Matthieu Bonnivard
email: bonnivard@math.univ-paris-diderot.fr

Anne-Laure Dalibard
email: laure.dalibard@ens.fr

David G Érard-Varet
email: gerard-varet@math.jussieu.fr

COMPUTATION OF THE EFFECTIVE SLIP OF ROUGH HYDROPHOBIC SURFACES VIA HOMOGENIZATION

Keywords: Wall laws, homogenization, effective slip AMS Subject Classification: 35B27, 76D07, 76M50

We present a quantitative analysis of the effect of rough hydrophobic surfaces on viscous newtonian flows. We use a model introduced by Ybert and coauthors in Ref. 20, in which the rough surface is replaced by a flat plane with alternating small areas of slip and noslip. We investigate the averaged slip generated at the boundary, depending on the ratio between these areas. This problem reduces to the homogenization of a non-local system, involving the Dirichlet to Neumann map of the Stokes operator, in a domain with small holes. Pondering on the works of Allaire (see Ref. 2, 3) we compute accurate scaling laws of the averaged slip for various types of roughness (riblets, patches). Numerical computations complete and confirm the analysis.

Introduction

With the development of microfluidics, drag reduction for low Reynolds number flows, notably at solid walls, has become a stimulating issue. Therefore, the interaction between a fluid and a solid boundary has been investigated thoroughly, both at the experimental and theoretical levels. A special attention has been paid to the detection of slip, for various types of flows and solid walls. We refer to Ref. [START_REF] Lauga | Microfluidics: The no-slip boundary condition, Handbook of Experimental Fluid Dynamics[END_REF] for a review.

As a result of this activity, the idea that rough boundaries could generate a substantial slip has spread out. This idea has developped on the basis of both experimental and theoretical works, notably on wall laws. We remind that in the context of roughness effects, a wall law is an effective boundary condition imposed at a smoothened boundary, reflecting the overall impact of the real rough boundary.

In particular, if one describes the rough boundary through an oscillation of small amplitude and wavelength , one can show rigorously that a no-slip condition at the rough boundary can be replaced by a wall law of Navier type, with slip length of order . We refer for instance to articles 1, 14, 7 for more precise statements.

However, these seemingly favorable results must be considered with care. For instance, at the experimental level, one must ensure that the slip is not measured too far away from the boundary. Also, as regards the theoretical works on wall laws, the position of the artificial boundary at which the law is prescribed is crucial. Indeed, when the artificial boundary is moved upwards by a height h = O( ), the effective slip is also increased by h. Let us emphasize that all forementioned works consider artifical boundaries that are at the top of the roughness. As a result, the flow rate in the smoothened domain does not equal the averaged flow rate in the rough domain, making comparisons inaccurate. In fact, in the case of rough wetting surfaces (endowed with a no-slip condition), one can even show the following: if one puts the artificial boundary in a way that the flow rates are the same, then the flat boundary is optimal with respect to drag minimization. We refer to Ref. [START_REF] Bucur | Wall laws for viscous fluids near rough surfaces[END_REF] for detailed statements and proofs. Hence, the possibility of decreasing drag through roughness is not so clear, especially for rough wetting surfaces.

Still, in the recent years, promising results have been obtained concerning a class of rough hydrophobic surfaces, see for instance Ref. [START_REF] Vinogradova | Surface roughness and hydrodynamic boundary conditions[END_REF]. Indeed, by the combination of the chemical and geometrical properties of these surfaces, the hollows of the roughness get filled with gas. Hence, the viscous fluid above does not penetrate: it slips above the hollows, and only sticks at the bumps, reaching the so-called Cassie or fakir state.

The aim of this paper is to study the slip generated by such configurations, both in a rigorous and quantitative manner. We focus on a model proposed in article 20, in which the rough boundary is replaced by a flat plane, divided in small periodic cells (say of side 1). Each cell is divided in two zones:

• A no-slip zone, corresponding to a plane projection of the sticky part of the roughness (bumps). • A slip-zone, corresponding to a plane projection of the slippery part.

Using homogenization techniques, we derive an effective boundary condition as goes to zero, depending on the characteristic scale a of the no-slip zones. We February 19, 2015 18:39 WSPC/INSTRUCTION FILE

BDGV-rev2

Computation of the effective slip of rough hydrophobic surfaces via homogenization 3 provide in this way scaling laws for the slip coefficients, for various configurations (patches, riblets). Such laws are in global agreement with the formal computations led in Ref. [START_REF] Ybert | Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries[END_REF]. One shows notably that the riblet configuration is less effective than patches one (see Remark 2.2). All our theoretical results are grounded by numerical computations at the end of the paper.

