Asymptotic analysis of the Navier-Stokes equations in a thin domain with power law slip boundary conditions

Asymptotic analysis of the Navier-Stokes equations in a thin domain with power law slip boundary conditions Mar´ıa ANGUIANO∗and Francisco Javier SU´ AREZ-GRAU† Abstract This theoretical study deals with the Navier-Stokes equations posed in a 3D thin domain with thickness 0 < ε 1, assuming power law slip boundary conditions, with an anisotropic tensor, on the bottom. This condition, introduced in (Djoko et al. Comput. Math. Appl. 128 (2022) 198–213), represents a generalization of the Navier slip boundary condition. The goal is to study the influence of the power law slip boundary conditions with an anisotropic tensor of order εγ s, with γ∈Rand flow index 1 < s < 2, on the behavior of the fluid with thickness εby using asymptotic analysis when ε→0, depending on the values of γ. As a result, we deduce the existence of a critical value of γgiven by γ∗ s= 3 −2sand so, three different limit boundary conditions are derived. The critical case γ=γ∗ scorresponds to a limit condition of type power law slip. The supercritical case γ > γ∗ scorresponds to a limit boundary condition of type perfect slip. The subcritical case γ < γ∗ s corresponds to a limit boundary condition of type no-slip. AMS classification numbers: 35Q35, 76A20, 76A05, 76M50. Keywords: Thin domain; homogenization; power law slip boundary conditions; Navier slip boundary conditions; Navier-Stokes. 1 Introduction The stationary Navier-Stokes equations in a domain Ω reads as follows −2νdiv(D[u]) + (u· ∇)u+∇p=fand div(u)=0,(1.1) where udenotes the velocity field, D[u] = 1 2(Du + (Du)T) the deformation tensor associated with the velocity field u,pthe scalar pressure, fthe external forces and ν > 0 the viscosity. Concerning the boundary conditions, it is commonly accepted that viscous fluids adhere to surfaces, and so the no-slip condition at the surfaces of a domain, given by u= 0 on ∂Ω, is widely used. Under suitable regularity conditions on the domain and f, this problem is well studied mathematically, see for instance Boyer & Fabrie [11], Galdi [18] or Temam [26]. However, this condition ∗Departamento de An´alisis Matem´atico. Facultad de Matem´aticas. Universidad de Sevilla. 41012-Sevilla (Spain) [email protected] †Departamento de Ecuaciones Diferenciales y An´alisis Num´erico. Facultad de Matem´aticas. Universidad de Sevilla. 41012-Sevilla (Spain) [email protected] 1 Mar´ıa Anguiano and Francisco J. Su´arez-Grau does not seem always valid physically, indeed some fluids melt and solutions slip against the surface. Also, sometimes the no-slip condition is not good enough because it is not possible to describe the behavior of the fluid near the boundary. Therefore, it is necessary to introduce other type of boundary conditions to describe this behavior. In this sense, Navier [22] proposed the Navier slip boundary conditions in which it is assumed a thin layer of a fluid near of the boundary and the tangential component of the strain tensor should be proportional to the tangential component of the fluid velocity on a part of the boundary Γ ⊂∂Ω, that is 2ν[D[u]n]τ=−λ[u]τ, u ·n = 0,on Γ,(1.2) where n denotes the outside unitary normal vector to Ω on Γ, λ > 0 is the friction coefficient and the subscript τdenotes the orthogonal projection on the tangent space of Γ, i.e. [u]τ=u−(u·n)n. Problem (1.1) with Navier slip boundary conditions (1.2) has been studied by many authors in different contexts, see for example Amrouche & Rejaiba [4], Clopeau et al. [14] and Solonnikov & ˇ Sˇcadilov [23]. Notice that depending on the value of λin (1.2), we shall consider the following type of boundary conditions: •Perfect slip when λ= 0, i.e. 2ν[D[u]n]τ= 0, u ·n = 0,on Γ,(1.3) •Partial slip when λ∈(0,+∞), 2ν[D[u]n]τ=−λ[u]τ, u ·n = 0,on Γ,(1.4) •No-slip when λ= +∞, i.e. [u]τ= 0, u ·n = 0,on Γ, which implies u= 0 on Γ. Related to this, we refer to Acevedo et al. [1] for the study the limiting behavior of the solution (uλ, pλ) of problem (1.1) with Navier