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

Roughness-induced effects on the thermomicropolar fluid flow through a thin domain Igor PAˇ ZANIN∗and Francisco Javier SU´ AREZ-GRAU† Abstract 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. AMS classification numbers: 35B27, 35Q35, 76A05, 76M50. Keywords: thermomicropolar fluid; thin domain; rough boundary; cooling condition; homogenization. 1 Introduction The model of micropolar fluid, proposed by Eringen [17] 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 [18], 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. [19], [24],[36]):                          1 Pr ((u· ∇)u+∇p)=∆u+N 1−N(2∇⊥w+ ∆u) + Ra Te2+f, div(u)=0, M Pr (u· ∇w) = L∆w+2N 1−N(rot(u)−2w) + g, u· ∇T= ∆T+D∇⊥w· ∇T. (1.1) ∗Department of Mathematics, Faculty of Science, University of Zagreb, Bijeniˇcka 30, 10000 Zagreb (Croatia) [email protected] †Departamento de Ecuaciones Diferenciales y An´alisis Num´erico. Facultad de Matem´aticas. Universidad de Sevilla. 41012-Sevilla (Spain) [email protected] 1 Igor Paˇzanin and Francisco J. Su´arez-Grau In the above system, the velocity vector field is denoted by u, the pressure by p,wrepresents the microrotation and Tis the temperature of the fluid. The external sources of linear and angular momentum are given by the functions f= (f1, f2) and g, respectively. We denote by e2= (0,1) ∈R2the unit upward vector, whereas the positive constants appearing in (1.1) represent the following (see e.g. [23]): •Nis the coupling parameter, i.e. the relation between the Newtonian and microrotation viscosities, •Mis the relation between the moment of inertia and geometry, •Lis the couple stress parameter, i.e. the relation between the geometry and the properties of the fluid, •Dis the micropolar heat conduction parameter, i.e. the relation between the micropolar thermal conduction and the geometry, •Pr 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. [6], [7], [8], [15], [16], [28], [29]. Although there have been a number of recent papers concerning engineering applications of the thermomicropolar fluid model (see e.g. [12], [20], [21], [31]), the rigorous treatments for such models are very sparse. Most recently, the system (1.1) has been studied in [25] for the thermicropolar flow through a thin channel with smooth walls, namely: Ωε=(x1, x2)∈R2:x1∈ω, 0< x2< 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−1/2and q1/2, 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 f1and gonly depends on the horizontal variable and after a dilatation in the vertical variable, it is proved that the average velocity Uav = (Uav 1, Uav 2) and the microrotation Wav at the main-order term are respectively given by: Uav 1=1 12 1−N Pr q−1/2−q1/2+P r Z1/2 −1/2 f1(ξ)dξ!, Uav 2= 0, Wav =1 12 1 Lg(x1),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 Land 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 [25] to a case of a thin channel with an irregular upper wall described by x2=ηεhx1 ε, where ηεis the thickness of the roughness, εis the period of the roughness and his a positive and periodic function (see Section 2). This kind of thin rough domain has been extensively studied for the isothermal flows, see [5], [26] for the classical Newtonian fluid flow, [3] for the flow of the generalized Newtonian fluid and [34] for the micropolar fluid flow. In these papers, a critical size has been found between the thickness of the domain ηε and the period of the roughness ε, which is given by λ= lim ε→0 ηε 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, 2 Igor Paˇzanin and Francisco J. Su´arez-Grau 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. [34]) 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 εand ηε, we rigorously derive the following expressions for the average velocity and microrotation: Uav 1=aλ 1−N Pr q−1/2−q1/2+P r Z1/2 −1/2 f1(ξ)dξ!, Uav 2= 0, Wav =bλ 1 Lg(x1),in ω. (1.3) where aλ, bλ∈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 aλ, bλare computed through local PDE problems (see Section 4, Theorem 4.3). However, in the subcritical case λ= 0, the parameters a0, b0can be explicitly computed (see Section 5, Theorem 5.3). In both cases, we obtain the same expression for the pressure as in [25]. 