Main results

Let us first present the model under study. We consider a three-dimensional Stokes flow between two infinite plates:

-∆u + ∇p = f in Ω, div u = 0 in Ω, (2.1)
where Ω = T 2 × (0, 1) and T 2 = R 2 /Z 2 . We denote by x = (x 1 , x 2 , x 3 ) = (x h , x 3 ) the space variable. The function f ∈ L 2 (Ω) is a given source term. On the upper surface x 3 = 1, we enforce a "no-slip" boundary condition

u |x3=1 = 0. (2.2)
On the lower surface, we assume that u satisfies alternately "perfect slip" and "no slip" boundary conditions, corresponding respectively to the hollows and bumps of the rough hydrophobic surface. More precisely, let > 0 and

S := [0, ) 2 ∼ (R/ ( Z)) 2 ,
the elementary square of side . For simplicity, we shall assume all along that -1 is an integer. Let T be a Lipschitz subdomain of S , modeling an elementary no-slip zone. Details about T will be given right below. From this elementary no-slip zone, we define a global one inside [0, 1) 2 ∼ T 2 :

T := k∈[|0,..., -1 |] 2 ( k + T ) .
Finally, the boundary condition at x 3 = 0 is

u 3|x3=0 = 0, ∂ 3 u h|x3=0 = 0 on (T ) c × {0}, u h|x3=0 = 0 on T × {0}. (2.3) It is easily proved that (2.1)-(2.2)-(2.3) has a unique solution (u , p ) ∈ H 1 (Ω) × L 2 (Ω)/R.
This article is devoted to the asymptotic analysis of (u , p ), as → 0. We will distinguish between two types of no-slip pattern T :

• Patches: we assume that

T :=     2 2     + a T, (2.4)
where 2 2 is the center of the square S , and where the domain T is relatively compact in the square (-1/2, 1/2) 2 , and contains a disk of radius α > 0, centered in the origin (see Figure 1). The parameter a is a positive number such that a < . In this case, the no-slip zone is a union of periodically distributed patches.

• Riblets: we assume that

T := ( T) × 2 + a I . (2.5)
where

I ⊂ (-1 2 , 1 2
) is an open interval (see Figure 2). In this case, the no-slip zone is a union of stripes, invariant in the x 1 -direction. Of course, invariance in the x 2 direction could have been considered as well. Note that later on, addressing the case of riblets, we shall focus on two particular cases:

f = e 1 : riblets parallel to the flow; f = e 2 : riblets perpendicular to the flow.

The issue is to derive a wall law for the system (2.1)-(2.2)-(2.3), that is, to replace the mixed boundary condition (2.3) at x 3 = 0 by a condition which does not depend on . We will show that u behaves asymptotically like the solution ū in H 1 of (2.1)-(2.2), endowed either with a Navier boundary condition

u 3 = 0 at x 3 = 0, ∂ 3 u h = M u h at x 3 = 0, (2.6)
or with a Dirichlet boundary condition

u |x3=0 = 0. (2.7)
In (2.6), M is a 2 × 2 non-negative matrix, whose eigenvalues have the dimension of the inverse of a length. If M = λId, the number λ -1 is called the "slip length". In the general case, the inverse of the eigenvalues provide the slip lengths in the directions of the eigenvectors. We shall denote ūM the solution of (2.1)-(2.2)-(2.6).

We will write ū0 in the special case M = ( 0 0 0 0 ). Eventually, we shall denote ū∞ the solution of (2.1)-(2.2)-(2.7).

With the previous notation, we can state our first result: Theorem 2.1. (Asymptotic behavior for patches) Assume that T := ( 2 , 2 ) + a T , where T (-1/2, 1/2) 2 contains a disc of radius α > 0 centered in the origin. Let u ∈ H 1 (T 2 × (0, 1)) be the solution of (2.1), (2.2), (2.3). One must distinguish between three cases:

(1) Sub-critical case: if a 2 , then u ū0 in H 1 (T 2 × (0, 1)); (2) Super-critical case: if a 2 , then u ū∞ in H 1 (T 2 × (0, 1)); (3
) Critical case: there exists a symmetric, positive definite matrix M 0 such that if a / 2 → C 0 > 0, then u ūC0M0 in H 1 (T 2 × (0, 1)).

Computation of the effective slip of rough hydrophobic surfaces via homogenization 5

, (k 1 + 1)) × [ k 2 , (k 2 + 1)) is defined by k + T = k + a T .
A similar result holds for riblets. Let us merely state the theorem in the critical case: Theorem 2.2. (Asymptotic behaviour for riblets) Assume that T := ( T) × (a I), where I ⊂ (-1/2, 1/2) is an open interval. Suppose that lim →0 -ln(a ) = C 0 > 0, and furthermore that f does not depend on x 1 .

Then, u ūM rib , where

M rib = π C0 0 0 2π C0 . (2.8)
Additionally, when f = e 1 or f = e 2 , the limit system can be simplified:

, (k 1 + 1)) × [ k 2 , (k 2 + 1)) is defined by k + T = k + ( T) × ( 2 + a I).
• if f = e 1 (riblets parallel to the main flow), then ūM rib ,2 = ūM rib ,3 = 0 and ūM rib ,1 satisfies

∂ 3 ūM rib ,1 = π C 0 ūM rib ,1 at x 3 = 0.
Hence, the slip length is C 0 /π; • if f = e 2 (riblets perpendicular to the main flow), then ūM rib ,1 = 0 and ūM rib ,2 satisfies

∂ 3 ūM rib ,2 = 2π C 0 ūM rib ,2 at x 3 = 0.
Hence, the slip length is C 0 /(2π).