slip boundary conditions (1.2), when the friction coefficient λ goes to 0 or ∞. In fact, they proved that (uλ, pλ) weakly converges to (u0, p0) when λ→0 in suitable Sobolev spaces, where (u0, p0) is the solution of the Navier-Stokes system with Navier slip boundary conditions corresponding with λ= 0. Also, it holds that (uλ, pλ) weakly converges to (u∞, p∞) when λ→ ∞, where (u∞, p∞) is the solution of the Navier-Stokes system with no-slip boundary conditions, i.e. the Navier boundary conditions corresponding with λ= +∞. In this work, we are interested in a generalization of the Navier slip condition recently introduced by Djoko et al. [16] (see also Aldbaissy et al. [2, 3] and Djoko et al. [17]), which arises when the contact surface is lubricated with a thin layer of a non-Newtonian fluid. This condition is called power law slip boundary condition and reads as follows 2ν[D[u]n]τ=−|K[u]τ|s−2K2[u]τ, u ·n = 0,on Γ,(1.5) where |v|2=v·vis the Euclidean norm. We observe that in this condition, the tangential shear is a power law function of the tangential velocity, where K∈R2×2is an anisotropic tensor, assumed to be uniformly positive definite, symmetric and bounded, and sis a real, strictly positive number representing the flow behavior index. 2 Mar´ıa Anguiano and Francisco J. Su´arez-Grau The boundary condition (1.5) represents a generalization of the Navier slip boundary condition (1.2), since for s= 2 and K=λ1 2Iwith λ > 0, then the power slip boundary condition (1.5) reduces to Navier slip boundary condition (1.2). We also mentioned that the power slip boundary condition (1.5) is present in the context of laminar flows of Newtonian liquids (e.g. water) over complex surfaces, also when a rough or structured boundary surface is anisotropic, e.g. when it has rows of riblets, pillars or periodic patterns, the effective slip condition is anisotropic, i.e., direction dependent. When the surface is heterogeneous, the effective slip is also position-dependent. This can occur, for example, when the boundary has a varying degree of roughness or when the boundary is a smooth surface with a varying hydrophobic/hydrophilic composition. For instance, we refer to the derivation of effective slip boundary conditions coming from rough boundaries in Bonnivard et al. [8], Bonnivard & Su´arez-Grau [9, 10], Bucur [12], Bucur et al. [13], Dalibard & G´erard-Varet [15] and Su´arez-Grau [24, 25]. The existence of solutions of the Stokes and Navier-Stokes equations with power law boundary conditions (1.5) on a part of the boundary was studied in [16] for 1 <s<2, which corresponds to the tangential shear thinning. In the case s > 2 the existence of solutions is not proven, it was not able to prove a inf-sup conditon, which is the key point to obtain the pressure. In the case s= 2, repeating the classical proof of the existence of solution of the Stokes and Navier-Stokes problem with homogeneous Dirichlet conditions (see for instance [1, Theorem 2.3], [19, Theorem 7.1] and [26, Theorem 10.1]) gives the existence of solution of the Stokes and Navier-Stokes equations with Navier slip boundary conditions. Our main interest in this paper is to study a lubrication problem corresponding to the asymptotic influence of the power law boundary condition (1.5), imposed on a part of the boundary with 1 <s<2, on the behavior of the Navier-Stokes equations through a thin domain Ωε, where the small parameter 0< ε 1 represents the thickness of the domain. More precisely, we consider the following 3Dthin domain (see Figure 1) Ωε={(x1, x2, x3)∈R2×R: (x1, x2)∈ω, 0< x3< εh(x1, x2)}, where ωis a smooth, connected open set of R2and his a smooth and positive function (see Section 2 for more details). To study the influence of the slip boundary conditions on the behavior of the Navier-Stokes equations in the thin domain Ωε(the subscript εis added to the unknowns to stress the dependence of the solution on the small parameter) (−ν∆uε+ (uε· ∇)uε+∇pε=fεin Ωε, div(uε) = 0 in Ωε, where we consider the case of the power law slip boundary conditions (1.5) on Γ0with an anisotropic tensor Kε, depending on ε, of the form Kε=εγ sK, with 1 <s<2 and γ∈R, where Kis assumed to be uniformly positive definite, symmetric and bounded. Thus, the power slip boundary condition, where the tangential shear is a power law function of the tangential velocity with a coefficient depending on ε, is given by 2[D[uε]n]τ=−εγ|K[uε]τ|s−2K2[uε]τ, uε·n=0,on Γ0,(1.6) and no-slip condition on the rest of the boundary, i.e. uε= 0 on ∂Ωε\Γ0. 