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. 2 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. 2.1 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 Ωε={x= (x1, x2)∈R2:x1∈ω, 0< x2< hε(x1)},(2.4) Here, the function hε(x1) = ηεh(x1/ε) represents the real gap between the two surfaces. The small parameter ηε is related to the film thickness and the small parameter εis the wavelength of the roughness. Here, we consider that ηεis of order smaller or equal than ε, i.e. we consider ηε≈εor ηεε. (2.5) Function h∈W1,∞(R), Z0-periodic with Z0= (−1/2,1/2) the cell of periodicity in R, and there exist hmin and hmax such that 0< hmin = min z1∈Z0h(z1), hmax = max z1∈Z0h(z1). We define the boundaries of Ωεas follows Γ0=(x1, x2)∈R2:x1∈ω, x2= 0,Γε 1=(x1, x2)∈R2:x1∈ω, x2=hε(x1) Σε i=(x1, x2)∈R2:x1=i, 0< x2< hε(x1), i =−1/2,1/2. We also define the respective rescaled sets e Ωε=ω×(0, h(x1/ε)),e Γε 1=ω× {h(x1/ε)}and e Σε i={i} × (0, h(i/ε)), i =−1/2,1/2. 3 Igor Paˇzanin and Francisco J. Su´arez-Grau Related to the microstructure of the periodicity of the boundary, we consider that the domain ωis divided by a mesh of size ε: for k0∈Z, each cell Z0 k0,ε =εk0+εZ0. We define Tε={k0∈Z:Z0 k0,ε ∩ω6=∅}. In this setting, there exists an exact finite number of periodic sets Z0 k0,ε such that k0∈Tε. Also, we define Zk0,ε =Z0 k0,ε ×(0, h(z1)) and Z=Z0×(0, h(z1)), which is the reference cell in R2. We define the boundaries ˆ Γ0=Z0× {0},ˆ Γ1=Z0× {h(z1)},ˆ Σi={i} × {i} × (0, h(i)), i=−1/2,1/2. The quantity hmax allows us to define the extended sets Ω = ω×(0, hmax) and Γ1=ω× {hmax}. In order to apply the unfolding method, we will use the following notation. For k0∈Z, we define κ:R→Zby κ(x1) = k0⇐⇒ x1∈Zk0,1.(2.6) Remark that κis well defined up to a set of zero measure in R(the set ∪k∈Z∂Yk,1). Moreover, for every ε > 0, we have κx1 ε=k0⇐⇒ x1∈Zk0,ε . We denote by Ca generic constant which can change from line to line. We use the following notation for the partial differential operators: ∆Φε=∂2Φε 1 ∂x2 1 +∂2Φε 1 ∂x2 2e1+∂2Φε 2 ∂x2 1 +∂2Φε 2 ∂x2 2e2,∆ϕε=∂2ϕε ∂x2 1 +∂2ϕε ∂x2 2 , div(Φε) = ∂Φε 1 ∂x1 +∂Φε 2 ∂x2 ,rot(Φε) = ∂Φε 2 ∂x1 −∂Φε 1 ∂x2 ,∇⊥ϕε=∂ϕε ∂x2 ,−∂ϕε ∂x1, where Φε= (Φε 1,Φε 2) is a vector function and ϕεis a scalar function defined in Ωε. Moreover, for ˜ Φε= (˜ Φε 1,˜ Φε 2) a vector function and ˜ϕεa scalar function defined in e Ωε, after a dilatation in the vertical variable, we will use the following operators ∆ηε˜ Φε= ∂2˜ Φε 1 ∂x2 1 +1 η2 ε ∂2˜ Φε 1 ∂z22!e1+ ∂2˜ Φε 2 ∂x2 1 +1 η2 ε ∂2˜ Φε 2 ∂z22!e2,∆ηε˜ϕε=∂2˜ϕε ∂x2 1 +1 η2 ε ∂2˜ϕε ∂z22, divηε(˜ Φε) = ∂˜ Φε 1 ∂x1 +1 ηε ∂˜ Φε 2 ∂z2 ,rotηε(˜ Φε) = ∂˜ Φε 2 ∂x1 −1 ηε ∂˜ Φε 1 ∂z2 ,∇⊥ ηε˜ϕε=1 ηε ∂˜ϕε ∂z2 ,−∂˜ϕε ∂x1. For ϕ= (ϕ1, ϕ2) and ψ= (ψ1, ψ2), we define ˜ ⊗by (ϕ˜ ⊗ψ)ij =ϕiψj, i = 1, j = 1,2.(2.7) Finally, we introduce some functional spaces. Lq 0is the space of functions of Lqwith zero mean value. Let C∞ #(Z) be the space of infinitely differentiable functions in R3that are Z0-periodic. By Lq #(Z) (resp. W1,q #(Z)), 1< q < +∞, we denote its completion in the norm Lq(Z) (resp. W1,q(Z)) and by Lq 0,#(Z) the space of functions in Lq #(Z) with zero mean value. 4 Igor Paˇzanin and Francisco J. Su´arez-Grau 2.2 The problem The governing equations in dimensionless form are given by                          1 Pr ((uε· ∇)uε+∇pε) = ∆uε+N 1−N(2∇⊥wε+ ∆uε) + Ra Tεe2+fεin Ωε, div(uε) = 0 in Ωε, M Pr (uε· ∇wε) = L∆wε+2N 1−N(rot(uε)−2wε) + gεin Ωε, uε· ∇Tε= ∆Tε+D∇⊥wε· ∇Tεin Ωε. (2.8) We complete the above system with the following boundary conditions on the bottom uε= 0, wε= 0, Tε= 0 on Γ0,(2.9) the following conditions on the lateral boundaries uε·e2= 0, wε= 0, Tε= 0, pε=1 η2 ε qion Σε i, i ={−1/2,1/2},(2.10) and the following boundary conditions on the top boundary uε= 0, wε= 0,∇Tε·n=Nus(Gε−Tε) on Γε 1.