Remark 2.1. Notice that in the critical and supercritical cases, the slip length is respectively of order one and infinite in the limit. Therefore large slip is achieved in the limit, which differs from previous papers on the subject (see Ref. Remark 2.4. Theorems 1 and 2 do not support the idea that rough hydrophobic surfaces can generate a substantial slip. Indeed, to obtain an effective slip law, the surface fraction of no-slip has to be very small. Back to wall roughness, this would correspond to narrow peaks separated by (comparatively) large hollows. It seems far from the roughness characteristics used experimentally to obtain a hydrophobic Cassie state.

The proofs of theorems 1 and 2, that rely strongly on the papers 2, 3 by Allaire, are given in Section 3 and 4 respectively. We then present in Section 5 numerical simulations that confirm the asymptotic results, and clarify the influence of the shape of patches on the slip length, i.e. on the eigenvalues of the matrix M .

Asymptotic study of "patch" designs

This section is devoted to the proof of Theorem 2.1. Let (u , p ) ∈ H 1 (Ω)×L 2 (Ω)/R be the solution of (2.1), (2.2), (2.3). By classical arguments, the sequence (u , p ) is uniformly bounded in H 1 (Ω) × L 2 (Ω)/R, and consequently, there exists a couple (u, p)

∈ H 1 (Ω) × L 2 (Ω)/R such that u u weakly in H 1 (Ω), p p weakly in L 2 (Ω)/R.
Using the weak formulation of Eqs. (2.1) and the continuity of the trace operator, one obtains easily that the weak-limit (u, p) satisfies Eqs. (2.1) and boundary condition (2.2) on x 3 = 1. On x 3 = 0, the boundary condition satisfied by the vertical component is preserved in the limit, and we obtain u 3|x3=0 = 0. To describe the boundary condition satisfied by the horizontal components (u 1 , u 2 ) on x 3 = 0, we need to distinguish between the so-called super-critical, critical and sub-critical cases.

Notation.

For every k = (k 1 , k 2 ) ∈ [|0, -1 |] 2
, we denote the elementary squares, cubes and half-cubes as follows:

S k := k + S , P k := S k × (- 2 , 2 ), P ,+ k := P k ∩ R 3 + . (3.1)
We shall use that notation throughout the paper.

Super-critical case: a 2 . The proof in the super-critical case relies on a quantitative Poincaré inequality: we claim that there exist 0 > 0 and a positive function η( ) such that η( ) → 0 as → 0, and such that

T 2 ×{0} |u | 2 ≤ η( ) Ω |∇u | 2 ∀ ∈ (0, 0 ). (3.2)
We provide a proof of this inequality in the Appendix.

Since u is bounded in H 1 (Ω), we immediately infer that ū|x3=0 vanishes in L 2 (T 2 ). Thus (ū, p) is a solution of the Stokes system with homogeneous Dirichlet boundary conditions at x 3 = 0 and x 3 = 1, i.e. ū = ū∞ . Critical and sub-critical cases: a 2 . We follow here the strategy of articles 2, 3 by Allaire. These articles deal with the homogenization of the Stokes equations across a network of balls, with a Dirichlet condition at the surface of the balls. Notably, in section 4 of article 2, the balls are assumed to be distributed along a hypersurface (for instance, 3d balls with centers periodically located on a plane). In the setting considered here, the rough idea is to extend the Stokes solution to the lower half-space by appropriate symmetry: our problem is then reduced to the homogenization of the Stokes equations across a planar network of patches. Hence, the ideas of Ref. 2, devoted to a planar network of balls, essentially apply. They are based on the construction of correctors and the method of oscillating test functions. We start with Lemma 3.1 (Existence of correctors). Assume that a 2 . For every > 0,

there exist W = (W i,j ) 1≤i,j≤3 ∈ H 1 (Ω) 9 , q = (q j ) 1≤j≤3 ∈ L 2 (Ω) 3 , supported in T 2 × [-/2, / 2
], which satisfy the following properties: 3 and every sequence ψ ∈ H 1 (Ω) 3 satisfying the boundary conditions

(i) W 0 weakly in H 1 (Ω), q 0 weakly in L 2 (Ω); (ii) for every j = 1 . . . 3, i ∂ i W ij = 0 in Ω; (iii) for 1 ≤ i, j ≤ 3, W 3j = W i3 = 0 on T 2 ×{0}, and for 1 ≤ i, j ≤ 2, W ij = δ ij on T × {0}; (iv) For every φ ∈ C ∞ (Ω) 3 , every ψ ∈ H 1 (Ω)
ψ = 0 on (T × {0}) ∪ (T 2 × {1}), ψ 3 = 0 on T 2 × {0, 1}, (3.3)
and converging weakly to ψ in H 1 (Ω) 3 , the following relation holds: if

lim →0 a / 2 = C 0 ≥ 0, then lim →0 1≤i,j≤3 Ω ∇W ij • ∇ψ i φ j - Ω ∂ i ψ i q j φ j = -C 0 T 2 ×{0} M 0 ψ h • φ h . (3.4) where M 0 ∈ M 2 (R)
is the symmetric definite positive matrix given by formula (3.19).

