1 dz - 1 P r ω×Z η 2 ε ε -1 pε 1 ∂ z1 ϕ 1 dx 1 dz = η 2 ε ε -1 P r ω×Z ûε 1 ûε 1 ∂ z1 ϕ 1 dx 1 dz - η ε P r ω×Z ∂ z2 ûε 2 ûε 1 ϕ 1 dx 1 dz + ω×Z ûε 2 ∂ z2 ûε 1 ϕ 1 dx 1 dz + 2N 1 -N ω×Z η ε ∂ z2 ŵε ϕ 1 dx 1 dz + ω×Z f 1 ϕ 1 dx 1 dz + O ε , (5.115)
where O ε is devoted to tends to zero when ε → 0. Below, let us pass to the limit when ε tends to zero in each term of the previous variational formulation:

• For the first two terms in the left-hand side of (5.115), taking into account convergence (5.102) and that η ε /ε → 0, we get that

1 1 -N ω×Z η 2 ε ε -2 ∂ z1 ûε 1 ∂ z1 ϕ 1 dx 1 dz → 0, 1 1 -N ω×Z ∂ z2 ûε 1 ∂ z2 ϕ 1 dx 1 dz → 1 1 -N ω×Z ∂ z2 û1 ∂ z2 ϕ 1 dx 1 dz.
• For the third term on the left hand side of (5.115), taking into account that convergence of the pressures (5.110) and η ε /ε → 0, we have the following convergence s

1 P r ω×Z η 2 ε ε -1 ∂ z1 pε 0 ϕ 1 dx 1 dz → 1 P r ω×Z (∂ x1 p + ∂ z1 p0 ) ϕ 1 dx 1 dz, - 1 P r ω×Z η 2 ε ε -1 pε 1 ∂ z1 ϕ 1 dx 1 dz → 0.
• For the first three terms in the right-hand side of (5.115), by taking into account the estimates (3.57), we get

η 2 ε ε -1 P r ω×Z ûε 1 ûε 1 ∂ z1 ϕ 1 dx 1 dz ≤ η 2 ε ε -1 ûε 2 L 2 (ω×Z) 2 ∂ z1 ϕ L ∞ (ω×Z) 2 ≤ Cη 2 ε ε -1 → 0, - η ε P r ω×Z ∂ z2 ûε 2 ûε 1 ϕ 1 dx 1 dz + ω×Z ûε 2 ∂ z2 ûε 1 ϕ 1 dx 1 dz ≤ η ε ûε L 2 (ω×Z) 2 ∂ z2 ûε L 2 (ω×Z) 2 ϕ L ∞ (ω×Z) 2 ≤ Cη ε → 0.
Then, we deduce that the convective terms satisfy

η 2 ε ε -1 P r ω×Z ûε 1 ûε 1 ∂ z1 ϕ 1 dx 1 dz - η ε P r ω×Z ∂ z2 ûε 2 ûε 1 ϕ 1 dx 1 dz + ω×Z ûε 2 ∂ z2 ûε 1 ϕ 1 dx 1 dz → 0.
• For the fourth term in the right-hand side of (5.115), by taking into account convergence (5.106), so we have

2N 1 -N ω×Z η ε ∂ z2 ŵε ϕ 1 dx 1 dz → 0.
Therefore, by previous convergences, we deduce that the limit variational formulation is given by the following one

1 1 -N ω×Z ∂ z2 û1 ∂ z2 ϕ 1 dx 1 dz + 1 P r ω×Z ∂ x1 p ϕ 1 dx 1 dz + 1 P r ω×Z ∂ z1 p0 ϕ 1 dx 1 dz = ω×Z f 1 ϕ 1 dx 1 dz.
(5.116) By density, (5.116) holds for every function ϕ in the H 1 (0, h(z 1 ); L 2 # (ω × Z )) and is equivalent to problem (5.112) 1 . We remark that (5.116) admits a unique solution, and then the complete sequences converge.