3 Mar´ıa Anguiano and Francisco J. Su´arez-Grau After the homogenization process (under assumptions given in Section 2) when ε→0 depending on the value of γ, we deduce (see Theorem 4.7) that the limit velocity eu= (eu0,0) and limit pressure ep satisfies the reduced 2D-Stokes system                          −ν∂2 z3eu0(z) = f0(z0)− ∇z0ep(z0) in Ω = {z∈R3:z0∈ω, 0< z3< h(z0)}, divz0 Zh(z0) 0eu0(z)dz3!= 0 in ω, Zh(z0) 0eu0(z)dz3!·n = 0 on ∂ω, eu0= 0 on Γ1=ω× {h(z0)}, (1.7) where eu0= (u1, u2) and f0= (f1, f2). Moreover, we prove the existence of a critical value for γgiven by γ∗ s= 3 −2s, with 1 <s<2,(1.8) which let us derive three different boundary conditions for eu0on the bottom Γ0: •If γ=γ∗ s, then the effective boundary condition on Γ0is a power slip boundary condition with anisotropic tensor K, i.e. −ν∂z3eu0=−|Keu0|s−2K2eu0on Γ0. Thus, to take into account the anisotropy, then Kεhas to be of order O(εγs∗ s). •If γ > γ∗ s, then the effective boundary condition on Γ0is the perfect slip boundary condition, i.e. −ν∂z3eu0= 0 on Γ0. This means that for an anisotropy tensor of order smaller than O(εγ∗ s s), then the fluid does not take into account anisotropy and slides perfectly. •If γ < γ∗ s, then the effective boundary condition on Γ0is the no-slip condition, i.e. eu0= 0 on Γ0. This means that for an anisotropy tensor of order greater than O(εγ∗ s s), then the anisotropy is so strong that the fluid is stopped on the boundary. Observe that for s= 2 and Kε=εγ 2λ1 2I, with λ > 0, where the power slip condition (1.6) reduces to the Navier slip condition with friction parameter λεγ, it holds that the critical value γ∗ 2=−1, which is the critical value for the case of Navier slip boundary condition. Namely, to take into account the friction coefficient λin the effective boundary condition, i.e. −ν∂z3eu0=−λeu0on Γ0, the original friction coefficient has to be of order O(ε−1). If the original friction coefficient λεγis of order smaller than O(ε−1), then the fluid behaves on the boundary as if there were no friction (perfect slippage), i.e. −ν∂z3eu0= 0 on Γ0. 4 Mar´ıa Anguiano and Francisco J. Su´arez-Grau Finally, if the friction coefficient λεγis of order greater than O(ε−1), then the friction coefficient is so strong that the fluid is stopped on the boundary (no-slip condition), i.e. eu0= 0 on Γ0. To prove these results, we first use the multiscale expansion method, which is a formal but powerful tool to analyse homogenization problems, see for instance the application of this method in Bayada & Chambat [7] and Mikeli´c [20]. Next, once the results have been understood, we rigorously justify them by means of the derivation of a priori estimates and some compactness results. As far as the authors know, the flow of a Newtonian fluids with power law slip boundary conditions has not been yet considered in the above described lubrication framework, which represents the main novelty of the paper. We observe that the obtained findings are amenable for the numerical simulations with a considerable simplification with respect to the original problem (which is computationally more expensive), since the effective system (1.7) is a two dimensional ordinary differential system with respect to z3. Therefore, we believe that it could prove useful in the engineering practice as well. The paper is structured as follows. In Section 2, we introduce the statement of the problem. In Section 3, we consider the formal derivation, and in Section 4 we will rigorously justify the results. We finish the paper with a section of references. 