Proof. [Proof of Lemma 3.1] This lemma is the analogue of Proposition 4.1.6 in Ref. 3 (see also section 2.3 in Ref. 2). As mentioned before, we do not claim any major novelty in the proof. Nevertheless, with regards to quantitative aspects, notably the exact expression of the slip matrix C 0 M 0 , we feel necessary to reproduce its main steps.

The starting idea is to consider a base flow (W, q) in the vicinity of T , which, after proper rescaling, will describe accurately the corrector behavior near a single patch. We shall then truncate it and periodize so as to obtain an appropriate global corrector. Namely, we introduce the solution (W, q), with W = (W ij ) 1≤i,j≤3 , q = Computation of the effective slip of rough hydrophobic surfaces via homogenization 9 (q i ) 1≤i≤3 , of the following problem:

-∆W ij + ∂ i q j = 0 in R 3 + , 1 ≤ i, j ≤ 3, (3.5) 3 i=1 ∂ i W ij = 0 in R 3 + , 1 ≤ j ≤ 3, (3.6)
completed with the boundary conditions

W ij = δ ij on T × {0}, 1 ≤ i, j ≤ 2, (3.7)
W i3 = W 3j = 0 on R 2 × {0}, 1 ≤ i, j ≤ 3, (3.8)
∂ 3 W ij = 0 on (R 2 \ T ) × {0}, 1 ≤ i, j ≤ 2. (3.9)
as well as lim |x|→∞ W = 0. Of course, for j = 3, we have W i3 ≡ 0 and q 3 ≡ 0. For j = 1, 2, the existence of a unique weak solution

(W •j , q j ) ∈ D 1,2 (R 3 + ) 3 × L 2 loc (R 3 + )/R follows from Lax-Milgram theorem. Following Ref. 11, we remind that D 1,2 (R 3 + ) is the closure of D(R 3 + ) in Ḣ1 (R 3 + ).
Asymptotic behaviour of W ij , q j . For j = 1 . . . 3, we extend q j , W 1j and W 2j into even functions of x 3 , and W 3j into an odd function of x 3 . We obtain in this way solutions of the Stokes equations on R 3 \ T . Proceeding exactly as in page 255 of Ref. 2, we obtain the following asymptotic expansions

W •j (x) = 1 8π F j |x| + (x • F j )x |x| 3 + O 1 |x| 2
as |x| → ∞.

(3.10)

q j (x) = 1 4π x • F j |x| 3 + O 1 |x| 3 as |x| → ∞, (3.11)
In formulas (3.10)-(3.11), the notation F j corresponds to the drag force, which is defined by (here, n + := e 3 , n -:= -e 3 ):

F j = - T ×{0 + } ∂W •j ∂n + - T ×{0 -} ∂W •j ∂n - + T ×{0 + } q j n + + T ×{0 -} q j n - = -2 T ×{0 + } ∂ 3 W •j .
(3.12)

Construction of W and q . Using the extended W and q, defined in the whole of R 3 , we can then proceed exactly as in Ref. 2, 3 to construct the correctors W and q . Therefore, we consider the following decomposition of P k (see definition (3.1)):

P k = C k ∪ D k ∪ K k ,
where C k is the ball of radius /4 centered in the cube, D k is the ball of radius /2, with same center, perforated by C k , and K k is the remaining part of the cube, that is K k = P k \ D k (see Figure 3). We denote by c k the center of cube P k . In each part of the cube, we define W •j and q j as follows:

W •j (x) = W •j ( x-c k a ) q j (x) = 1 a q j ( x-c k a ) ∀x ∈ C k , , ∇q j -∆W •j = 0 div W •j = 0 in D k , W •j = 0 q j = 0 in K k .
Moreover, we impose D k q j = 0 and W •j ∈ H 1 (P k ) 3 (so that there is no jump of W across ∂D k , ∂C k ).

Estimates on W and q . We use again the decomposition

P k = C k ∪ (P k \ C k ).
The estimates in C k follow from the asymptotic expansions (3.10)-(3.11) and a scaling argument: for every > 0,

∇W •j 2 L 2 (C k ) ≤ Ca , q j 2 L 2 (C k ) ≤ Ca , W •j 2 L 2 (C k ) ≤ Ca 2 ,
where C > 0 is a constant. To treat the remaining part P k \ C k , we use a properly rescaled version of standard estimates for the homogeneous Stokes equations: basically, the L 2 norm, resp. H 1 norm of the solution is controlled by the L 2 norm, resp. H 1/2 norm of the boundary data (see for instance Ref. 18). Since the velocity fields W •j satisfy the following pointwise asymptotics as vanishes

W •j = O a on ∂C k ∩ ∂D k , ∇W •j = O a 2 on ∂C k ∩ ∂D k ,
using a scaling argument, we obtain the following estimates for a given constant C > 0. Since 0 < a < , we deduce

∇W •j 2 L 2 (P k \C k ) ≤ C a 2 , q j 2 L 2 (P k \C k ) ≤ C a 2 , W •j 2 L 2 (P k \C k ) ≤ Ca
∇W •j 2 L 2 (P k ) ≤ Ca , q j 2 L 2 (P k ) ≤ Ca , W •j 2 L 2 (P k ) ≤ Ca 2 .
As a result, summing over k ∈ [|0, -1 |] 2 , we obtain the following asymptotics as vanishes 3 and let ψ ∈ H 1 (Ω) 3 be a sequence of vector fields satisfying the boundary conditions (3.3), and converging weakly to ψ in H 1 (Ω) 3 . In the sub-critical case a 2 , the asymptotics (3.14) imply that W → 0 strongly in H 1 (Ω) 9 , q → 0 strongly in L 2 (Ω).

