Francisco Javier Suárez-Grau

Juan Casado-Díaz

Manuel Luna-Laynez

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 348 (2010) 967–971Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comPartial Differential Equations/Mathematical PhysicsA viscous fluid in a thin domain satisfying the slip condition on a slightlyrough boundaryFluide visqueux dans un domaine de faible épaisseur vérifiant la condition de glissementsur une frontière lègèrement rugueuseJuan Casado-Díaz, Manuel Luna-Laynez, Francisco Javier Suárez-GrauDpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spaina r t i c l e i n f o a b s t r a c tArticle history:Received 29 June 2010Accepted after revision 26 July 2010Available online 11 August 2010Presented by Evariste Sanchez-PalenciaWe consider a viscous fluid of small height ε on a periodic rough bottom Γε of periodrε and amplitude δε , δε  rε  ε, where we impose the slip boundary condition. When εtends to zero we obtain a Reynolds system depending on the limit λ of (δε√ε )/(rε√rε ).If λ = +∞, the fluid behaves as if we would impose the adherence condition on Γε . Thisjustifies why this is the usual boundary condition for viscous fluids. If λ = 0 the fluidbehaves as if Γε was plane. Finally, for λ ∈ (0,+∞) it behaves as if Γε was flat but with ahigher friction coefficient.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éOn considère un fluide visqueux de faible épaisseur ε sur un fond rugueux Γε , périodiquede période rε et amplitude δε , δε  rε  ε, où on impose la condition de glissement.Quand ε converge vers zéro on obtient un système de type Reynolds qui dépend de lalimite λ de (δε√ε )/(rε√rε ). Si λ = +∞, le fluide se comporte comme si on aurait imposéla condition d’adhérence sur Γε . Ceci justifie la condition usuelle pour un fluide visqueux.Si λ = 0 le fluide se comporte comme si Γε était plate. Enfin, pour λ ∈ (0,+∞), tout sepasse comme si Γε était plate, mais avec un coefficient de frottement plus élevé.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéePour un fluide visqueux dans un ouvert de R3 à frontière rugueuse, on sait que la condition de glissement et la conditiond’adhérence sont asymptotiquement équivalentes. Ceci donne une justification mathématique de l’imposition usuelle de lacondition d’adhérence pour les fluides visqueux. L’équivalence entre la condition de glissement et la condition d’adhérence aété montrée dans [10] dans le cas d’une frontière rugueuse de période ε et amplitude ε. Une extension au cas de frontièresnon périodiques a été obtenue dans [8]. Dans [11], on a considéré le cas d’une rugosité faible, plus exactement, la frontièreest décrite par une fonction périodique de periode ε mais d’amplitude δε , où δε tend vers zéro. Alors, on a établie que siδε/ε3/2 tend vers l’infini, l’équivalence entre la condition de glissement et la condition d’adhérence est maintenue, maissi δε/ε3/2 converge vers zéro le fluide se comporte comme si la frontière était plate. Dans le cas où δε ∼ ε3/2 la rugositéE-mail addresses: jcasadod@us.es (J. Casado-Díaz), mllaynez@us.es (M. Luna-Laynez), fjsgrau@us.es (F.J. Suárez-Grau).1631-073X/$ – see front matter © 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2010.07.023968 J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 967–971n’est pas assez grande pour impliquer la condition d’adhérence, mais elle est assez grande pour augmenter le coefficient defrottement. Un rèsultat général sur la forme de la limite du système de Navier–Stokes avec des conditions de glissement surune frontière non nécessairement périodique a été obtenue dans [7].Dans cette Note, on généralise les résultats obtenus dans [11] au cas d’un domaine de hauteur ε. Plus exactement, pourun ouvert borné lipschitzien ω ⊂ R2 et une fonction Ψ ∈ W 2,∞loc (R2), périodique de période Z ′ = (−1/2,1/2)2, on définitΩε parΩε ={x = (x′, x3) ∈ ω × R: −δεΨ(x′rε)< x3 < ε},où les paramètres rε , δε vérifient limε→0 rεε = 0, limε→0 δεrε = 0. On considère alors un fluide satisfaisant le système deStokes