2 Formulation of the problem and preliminaries In this section, we first define the thin domain and some sets necessary to study the asymptotic behavior of the solutions. Next, we introduce the problem considered in the thin domain and also, the rescaled problem posed in the domain of fixed height, together with the respective weak variational formulations. The domain and some notation. Along this paper, the points x∈R3will be decomposed as x= (x0, x3) with x0∈R2,x3∈R. We also use the notation x0to denote a generic vector of R2. We consider ωas an open, smooth, bounded and connected set of R2, and a 3D thin domain given by Ωε={(x0, x3)∈R2×R:x0∈ω, 0< x3< hε(x0)}, Here, the function hε(x0) = εh(x0) represents the real gap between the two surfaces. The small parameter εis related to the film thickness. Function his positive and smooth C1bounded function defined for x0. We define the bottom, top and lateral boundaries of Ωεas follows (see Figure 1) Γ0=ω× {0},Γε 1=(x0, x3)∈R3:x0∈ω, x3=hε(x0),Γε `=∂Ωε\(Γ0∪Γε 1). Let us now introduce some notation which will be useful in the following. For a vectorial function ϕ= (ϕ0, ϕ3) and a scalar function φ, we introduce the operators ∆, div, Dand ∇by ∆ϕ= ∆x0ϕ+∂2 x3ϕ, div(eϕ) = divx0(ϕ0) + ∂x3ϕ3, (Dϕ)ij =∂xjϕifor i= 1,2,3, j = 1,2,3, ∇φ= (∇x0φ, ∂x3φ)t. (2.9) We denote by Oεa generic real sequence which tends to zero with εand can change from line to line. We denote by Ca generic positive constant which can change from line to line. 5 Mar´ıa Anguiano and Francisco J. Su´arez-Grau <latexit sha1_base64="q8UZYE0LJFwQOl3P+gAHh2qqQrw=">AAAB93icbVC7TsNAEDzzDOEVHh2NRYREFdkoAsoICiiDRB5SbKL1ZZ2ccmdbd+dIwcq3QIWAjv/gB/gbziEFJEw1OzMr7U6QcKa043xZS8srq2vrhY3i5tb2zm5pb7+p4lRSbNCYx7IdgELOImxopjm2E4kgAo6tYHid+60RSsXi6F6PE/QF9CMWMgraSN3SoXcDQkDXffBGIDFRjOdy2ak4U9iLxJ2RMpmh3i19er2YpgIjTTko1XGdRPsZSM0ox0nRSxUmQIfQx46hEQhUfja9fmKfhLG09QDt6fw7m4FQaiwCkxGgB2rey8X/vE6qw0s/Y1GSaoyoiRgvTLmtYzsvwe4xiVTzsSFAJTNX2nQAEqg2VRXN++78s4ukeVZxzyvVu2q5djUrokCOyDE5JS65IDVyS+qkQSh5JM/kjbxbY+vJerFef6JL1mzngPyB9fENSnOS3Q==</latexit> " 1 <latexit sha1_base64="RvWYISXg+4oWjD3o/hMvvK4sJek=">AAAB6XicbZDNTsJAFIVv8Q/xD3XpppGYuCKtMeqS6EKXmMhPAg25HW5hwkzbzExNCOEhdGXUna/jC/g2DtiFgmf1zT1nkntumAqujed9OYWV1bX1jeJmaWt7Z3evvH/Q1EmmGDVYIhLVDlGT4DE1DDeC2qkilKGgVji6mfmtR1KaJ/GDGacUSBzEPOIMjR21u7coJfa8XrniVb253GXwc6hArnqv/NntJyyTFBsmUOuO76UmmKAynAmalrqZphTZCAfUsRijJB1M5vtO3ZMoUa4Zkjt//85OUGo9lqHNSDRDvejNhv95ncxEV8GEx2lmKGY2Yr0oE65J3Fltt88VMSPGFpApbrd02RAVMmOPU7L1/cWyy9A8q/oX1fP780rtOj9EEY7gGE7Bh0uowR3UoQEMBDzDG7w7I+fJeXFef6IFJ/9zCH/kfHwDZzGNNg==</latexit> 0 <latexit sha1_base64="8gAeHFwLRWPRX5l0TyaYKOambTc=">AAAB7HicbZDNSsNAFIVv6l+tf1WXboJFcFUSEXVZdOOygv2BNJTJ9KYdOpkJM5NCCX0LXYm682l8Ad/Gac1CW8/qm3vOwD03SjnTxvO+nNLa+sbmVnm7srO7t39QPTxqa5kpii0quVTdiGjkTGDLMMOxmyokScSxE43v5n5ngkozKR7NNMUwIUPBYkaJsaOgNyEKU824FP1qzat7C7mr4BdQg0LNfvWzN5A0S1AYyonWge+lJsyJMoxynFV6mcaU0DEZYmBRkAR1mC9WnrlnsVSuGaG7eP/O5iTReppENpMQM9LL3nz4nxdkJr4JcybSzKCgNmK9OOOuke68uTtgCqnhUwuEKma3dOmIKEKNvU/F1veXy65C+6LuX9UvHy5rjdviEGU4gVM4Bx+uoQH30IQWUJDwDG/w7gjnyXlxXn+iJaf4cwx/5Hx8A5iXjyU=</latexit> " Figure 1: Thin domain Ωε, top boundary Γε 1and bottom boundary Γ0 The problem and the rescaling. As stated in the introduction, we consider the 3D stationary Navier-Stokes equations by setting uε= (u0 ε(x), u3,ε(x)), pε=pε(x), at a point x∈Ωε, which is given by        −ν∆uε+ (uε· ∇)uε+∇pε=fεin Ωε, div(uε) = 0 in Ωε, uε= 0 on Γε 1∪Γε `, (2.10) where ν > 0, and the power law slip boundary conditions prescribed on Γ0given by −ν∂x3u0 ε=−εγ|Ku0 ε|s−2K2u0 ε, uε,3= 0,on Γ0,(2.11) where 1 <s<2 and γ∈R. Remark 2.1. Here, we have assumed the following: •Since div(uε) = 0, it holds 2 div(D[uε]) = ∆uε. Then, it also holds 2ν[D[uε] n]τ=ν[Duεn]τ. •Since the bottom boundary Γ0is flat, the outside normal vector n = −e3, where {ei}3 i=1 is the canonical basis in R3, and so uε·n = 0 implies uε,3= 0. Also, the orthogonal projection of a function uεon the tangent space of Γ0is [uε]τ=u0 ε, where u0 ε= (uε,1, uε,2), and then, ν[Duεn]τ=−ν∂x3u0 ε. •Due to the thickness of the domain, it is usual to assume