∇W 2 L 2 (Ω) = O a 2 , q 2 L 2 (Ω) = O a 2 , W 2 L 2 (Ω) = O a 2 . (3.14) Conclusion of the proof. Let φ ∈ C ∞ (Ω) 3 , ψ ∈ H 1 (Ω)
Consequently, the following relation holds:

lim →0 1≤i,j≤3 Ω ∇W ij • ∇ψ i φ j - Ω ∂ i ψ i q j φ j = 0.
Thus, relation (3.4) holds with M = 0.

In the critical case lim →0 a 2 = C 0 > 0, we define Ω = T 2 × (-1, 1), and we extend φ j and ψ j into even functions of x 3 on Ω for j = 1, 2, and φ 3 and ψ 3 into odd functions of x 3 . First, asymptotics (3.14) imply that W is bounded in H 1 , and therefore converges weakly in H 1 , up to a subsequence. Since W vanishes in L 2 (Ω), we obtain ∇W 0 weakly in L 2 (Ω) 9 . From (3.14), we also infer q j 0 weakly in L 2 (Ω), and thus the following identity holds for every 1

≤ i ≤ 3, 1 ≤ j ≤ 2 Ω ∇W ij • ∇ψ i φ j -q j (∂ i ψ i )φ j = 1 2 Ω ∇W ij • ∇ψ i φ j -q j (∂ i ψ i )φ j = 1 2 Ω ∇W ij • ∇(ψ i φ j ) -q j ∂ i (ψ i φ j ) + o(1), as → 0.
Moreover,

Ω ∇W ij • ∇(φ j ψ i ) -q j ∂ i (φ j ψ i ) = k P k ∇W ij • ∇(φ j ψ i ) -q j ∂ i (φ j ψ i ) = k C ,+ k ∇W ij • ∇(φ j ψ i ) -q j ∂ i (φ j ψ i ) + k C ,- k ∇W ij • ∇(φ j ψ i ) -q j ∂ i (φ j ψ i ) + k P k \C k ∇W ij • ∇(φ j ψ i ) -q j ∂ i (φ j ψ i ),
k P k \C k ∇W ij • ∇(φ j ψ i ) -q j ∂ i (φ j ψ i ) ≤ C ∇W L 2 (∪ k P k \C k ) + q L 2 (∪ k P k \C k ) ≤ C a 2 1 2 1/2 1.
At this stage the proof differs slightly from the one of Ref. 3, because of the mixed boundary conditions at x 3 = 0. Indeed, since (W , q ) satisfies the Stokes system in C ,± k , we have

C ,± k ∇W ij • ∇(φ j ψ i ) - C ,± k q j ∂ i (φ j ψ i ) = ∂C ,± k ∂W ij ∂n -q j n • e i φ j ψ i ,
where n denotes the outer normal to the set C ,± k . In particular, due to the symmetry properties of W , q , φ, ψ , there holds

C ,+ k ∇W ij • ∇(φ j ψ i ) - C ,+ k q j ∂ i (φ j ψ i ) + C ,- k ∇W ij • ∇(φ j ψ i ) - C ,- k q j ∂ i (φ j ψ i ) = ∂C k ∂W ij ∂n -q j n • e i φ j ψ i -2 C k ∩{z=0} ∂ 3 W ij φ j ψ i . By definition of W , ∂ 3 W ij = 0 on (C k ∩ {x 3 = 0}) \ (T × {0}
). On the other hand, since ψ satisfies (3.3), ψ i = 0 on T × {0}. Therefore, the r.h.s. reduces to the integral on ∂C k . From now on, in order to avoid confusion, we denote by n k the normal vector to the ball C k . Using the asymptotic expansions (3.10)-(3.11) and the expression of W ij , q j in C k , we obtain, on ∂C k ,

∂W ij ∂n k -q j n k • e i = - a 2 2 π F ij + 6 π e i • n k F j • n k + a 2 2 R ij ,
where R ij is a function of x, satisfying R ij (x) = O(1) as → 0, uniformly in x and k. This leads to the following decomposition

Ω ∇W ij • ∇(φ j ψ i ) - Ω q j ∂ i (φ j ψ i ) = - a 2 k ∂C k 2 π F ij + 6 π e i • n k F j • n k φ j ψ i + a 2 2 k ∂C k R ij φ j ψ i + o(1).
Let δ ∂C k be the unit mass concentrated on ∂C k . We use the following Lemma, proved by Allaire a :

a Notice that in the paper of Allaire, the periodicity of the pattern is 2 , rather than as in the present paper. Hence the constant in front of the Dirac mass in the right-hand side is π/16, rather than π/64 for the first line, and π/48 rather than π/192 in the second line.