Step 2. Next, we prove that ŵ satisfies problem (5.113). Below, let us pass to the limit when ε tends to zero in each term of the previous variational formulation (3.72):

• For the first two terms in the left-hand side of (3.72), by using convergence (4.80) and η ε /ε → 0, we get

L ω×Z η 2 ε ε -2 ∂ z1 ŵε ∂ z1 ψ dx 1 dz → 0, L ω×Z ∂ z2 ŵε ∂ z2 ψ dx 1 dz → L ω×Z ∂ z2 ŵε • ∂ z2 ψ dx 1 dz,
and so,

L ω×Z η 2 ε ε -2 ∂ z1 ŵε ∂ z1 ψ dx 1 dz + L ω×Z ∂ z2 ŵε ∂ z2 ψ dx 1 dz → L ω×Z ∂ z2 ŵ ∂ z2 ψ dx 1 dz.
• For the third term of the left-hand side of (3.72), by using convergence (4.80), we have

4N 1 -N ω×Z η 2 ε ŵε ψ dx 1 dz → 0.
• For the first two terms in the right-hand side of (3.72), by using estimates (3.57) and (3.58), we get

- M P r ω×Z η 2 ε ε -1 ûε 1 ∂ z1 ŵε ψ dx 1 dz ≤ Cη 2 ε ε -1 ûε L 2 (ω×Z) 2 ∂ z1 ŵε L 2 (ω×Z) ≤ Cη ε , - M P r ω×Z η ε ûε 2 ∂ z2 ŵε ψ dx 1 dz ≤ Cη ε ûε L 2 (ω×Z) 2 ∂ z2 ŵε L 2 (ω×Z) ≤ Cη ε .
Thus, we get -M P r ω×Z η 2 ε ε -1 ûε 1 ∂ z1 ŵε ψ dx 1 dz -M P r ω×Z η ε ûε 2 ∂ z2 ŵε ψ dx 1 dz → 0.

• For the third and fourth terms in the right-hand side of (3.72), by using estimates (3.57) and (3.58), we get 2N 1

-N ω×Z η 2 ε ε -1 ∂ z1 ûε 2 ψ dx 1 dz ≤ Cη 2 ε ε -1 ∂ z1 ûε L 2 (ω×Z) 2 ≤ Cη ε , - 2N 1 -N ω×Z η ε ∂ z2 ûε 1 ψ dx 1 dz ≤ Cη ε ∂ z2 ûε L 2 (ω×Z) 2 ≤ Cη ε .
Thus, we have

2N 1 -N ω×Z η 2 ε ε -1 ∂ z1 ûε 2 ψ dx 1 dz - 2N 1 -N ω×Z η ε ∂ z2 ûε 1 ψ dx 1 dz → 0.
Then, from the above convergences, we get that the limit variational formulation for ŵ is given by

L ω×Z ∂ z2 ŵ ∂ z2 ψ dx 1 dz = ω×Z g ψ dx 1 dz.
(5.117) By density (5.117) holds for every function ψ in H 1 (ω; L 2 # (ω × Z )) and is equivalent to problem (5.113) 1 . We remark that (5.117) admits a unique solution, and then the complete sequence converges.

Step 3. Next, we prove that T satisfies problem (5.114). we take into account that the variational formulation (3.73) can be written as follows

η 2 ε ε -2 ω×Z η -2 ε ∂ z1 T ε ∂ z1 φ dx 1 dz + η -2 ε ω×Z ∂ z2 T ε • ∂ z2 φ dx 1 dz = -η ε ω×Z ûε • ∇ ηε,ε T ε φ dx 1 dz +D ω×Z ∇ ⊥ ηε,ε ŵε • ∇ ηε,ε (η -2 ε T ε )φ dx 1 dz + k ω× Γ1 G φ dx 1 dσ + O ε , (5.118)
where we use the operators ∇ ηε,ε = (η ε ε -1 ∂ z1 , ∂ z2 ) and ∇ ⊥ ηε,ε = (∂ z2 , -η ε ε -1 ∂ z1 ). Below, we pass to the limit in every terms:

• For the first two terms in the left-hand side of (3.73), by using convergence (5.108) and η ε /ε → 0, we get

η 2 ε ε -2 ω×Z η -2 ε ∂ z1 T ε ∂ z1 φ dx 1 dz → 0, η -2 ε ω×Z ∂ z2 T ε ∂ z2 φ dx 1 dz → ω×Z ∂ z2 T ∂ z2 φ dx 1 dz,
and so,