that the vertical components of the external forces can be neglected and, moreover, the forces can be considered independent of the vertical variable. Thus, for sake of simplicity, given f0= (f1, f2)∈L2(ω)2, along the paper we consider the following type of external forces fε(see for instance [24, 25]): fε= (f0(x0),0)t. Definition 2.2. For ε > 0, we say that (uε, pε)defined on Ωεis a weak solution of problem (2.10)– (2.11) if and only if the functions (uε, pε)∈V(Ωε)×L2 0(Ωε), where the corresponding functional space for velocity is V(Ωε) = ϕ∈H1(Ωε)3:ϕ= 0 on ∂Ωε\Γ0, ϕ3= 0 on Γ0, 6 Mar´ıa Anguiano and Francisco J. Su´arez-Grau and the space for pressure L2 0is the space of functions of L2with zero mean value, and satisfy νZΩε Duε:Dϕ dx +ZΩε (uε· ∇)uε·ϕ dx +εγZΓ0 |Ku0 ε|s−2Ku0 ε·Kϕ0dσ −ZΩε pεdiv(ϕ)dx =ZΩε f0·ϕ0dx, ∀ϕ∈V(Ωε), (2.12) and ZΩε div(uε)ψ dx = 0,∀ψ∈L2(Ωε).(2.13) Remark 2.3. Under previous assumptions, for every ε > 0, we have that reference [16, Proposition 2.2] gives the existence of at least one weak solution (uε, pε)of problem (2.10)–(2.11). The objetive of this paper is to study the asymptotic problems for the behavior of the sequence of solutions (uε, pε) of previous problems, when εtends to zero depending on the value of γ. To do that, we introduce a classical change of variables in thin domains, the dilatation z0=x0, z3=ε−1x3.(2.14) This change transforms Ωεinto a fixed domain Ω, defined by Ω = (z0, z3)∈R2×R:z0∈ω, 0< z3< h(z0).(2.15) The boundary of Ω is denoted by ∂Ω, where the top and lateral boundaries of the rescaled domain Ω is defined by Γ1=(z0, z3)∈R2×R:z0∈ω, z3=h(z0),Γ`=∂Ω\(Γ0∪Γ1).(2.16) Accordingly, we define the functions euεand epεby euε(z) = uε(z0, εz3),epε(z) = pε(z0, εz3) a.e. z∈Ω.(2.17) Observe that according to the assumption on fε, it holds e fε(z)=(f0(z0),0) a.e. z∈Ω. Let us now introduce some notation which will be useful in the following. For a vectorial function eϕ= (eϕ0,eϕ3) and a scalar function e φobtained respectively from functions ϕand φby using the change of variables (2.14), we introduce the operators ∆ε, divε,Dεand ∇εby ∆εeϕ= ∆z0eϕ+ε−2∂2 z3eϕ, divε(eϕ) = divz0eϕ0+ε−1∂z3eϕ3, (Dεeϕ)ij =∂zjeϕifor i= 1,2,3, j = 1,2,(Dεeϕ)i3=ε−1∂z3eϕifor i= 1,2,3, ∇εe φ= (∇z0e φ, ε−1∂z3e φ). (2.18) Thus, using the change of variables (2.14), the system (2.10)–(2.11) can be rewritten as        −ν∆εeuε+ (euε· ∇ε)euε+∇εepε=e fεin Ω, divε(euε) = 0 in Ω, euε= 0 on Γ1∪Γ`, (2.19) 7 Mar´ıa Anguiano and Francisco J. Su´arez-Grau with power law slip boundary conditions −ν ε∂z3eu0 ε=−εγ|Keu0 ε|s−2K2eu0 ε,euε,3= 0 on Γ0.(2.20) According to the change of variables (2.14) applied to the weak variational formulations given in Definition 2.2, then, for ε > 0, a rescaled weak solution (euε,epε)∈V(Ω)×L2 0(Ω), where the corresponding functional space for velocity is V(Ω) = ϕ∈H1(Ω)3:ϕ= 0 on ∂Ω\Γ0, ϕ3= 0 on Γ0, satisfies νZΩ Dεeuε:Dεeϕ dz +Ze Ωε (euε· ∇ε)euε·eϕ dz +εγ−1ZΓ0 |Keu0 ε|s−2Keu0 ε·Keϕ0dσ −ZΩepεdivε(eϕ)dz =ZΩ f0·eϕ0dz, (2.21) and ZΩ divε(euε)e ψ dz = 0,(2.22) for every eϕ∈V(Ω) and e ψ∈L2(Ω) obtained from (ϕ, ψ) by the change of variables (2.14). Now, the goal is to study the asymptotic behavior of a sequence of solution (euε,epε) of problem (2.19)–(2.20). In the next section, we will study the asymptotic analysis in a formal way, and in Section 4, we develop the rigorous analysis. 3 The formal asymptotic expansion In this section, we apply the asymptotic expansion method (see for instance [7, 20]) to the Navier-Stokes equations (2.19) with power slip boundary conditions (2.20). We will derive a reduced Stokes system with no-slip condition on the top boundary Γ1and different boundary conditions on Γ0depending on the value of γ. The idea is to assume an expansion in εof the solution (euε,epε) given by euε(z) = εβv0(z) + εv1(z) + ε2v2(z) + · · · ,epε(z) = p0(z) + εp1(z) + ε2p2(z) + ε3p3(z) + · · · (3.23) To determine the effective problem given by functions (v0, p0), the expansion (3.23) is plugged into the PDE, we identify the various powers of εand we obtain a cascade of equations