Computation of the effective slip of rough hydrophobic surfaces via homogenization 13

Lemma 3.2 (see Lemma 4.2.1 in Ref 3). k δ ∂C k → π 16 δ T 2 ×{0} strongly in H -1 ( Ω), k e i • n k n k δ ∂C k → π 48 e i δ T 2 ×{0} strongly in H -1 ( Ω). (3.15)
Let us write

k ∂C k φ j ψ i = k∈K δ ∂C k , φ j ψ i H -1 ( Ω)×H 1 0 ( Ω) , k ∂C k e i • n k n k φ j ψ i = k e i • n k n k δ ∂C k , φ j ψ i H -1 ( Ω)×H 1 0 ( Ω) ; (3.16) consequently, since φ j ψ i → φ j ψ i weakly in H 1 ( Ω) and a 2 → C 0 , we obtain lim →0 a 2 k ∂C k 2 π F ij + 6 π F j • n k n k • e i φ j ψ i = 1 4 C 0 T 2 ×{0} F ij φ j ψ i . (3.17)
Moreover, since R ij is uniformly bounded in L ∞ ( Ω), we use the following comparison principle: If β converges strongly to zero in H -1 (Ω), then so does α .

Whence we deduce from (3.15)

that k R ij δ ∂C k → 0 strongly in H -1 ( Ω).
Using (3.16), we obtain the following convergence:

a 2 k ∂C k R ij φ j ψ i → 0 as → 0. (3.18)
Gathering the convergence results (3.17)-(3.18), we obtain relation (3.4), where the matrix M 0 is defined by

M 0,ij = 1 8 F ij , F •j given by (3.

12) (3.19)

There only remains to prove that the matrix (F ij ) 1≤i,j≤2 is negative definite. To that end, we go back to system (3.5)-(3.9). We multiply by W •i the system satisfied by W •j , and we obtain In particular, for all η ∈ R 2 , 1≤i,j≤2

F ij = -2 T ×{0} ∂ 3 W ij = -2 R 3 + ∇W •i : ∇W •j .
η i η j F ij = -2 R 3 + |∇(η 1 W •1 + η 2 W •2 )| 2 ≤ 0,
and the right-hand side above vanishes if and only if

η 1 W •1 + η 2 W •2 = 0 a.e. in R 3 + .
In view of the boundary conditions (3.7), this implies η 1 = η 2 = 0. This concludes the proof of Lemma 3.1.

To complete the proof of Theorem 2.1, we rely on Lemma 3.1, as follows. Let φ ∈ C ∞ (Ω

3 ) satisfying the no-slip condition φ = 0 on the upper boundary 2 × {1}, and the non-penetration condition φ 3 = 0 on the lower boundary T 2 × {0}. Let W ∈ H 1 (Ω) 9 , q ∈ L 2 (Ω) 3 be the sequences introduced in Lemma 3.1. We define the following test functions for the weak formulation associated to system (2.1)-(2.3):

φ = (I 3 -W )φ, r = q φ,
where I 3 is the identity matrix in M 3 (R). We deduce the following relation:

Ω ∇u : ∇φ - Ω p div φ = Ω f φ (3.20) Ω r div u = 0. (3.21)
Since W converges weakly to 0 in H 1 (Ω) 9 , and strongly to 0 in L 2 (Ω) 9 , we readily obtain

Ω ∇u : ∇φ = Ω ∇u : ∇φ - Ω ∇u i ∇W ij φ j - Ω ∇u i ∇φ j W ij = Ω ∇u : ∇φ - Ω ∇u i ∇W ij φ j + o(1), as → 0, - Ω p div φ = - Ω p (I 3 -W ) : ∇φ = - Ω p div φ + o(1), as → 0.
Consequently, summing relations (3.20) and (3.21), we deduce the asymptotic relation

Ω ∇u : ∇φ - Ω p div φ - Ω ∇u i ∇W ij φ j + Ω q φ div u = o(1), as → 0.
Applying Lemma 3.1 with ψ = u and ψ = u , we obtain the following relation: where Allaire considers slip conditions on volumic obstacles (for which an extension like the one mentioned above cannot be performed).

Ω ∇u : ∇φ - Ω p div φ + C 0 T 2 ×{0} M 0 u • φ = 0, ( 3

Asymptotic study of "riblet" designs

This section is devoted to the proof of Theorem 2.2. In the case of riblets, we recall that T is invariant by translation in x 1 . Since f = (f 1 , f 2 , f 3 ) is also independent on the x 1 variable, the solution (u , p ) of system (2.1)-(2.2)-( 2.3) depends only on (x 2 , x 3 ). As a result, the first component of u satisfies:

-∆ 2,3 u 1 = f 1 in T × (0, 1)
,

u 1 = 0 on T × {1}, ∂ 3 u 1 = 0 on (T × {0}) \ (ΠT ), u 1 = 0 on ΠT , (4.1)
where ∇ 2,3 and ∆ 2,3 stand for the gradient (resp. the Laplacian) with respect to the (x 2 , x 3 ) variables, T 1 = R/Z and where we have denoted Π the projection operator defined by Π(x 1 , x 2 , 0) = (x 2 , 0). In the same fashion, (u 2 , u 3 ), p satisfy the following Stokes problem:

-∆ 2,3 u 2 u 3 + ∇ 2,3 p = f 2 f 3 in T × (0, 1), ∇ 2,3 • u 2 u 3 = 0 in T × (0, 1), u 2 = u 3 = 0 on T × {1}, u 3 = 0 on T × {0}, ∂ 3 u 2 = 0 on (T × {0}) \ (ΠT ), u 2|x3=0 = 0 on ΠT . (4.2)
Hence, the original 3d problem reduces to the study of two independent systems (with a Laplace and a Stokes equations), set in the 2d domain T×(0, 1). This change from a 3d to a 2d setting explains the change of scalings between Theorem 2.1 and Theorem 2.2.

