Asymptotic behavior of a viscous fluid near a rough boundary

Asymptotic behavior of a viscous fluid near a rough boundary Casado-D´ıaz, J.1, Luna-Laynez, M.1and Su´arez-Grau, F.J.1 Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, c/Tarfia s/n, 41012 Sevilla, Spain [email protected], [email protected], [email protected] Summary. The purpose of this paper is to study the asymptotic behavior of a viscous fluid satisfying Navier’s condition on a slightly rugous boundary. We consider the case of a fluid of height 1 and the case of the fluid small height ε. In both cases we show that three different behaviors are possible. 1 Introduction For a viscous fluid in a three-dimensional domain with a rugous boundary, it is known that if the normal velocity vanishes on the boundary (slip 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 above assertion was proved in [CFS03] for a boundary described by the equation x3=−εΨx1 ε,x2 ε∀(x1, x2)∈ω, (1) with ε > 0 devoted to converge to zero, ωa Lipschitz bounded open set of R2 and Ψ a smooth periodic function such that Span({∇Ψ(z0) : z0∈R2}) = R2.(2) An extension to non-periodic boundaries was obtained in [BFN10]. Our aim in Section 2 is to generalize the result given in [CFS03] to the case of weak rugosities of small period εand amplitude δεdescribed by Γε=nx= (x1, x2, x3)∈ω×R:x3=−δεΨx1 ε,x2 εo,(3) where ω⊂R2is a Lipschitz bounded open set, Ψ in W2,∞ loc (R2) is a periodic function of period Z0= (−1/2,1/2)2and δε>0 satisfies lim ε→0 δε ε= 0. Taking the oscillating domain Ωεby 2 Casado-D´ıaz, J., Luna-Laynez, M. and Su´arez-Grau, F.J. Ωε=nx= (x1, x2, x3)∈ω×R:−δεΨx1 ε,x2 ε< x3<1o,(4) we show that if the limit of δε/ε3 2tends to infinity and (2) holds, the slip and the adherence boundary conditions are still asymptotically equivalent. However, this result does not hold if the limit λof δε/ε3 2belongs to (0,+∞). In this case we do not have the adherence condition in the limit but the rugosity is large enough to enlarge the friction coefficient in the limit. When δε/ε3 2 converges to zero, we prove that the rugosity is so small that it has no effect on the limit problem. For a related result we refer to [BFNW08]. In Section 3 we will generalize the results obtained in Section 2 to a thin domain of small height ε. Taking ωand Ψ as above, our aim is to study the behavior of the fluid near the rugose periodic boundary of period rεand amplitude δεdefined by Γthin ε=x= (x1, x2, x3)∈ω×R:x3=−δεΨx1 rε ,x2 rε,(5) with rε, δε>0 satisfying lim ε→0 rε ε= 0,lim ε→0 δε rε = 0. Defining Ωthin εby Ωthin ε=x= (x1, x2, x3)∈ω×R:−δεΨx1 rε ,x2 rε< x3< ε,(6) we show analogous results to those presented in Section 2, but in this case the behavior of the fluid near Γthin depends on the limit of a δε rεrε rε . That is, the behavior only depend on the small height of the domain, and not only on the parameters describing the rugose boundary. When εgoes to 1 we recover the previous case. The proofs of the results given in Section 2 can be found in [CLS10] while the generalization to a thin domains given in Section 3 is in [CLS210]. 