from which we retain only the leading ones that constitute the effective problem. We remark that the value βin the expansion (3.23) will be determined in the next section, by deriving a priori estimates for (euε,epε) (see Lemmas 4.4 and 4.5). Moreover, the effective problem will be justified by corresponding compactness results (see Lemma 4.6 and Theorem 4.7). Theorem 3.1. Assume 1<s<2and define γ∗ s= 3 −2s. Consider (euε,epε)a sequence of solutions of problem (2.19)–(2.20). Assuming the asymptotic expansion of the unknown (euε,epε)in the following form euε(z) = ε2v0(z) + ε3v1(z) + ε4v2(z) + · · · ,epε(z) = p0(z) + εp1(z) + ε2p2(z) + ε3p3(z) + · · · (3.24) 8 Mar´ıa Anguiano and Francisco J. Su´arez-Grau for a.e. z∈Ω, where vi= (¯vi, vi 3)with ¯vi= (vi 1, vi 2), i = 0,1,..., we deduce that the main order pair of functions (v0, p0), with v0 3≡0and p0=p0(z0), satisfies the following effective reduced Stokes problem              −ν∂2 z3¯v0(z) = f0(z0)− ∇x0p0(z0)in Ω, divz0 Zh(z0) 0 ¯v0(z)dz3!= 0 in ω, ¯v0= 0 on Γ1, (3.25) Moreover, ¯v0satisfies the following boundary condition on the bottom Γ0depending on the value of γ: •If γ=γ∗ s, then it holds a power law slip boundary condition −ν∂z3¯v0=−|K¯v0|s−2K2¯v0on Γ0.(3.26) •γ > γ∗ s, then it holds a perfect slip boundary condition −ν∂z3¯v0= 0 on Γ0.(3.27) •γ < γ∗ s, then it holds a no-slip boundary condition ¯v0= 0 on Γ0.(3.28) Proof. We first prove system (3.25). For this, we assume the asymptotic expansion of the unknowns (euε,epε) given by (3.24). Then, substituting the expansion into the problem (2.19)1,2, we get −νε2∆z0(¯v0+O(ε)) −ν∂2 z3(¯v0+O(ε)) + ε3(v0 3+O(ε))∂z3(¯v0+O(ε)) + ∇z0(epε+O(ε)) = f0, −νε2∆z0(v0 3+O(ε)) −ν∂2 z3(v0 3+O(ε)) + ε3(v0 3+O(ε))∂z3(v0 3+O(ε)) + 1 ε∂z3(epε+O(ε)) = 0, ε2divz0(¯v0+O(ε)) + ε∂z3(v0 3+εv1 3+O(ε2)) = 0. (3.29) Collecting the terms of the same order with respect to ε, we have – The main order terms in (3.29)1,2are 1 : −ν∂2 z3¯v0+∇z0p0=f0in Ω, 1 ε:∂z3p0= 0 in Ω.(3.30) – The main and next order terms in (3.29)3are ε:∂z3v0 3= 0 in Ω, ε2: divx0¯v0+∂z3v1 3= 0 in Ω,(3.31) 9 Mar´ıa Anguiano and Francisco J. Su´arez-Grau –Pressure. Estimate (4.46)1implies, up to a subsequence, the existence of ep∈L2(Ω) such that convergence (4.53) holds. Also, from (4.46)2, by noting that ε−1∂z3epεalso converges weakly in H−1(Ω), we deduce ∂z3ep= 0. Then, epis independent of z3. To finish, it remains to prove that p∈L2 0(Ω). Passing to the limit when εtends to zero in ZΩepεdz = 0, we respectively deduce ZΩep(z0)dz =Zω h(z0)ep(z0)dz0= 0, and so that ephas null mean value in Ω. This ends the proof. Next, we prove the main result of this paper. Theorem 4.7 (Main result).The limit pair of functions (eu, ep)∈Vz3×L2 0(Ω), with eu3≡0and ep=ep(z0), given in Lemma 4.6 satisfies the following system in each case:                          −ν∂2 z3eu0(z) = f0(z0)− ∇z0ep(z0)in Ω, divz0 Zh(z0) 0eu0(z)dz3!= 0 in ω, Zh(z0) 0eu0(z)dz3!·n=0 on ∂ω, eu0= 0 on Γ1, (4.54) with ν > 0. Moreover, eu0satisfies the following boundary condition on the bottom Γ0depending on the value of γ: •γ=γ∗ s, then it holds a power law slip boundary condition −ν∂z3eu0=−|Keu0|s−2K2eu0on Γ0,(4.55) where 1< s < 2and the anisotropic tensor K∈R2×2is uniformly positive definite, symmetric and bounded. •γ > γ∗ s, then it holds a perfect slip boundary condition −ν∂z3eu0= 0 on Γ0.(4.56) •γ < γ∗ s, then it holds a no-slip boundary condition eu0= 0 on Γ0.(4.57) Remark 4.8. By uniqueness of solutions of problems given in Theorem 4.7, we observe that the pair of functions (eu, ep)are the same as those functions (v0, p0)obtained in Theorem 3.1 by formal arguments. 