To handle the Stokes equations (4.2), we proceed like in the previous section: in short, we adapt the homogenization techniques of Ref. 2, 3, dedicated to the Stokes flow across a periodic network of balls, set along an hypersurface. As mentioned before, the difference is the dimension of the domain. One must this time consider the 2d results of Ref. 3, about periodic network of disks along a line. For brevity, we do not give further details. We eventually obtain the following limit system:

-∆ 2,3 u 2 u 3 + ∇ 2,3 p = f 2 f 3 in T 1 × (0, 1), ∇ 2,3 • u 2 u 3 = 0 in T 1 × (0, 1), u 2 = u 3 = 0 on T 1 × {1}, u 3 = 0 on T 1 × {0}, ∂ 3 u 2 = 2π C 0 u 2 on T 1 × {0}, (4.3)
where we recall that C 0 := lim →0 -ln |a |. As regards the Laplace equation (4.1), the idea is exactly the same. Actually, the situation is even simpler, and has been analysed for a longer time. Namely, one may start from the work of Cioranescu and Murat (see Ref. 9), instead of section 4 in Ref. 2. Again, we leave the details to the reader. In our setting, the limit system is

-∆ 2,3 ū1 = f 1 in T 1 × (0, 1), ū1 = 0 on T 1 × {1}, ∂ 3 ū1 = π C 0 ū1 on T 1 × {0}. (4.4)
We deduce from systems (4.4) and (4.3) that u = u M riblets , M riblets being given by (2.8). The sub-cases where f = e 1 or f = e 2 follow easily.

Numerical simulations

This section is devoted to simulations of system (2.1)-(2.2)-(2.3). For simplicity, we shall restrict to constant source term (average pressure gradient), say f = 2e, e ∈ span(e 1 , e 2 ).

The idea is to recover numerically the scalings for the slip length given in Theorems 1 and 2. However, to observe significant slip implies to consider very small scales: patches of size less than 2 , in a grid of side . This forbids direct computations. To overcome this difficulty, we shall rely on a boundary layer approximation of the Stokes flow. Such approximation, often implicitly used in physics papers, has been fully justified in the context of wall laws: see References 14, 10, 5 among many others.

The starting point is to write the exact solution u as

u (x) = u P (x) + v (x/ )
where u P is the reference Poiseuille flow, satisfying (2.1) with Dirichlet condition at both planes. Remind that Hence, v = (v h (y), v 3 (y)) satisfies

u P (x) = -x 3 (x 3 -1)e.
-∆v + ∇p = 0, in T 2 × (0, -1 ), div v = 0, in T 2 × (0, -1 ), v = 0, y 3 = -1 , v 3 = 0, y 3 = 0, v h = 0, y ∈ -1 T × {0}, ∂ y3 v h = -e, y ∈ -1 (T ) c × {0} (5.1)
Note that no approximation has been made so far. It is then tempting to put the roof y 3 = -1 at infinity replacing T 2 × (0, -1 ) by T 2 × R + . However, it is wellknown that the resulting problem is overdetermined. Namely, the boundary layer field v ,bl satisfying

-∆v + ∇p = 0, in T 2 × R + , div v = 0, in T 2 × R + , v 3 = 0, y 3 = 0, v h = 0, y ∈ -1 T × {0}, ∂ y3 v h = -e, y ∈ -1 (T ) c × {0} (5.2) has constant horizontal average: v ,∞ h := T 2 v ,bl h (y)dy 1 dy 2
with respect to y 3 . More precisely, it can be shown that v ,bl → (v ,∞ h , 0) exponentially fast as y 3 goes to infinity. Furthermore, by linearity of (5.2), one may denote v ,∞ h = V ,∞ e for a two by two matrix V ,∞ . Then, one can show that V ,∞ is symmetric positive definite, with

V ,∞ e • e = T 2 ×R+ |∇v ,bl | 2 .
Note that everything depends on , through the rescaled domain -1 T in (5.2).