(2.11) Here, uε= (uε 1, uε 2) represents the velocity field, pεthe pressure, wεthe microrotation and Tεthe temperature. The external body forces are given by fε= (fε 1, fε 2) and the external body torque by gε. We make the following assumptions: – The Robin boundary condition (2.11)3comes 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 Gε=G(x1/ε) with G∈L2(Z0) a given Z0- periodic and bounded function depending only on the longitudinal variable. – Following previous result [25], we consider that the Nusselt number Nus depends on ε, whereas all the other characteristic numbers are kept independent of ε. In fact, we compare the Nusselt number Nus to the small parameter of the height ηε. Namely, we consider the following scaling of the Nusselt number Nus =ηεk, k =O(1).(2.12) – We assume that the external source functions are independent of the variable x2and take the following scaling fε=1 η2 ε (f1(x1),0), gε=1 η2 ε g(x1),with f1, g ∈L2(ω).(2.13) Under the previous assumptions, the well-posedness of the problem (2.8)-(2.11) can be established using the methods from [23] (see also [22]) and prove that there exists a unique weak solution (uε, wε, pε, T ε)∈H1(Ωε)2× H1 0(Ωε)×L2 0(Ωε)×H1(Ωε). 5 Igor Paˇzanin and Francisco J. Su´arez-Grau Our aim is to study the asymptotic behavior of uε,wε,pεand Tεwhen εand ηεtend to zero and identify homogenized models coupling the effects of the thickness of the domain and the roughness of the boundary. For this, we use the dilatation in the variable x2given by z2=x2 ηε ,(2.14) in order to have the functions defined in the open set with fixed height e Ωεand the rescaled boundaries e Γε 1and e Σε 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                          1 Pr ((˜ uε· ∇ηε)˜ uε+∇ηε˜pε)=∆ηε˜ uε+N 1−N(2∇⊥ ηε˜wε+ ∆ηε˜ uε) + Ra ˜ Tεe2+fεin e Ωε, divηε(˜ uε) = 0 in e Ωε, M Pr (˜ uε· ∇ηε˜wε) = L∆ηε˜wε+2N 1−N(rotηε(˜ uε)−2 ˜wε) + gεin e Ωε, ˜ uε· ∇ηε˜ Tε= ∆ηε˜ Tε+D∇⊥ ηε˜wε· ∇ηε˜ Tεin e Ωε, (2.15) with the boundary conditions ˜ uε= 0,˜wε= 0,˜ Tε= 0 on Γ0,(2.16) ˜ uε·e2= 0,˜wε= 0,˜ Tε= 0,˜pε=1 η2 ε qion e Σε i, i ={−1/2,1/2},(2.17) ˜ uε= 0,˜wε= 0,∇ηε˜ Tε·n=ηεk(Gε−˜ Tε) on e Γε 1,(2.18) The unknown functions in the above system are given by ˜ uε(x1, z2) = uε(x1, ηεz2), ˜pε(x1, z2) = pε(x1, ηεz2), ˜wε(x1, z2) = wε(x1, ηεz2) and ˜ Tε(x1, z2) = Tε(x1, ηεz2) for a.e. (x1, z2)∈e Ωε. Our goal then is to describe the asymptotic behavior of this new sequences ˜ uε, ˜wε, ˜pεand ˜ Tεwhen εand ηεtend 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 (ηε≈ε) in Section 4 and of the sub-critical case (ηεε) in Section 5. 3A 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. 3.1 Estimates for velocity, microroration and temperature To derive the desired estimates, let us recall some well-known technical results (see, e.g. [25]). Lemma 3.1 (Poincar´e and Ladyzhenskaya inequalities).For all ϕ∈H1(Ωε)such that ϕ= 0 on Γ0, there hold the following inequalities kϕkL2(Ωε)≤Cηεk∇ϕkL2(Ωε)2,kϕkL4(Ωε)≤Cη 1 2 εk∇ϕkL2(Ωε)2.(3.19) Moreover, from the change of variables (2.14), there hold the following rescaled estimates k˜ϕkL2( e Ωε)≤Cηεk∇ηε˜ϕkL2( e Ωε)2,k˜ϕkL4( e Ωε)≤Cη 3 4 εk∇ηε˜ϕkL2( e Ωε)2.(3.20) 6 Igor Paˇzanin and Francisco J. Su´arez-Grau Lemma 3.2 (Trace estimates).For all ϕ∈H1(Ωε)such that ϕ= 0 on Γ0, there hold the following trace estimates: kϕkL2(Γε 1)≤Cη 1 2 εk∇ϕkL2(Ωε)2,(3.21) Moreover, for every case, the rescaled function satisfies the following estimate k˜ϕkL2( e Γε 1)≤Cηεε−1 2k∇ηε˜ϕkL2( e Ωε)2,(3.22) Proof. Since the upper boundary Γε 1is not flat, one needs to take into account the variations of the normal direction nin order to estimate the L2-norm of the trace of a function ϕ∈H1(Ωε). Intregrating on vertical lines, we obtain ZΓε 1 |ϕ|2dσ =Zω |ϕ(x1, hε(x1))|2r1 + ηε ε2h0x1 ε 2dx1 ≤C1 + η2 εε−21 2Zω |ϕ(x1, hε(x1))|2dx1 ≤C1 + η2 εε−21 2ZωZhε(x1) 0 ∂x2ϕ(x1, x2)dx2 2 dx1 ≤C1 + η2 εε−21 2ηεZΩε |∂x2ϕ(x1, x2)|2dx1dx2. Then, since ηεεor ηε≈ε, it holds ZΓε 1 |ϕ|2dσ ≤CηεZΩε |∂x2ϕ(x1, x2)|2dx1dx2, which implies (3.21). For the rescaled function, proceeding analogously, we get Ze Γε 1 |˜ϕ|2dσ =Zω |˜ϕ(x1, h(x1/ε))|2s1 + 1 ε2h0x1 ε 2dx1 ≤C1 + ε−21 2Zω |˜ϕ(x1, h(x1/ε))|2dx1 ≤C1 + ε−21 2ZωZh(x1/ε) 0 ∂x2˜ϕ(x1, x2)dx2 2 dx1 ≤C1 + ε−21 2η2 εZe Ωε |η−1 ε∂x2˜ϕ(x1, x2)|2dx1dx2. Then, we have that Ze Γε 1 |˜ϕ|2dσ ≤Cη2 εε−1Ze Ωε |η−1 ε∂x2˜ϕ(x1, x2)|2dx1dx2, which implies (3.22). Corollary 3.3. For Gε(x1) = G(x1/ε)with G∈L2(Z0)and Z0-periodic function, we have the following estimate kGεkL2(Γε 1)≤C . (3.23) Proof. The proof follows the line of estimates (3.21) and (3.21), just taking into account that G∈L2(Z0). 