16 Mar´ıa Anguiano and Francisco J. Su´arez-Grau Proof of Theorem 4.7. We will divide the proof in three steps. Step 1. Let us first consider the case γ > γ∗ s. From Lemma 4.6, to prove (4.54), it just remains to prove the equations (4.54)1. To do this, we consider eϕ∈C1 c(ω×(0, h(z0)))3such that eϕ3≡0 and eϕ= 0 on Γ1. Thanks to eϕequaling zero for z0outside a compact subset of ω, then eϕ= 0 on ∂Ω\Γ0and eϕ3= 0 on Γ0, and so, we can take it as test function in (2.21), which is given by νZΩ Dεeuε:Dεeϕ dz +ZΩ (euε· ∇ε)euε·eϕ dz +εγ−1ZΓ0 |Keu0 ε|s−2Keu0 ε·Keϕ0dσ −ZΩepεdivε(eϕ)dz =ZΩ f0·eϕ0dz. (4.58) Let us now pass to the limit when εtends to zero in every terms of (4.58): – First term on the left-hand side. Taking into account convergence (4.49), we get νZΩ Dεeuε:Dεeϕ dz =νZΩ ε−2∂z3eu0 ε·∂z3eϕ0dz +Oε=νZΩ ∂z3eu0·∂z2eϕ0dz +Oε. – Second term on the left-hand side. Taking into account the regularity of ϕ0, applying CauchySchwarz’s inequality and estimates (4.43), we get ZΩ (euε· ∇ε)euεeϕ dz≤ keuεkL2(Ω)3kDεeuεkL2(Ω)3×3keϕkL∞(Ω)3≤Cε3,(4.59) which implies ZΩ (euε· ∇ε)euε·eϕ dz →0. – Third term on the left-hand side. We observe that since Γ0is flat, then the surface measure associated to Γ0given by dσ =dz0. From H¨older’s inequality, Kis bounded, the Sobolev embedding L2,→Ls, the trace estimate (4.40)1applied to eϕ0, the trace estimate (4.40)2applied to eu0 ε, and the estimate (4.43) for Dεeu0 ε, we get εγ−1ZΓ0 |Keu0 ε|s−2Keu0 ε·Keϕ0dσ≤εγ−1kKeu0 εks−1 Ls(Γ0)2kKeϕ0kLs(Γ0)2 ≤εγ−1kKks L∞(Γ0)2×2keu0 εks−1 Ls(Γ0)2keϕ0kLs(Γ0)2 ≤Cεγ−1keu0 εks−1 L2(Γ0)2keϕ0kL2(Γ0)2 ≤Cεγ−1εs−1kDεeu0 εks−1 L2(Γ0)3×2kDzeϕ0kL2(Ω)3×2 ≤Cεγ−γ∗ skDzeϕ0kL2(Γ0)3×2 ≤Cεγ−γ∗ s, which tends to zero because γ > γ∗ s. Then, we get εγ−1ZΓ0 |Keu0 ε|s−2Keu0 ε·Keϕ0dσ →0. 17 Mar´ıa Anguiano and Francisco J. Su´arez-Grau – Fourth term on the left-hand side of (4.58). Taking into account that eϕ3≡0 and convergence (4.53), we get ZΩepεdivε(eϕ)dz =ZΩepεdivx0(eϕ0)dz =ZΩepdivx0(eϕ0)dz +Oε. Finally, from the above convergences when ε→0, we derive the following limit system νZΩ ∂z3eu0·∂z3eϕ0dz −ZΩepdivz0(eϕ0)dz =ZΩ f0·eϕ0dz, (4.60) for every eϕ0∈C1 c(ω×(0, h(z0)))2with ϕ0= 0 on Γ1. By density, this equality holds true for every eϕ0∈H1(0, h(z0); L2(ω)2) such that eϕ0= 0 on Γ1. We observe that problem (4.60) has a unique solution (eu0,ep) and the problem is equivalent to (4.54)1with boundary condition (4.56). Uniqueness of solution for (4.63) implies that limit does not depend on the subsequence. Step 2. Next, we consider the case γ < γ∗ s. According to Lemma 4.6, we proceed similarly to the Step 1, but here we also consider eϕ= 0 on Γ0, i.e. we consider a test function eϕ∈C1 c(Ω)3with eϕ3≡0. This means that there is no boundary term in the variational formulation (4.58). Thus, proceeding as Step 1, we deduce the limit variational formulation (4.60), which holds for every eϕ0∈H1 0(Ω)2. This problem has a unique solution and is equivalent to problem (4.54)1with boundary condition (4.57). Uniqueness of solution for (4.63) implies that limit does not depend on the subsequence. Step 3. Finally, we consider the case γ=γ∗ s. Due to the nonlinear boundary term in (4.58), we need to use monotonicity arguments to pass to the limit. For this, to simplify the notation, we define the application ev7→ Aε(ev) as follows (Aε(ev),ew) = νZΩ Dεev:Dεew dz +εγ−1ZΓ0 |Kev0|s−2Kev0·Kew0dσ, for all ev, ew∈H1(Ω)3such that eϕ=ew= 0 on ∂Ω\Γ0and eϕ3≡ew3≡0 on Γ0. From [16, Lemma 2.3], for every ε > 0, the mapping Aεis strictly monotone, i.e. (Aε(v)− Aε(w), v −w)≥0.(4.61) According to Lemma 4.6 and similar to Step 1, we consider eϕ∈C1 c(ω×(0, h(z0)))3with eϕ3≡0 and ϕ= 0 on Γ1, and we choose evεdefined by evε=eϕ−euε, as test function in (4.58). So we get (Aε(euε),evε)−ZΩepεdivε(evε)dz =ZΩ f0·ev0 εdz −ZΩ (euε· ∇ε)euε·evεdz, which is equivalent to (Aε(eϕ)− Aε(euε),evε)−(Aε(eϕ),evε) + ZΩepεdivε(evε)dz =−ZΩ f0·ev0 εdz +ZΩ (euε· ∇ε)euε·evεdz. 