η 2 ε ε -2 ω×Z η -2 ε ∂ z1 T ε ∂ z1 φ dx 1 dz + η -2 ε ω×Z ∂ z2 T ε ∂ z2 φ dx 1 dz → ω×Z ∂ z2 T ∂ z2 φ dx 1 dz.
• For the first term in the left-hand side of (3.73), by using estimates (3.57) and (3.59), we get

-η ε ω×Z ûε • ∇ ηε,ε T ε φ dx 1 dz ≤ Cη ε ûε L 2 (ω×Z) 2 ∇ ηε,ε T ε L 2 (ω×Z) ≤ Cη 3 ε , so we have -η ε ω×Z ûε • ∇ ηε,ε T ε φ dx 1 dz → 0.
• For the second term in the right-hand side of (3.73), by using convergences (5.108), the strong convergence of ∇ ⊥ Then, using previous convergences, we get that the limit variational formulation for T is given by

ω×Z ∂ z2 T ∂ z2 φ dx 1 dz = k ω× Γ1
G(z 1 )φ dx 1 dσ.

(5.119)

By density (4.92) holds for every function φ in H 1 (ω; L 2 # (ω × Z )). We remark that (4.92) admits a unique solution, and then the complete sequence converges.

Finally, we give the main result concerning the homogenized flow. We have the following:

• The average velocity is given by U av 1 = a 0 1 -N P r q -1/2 -q 1/2 + P r where, by using (5.112) 2 , then π bl ∈ L 2 # (Z )/R is the solution of the second order ordinary differential equation with respect to z 1 with periodic boundary conditions on Z , given by      h 3 (z 1 )∂ 2 z1 π bl (z 1 ) -3h 2 (z 1 ) dh dz 1 (z 1 )∂ z1 π bl (z 1 ) = -3h 2 (z 1 ) dh dz 1 (z 1 ) in Z , π bl (-1/2) = π bl (1/2).

(5.131)

Solving this equation, we obtain an expression for π bl , up to a constant,

π bl (z 1 ) = - 1/2 -1/2 h 3 (ξ) dξ -1 z1 -1/2 h 3 (ξ) dξ + z 1 + 1/2 + C, C ∈ R, z 1 ∈ Z .
This implies that

∂ z1 π bl (z 1 ) = - 1/2 -1/2 h 3 (ξ) dξ -1 h 3 (z 1 ) + 1, z 1 ∈ Z ,
and so, from (5.130), we get

a 0 = - 1 12
1/2 -1/2 h 3 (z 1 )   2 -h 3 (z 1 ) 1/2 -1/2 h 3 (ξ) dξ -1   dz 1 .
(5.132)

From condition (5.101), by taking into account the expression of U av 1 and the boundary conditions of p, we get the expression for pressure p given in (5.122).

Finally, taking into account the expressions of (5.122), (5.127) and (5.132), then the average velocity can be written as (5.120)-(5.121).

Next, we focus on the microrotation. We eliminate the microscopic variable z in the effective linear problem (5.113). To do that, as in the critical case, we consider the following identification ŵ(x 1 , z) = g(x 1 ) L w bl (z), where w bl ∈ H 1 # (Z) is the solution of the local problem -∂ 2 z2 w bl = -1 in Z, ŵ = 0 on Γ0 ∪ Γ1 .

(5.133)

This implies that

w bl (z) = - 1 2 z 2 2 -h(z 1 )z 2 ,
and taking into account that h(z1) 0 w bl dz 2 = h 3 (z 1 )/12 and that W av (x 1 ) = Z ŵ dz, we get

W av = b 0 g(x 1 ) L , b 0 = 1 12 Z h 3 (z 1 ) dz 1 ,
which is (5.123)-(5.124).

Finally, we obtain the expression of the average of the temperature. To do this, we solve the problem (5.114), which gives the expression for T T (x 1 , z) = kG(z 1 )z 2 , in ω × Z.

Taking into account that T av (x 1 ) = Z T dz, we easily get (5.125)-(5.126).