7 Igor Paˇzanin and Francisco J. Su´arez-Grau Lemma 3.4 (A priori estimates).Let (uε, wε, Tε)be the solution of the problem (2.8)-(2.11). Then there hold the following estimates kuεkL2(Ωε)2≤Cη 1 2 ε,k∇uεkL2(Ωε)2×2≤Cη−1 2 ε,(3.24) kwεkL2(Ωε)≤Cη 1 2 ε,k∇wεkL2(Ωε)2≤Cη−1 2 ε,(3.25) kTεkL2(Ωε)≤Cη 5 2 ε,k∇TεkL2(Ωε)2≤Cη 3 2 ε.(3.26) Moreover, from the change of variables (2.14), there hold the following estimates for the rescaled unknowns k˜ uεkL2( e Ωε)2≤C, k∇ηε˜ uεkL2( e Ωε)2×2≤Cη−1 ε,(3.27) k˜wεkL2( e Ωε)≤C, k∇ηε˜wεkL2( e Ωε)2≤Cη−1 ε,(3.28) k˜ TεkL2( e Ωε)≤Cη2 ε,k∇ηε˜ TεkL2( e Ωε)2≤Cηε.(3.29) Proof. We divide the proof in four steps. Step 1. First, we multiply (2.8)3by wε, integrate over Ωεto obtain LZΩε |∇wε|2dx +4N 1−NZΩε |wε|2dx =−M Pr ZΩε (uε· ∇wε)wεdx +2N 1−NZΩε rot(uε)wεdx +1 η2 εZΩε g wεdx. (3.30) For the first term on the right-hand side, since div(uε) = 0 and wε= 0 on ∂Ωε, we get ZΩε (uε· ∇wε)wεdx =1 2ZΩε uε· ∇|wε|2dx =−1 2ZΩε |wε|2div(uε)dx = 0.(3.31) For the rest of the terms of the right-hand side, using the Cauchy-Schwarz inequality and the Poincar´e inequality (3.19), we get ZΩε rot(uε)wεdx≤ k∇uεkL2(Ωε)2×2kwεkL2(Ωε)≤Cηεk∇uεkL2(Ωε)2×2k∇wεkL2(Ωε)2,  1 η2 εZΩε g wεdx1dx2≤η−2 εkgkL2(Ωε)kwεkL2(Ωε)≤Cη−1 2 εk∇wεkL2(Ωε)2. (3.32) Then, taking into account (3.31) and (3.32) in (3.30), we get k∇wεkL2(Ωε)≤Cηεk∇uεkL2(Ωε)+η−1 2 ε.(3.33) Step 2. We multiply (2.8)4by Tε, integrate over Ωεto obtain ZΩε |∇Tε|2dx +ηεkZΓε 1 |Tε|2dσ =−ZΩε (uε· ∇)TεTεdx +DZΩε (∇⊥wε· ∇)TεTεdx +ηεkZΓε 1 GεTεdσ. (3.34) For the first term on the right-hand side of (3.34), since uε= 0 on ∂Ωε, div(uε) = 0 in Ωεand Tε= 0 on Σε i, i=−1/2,1/2, we get ZΩε (uε· ∇)TεTεdx =1 2ZΩε uε· ∇|Tε|2dx =−1 2ZΩε |Tε|2divuεdx = 0.(3.35) 8 Igor Paˇzanin and Francisco J. Su´arez-Grau For the second term on the right-hand side of (3.34), we have ZΩε ∇⊥wε· ∇TεTεdx =1 2ZΩε ∇⊥wε· ∇(Tε)2dx =−1 2ZΩε ∇wε×(∇(Tε)2)dx =1 2ZΩε wεrot(∇(Tε)2)dx −1 2ZΩε rot(wε∇(Tε)2)dx =−1 2ZΩε n×(wε∇(Tε)2)dx =−1 2ZΩε wε∂Tε ∂x2 Tεn1−∂Tε ∂x1 Tεn2dx = 0, (3.36) where we have used that wε= 0 on ∂Ωε, the identity rot(∇(Tε)2) = rot 2∂T ε ∂x1 Tε,2∂Tε ∂x2 Tε= 2 ∂2Tε ∂x2∂x1 Tε+ 2∂Tε ∂x2 ∂Tε ∂x1 −2∂2Tε ∂x1∂x2 Tε−2∂Tε ∂x1 ∂Tε ∂x2 = 0. and ZΩε ∂wε ∂x2 Tεdx1dx2=−ZΩε wε∂Tε ∂x2 dx1dx2. 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 Tεand Corollary 3.3, we get ηεkZΓε 1 GεTεdσ≤ηεkGεkL2(Γε 1)kTεkL2(Γε 1)≤Cη 3 2 εk∇TεkL2(Ωε)2.(3.37) Then, taking into account (3.35) - (3.37) in (3.34), we have k∇TεkL2(Ωε)≤Cη 3 2 ε,(3.38) which is the second estimate in (3.26). From the Poincar´e inequality (3.19), we get the first estimate in (3.26). Step 3. We multiply (2.8)1by uε, integrate over Ωεto obtain 1 1−NZΩε |∇uε|2dx =−1 Pr ZΩε (uε· ∇)uεuεdx +2N 1−NZΩε ∇⊥wε·uεdx +Ra ZΩε Tε(e2·uε)dx +1 η2 εZΩε f1(e1·uε)dx, +1 Pr 1 η2 ε q−1/2ZΣε −1/2 ϕ·e1dx2−1 Pr 1 η2 ε q1/2ZΣε 1/2 ϕ·e1dx2. (3.39) The first term on the right-han side of (3.39) satisfies ZΩε (uε· ∇)uεuεdx1dx2=1 2ZΩε uε· ∇|uε|2dx1dx2=−1 2ZΩε |uε|2div uεdx1dx2= 0.(3.40) We estimate the rest of the terms on the right-hand side of (3.39) by using the Poincar´e inequality (3.19) and using (3.33), we get ZΩε ∇⊥wεuεdx≤ k∇wεkL2(Ωε)kuεkL2(Ωε)≤Cηεk∇wεkL2(Ωε)2k∇uεkL2(Ωε)2×2 ≤Cη2 εk∇uεk2 L2(Ωε)2×2+η 1 2 εk∇uεkL2(Ωε)2×2, ZΩε Tε(e2·uε)dx≤ kTεkL2(Ωε)kuεkL2(Ωε)2≤CηεkTεkL2(Ωε)k∇uεkL2(Ωε)2×2 ≤Cη 7 2 εk∇uεkL2(Ωε)2×2, η−2 εZΩε f1uεdx≤η−2 εkf1kL2(Ωε)kuεkL2(Ωε)2≤Cη−1 2 εk∇uεkL2(Ωε)2×2, (3.41) 9 Igor Paˇzanin and Francisco J. Su´arez-Grau 4 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 λthe proportionality constant, that is ηε≈ε, with ηε/ε →λ, 0 <λ<+∞. Let us introduce some notation which will be useful along this section. For a vectorial function v= (v1, v2) and a scalar w, we introduce the operators ∇λ, ∆λ, divλand by (∇λv)i,1=λ∂z1vi,(∇λv)i,2=∂z2vifor i= 1,2, ∆λv=λ2∂2 z1v+∂2 z2v,∇λw= (λ∂z1w, ∂z2w)t, divλv=λ∂z1v1+∂z2v2,∇⊥ λwε= (∂z2w, −λ∂z1w)t. Next, we give some compactness results about the behavior of the sequences ( ˜ Uε,˜ Wε,˜ θε, pε 0,˜pε 1) and the related unfolding functions (ˆ uε,ˆwε,ˆ Tε,ˆpε 0,ˆpε 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 εstill denote by ε, we have the following convergence results: (i) (Velocity) There exist ˜ U= ( ˜ U1,˜ U2)∈H1(0, hmax;L2(ω)2), with ˜ U= 0 on z2={0, hmax}and ˜ U2= 0, such that ˜ Uε*˜ Uin H1(0, hmax;L2(ω)2),(4.74) ∂x1 Zhmax 0 ˜ U1(x1, z2)dz2!