2 Asymptotic behavior of a viscous fluid in a rugous domain. From now on, the points xof R3are supposed to be decomposed as x= (x0, x3) with x0∈R2,x3∈R. We also use the notation x0to denote a generic vector of R2. Given a bounded connected Lipschitz open set ω⊂R2and Ψ ∈W2,∞ loc (R2), periodic of period Z0= (−1/2,1/2)2, we define the domain Ωεby (4) and its rugose boundary Γεby (3). Then, for f∈L2(ω×(−1,1))3, we consider the Navier-Stokes system in Ωε, Asymptotic behavior of a viscous fluid near a rough boundary 3          −µ∆uε+∇pε+ (uε·∇)uε=fin Ωε,div uε= 0 in Ωε, uε= 0 on ∂Ωε\Γε, uε·ν= 0 on Γε,∂uε ∂ν parallel to νon Γε. (7) Here, µ > 0 corresponds to the viscosity of the fluid and νdenotes the unitary outside normal vector to Ωεon Γε. It is well known that (7) has at least a 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 kuεkH1(Ωε)3+kpεkL2(Ωε)≤C, ∀ε > 0.(8) Our problem is to describe the asymptotic behavior of the sequences uε and pεwhen εtends to zero. This is given by the following theorem which is the main result of this section. Theorem 2.1 We assume that (uε, pε)is a solution of (7). Then, there exists (u, p)∈H1(Ω)3×L2 0(Ω), such that, up to a subsequence, uε* u in H1(Ω)3, pε* p in L2(Ω),where Ω = ω×(0,1).(9) The pair (u, p)satisfies the Navier-Stokes system −µ∆u+∇p+ (u·∇)u=fin Ω,div u= 0 in Ω,(10) the adherence condition u= 0 on ∂Ω\Γand the vertical component of the limit velocity satisfies u3= 0 on Γ, where Γ = ω× {0}. Moreover, denoting (this limit exists at least for a subsequence) λ= lim ε→0 δε ε3 2∈[0,+∞],(11) the tangential component of the limit velocity, u0, also satisfies the following boundary condition on Γ i) If λ= 0, then ∂3u0= 0 on Γ.(12) ii) If λ∈(0,+∞), then defining (b φi,bqi),i= 1,2as a solution of                    −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,+∞)), (13) 4 Casado-D´ıaz, J., Luna-Laynez, M. and Su´arez-Grau, F.J. and R∈R2×2by Rij =µZZ0×(0,+∞) Dzb φi:Dzb φjdz, ∀i, j ∈ {1,2}(14) we have −µ∂3u0+λ2Ru0= 0 on Γ.(15) iii) If λ= +∞, then defining W=Span({(∇Ψ(z0),0) : z0∈Z0}),(16) we have u0∈W⊥on Γ, ∂3u0∈W. (17) Remark 2.2 For λ= 0, the rugosity of Γεis very slight and the solution (uε, pε)of (7) behaves as if Γεcoincides with the plane boundary Γ. For 0< λ < +∞(critical size), the boundary condition satisfied by the limit uof uεon the tangent space to Γcontains the new term λ2Ru0. The effect of the rugosity of the wall Γεis not worthless in this case. Finally, for λ= +∞the rugosity is so strong that the limit uof uεdoes not only satisfies the condition u3= 0 on Γ, but it is also such that its tangent velocity on Γ,u0, is in W⊥, for every z0∈Z0. In particular, if the linear space spanned by Whas dimension 2 (this holds if and only if Ψis not constant in any straight line of R2, see [CFS03]), we get that usatisfies the adherence condition u= 0 on Γ, i.e. although we have imposed a slip condition on Γε, the rugosity forces uto satisfy a no-slip (adherence) condition on Γ. This result extends to the case where lim ε→0 δε ε= 0,lim ε→0 δε ε3 2 = +∞, the results obtained in [CFS03] for δε=ε(see also [BFNW08] for the nonperiodic case). The limit equation (15) corresponding to the critical size λ∈(0,+∞)can be considered as the general one. In fact, if λis tending to zero or +∞in (15) we get (12) and (17) respectively. Remark 2.3 In the cases λ= 0 or +∞, we can prove that the convergences in (9) are strong. In fact, assuming ωsmooth enough (for example C2), we can show that we have ZΩε|uε−u|2dx →0,ZΩε|D(uε−u)|2dx →0,ZΩε|pε−p|2dx →0. In the critical case λ∈(0,+∞), defining ¯uεand ¯pεby ¯uε(x) = u(x) + λ√εu1(x0,0)b φ1(x ε) + u2(x0,0)b φ2(x ε), ¯pε(x) = p(x) + λ √εu1(x0,0)bq1(x ε) + u2(x0,0)bq2(x ε), then the above assertion still holds by replacing uand pby ¯uεand ¯pε, respectively. Asymptotic behavior of a viscous