18 Mar´ıa Anguiano and Francisco J. Su´arez-Grau Due to (4.61), we can deduce (Aε(eϕ),evε)−ZΩepεdivε(evε)dz ≥ZΩ f0·ev0 εdz −ZΩ (euε· ∇ε)euε·evεdz, i.e. using the expression of Aε, we have νZΩ Dεeϕ:Dεevεdz +εγ−1ZΓ0 |Keϕ0|s−2Keϕ0·Kev0 εdσ −ZΩepεdivε(evε)dz ≥ZΩ f0·ev0 εdz −ZΩ (euε· ∇ε)euε·evεdz. (4.62) Since eϕ3≡0 and divε(euε) = 0 in Ω, it holds ZΩepεdivε(evε)dz =ZΩepεdivz0(eϕ0)dz, and from RΩ(euε· ∇εeuε)euε·euεdz = 0, we deduce that (4.62) reads νZΩ Dεeϕ:Dε(eϕ−euε)dz +εγ−1ZΓ0 |Keϕ0|s−2Keϕ0·K(eϕ0−eu0 ε)dσ −ZΩepεdivz0(eϕ0)dz ≥ZΩ f0·(eϕ0−eu0 ε)dz −ZΩ (euε· ∇ε)euε·eϕ dz. Replacing ϕby ε2ϕand dividing by ε2gives νZΩ ε2Dεeϕ:Dε(eϕ−ε−2euε)dz +εγ−γ∗ sZΓ0 |Keϕ0|s−2Keϕ0·K(eϕ0−ε−2eu0 ε)dσ −ZΩepεdivz0(eϕ0)dz ≥ZΩ f0·(eϕ0−ε−2eu0 ε)dz −ZΩ (euε· ∇ε)euε·eϕ dz. Next, we pass to the limit when εtends to zero: – First term in the left-hand side. From convergence (4.49), we get νZΩ ε2Dεeϕ:Dε(eϕ−ε−2euε)dz =νZΩ ∂z3eϕ0·∂z3(eϕ−ε−2eu0 ε)dz +Oε =νZΩ ∂z3eϕ0·∂z3(eϕ−eu0)dz +Oε. – Second term in the left-hand side. Since γ=γ∗ s, from the continuous embedding of H1(0, h(z0); L2(ω)) into L2(Γ0) and convergence (4.49), we get εγ−γ∗ sZΓ0 |Keϕ0|s−2Keϕ0·K(eϕ0−ε−2eu0 ε)dσ =ZΓ0 |Keϕ0|s−2Keϕ0·K(eϕ0−ε−2eu0 ε)dσ +Oε =ZΓ0 |Keϕ0|s−2Keϕ0·K(eϕ0−eu0)dσ +Oε. 19 Mar´ıa Anguiano and Francisco J. Su´arez-Grau – Third term in the left-hand side. From convergence (4.53), we get ZΩepεdivz0(eϕ0)dz =ZΩepdivz0(eϕ0)dz +Oε, Moreover, since epis independent of z3and from condition (4.50), we have ZΩep(z0) divz0(eu0)dz =Zωep(z0) divz0 Zh(z0) 0eu0dz3!dz0= 0, so the third term is written as follows ZΩepεdivz0(eϕ0)dz =ZΩepdivz0(eϕ0−eu0)dz +Oε. – First term in the right-hand side. From convergence (4.49), we have ZΩ f0·(eϕ0−ε−2eu0 ε)dz =ZΩ f0·(eϕ0−eu0)dz +Oε. – Second term in the right-hand side. From H¨older’s inequality and estimates (4.43), we get ZΩ (euε· ∇ε)euεeϕ dz≤ keuεkL2(Ω)3kDεeuεkL2(Ω)3×3keϕkL∞(Ω)3≤Cε3, which implies ZΩ (euε· ∇ε)euεeϕ dz →0. Finally, from previous convergences, we deduce the following limit variational inequality νZΩ ∂z3eϕ0·∂z3(eϕ−eu0)dz +ZΓ0 |Keϕ0|s−2Keϕ0·K(eϕ0−eu0)dσ −ZΩepdivz0(eϕ0−eu0)dz ≥ZΩ f0·(eϕ0−eu0)dz. Since eϕ0is arbitrary, by Minty’s lemma, see [19, Chapter 3, Lemma 1.2], we deduce νZΩ ∂z3eu0·∂z3eϕ0dz +ZΓ0 |Keu0|s−2Keu0·Keϕ0dσ −ZΩepdivx0(eϕ0)dz =ZΩ f0·eϕ0dz, (4.63) for every eϕ0∈C1 c(ω×(0, h(z0)))2such that eϕ0= 0 on ∂Ω\Γ0. By density, this equality holds true for every eϕ0∈Vz3such that eϕ0= 0 on ∂Ω\Γ0. From [16, Theorem 2.1], problem (4.63) has a unique solution (eu0,ep) and is equivalent to (4.54)1with boundary condition (4.55). Uniqueness of solution for (4.63) implies that limit does not depend on the subsequence. 20 Mar´ıa Anguiano and Francisco J. Su´arez-Grau Remark 4.9. In the case s= 2 and Kε=εγ 2λ1 2I, with λ > 0, where the power slip condition (1.6) reduces to the following Navier slip condition, with friction parameter depending on ε, −ν∂x3u0 ε=−εγu0 ε, uε,3= 0,on Γ0.(4.64) Repeating the classical proof of the existence of solution of the Navier-Stokes problem with homogeneous Dirichlet conditions (see for instance [1, Theorem 2.3], [19, Theorem 7.1] and [26, Theorem 10.1]) gives the existence of at least a weak solution (uε, pε)∈V(Ωε)×L2 0(Ωε)of problem (2.10) and (4.64). Proceeding similarly to the proof of Theorem 4.7, but without monotonicity arguments, we can prove that the limit system satisfied by (eu0,ep)is (4.54), and that there exists a critical value of γis −1, which agrees with γ∗ 2, such that the effective limit conditions on Γ0are the following ones •If γ=−1, then −ν∂z3eu0=−λeu0on Γ0. •If γ > −1, then −ν∂z3eu0= 0 on Γ0. •If γ < −1, then eu0= 0 on Γ0. By classical arguments of the existence and uniqueness of solution for the Stokes problem with homogeneous Dirichlet conditions, there exists a unique weak solution (eu0,ep)of problem (4.54) with corresponding boundary conditions on Γ0depending on the value of γgiven above. Acknowledgments We would like to thank all those people (publishers, editors, referees and researchers) for all the support received for the development of our lines of research. In particular, Mar´ıa would like to dedicate this article to her father, Julio, for his unconditional support. Conflict of interest The authors confirm that there is no conflict of interest to report. Data availability statement Data sharing not applicable to this article as no datasets were generated or analysed during the current study. 21 Mar´ıa Anguiano and Francisco J. Su´arez-Grau References [1] P. Acevedo, C. Amrouche, C. Conca, and A. Ghosh, Stokes and Navier–Stokes equations with Navier boundary conditions, J. Differ. 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