= 0 in ω, (4.75) and ˆ u= (ˆu1,ˆu2)∈L2(ω;H1 #(Z))2, with ˆ u= 0 on z2={0, h(z1)}such that it hold RZˆ u(x1, z)dz = Rhmax 0˜ U(x1, z2)dz2with RZˆu2(x1, z)dz = 0, and moreover ˆ uε*ˆ uin L2(ω;H1(Z)2),(4.76) divλˆ u= 0 in ω×Z, (4.77) ∂x1ZZ ˆu1(x1, z)dz= 0 in ω . (4.78) (ii) (Microrotation) There exist ˜ W∈H1(0, hmax;L2(ω)), with ˜ W= 0 on z2={0, hmax}, such that ˜ Wε*˜ Win H1(0, hmax;L2(ω)),(4.79) and ˆw∈L2(ω;H1 #(Z)), with ˆw= 0 on z2={0, h(z1)}such that RZˆw(x1, z)dz =Rhmax 0˜ W(x1, z2)dz2, and moreover ˆwε*ˆwin L2(ω;H1(Z)).(4.80) (iii) (Temperature) There exist ˜ θ∈H1(0, hmax;L2(ω)), with ˜ θ= 0 on z2={0}, such that η−2 ε˜ θε*˜ θin H1(0, hmax;L2(ω)),(4.81) and ˆ T∈L2(ω;H1 #(Z)), with ˆ T= 0 on z2={0}, such that RZˆ T(x1, z)dz =Rhmax 0˜ θ(x1, z2)dz2, and moreover η−2 εˆ Tε*ˆ Tin L2(ω;H1(Z)).(4.82) 16 Igor Paˇzanin and Francisco J. Su´arez-Grau (iv) (Pressure) There exist three functions ˜p∈L2 0(ω)∩H1(ω), independent of z2with ˜p(i) = qi,i=−1/2,1/2, ˆp0∈L2(ω;H1 #(Z0)) and ˆp1∈L2(ω;L2 #(Z)) such that η2 εpε 0*˜pin H1(ω),(4.83) η2 εε−1∂z1ˆpε 0* ∂z1˜p+∂z1ˆp0in L2(ω;L2(Z0)), ηεˆpε 1*ˆp1in L2(ω;L2(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 [3]. We start with the extension ˜ Uε. Estimates (3.44) imply the existence of ˜ U∈H1(0, hmax;L2(ω)2) such that convergence (4.74) holds, and the continuity of the trace applications from the space of ˜ Usuch that k˜ UkL2and k∂z2˜ UkL2are bounded to L2(Γ1) and to L2(Γ0) implies ˜ U= 0 on Γ1and Γ0. Next, from the free divergence condition divηε(˜ Uε) = 0, it can be deduced that ˜ U2is independent of z2, which together with the boundary conditions satisfied by ˜ U2on z2={0, hmax}implies that ˜ U2= 0. Finally, from the free divergence condition and the convergence (4.74) of ˜ Uε, it is straightforward the corresponding free divergence condition in a thin domain given in (4.75). Concerning ˆ uε, estimates given in (3.57) imply the existence of ˆu∈L2(ω;H1(Z)3) such that convergence (4.76) holds. It can be proved the Z0-periodicity of ˆu, and applying the unfolding change of variables to the free divergence condition divηε˜uε= 0, passing to the limit, we get divergence condition (4.77). Finally, it can be proved that RZˆ u(x1, z)dz =Rhmax 0˜ U(x1, z2)dz2which together with ˜ U2= 0 implies Rhmax 0˜ U2(x1, z2)dz2= 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 z2={0, hmax}are obtained straighfordward. The proofs of the convergence of Wεand identity RZˆw dz =Rhmax 0˜ W dz2are similar to the ones of ˆ uε. 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 pε 0and ˆpε 0, and the classical compactness result for the unfolding method for a bounded sequence in H1, we get convergences for (4.83) and (4.84)1. Estimate for ˆpε 1implies convergence (4.84)2. From the boundary conditions of ˜pεon x1={−1/2,1/2}, the decomposition of the pressure and the convergences of ˆpε 0and ˆpε 1, it holds the boundary conditions for ˜p. Finally, since ˜pεhas mean value zero, from the decomposition of the pressure, we have 0 = Ze Ωε η2 ε˜pεdx1dz2=Zω h(x1/ε)η2 εpε 0dx1+Ze Ωε η2 ε˜pε 1dx1dz2. Taking into account that his x1-periodic, the convergence of η2 εpε 0to ˜pand that Ze Ωε η2 ε˜pε 1dx1dz2≤Cηε→0, we get ZZ0 h dz1Zω ˜p dx1= 0, and so that ˜phas null mean value in ω. Using previous convergences, in the following theorem we give the two-pressured homogenized system satisfied by (ˆ u,ˆw, ˜ P, ˆ T). 