  .42) Taking into account (3.40) and estimates (3.41) in (3.39), we get

  .46) Proof. Estimates for the extension of Ũε and W ε are obtained straightforward from (3.27) and (3.28), respectively. For the extension of the temperature θε , from (3.29) and (3.43), we deduce (3.46).

  .61) Thus, combining previous estimates of φε with estimates for ũε , wε , T ε given in Lemma 3.4 and estimate for pε 1 given in Corollary 3.7, we respectively get (3.57), (3.58), (3.59) and (3.60).

Theorem 4 . 3 (

 43 Main result for the critical case). Consider ( Ũ, W , θ, p) given in Lemma 4.1. Let us define the average velocity, microrotation and temperature respectively by

)

  where b λ ∈ R is given byb λ = Z |∇ λ w bl (z)| 2 dz, with w bl ∈ H 1 # (Z) the solution of the local Laplace problem -∆ λ w bl = 1 in Z,w bl = 0 on z 2 = {0, h(z 1 )}.

.100) ∂ x1 hmax 0 Ũ1 (x 1 , y 2 ) 0 Ũ(x 1 , z 2 )

 012012 dy 2 = 0 in ω.(5.101)and û = (û 1 , û2 ) ∈ H 1 (0, h(z 1 ); L 2 # (ω × Z ) 2 ), with û = 0 on z 2 = {0, h(z 1 )} and û2 = 0, such that it hold Z û(x 1 , z)dz = hmax dz 2 with Z û2 (x 1 , z) dz = 0, and moreover ûε û in H 1 (0, h(z 1 ); L 2 (ω × Z ) 1 , z) dz = 0 in ω .

2 ε

 2 ŵε to (∂ z2 ŵ, 0) (it can be proved as in the critical case) and the weak convergence of ∇ ηε,ε (η-ŵε • ∇ ηε,ε (η -2 ε T ε )φ dx 1 dz → 0.

Theorem 5 . 3 (Ũ(x 1 , z 2 )W (x 1 , z 2 )

 531212 Main result for the subcritical case). Consider ( Ũ, W , θ, p) given in Lemma 5.1. Let us define the average velocity, microrotation and temperature respectively byU av (x 1 ) = hmax 0 dz 2 , W av (x 1 ) = hmax 0 dz 2 , T av (x 1 ) = hmax 0 θ(x 1 , z 2 ) dz 2 .

1 / 2 - 1 / 2 f 1 12 1/ 2 - 1 / 2 h 3 (z 1 ) 2 -h 3 (z 1 ) 1 / 2 - 1 / 2 h 3 1 

 121211221231231121231 (ξ) dξ , U av 2 = 0 in ω,(5.120)where a 0 ∈ R is given bya 0 = 1 (ξ) dξ dz 1 .(5.121)

  .14) in order to have the functions defined in the open set with fixed height Ω ε and the rescaled boundaries Γ ε

	1 and
	Σ ε i , i = -1/2, 1/2. Then, using the change of variables (2.14) in (2.8)-(2.11), we obtain the following rescaled
	system

Acknowledgements. The first author of this paper has been supported by the Croatian Science Foundation under the project Multiscale problems in fluid mechanics -MultiFM (IP-2019-04-1140).

• The pressure p is given by p(x 1 ) = q -1/2 -q -1/2 -q 1/2 + P r

f 1 (ξ) dξ in ω.

(5.122)

• The average microrotation is given by

where b 0 ∈ R is given by

(5.124)

• The average temperature is given by

where c 0 ∈ R is given by

(5.126)

Proof. First, we start with the velocity by proceeding to eliminate the microscopic variable z in the effective linear problem (5.112). To do that, as in the critical case, we consider the following identification

From the identities for velocity U av 1 = hmax 0 Ũ1 (x 1 , z 2 ) dz 2 = Z û1 (x 1 , z) dz and û2 = 0 given in Lemma 5.1, by linearity we deduce that U av is given by

U av 2 = 0, in ω.

(5.127) with a 0 given by

where (u bl , π bl ) satisfies the following local reduced problem

(5.128)

Now, we observe that we can obtain more accurate expressions for a 0 , because problem (5.128) is an ordinary differential equation with respect to the variable z 2 and it can be solved. Thus, from the boundary conditions on the top and bottom, we get