fluid near a rough boundary 5 3 Asymptotic behavior of a viscous fluid in a rugous thin domain. In this section we will generalize the results given in Section 2 to the thin domain Ωthin εgiven by (6) with a rugose boundary Γthin εdescribed by (5). Then, for f= (f0, f3)∈L2(ω)3we consider the Navier-Stokes system          −µ∆uε+∇pε+ (uε·∇)uε=fin Ωthin ε,div uε= 0 in Ωthin ε, uε= 0 on ∂Ωthin ε\Γthin ε, uε·ν= 0 on Γthin ε,∂uε ∂ν parallel to νon Γthin ε. (18) This system has at least a solution (uε, pε)∈H1(Ωε)3×L2 0(Ωε). Moreover it satisfies − ZΩthin ε|uε|2dx ≤Cε4,− ZΩthin ε|Duε|2dx ≤Cε2,− ZΩthin ε|pε|2dx ≤C. (19) As in the previous section, our aim is to study the asymptotic behavior of uε and pεwhen εtends to zero. For this purpose, as usual, we use a dilatation in the variable x3in order to have the functions defined in an open set of fixed height. Namely, 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 = ω×(0,1).(20) Then, our problem is to describe the asymptotic behavior of these sequences ˜uε, ˜pε. This is given by the following theorem. Theorem 3.1 Let (uε, pε)∈H1(Ωε)3×L2 0(Ωε)be a solution of (18) and let ˜uε,˜pεbe defined by (20). Then, there exist v∈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, ˜uε ε*0in H1(Ω)3,˜uε ε2*(v, 0) in H1(0,1; L2(ω))3, ˜uε,3 ε3* w in H2(0,1; H−1(ω)), (21) ˜pε* p in L2(Ω),∂y3˜pε ε* f3in H−1(Ω).(22) According to the value of λthin defined by λthin = lim ε→0 δε rεrε rε∈[0,+∞],(23) the functions v,wand pare given by 6 Casado-D´ıaz, J., Luna-Laynez, M. and Su´arez-Grau, F.J. (i) If λthin = +∞, then denoting by PW⊥the orthogonal projection from R2 to the orthogonal of the space Wdefined by (16), we have that vis given by v(y) = (y3−1) 2µy3I+PW⊥(∇y0p(y0)−f0(y0)) ,a.e. y∈Ω, where psatisfies        −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 0 divy0v(y0, s)ds, in Ω.(24) (ii)If λ∈(0,+∞), then defining (b φi,bqi),i= 1,2, as solutions of the Stokes systems (13) and the matrix Rby (14), we have v(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 (24). (iii)If λ= 0, then v(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. Remark 3.2 The role of the parameter λthin in Theorem 3.1 is similar to the one of λin Theorem 2.1. Indeed, we remember that for ε= 1 both parameters agree (remark that the parameter εin Theorem 2.1 is now called rε). Asymptotic behavior of a viscous fluid near a rough boundary 7 Remark 3.3 In the cases λthin = 0 or +∞, we can prove that the convergences in (21)-(22) are strong. In fact, assuming ωsmooth enough, we prove that defining ¯uε,¯pεby ¯uε(x) = ε2v(x0,x3 ε),0,¯pε(x) = p(x0)a.e. x∈Ωthin ε, we have 1 ε4− ZΩε|uε−¯uε|2dx →0,1 ε2− ZΩε|D(uε−¯uε)|2dx →0,− ZΩε|pε−¯pε|2dx →0. In the critical case λthin ∈(0,+∞), the above assertion still holds replacing ¯uεby ¯uε(x) = ε2v(x0,x3 ε),0+λε√εrεv1(x0,0)b φ1(x rε ) + v2(x0,0)b φ2(x rε ). Acknowledgements This work has been partially supported by the projects MTM2008-00306/MTM of the “Ministerio de Ciencia e Innovaci´on” and FQM309 of the “Junta de Andaluc´ıa”. References [ABLS01] Amirat, Y., Bresch, D., Lemoine, J., Simon, J.: Effect of rugosity on a flow governed by stationary Navier-Stokes equations. Quart. Appl. Math., 59, 769–785 (2001). [ACFS01] Amirat, Y., Climent, B., Fern´andez-Cara, E. Simon, J.: The Stokes equations with Fourier boundary conditions on a wall with asperities. Math. Mod. Meth. Appl. Sci., 24, 255–276 (2001). 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