17 Igor Paˇzanin and Francisco J. Su´arez-Grau Theorem 4.2 (Limit unfolded problems).In the case ηε≈ε, with ηε/ε →λ,0<λ<+∞, then the functions ˆ u,ˆw, ˆ Tand ˜pgiven in Lemma 4.1 satisfy •(ˆ u,˜p)∈L2(ω;H1 #(Z)2)×(L2 0(ω)∩H1(ω)) is the unique solution of the two-pressure homogenized Stokes problem                                          −1 1−N∆λˆ u+1 Pr ∇λˆq=f1(x1)−1 Pr ∂x1˜p(x1)e1in ω×Z, divλˆ u= 0 in ω×Z, ˆ u= 0 on ω×(ˆ Γ0∪ˆ Γ1), ∂x1ZZ ˆu1(x1, z)dz= 0 in ω, ZZ ˆu2dz = 0 in ω, ˜p(i) = qii=−1/2,1/2, ˆq(x1, z)∈L2(ω;L2 #(Z)). (4.85) •ˆw∈L2(ω;H1 #(Z)) is the unique solution of the Laplace problem (−L∆λˆw=g(x1)in ω×Z, ˆw= 0 on ω×(ˆ Γ0∪ˆ Γ1),(4.86) •ˆ T∈L2(ω;H1 #(Z)) is the unique solution of the nonlinear problem        −∆λˆ T−D∇⊥ λˆw· ∇λˆ T= 0 in ω×Z, ˆ T= 0 on z2=ω׈ Γ0, ∇λˆ T·n=k G(z1)on ω׈ Γ1. (4.87) Proof. We only have to prove (4.85)1, (4.86)1and (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 ηε/ε →λ, 0 <λ<+∞. Step 1. To prove (4.85)1, we consider (3.71) with ϕreplaced by ¯ϕε= (λ(ε/ηε)ϕ1, ϕ2) with ϕ= (ϕ1, ϕ2)∈ D(ω;C∞ #(Z)2). This gives the following variational formulation: 1 1−NZω×Z ηεε−1λ∂z1ˆuε 1∂z1ϕ1dx1dz +1 1−NZω×Z η2 εε−2∂z1ˆuε 2∂z1ϕ2dx1dz +1 1−NZω×Z λ(ε/ηε)∂z2ˆuε 1∂z2ϕ1dx1dz +1 1−NZω×Z ∂z2ˆuε 2∂z2ϕ2dx1dz +1 Pr λ(ε/ηε)Zω×Z η2 εε−1∂z1ˆpε 0∂z1ϕ1dx1dz −1 Pr Zω×Z ληεˆpε 1∂z1ϕ1dx1dz −1 Pr Zω×Z ηεˆpε 1∂z2ϕ2dx1dz =η2 εε−1 Pr Zω×Z ˆuε˜ ⊗ˆuε∂z1¯ϕεdx1dz −ηε Pr Zω×Z ∂z2ˆuε 2ˆ uε·¯ϕεdx1dz +Zω×Z ˆuε 2∂z2ˆ uε¯ϕεdx1dz +2N 1−NZω×Z ηελ(ε/ηε)∂z2ˆwεϕ1dx1dz −2N 1−NZω×Z η2 εε−1∂z1ˆwεϕ2dx1dz +Ra Zω×Z η2 εˆ Tεϕ2dx1dz +Zω×Z λ(ε/ηε)f1ϕ1dx1dz +Oε, (4.88) 18 Igor Paˇzanin and Francisco J. Su´arez-Grau where Oεis devoted to tends to zero when ε→0. Below, let us pass to the limit when εtends 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 ηε/ε →λand λ(ε/ηε)→1, we get that 1 1−NZω×Z ηεε−1λ∂z1ˆuε 1∂z1ϕ1dx1dz +1 1−NZω×Z η2 εε−2∂z1ˆuε 2∂z1ϕ2dx1dz +1 1−NZω×Z λ(ε/ηε)∂z2ˆuε 1∂z2ϕ1dx1dz +1 1−NZω×Z ∂z2ˆuε 2∂z2ϕ2dx1dz converges to 1 1−NZω×Z λ2∂z1ˆ uε·∂z1ϕ dx1dz +1 1−NZω×Z ∂z2ˆ uε·∂z2ϕ dx1dz =1 1−NZω×Z ∇λˆ u· ∇λϕ dx1dz. •For the fifth to eighth terms in the left hand side of (4.88), taking into account that λ(ε/ηε)→1 and the convergences (4.83) and (4.84), we have the following convergences 1 Pr λ(ε/ηε)Zω×Z η2 εε−1∂z1ˆpε 0ϕ1dx1dz →1 Pr Zω×Z (∂x1˜p+∂z1ˆp0)ϕ1dx1dz, −1 Pr Zω×Z ληεˆpε 1∂z1ϕ1dx1dz −1 Pr Zω×Z ˆηεpε 1∂z2ϕ2dx1dz → − 1 Pr Zω×Z ˆp1divλϕ dx1dz. •For the first three terms in the right-hand side of (4.88), by taking into account the estimates (3.57), we get  η2 εε−1 Pr Zω×Z ˆ uε˜ ⊗ˆ uε∂z1¯ϕεdx1dz≤Cη2 εε−1kˆ uεk2 L2(ω×Z)2k∂z1ϕkL∞(ω×Z)2≤Cη2 εε−1→0, and  ηε Pr Zω×Z ∂z2ˆuε 2ˆ uε·¯ϕεdx1dz +Zω×Z ˆuε 2∂z2ˆ uε¯ϕεdx1dz ≤ηεkˆ uεkL2(ω×Z)2k∂z2ˆ uεkL2(ω×Z)2kϕkL∞(ω×Z)2 ≤Cηε→0. Then, we deduce that the convective terms satisfy η2 εε−1 Pr Zω×Z ˆ uε˜ ⊗ˆ uε∂z1¯ϕεdx1dz −ηε Pr Zω×Z ∂z2ˆuε 2ˆ uε·¯ϕεdx1dz +Zω×Z ˆuε 2∂z2ˆ uε¯ϕεdx1dz→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−NZω×Z ηελ(ε/ηε)∂z2ˆwεϕ1dx1dz −2N 1−NZω×Z η2 εε−1∂z1ˆwεϕ2dx1dz →0. •For the sixth term in the right-hand side of (4.88), by taking into account convergence (4.82), we get Ra Zω×Z η2 εˆ Tεϕ2dx1dz →0. •For the last term in the right-hand side of (4.88), by taking into account that λ(ε/ηε)→1, we get Zω×Z λ(ε/ηε)f1ϕ1dx1dz →Zω×Z f1ϕ1dx1dz. 19 Igor Paˇzanin and Francisco J. Su´arez-Grau Therefore, by previous convergences, and taking ˆq:= ˆp0/λ + ˆp1∈L2(ω;L2 #(Z)), we deduce that the limit variational formulation is given by the following one 1 1−NZω×Z ∇λˆ u· ∇λϕ dx1dz +1 Pr Zω×Z ∂x1˜p(e1·ϕ)dx1dz −1 Pr Zω×Z ˆqdivλϕ dx1dz =Zω×Z f1(e1·ϕ)dx1dz. (4.89) By density, (4.89) holds for every function ϕ∈L2(ω;H1 #(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 εtends 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 ηε/ε →λ, we get LZω×Z η2 εε−2∂z1ˆwε∂z1ψ dx1dz +LZω×Z ∂z2ˆwε∂z2ψ dx1dz →LZω×Z ∇λˆw· ∇λψ dx1dz. •For the third term in the left-hand side of (3.72), by using convergence (4.80), we have 4N 1−NZω×Z η2 εˆwεψ dx1dz →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 Pr Zω×Z η2 εε−1ˆuε 1∂z1ˆwεψ dx1dz≤Cη2 εε−1kˆ uεkL2(ω×Z)2k∂z1ˆwεkL2(ω×Z)≤Cηε, −M Pr Zω×Z ηεˆuε 2∂z2ˆwεψ dx1dz≤Cηεkˆ uεkL2(ω×Z)2k∂z2ˆwεkL2(ω×Z)≤Cηε. Thus, we get −M Pr Zω×Z η2 εε−1ˆuε 1∂z1ˆwεψ dx1dz −M Pr Zω×Z ηεˆuε 2∂z2ˆwεψ dx1dz →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−NZω×Z η2 εε−1∂z1ˆuε 2ψ dx1dz≤Cη2 εε−1k∂z1ˆ uεkL2(ω×Z)2≤Cηε, −2N 1−NZω×Z ηε∂z2ˆuε 1ψ dx1dz≤Cηεk∂z2ˆ uεkL2(ω×Z)2≤Cηε. Thus, we have 2N 1−NZω×Z η2 εε−1∂z1ˆuε 2ψ dx1dz −2N 1−NZω×Z ηε∂z2ˆuε 1ψ dx1dz →0. Then, from the above convergences, we get that the limit variational formulation for ˆwis given by LZω×Z ∇λˆw· ∇λψ dx1dz =Zω×Z g ψ dx1dz. (4.90) 20 Igor Paˇzanin and Francisco J. Su´arez-Grau By density, (5.117) holds for every function ψin L2(ω;H1 #(Z)) and is equivalent to problem (4.86). We remark that (5.117) admits a unique solution, and then the complete sequence ˆwεconverges to the unique solution ˆw(x1, z). Before passing to the next step, we need to prove that ∇ηε,ε ˆwεconverges strongly to ∇λˆwin L2(ω×Z)2. To do this, we take ˆwεas test function in (3.72) and ˆwin (5.117). Then, it is easy to prove that lim ε→0Zω×Z |∇ηε,ε ˆwε|2dx1dz =1 LZω×Z gˆw dx1dz =Zω×Z |∇λˆw|2dx1dz This together with the weak convergence of ∇ηε,ε ˆwεto ∇λˆwin L2(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 η2 εε−2Zω×Z η−2 ε∂z1ˆ Tε∂z1φ dx1dz +η−2 εZω×Z ∂z2ˆ Tε∂z2φ dx1dz =−ηεZω×Zˆ uε· ∇ηε,εˆ Tεφ dx1dz +DZω×Z ∇⊥ ηε,ε ˆwε· ∇ηε,ε(η−2 εˆ Tε)φ dx1dz +kZω׈ Γ1 G φ dx1dσ +Oε, (4.91) where Oεtends 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 η2 εε−2Zω×Z η−2 ε∂z1ˆ Tε∂z1φ dx1dz +η−2 εZω×Z ∂z2ˆ Tε∂z2φ dx1dz →Zω×Z ∇λˆ T· ∇λφ dx1dz. •For the first term in the right-hand side of (5.118), by using estimates (3.57) and (3.59), we get −ηεZω×Zˆ uε· ∇ηε,εˆ Tεφ dx1dz≤Cηεkˆ uεkL2(ω×Z)2k∇ηε,ε ˆ TεkL2(ω×Z)≤Cη3 ε, so we have −ηεZω×Zˆ uε· ∇ηε,εˆ Tεφ dx1dz →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 ˆwεto ∇⊥ λˆw, we get DZω×Z ∇⊥ ηε,ε ˆwε· ∇ηε,ε(η−2 εˆ Tε)φ dx1dz →DZω×Z ∇⊥ λˆw· ∇λˆ Tφ dx1dz. Then, using previous convergences, we get that the limit variational formulation for ˆ Tis given by Zω×Z ∇λˆ T· ∇λφ dx1dz =DZω×Z ∇⊥ λˆw· ∇λˆ Tφ dx1dz +kZω׈ Γ1 G(z1)φ dx1dσ. (4.92) By density, (4.92) holds for every function φin L2(ω;H1 #(Y)) and is equivalent to problem (4.87). We remark that (4.92) admits a unique solution, and then the complete sequence ˆ Tεconverges to the unique solution ˆ T(x1, z). Finally, we give the main result concerning the homogenized flow. 21 Igor Paˇzanin and Francisco J. Su´arez-Grau Theorem 4.3 (Main result for the critical case).Consider (˜ U,˜ W, ˜ θ, ˜p)given in Lemma 4.1. Let us define the average velocity, microrotation and temperature respectively by Uav(x1) = Zhmax 0 ˜ U(x1, z2)dz2, Wav(x1) = Zhmax 0 ˜ W(x1, z2)dz2, Tav(x1) = Zhmax 0 ˜ θ(x1, z2)dz2. We have the following: •The average velocity is given by Uav 1=aλ 1−N Pr q−1/2−q1/2+P r Z1/2 −1/2 f1(ξ)dξ!, Uav 2(x1)=0 in ω, (4.93) where aλ∈Ris given by aλ=ZZ |∇λubl(z)|2dz, with (ubl, πbl)∈H1 #(Z)2×L2 #(Z)the solution of the local Stokes problem                  −∆λubl +∇λπbl =e1in Z, divλubl = 0 in Z, ubl = 0 on z2={0, h(z1)}, ZZ ubl 2(z)dz = 0. (4.94) •The pressure ˜pis given by ˜p(x1) = q−1/2− q−1/2−q1/2+Pr Z1/2 −1/2 f1(ξ)dξ!x1+1 2+Pr Zx1 −1/2 f(ξ)dξ in ω. (4.95) •The average microrotation is given by Wav(x1) = bλ 1 Lg(x1)in ω, (4.96) where bλ∈Ris given by bλ=ZZ |∇λwbl(z)|2dz, with wbl ∈H1 #(Z)the solution of the local Laplace problem (−∆λwbl = 1 in Z, wbl = 0 on z2={0, h(z1)}.(4.97) •The average temperature is given by Tav(x1) = ZZ Tbl(x1, z)dz in ω, (4.98) with Tbl ∈L2(ω;H1 #(Z)) the unique solution of the nonlinear local problem          −∆λTbl −D Lg(x1)(∇⊥ λwbl · ∇λ)Tbl = 0 in ω×Z, Tbl = 0 on ω׈ Γ0, ∇λTbl ·n=k G(z1)on ω׈ Γ1. 22 Igor Paˇzanin and Francisco J. Su´arez-Grau Proof. First, we proceed to eliminate the microscopic variable zin the effective linear problems (4.85) and (4.86). To do that, we consider the following identification ˆ u(x1, z) = (1 −N)f1(x1)−1 P r ∂x1˜ P(x1)ubl(z),ˆq(x1, z) = Pr f1(x1)−1 P r ∂x1˜ P(x1)πbl(z), ˆw(x1, z) = g(x1) Lwbl(z), where (ubl, πbl) and wbl satisfies (4.94) and (4.97), respectively. From the identities for velocity RZˆu1(x1, z)dz =Rhmax 0˜ U1(x1, z2)dz2and RZˆu2dz = 0, and for the microrotation RZˆw(x1, z)dz =Rhmax 0˜ W(x1, z2)dz2given in Lemma 4.1, by linearity we deduce that Uav is given by Uav 1=aλ(1 −N)f1(x1)−1 Pr ∂x1˜p(x1)Uav 2= 0,in ω, and Wav is given by (4.96). Next, the divergence condition with respect to the variable x1given in (4.75) together with the expression of Uav 1gives that Uav 1=aλ(1 −N)f1(x1)−1 P r ∂x1˜p(x1)=C1, C1∈R.