Suárez Grau, Francisco Javier

idUS. Depósito de Investigación Universidad de Sevilla

On the Navier boundary condition forviscous fluids in a thin domaincovered by very small asperitiesF.J. Sua´rez-GrauDepartamento de Matema´ticas, Universidad de Huelva, Spainfjsgrau@us.es1. IntroductionFOR a viscous fluid in an open set of R3 with a rugous boundary, it is known that if thenormal velocity vanishes on the boundary (Navier condition), then the fluid behaves asif the whole velocity vector vanishes on the boundary (adherence condition). This gives amathematical explanation of why it is usual for a viscous fluid to impose the adherence con-dition. The equivalence between the Navier and adherence conditions was proved in [2] for aperiodic rough boundary of small period ε and amplitude ε. In [3] it was considered the caseof a weak roughness, namely the boundary was described by a periodic function of smallperiod ε and amplitude δε, with δε/ε converging to zero.Our aim in the present work is to study the relation between the Navier and adherence con-ditions in the case of a domain of small height ε. Namely, for a Lipschitz bounded open setω ⊂ 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 satisfyinglimε→0rεε= 0, limε→0δεrε= 0, i.e. δε  rε  ε.We consider a fluid satisfying the Stokes system in Ωε, the Navier condition on the roughboundaryΓε ={x = (x′, x3) ∈ ω × R : x3 = −δεΨ(x′rε)}(2)and (to simplify) the adherence condition on the rest of the boundary ∂Ωε \ Γε,−µ∆uε +∇pε = f in Ωε, divuε = 0 in Ωε,uε = 0 on ∂Ωε \ Γε,uε · ν = 0 on Γε, µ∂uε∂νparallel to ν on Γε.(3)Here f = (f ′, f3) ∈ L2(ω)3, ν denotes the unitary outside normal vector to Ωε in Γε and µ > 0corresponds to the viscosity of the fluid.It is well known that (3) has a unique solution (uε, pε) ∈ H1(Ωε)3 × L20(Ωε) (L20(Ωε) denotesthe space of functions in L2(Ωε) whose integral in Ωε is zero). Moreover, we can show thefollowing estimates−∫Ωε|uε|2dx ≤ Cε4, −∫Ωε|Duε|2dx ≤ Cε2, −∫Ωε|pε|2dx ≤ C. (4)Our purpose is to study the asymptotic behavior of this system when ε tends to zero. Weshow that it depends onλ = limε→0δεr32ε√ε ∈ [0,+∞]. (5)2. Changes of variablesOUR aim is to study the asymptotic behavior of uε and pε when ε tends to zero. For thispurpose, we use a suitable combination of two changes of variables:(1) Far of the rough boundary Γε we use a dilatation in the variable x3 in order to have thefunctions defined in an open set of fixed height. Namely, we take Ω = ω× (0, 1) and we defineu˜ε ∈ H1(Ω)3, p˜ε ∈ L20(Ω) byu˜ε(y) = uε(y′, εy3), p˜ε(y) = pε(y′, εy3), a.e. y ∈ Ω. (6)(2) Near Γε we use an original adaptation ([3]) of the Unfolding Method ([1], [5]), which isvery related to the two-scale convergence method.3. Main ResultLet (uε, pε) ∈ H1(Ωε)3 × L20(Ωε) be the solution of the Stokes system (3) and let u˜ε, p˜ε bedefined by (6). Then, there exist 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,u˜′εε2⇀ v′ in H1(0, 1;L2(ω))2,u˜ε,3ε3⇀ w in H2(0, 1;H−1(ω)),p˜ε ⇀ p in L2(Ω).According to the value of λ defined by (5), we obtain the different expressions for v′ and wdepending on p which satisfies a Reynolds equation:(i) If λ = +∞, then denoting by PW⊥ the orthogonal projection from R2 to the orthogonal ofthe space W = {∇z′Ψ(z′) ∈ R2 : z′ ∈ Z ′}, we have that v′ and p are given byv′(y) = (y3 − 1)2µ(y3I + PW⊥) (∇y′p(y′)− f ′(y′)) , a.e. y ∈ Ω,−divy′((13I + PW⊥)(∇y′p− f ′))= 0 in ω,(13I + PW⊥)(∇y′p− f ′) · ν = 0 on ∂ω.Moreover, the distribution w is given by w(y) = −∫ y30divy′v(y′, s)ds, in Ω. (7)(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) + ∂ziΨ(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 by Rij = µ∫Z ′×(0,+∞)Dzφ̂i : Dzφ̂j dz, ∀ i, j ∈ {1, 2}, we havev′(y) = (y3 − 1)2µy3I +(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 (7).(iii) If λ = 0, then v′(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.4. Conclusions• For λ = +∞, the main result shows that uε, pε behave as if in (3) we had assumed that Γεwas the plane boundary {x3 = 0} and that the boundary condition on Γε wasuε ∈ W⊥ × {0} on Γε, ∂3u′ε ∈ W. (8)In particular, if W agrees with R2 we deduce that the Navier condition in (3) is equivalent tothe adherence condition uε = 0 on {x3 = 0}.• For λ ∈ (0,+∞), the result shows that the asymptotic behavior of uε and pε is the same thatif Γε was the plane boundary {x3 = 0} and the boundary condition on Γε wasuε,3 = 0 on Γε, −µ∂3u′ε + λ2Ru′ε = 0 on Γε, (9)i.e. although the roughness is not strong enough to deduce that the Navier condition on Γε isequivalent to (8), it is sufficient to provide the friction coefficient λ2Ru′ε in (9).• 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 = 0and λ = +∞ can be obtained from this one by taking the limit when λ tends to zero andinfinity respectively.References[1] 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.[2] J. Casado-Dı´az, E. Ferna´ndez-Cara, J. Simon, Why viscous fluids adhere to rugose walls: A mathematical explanation, J. Differential Equations 189 (2003) 526-537.[3] J. Casado-Dı´az, M. Luna-Laynez, F.J. Sua´rez-Grau, Asymptotic behavior of a viscous fluid with slip boundary conditions on a slightly rough wall, M3AS 20 (2010) 121-156.[4] J. Casado-Dı´az, M. Luna-Laynez, F.J. Sua´rez-Grau, A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary, C. R. Acad. Sci. Paris, Ser. I (2010).[5] D. Cioranescu, A. Damlamian, G. Griso, Periodic unfolding and homogenization, C.R. Acad. Sci. Paris, Ser. I 335 (2002) 99-104.

On the Navier boundary condition for viscous fluids in a thin domain covered by very small asperities

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On the Navier boundary condition for viscous fluids in a thin domain covered by very small asperities

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