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

On the Navier boundary condition for viscous fluids in a thin domain covered by very small asperities F.J. Su´ arez-Grau Departamento de Matem´ aticas, Universidad de Huelva, Spain [email protected] 1. Introduction FOR a viscous fluid in an open set of R3with a rugous boundary, it is known that if the normal velocity vanishes on the boundary (Navier condition), then the fluid behaves as if the whole velocity vector vanishes on the boundary (adherence condition). This gives a mathematical explanation of why it is usual for a viscous fluid to impose the adherence condition. The equivalence between the Navier and adherence conditions was proved in [2] for a periodic rough boundary of small period εand amplitude ε. In [3] it was considered the case of a weak roughness, namely the boundary was described by a periodic function of small period εand amplitude δε, with δε/ε converging to zero. Our aim in the present work is to study the relation between the Navier and adherence conditions in the case of a domain of small height ε. Namely, for a Lipschitz bounded open set ω⊂R2and a function Ψin W2,∞ loc (R2), periodic of period Z0= (−1/2,1/2)2, we define Ωεby Ωε=x= (x0, x3)∈ω×R:−δεΨx0 rε< x3< ε,(1) where the parameters rε,δεare chosen non-negative and satisfying lim ε→0 rε ε= 0,lim ε→0 δε rε= 0, i.e. δεrεε. We consider a fluid satisfying the Stokes system in Ωε, the Navier condition on the rough boundary Γε=x= (x0, x3)∈ω×R:x3=−δεΨx0 rε (2) and (to simplify) the adherence condition on the rest of the boundary ∂Ωε\Γε,          −µ∆uε+∇pε=fin Ωε,divuε= 0 in Ωε, uε= 0 on ∂Ωε\Γε, uε·ν= 0 on Γε, µ∂uε ∂ν parallel to νon Γε. (3) Here f= (f0, f3)∈L2(ω)3,νdenotes the unitary outside normal vector to Ωεin Γεand µ > 0 corresponds to the viscosity of the fluid. It is well known that (3) has a unique solution (uε, pε)∈H1(Ωε)3×L2 0(Ωε)(L2 0(Ωε)denotes the space of functions in L2(Ωε)whose integral in Ωεis zero). Moreover, we can show the following estimates − ZΩε|uε|2dx ≤Cε4,− ZΩε|Duε|2dx ≤Cε2,− ZΩε|pε|2dx ≤C. (4) Our purpose is to study the asymptotic behavior of this system when εtends to zero. We show that it depends on λ= lim ε→0 δε r 3 2 ε √ε∈[0,+∞].(5) 2. Changes of variables OUR aim is to study the asymptotic behavior of uεand pεwhen εtends to zero. For this purpose, we use a suitable combination of two changes of variables: (1) Far of the rough boundary Γεwe use a dilatation in the variable x3in order to have the functions defined in an open set of fixed height. Namely, we take Ω = ω×(0,1) and we define ˜ uε∈H1(Ω)3,˜ pε∈L2 0(Ω) by ˜ uε(y) = uε(y0, εy3),˜ pε(y) = pε(y0, εy3),a.e. y∈Ω.(6) (2) Near Γεwe use an original adaptation ([3]) of the Unfolding Method ([1], [5]), which is very related to the two-scale convergence method. 3. Main Result Let (uε, pε)∈H1(Ωε)3×L2 0(Ωε)be the solution of the Stokes system (3) and let ˜ uε,˜ pεbe defined by (6). Then, there exist v0∈H1(0,1; L2(ω))2,w∈H2(0,1; H−1(ω)) and p∈L2 0(ω), where pdoes not depend on y3, such that, up to a subsequence, ˜ u0 ε ε2* v0in H1(0,1; L2(ω))2,˜ uε,3 ε3* w in H2(0,1; H−1(ω)), ˜ pε* p in L2(Ω). According to the value of λdefined by (5), we obtain the different expressions for v0and w depending on pwhich satisfies a Reynolds equation: (i)If λ= +∞, then denoting by PW⊥the orthogonal projection from R2to the orthogonal of the space W={∇z0Ψ(z0)∈R2:z0∈Z0}, we have that v0and pare given by v0(y) = (y3−1) 2µy3I+PW⊥∇y0p(y0)−f0(y0),a.e. y∈Ω, −divy01 3I+PW⊥(∇y0p−f0)= 0 in ω, 1 3I+PW⊥(∇y0p−f0)·ν= 0 on ∂ω. Moreover, the distribution wis given by w(y) = −Zy3 0divy0v(y0, s)ds, in Ω.(7) (ii)If λ∈(0,+∞), then defining (b φi,bqi),i= 1,2, as solutions of the Stokes systems          −µ∆zb φi+∇zbqi= 0 in R2×(0,+∞),divzb φi= 0 in R2×(0,+∞), b φi 3(z0,0) + ∂ziΨ(z0)=0, ∂z3(b φi)0(z0,0) = 0,b φi(., z3),bqi(., z3)periodic of period Z0, Dzb φi∈L2(Z0×(0,+∞))3×3,bqi∈L2(Z0×(0,+∞)), and R∈R2×2by Rij =µZZ0×(0,+∞) Dzb φi:Dzb φjdz, ∀i, j ∈ {1,2},we have v0(y) = (y3−1) 2µ y3I+ I+λ2 µR!−1 ∇y0p(y0)−f0(y0),a.e. y∈Ω, where psatisfies                −divy0  1 3I+ I+λ2 µR!−1 (∇y0p−f0) = 0 in ω,  1 3I+ I+λ2 µR!−1 (∇y0p−f0)·ν= 0 on ∂ω. Moreover, the distribution wis given by (7). (iii)If λ= 0, then v0(y) = (y2 3−1) 2µ(∇y0p(y0)−f0(y0)),a.e. y∈Ω, where psatisfies −∆y0p=−divy0f0in ω, ∂p ∂ν =f0·νon ∂ω. Moreover, the distribution wis zero. 4. Conclusions •For λ= +∞, the main result shows that uε,pεbehave as if in (3) we had assumed that Γε was the plane boundary {x3= 0}and that the boundary condition on Γεwas uε∈W⊥×{0}on Γε, ∂3u0 ε∈W. (8) In particular, if Wagrees with R2we deduce that the Navier condition in (3) is equivalent to the adherence condition uε= 0 on {x3= 0}. •For λ∈(0,+∞), the result shows that the asymptotic behavior of uεand pεis the same that if Γεwas the plane boundary {x3= 0}and the boundary condition on Γεwas uε,3= 0 on Γε,−µ∂3u0 ε+λ2Ru0 ε= 0 on Γε,(9) i.e. although the roughness is not strong enough to deduce that the Navier condition on Γεis equivalent to (8), it is sufficient to provide the friction coefficient λ2Ru0 εin (9). •For λ= 0, the roughness is so weak that uεand pεbehave as if Γεwas plane. The critical size λ∈(0,+∞)can be considered as the general one. In fact, the cases λ= 0 and λ= +∞can be obtained from this one by taking the limit when λtends to zero and infinity respectively. 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