dans Ωε et la condition de Navier (ou glissement) uε · ν = 0, ∂uε∂ν parallel to ν sur la frontière rugueuse,Γε ={x = (x′, x3) ∈ ω × R: x3 = −δεΨ(x′rε)},où uε = (u′ε, uε,3) est la vitesse et ν la normal extérieure à Ωε sur Γε . Pour simplifier on impose aussi la condition d’adhé-rence uε = 0 sur ∂Ωε \ Γε . Notre but est d’étudier le comportement asymptotique de ce système pour ε tendant vers zéro.On obtient à la limite un système de type Reynolds qui dépend de λ = limε→0 δε√εrε√rε. De là on déduit :Si λ = +∞, le fluide se comporte comme si on supposait Γε = {x3 = 0} avec la condition à la frontière sur Γε , uε ∈W ⊥ × {0}, ∂3u′ε ∈ W avec W = {∇z′Ψ (z′) ∈ R2: z′ ∈ Z ′}. En particulier, si W est de dimension 2, le fluide se comportecomme si on aurait imposé la condition d’adhérence dans Γε .Si λ = 0 on voit que la rugosité n’a pas d’effet à la limite.Si λ ∈ (0,+∞), le fluide se comporte comme si on supposait Γε = {x3 = 0} avec la condition à la frontière sur Γε , uε,3 =0, −μ∂3u′ε + λ2 Ru′ε = 0, où μ est la viscosité du fluide et R une matrix symétrique carrée non negative de dimension 2 quiest définie positive sur l’espace W . On observe que le nouveau terme λ2 Ru′ε est un terme de frottement. Cette conditionà la frontière peut être considerée comme la condition générale puisque quand λ tend vers zéro ou +∞, elle donne lesrésultats antérieurs.Ce résultat est semblable à celui obtenu dans [11] pour un fluide de hauteur fixe, mais la taille critique est différente deδε ∼ r3/2ε qui serait la taille correspondante à [11]. Ceci vient du fait que loin de la frontière rugueuse le comportement dufluide est différent. Dans notre cas on montre que la vitesse est d’ordre ε2 et la pression est d’ordre 1 et ne dépend pas dela profondeur (en une prémière approximation).Pour finir on trouve dans les références [1,2,4–6,13], sur l’étude du comportement des fluides visqueux dans des do-maines à frontière rugueuse, avec des conditions aux limites différentes de celles considérées dans cette Note.1. IntroductionFor a viscous fluid in an open set of R3 with a rugous boundary, it is known that if the normal velocity vanishes onthe boundary (slip condition), then the fluid behaves as if the whole velocity vector vanishes on the boundary (adherencecondition). This gives a mathematical explanation of why it is usual for a viscous fluid to impose the adherence condition.The equivalence between the slip and adherence conditions was proved in [10] for a periodic rough boundary of smallperiod ε and amplitude ε. An extension to non-periodic boundaries was obtained in [8]. In [11] it was considered the caseof a weak roughness, namely the boundary was described by a periodic function of small period ε and amplitude δε , withδε/ε converging to zero. It was proved that if δε/ε3/2 tends to infinity, then the adherence and the slip conditions are stillequivalent, while if δε/ε3/2 tends to zero the fluid behaves as if the boundary was plane. In the critical case δε ∼ ε3/2 theroughness is not so large to imply the adherence condition but it is enough to increase the friction coefficient. A generalresult about the form of the limit equation for the Navier–Stokes system satisfying the slip condition on a (non-necessarilyperiodic) rough boundary has been obtained in [7].Our aim in the present Note is to generalize the results in [11] to the case of a domain of small height ε. Namely, for aLipschitz bounded open set ω ⊂ R2 and a function Ψ in W 2,∞loc (R2), periodic of period Z ′ = (−1/2,1/2)2, we define Ωε byΩε ={x = (x′, x3) ∈ ω × R: −δεΨ(x′rε)< x3 < ε}, (1)where the parameters rε , δε are chosen non-negative and satisfying limε→0 rεε = 0, limε→0 δεrε = 0. We consider a fluidsatisfying the Stokes system in Ωε , the Navier (or slip condition) on the rough boundaryΓε ={x = (x′, x3) ∈ ω × R: x3 = −δεΨ(x′rε)}, (2)and (to simplify) the adherence condition on the rest of the boundary ∂Ωε \ Γε . Our purpose is to study the asymptoticbehavior of this system when ε tends to zero. We show that it depends onJ. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 967–971 969λ = limε→0δεrε√εrε∈ [0,+∞]. (3)If λ = +∞ and the space W = {∇z′Ψ (z′) ∈ R2: z′ ∈ Z ′} agrees with R2, then the fluid behaves as if we impose theadherence condition on the whole ∂Ωε .If λ = 0, then the fluid behaves as if Γε agrees with the plane boundary {x3 = 0}.If λ ∈ (0,+∞) the fluid behaves as if we had considered a plane boundary and added a friction coefficient to the Naviercondition (see Theorem 2.1 and Remark 2).This is analogous to the result proved in [11] for a fluid with fixed height, but the critical size is not δε ∼ r3/2ε whichwould be the expected size from [11]. This is due to the fact that far of the rugous boundary the behavior of the fluid isdifferent from the corresponding one in [11]. Here one can show that the velocity is of order ε2 while the pressure is oforder 1 and does not depend on the depth (in a first approximation).To finish this introduction we refer to [1,2,4–6,13], for the study of viscous fluids in rugous domains satisfying differentboundary conditions from the ones considered in the present paper.2. Main results and some commentsAlong this section, the points x of R3 are supposed to be decomposed as x = (x′, x3) with x′ ∈ R2, x3 ∈ R. We also usethe notation x′ to denote a generic vector of R2.Given a bounded connected Lipschitz open set ω ⊂ R2 and Ψ ∈ W 2,∞loc (R2), periodic of period Z ′ , we define Ωε by (1)and Γε by (2). Then, for f = ( f ′, f3) ∈ L2(ω)3, we consider the Stokes system in Ωε⎧⎨⎩−μuε + ∇pε = f in Ωε, div uε = 0 in Ωε,uε = 0 on ∂Ωε \ Γε, uε · ν = 0 on Γε, ∂uε∂νparallel to ν on Γε.(4)Here ν denotes the unitary outside normal vector to Ωε in Γε and μ > 0 corresponds to the viscosity of the fluid. It is wellknown that (4) has a unique solution (uε, pε) ∈ H1(Ωε)3 × L20(Ωε) (L20(Ωε) denotes the space of functions in L2(Ωε) whoseintegral in Ωε is zero). Moreover, we can show the following estimates:−∫Ωε|uε|2 dx  Cε4, −∫Ωε|Duε|2 dx  Cε2, −∫Ωε|pε|2 dx  C . (5)Our aim is to study the asymptotic behavior of uε and pε when ε tends to zero. For this purpose, as usual, we usea dilatation in the variable x3 in order to have the functions defined in an open set of fixed height. Namely, we takeΩ = ω × (0,1) and we define ũε ∈ H1(Ω)3, p̃ε ∈ L20(Ω) byũε(y) = uε(y′, εy3), p̃ε(y) = pε(y′, εy3), a.e. y ∈ Ω. (6)Then, our problem is to describe the asymptotic behavior of these sequences ũε , p̃ε . This is given by the following theoremwhich is the main result of the present Note:Theorem 2.1. Let (uε, pε) ∈ H1(Ωε)3 × L20(Ωε) be the solution of the Stokes system (4) and let ũε , p̃ε be defined by (6). Then, thereexist v ∈ H1(0,1; L2(ω))2 , w ∈ H2(0,1; H−1(ω)) and p ∈ L20(Ω), where p does not depend on y3 , such that, up to a subsequence,ũεε⇀ 0 in H1(Ω)3,ũεε2⇀ (v,0) in H1(0,1; L2(ω))3, ũε,3ε3⇀ w in H2(0,1; H−1(ω)), (7)p̃ε ⇀ p in L2(Ω),∂y3 p̃εε⇀ f3 in H−1(Ω). (8)According to the value of λ defined by (3), the functions v, w and p are given by:(i) If λ = +∞, then denoting by P W ⊥ the orthogonal projection from R2 to the orthogonal of the space W = {∇z′Ψ (z′) ∈ R2: z′ ∈Z ′}, we have that v and p are given byv(y) = (y3 − 1)2μ(y3 I + P W ⊥)(∇y′ p(y′) − f ′(y′)), a.e. y ∈ Ω,−divy′((13I + P W ⊥)(∇y′ p − f ′))= 0 in ω,(13I + P W ⊥)(∇y′ p − f ′) · ν = 0 on ∂ω.Moreover, the distribution w is given by970 J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 967–971w(y) = −y3∫0divy′ v(y′, s)ds in Ω. (9)(ii) If λ ∈ (0,+∞), then defining (φ̂i, q̂i), i = 1,2, as solutions of the Stokes systems⎧⎪⎪⎨⎪⎪⎩−μzφ̂i + ∇zq̂i = 0 in R2 × (0,+∞), divz φ̂i = 0 in R2 × (0,+∞),φ̂i3(z′,0) + ∂zi Ψ (z′) = 0, ∂z3(φ̂i)′(z′,0) = 0, φ̂i(., z3), q̂i(., z3) periodic of period Z ′,Dzφ̂i ∈ L2(Z ′ × (0,+∞))3×3, q̂i ∈ L2(Z ′ × (0,+∞)),and R ∈ R2×2 byRij = μ∫Z ′×(0,+∞)Dzφ̂i : Dzφ̂ j dz, ∀i, j ∈ {1,2},we havev(y) = (y3 − 1)2μ(y3 I +(I + λ2μR)−1)(∇y′ p(y′) − f ′(y′)), a.e. y ∈ Ω,where p satisfies⎧⎪⎪⎪⎨⎪⎪⎪⎩−divy′((13I +(I + λ2μR)−1)(∇y′ p − f ′))= 0 in ω,(13I +(I + λ2μR)−1)(∇y′ p − f ′) · ν = 0 on ∂ω.Moreover, the distribution w is given by (9).