(4.99) Then, integrating with respecto to x1, and taking into account that ˜p(−1/2) = q−1/2, it holds ˜p(x1) = q−1/2−Pr aλ(1 −N)C1x1+1 2+Pr Zx1 −1/2 f(ξ)dξ. Finally, since ˜p(1/2) = q1/2, we deduce C1=aλ(1 −N) Pr q−1/2−q1/2+P r Z1/2 −1/2 f1(ξ)dξ!. This implies (4.95). Then, by using the expression of ˜p, we deduce that the average velocity Uav 1is given by (4.93). Finally, the formula for Tav follows from (4.93)3and the identity RZˆ T(x1, z)dz =Rhmax 0˜ θ(x1, z2)dz2by renaming Tbl ≡ˆ T. 5 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., ηεε, which is equivalent to λ= 0. We start by giving some compactness results about the behavior of the extended sequences ( ˜ Uε,˜ Wε,˜ θε, pε 0,˜pε 1) and the related unfolding functions (ˆ uε,ˆwε,ˆ Tε,ˆpε 0,ˆpε 1) satisfying the a priori estimates given in Lemmas 3.5 and Lemma 3.9 respectively. Lemma 5.1. For a subsequence of εstill denote by ε, we have the following convergence results: 23 Igor Paˇzanin and Francisco J. Su´arez-Grau (i) (Velocity) There exist ˜ U= ( ˜ U1,˜ U2)∈H1(0, hmax;L2(ω)2), with ˜ U= 0 on z2={0, hmax}and ˜ U2= 0, such that ˜ Uε*˜ Uin H1(0, hmax;L2(ω)2),(5.100) ∂x1 Zhmax 0 ˜ U1(x1, y2)dy2!= 0 in ω. (5.101) and ˆ u= (ˆu1,ˆu2)∈H1(0, h(z1); L2 #(ω×Z0)2), with ˆ u= 0 on z2={0, h(z1)}and ˆu2= 0, such that it hold RZˆ u(x1, z)dz =Rhmax 0˜ U(x1, z2)dz2with RZˆu2(x1, z)dz = 0, and moreover ˆ uε*ˆ uin H1(0, h(z1); L2(ω×Z0)2),(5.102) ∂z1 Zh(z1) 0 ˆu1dz2!= 0 in ω×Z0,(5.103) ∂x1ZZ ˆu1(x1, z)dz= 0 in ω . (5.104) (ii) (Microrotation) There exist ˜ W∈H1(0, hmax;L2(ω)), with ˜ W= 0 on z2={0, hmax}, such that ˜ Wε*˜ Win H1(0, hmax;L2(ω)),(5.105) and ˆw∈H1(0, h(z1); L2 #(ω×Z0)), with ˆw= 0 on z2={0, h(z1)}such that it hold RZˆw(x1, z)dz = Rhmax 0˜ W(x1, z2)dz2, and moreover ˆwε*ˆwin H1(0, h(z1); L2(ω×Z0)).(5.106) (iii) (Temperature) There exist ˜ θ∈H1(0, hmax;L2(ω)), with ˜ θ= 0 on z2={0}, such that η−2 ε˜ θε*˜ θin H1(0, hmax;L2(ω)),(5.107) and ˆ T∈H1(0, h(z1); L2 #(ω×Z0)), with ˆ T= 0 on z2={0}, such that RZˆ T(x1, z)dz =Rhmax 0˜ θ(x1, z2)dz2, and moreover η−2 εˆ Tε*ˆ Tin H1(0, h(z1); L2(ω×Z0)).(5.108) (iv) (Pressure) There exist three functions ˜p∈L2 0(ω)∩H1(ω), independent of z2with with ˜p(i) = qi,i= −1/2,1/2,ˆp0∈L2(ω;H1 #(Z0)) and ˆp1∈L2(ω;L2 #(Z)) such that η2 εpε 0*˜pin H1(ω),(5.109) η2 εε−1∂z1ˆpε 0* ∂z1˜p+∂z1ˆp0in L2(ω;L2(Z0)), ηεˆpε 1*ˆp1in L2(ω;L2(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 divηε(˜uε) = 0 and multiplying by ηε, we get ηε ε∂z1ˆuε 1+∂z2ˆuε 2= 0.(5.111) Passing to the limit, since ηεε, we get ∂z2ˆu2= 0, which means that ˆu2is independent of z2. Due to the boundary conditions on the top and bottom, it holds that ˆu2= 0. Now, multiplying (5.111) by εη−1 εϕwith ϕ independent of z2and integrating by parts, we get Zω×Z0 Zh(z1) 0 ˆuε 1dz2!∂z1ϕ dx1dz1= 0. 24 Igor Paˇzanin and Francisco J. Su´arez-Grau 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 [3] (see also [34]). 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 (ˆ u,ˆw, ˜ P, ˆ T). Theorem 5.2 (Limit unfolded problems).In the case ηεε, then the functions ˆ u,ˆw, ˆ Tand ˜pgiven in Lemma 5.1 satisfy •(ˆ u,˜p)∈H1(0, h(z1); L2 #(ω×Z0)) ×(L2 0(ω)∩H1(ω)) with ˆu2= 0 is the unique solution of the two-pressure homogenized reduced Stokes problem                                    −1 1−N∂2 z2ˆu1+1 Pr ∂z1ˆp0=f1(x1)−1 Pr ∂x1˜p(x1)in ω×Z, ∂z1 Zh(z1) 0 ˆu1dz2!= 0 in ω×Z0, ˆu1= 0 on ω×(ˆ Γ0∪ˆ Γ1), ∂x1ZZ ˆu1(x1, z)dz= 0 in ω, ˜p(i) = qii=−1/2,1/2, ˆp0∈L2 #(ω×Z0). (5.112) •ˆw∈L2(ω;H1 #(Z)) is the unique solution of the Laplace problem (−L∂2 z2ˆw=g(x1)in ω×Z, ˆw= 0 on ω×(ˆ Γ0∪ˆ Γ1),(5.113) •ˆ T∈L2(ω;H1 #(Z)) is the unique solution of the nonlinear problem        ∂2 z2 ˆ T= 0 in ω×Z, ˆ T= 0 on z2=ω׈ Γ0, ∂z2ˆ T=k G(z1)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(x0, z)∈ D(ω;C∞ #(Z)2) with ϕ2= 0 in ω×Z. 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