(iii) If λ = 0, thenv(y) = (y23 − 1)2μ(∇y′ p(y′) − f ′(y′)), a.e. y ∈ Ω,where p satisfies−y′ p = −divy′ f ′ in ω, ∂ p∂ν= f ′ · ν on ∂ω.Moreover, the distribution w is zero.Remark 1. An analogous result to Theorem 2.1 is proved in [11] where it is studied the Stokes and Navier–Stokes systemswith the slip condition on a rough boundary for an open set of R3 of fixed height. The functions (φ̂i, q̂i) are the samefunctions which appear in [11] to describe the behavior of the velocity and the pressure near the rough boundary. Moreover,it is proved there that Dzφ̂i , q̂i belong to Lr(Z ′ × (0,+∞))3×3 and Lr(Z ′ × (0,+∞)) respectively for every r  1 and haveexponential decay at infinity.Remark 2. For λ = +∞, Theorem 2.1 shows that uε , pε behave as if in (4) we had assumed that Γε was the plane boundary{x3 = 0} and that the boundary condition on Γε was:uε ∈ W ⊥ × {0} on Γε, ∂3u′ε ∈ W . (10)In particular, if W agrees with R2 (which is true except if Ψ (z1, z2) does not depend on z1 and/or z2) we deduce that theslip condition in (4) is equivalent to the adherence condition uε = 0 on {x3 = 0}.For λ ∈ (0,+∞), Theorem 2.1 shows that the asymptotic behavior of uε and pε is the same that if Γε was the planeboundary {x3 = 0} and the boundary condition on Γε wasuε,3 = 0 on Γε, −μ∂3u′ε + λ2 Ru′ε = 0 on Γε, (11)i.e. although the roughness is not strong enough to deduce that the slip condition on Γε is equivalent to (10), it is sufficientto provide the friction coefficient λ2 Ru′ε in (11).For λ = 0, the roughness is so weak that uε and pε behave as if Γε was plane.The critical size λ ∈ (0,+∞) can be considered as the general one. In fact, the cases λ = 0 and λ = +∞ can be obtainedfrom this one by taking the limit when λ tends to zero and infinity respectively.J. Casado-Díaz et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 967–971 971Remark 3. In the cases λ = 0 or +∞, we can prove that the convergences in (7)–(8) are strong. In fact, assuming ω smoothenough (for example C2), we can show that defining ūε , p̄ε by:ūε(x) =(ε2 v(x′, x3ε),0), p̄ε(x) = p(x′)a.e. x ∈ Ωε,we have1ε4−∫Ωε|uε − ūε|2 dx → 0, 1ε2−∫Ωε∣∣D(uε − ūε)∣∣2 dx → 0, −∫Ωε|pε − p̄ε|2 dx → 0.In the critical case λ ∈ (0,+∞), the above assertion still holds replacing ūε by:ūε(x) =(ε2 v(x′, x3ε),0)+ λε√εrε(v1(x′,0)φ̂1(xrε)+ v2(x′,0)φ̂2(xrε)).Remark 4. The proof of Theorem 2.1 is based on the unfolding method [3,9,12]. For a.e. x′ ∈ R2, we define κ(x′) ∈ Z2 byx′ ∈ κ(x′) + Z ′ . Then, to study the behavior of (uε, pε) near Γε , the idea is to study the behavior of the sequences ûε , p̂εdefined asûε(x′, z) = uε(rεκ(x′rε)+ rεz′, rεz3), p̂ε(x′, z) = pε(rεκ(x′rε)+ rεz′, rεz3),for a.e. x′ ∈ ω, z′ ∈ Z ′ , −(δε/rε)Ψ (z′) < z3 < 1/rε . This is similar to the idea used in [11], but here it is necessary to combinethis change of variables with (6).AcknowledgementsThe authors thank Didier Bresch for interesting discussions and many suggestions concerning this paper. This work waspartially supported by the projects MTM2008-00306/MTM of the “Ministerio de Ciencia e Innovación” and FQM309 of the“Junta de Andalucía”.References[1] Y. Amirat, D. Bresch, J. Lemoine, J. Simon, Effect of rugosity on a flow governed by stationary Navier–Stokes equations, Quart. Appl. Math. 59 (2001)769–785.[2] Y. Amirat, B. Climent, E. Fernández-Cara, J. Simon, The Stokes equations with Fourier boundary conditions on a wall with asperities, Math. ModelsMeth. Appl. Sci. 24 (2001) 255–276.[3] T. Arbogast, J. Douglas, U. Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal. 21(1990) 823–836.[4] G. Bayada, M. 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A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary

Casado-Díaz, Juan

Luna-Laynez, Manuel

Suárez-Grau, Francisco Javier

Numérisation de Documents Anciens Mathématiques

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2010.07.023.pdf

A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary

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