To correct the "boundary layer constant" at infinity, one must add a macroscopic Couette flow. One ends up with

u ≈ u P (x) + v ,bl (x/ ) -x 3 (V ,∞ e, 0)
Averaging in the small scale, we find

u h | x3=0 ≈ V ,∞ e, ∂ 3 u | x3=0 ≈ ∂ 3 u P | x3=0 ≈ e.
We end up with the approximate boundary condition

u h = V ,∞ ∂ 3 u h at x 3 = 0. (5.3)
On the basis of the previous reasoning, one can implement the following strategy for the numerical computation of the slip length:

• Compute numerically (say with e = e 1 and e = e 2 ) the solution of (5.2), in order to determine the matrix V ,∞ . • Check for the asymptotics of V ,∞ , for various shapes and sizes of the noslip zone T . This allows to make the comparison with theoretical results of Theorems 1 and 2. Indeed, sending to zero in (5.3) yields

ūh = lim →0 ( V ,∞ ) ∂ 3 ūh at x 3 = 0, (5.4)
so that the matrix M in the theorems satisfies

M -1 = lim →0 ( V ,∞ ).
Numerical approximation of the matrix V ,∞ . In the numerical simulations, we will solve the system (5.2) associated to different shapes of the no-slip zone T : circular or rectangular patches, and riblets parallel or orthogonal to the flow. Let us first notice that for such configurations, the matrix V ,∞ is diagonal. Indeed, since the domain -1 T is symmetric with respect to the axis {y 2 = 1/2}, if we denote by v the solution to system (5.2) with e = e 1 , then the vector field v * defined by

v * i (y 1 , y 2 , y 3 ) = v i (y 1 , 1 -y 2 , y 3 ), for i = 1, 3, and by v * 2 (y 1 , y 2 , y 3 ) = -v 2 (y 1 , 1 -y 2 , y 3 ), is also a solution. By uniqueness, we deduce that v 2 (y 1 , 1 - y 2 , y 3 ) = -v 2 (y 1 , y 2 , y 3 ) for a.e. (y 1 , y 2 , y 3 ) ∈ T 2 × R + , which yields V ,∞ e 1 • e 2 = 0. By symmetry of V ,∞ , we obtain also that V ,∞ e 2 • e 1 = 0
Consequently the boundary conditions satisfied by the horizontal components of the approximate solution to system (2.1)-(2.2)-(2.3) on x 3 = 0, simply writes:

u i = (V ,∞ e i • e i ) ∂ 3 u i at x 3 = 0, for i = 1, 2.
(5.5)

In the rest of this section, for i = 1, 2, the quantity V ,∞ e i • e i will be refered to as the average slip length associated to our problem, in the direction e i .

To compute an approximate value of the average slip length associated to system (5.2), we consider a truncated domain T 2 × (0, H), for a given H > 0, and we introduce the solution w to the following problem:

-∆w + ∇q = 0, in T 2 × (0, H), div w = 0, in T 2 × (0, H), ∂ y3 w -q e 3 = 0, y 3 = H, w 3 = 0, y 3 = 0, w h = 0, y ∈ -1 T × {0}, ∂ y3 w h = -e, y ∈ -1 (T ) c × {0} (5.6)
Using arguments developed in Ref. [START_REF] Jäger | Asymptotic analysis of the laminar viscous flow over a porous bed[END_REF], the difference between v ,bl and w can be estimated as follows. First, we claim that v ,bl satisfies the following H 1 bound:

∇v ,bl L 2 (T 2 ×R+) ≤ C a , (5.7)
where C is a constant which does not depend on . This bound follows from a quantitative trace inequality, whose proof is similar to the one of (3.2): there exists a constant C > 0 such that for all b ∈ (0, 1), for all v ∈ H 1 (T 2 × (0, 1)) such that v |x3=0 vanishes on a ball of radius b ,

v |x3=0 L 2 (T 2 ) ≤ C √ b ∇v L 2 (T 2 ×(0,1)) .
Then, we decompose v ,bl into horizontal Fourier series and we derive exponential decay bounds: for all s ∈ N, there exists a constant γ s > 0, which does not depend on , such that

v ,bl (•, x 3 ) -(V ,∞ e, 0) L 2 (T 2 ) ≤ C a exp(-γ 0 x 3 ), α∈N 3 ,|α|≤s ∇ α v ,bl (•, x 3 ) L 2 (T 2 ) ≤ C a exp(-γ s x 3 ).
(5.8)

As a consequence, v ,bl is a solution of (5.6) in T 2 × (0, H), with a slightly modified condition at y 3 = H, namely

∂ y3 v ,bl -p ,bl e 3 = F at y 3 = H, with F H s (T 2 ) ≤ C a exp(-γ s H) ∀s ∈ N and T 2 F = 0.
It follows that there exist constants C, γ > 0 such that

∇(v ,bl -w) 2 L 2 (T 2 ×(0,H)) ≤ C a exp(-γH).
Notice also that w, as v ,bl , has constant horizontal average and that

H 0 T 2 |∇w| 2 = T 2
w(y) dy 1 dy 2 .

We solve problem (5.6) by a finite element method. We use P 2 elements for the velocity and P 1 elements for the pressure. The three-dimensional mesh of the fluid domain T 2 × (0, H) is obtained by a constrained Delaunay tetrahedralization. The incompressibility condition is treated by a Lagrange multiplier (see Ref. [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF]Ref. 13).

Given two approximate solutions w 1 app , w 2 app of system (5.6), associated respectively to e = e 1 and e = e 2 , we define the numerical approximation V ,∞ app of the matrix V ,∞ , by the following formula:

V ,∞ app e i • e j := T 2
w i app (y 1 , y 2 , H) • e j dy 1 dy 2 , for i, j = 1, 2.

By analogy with formula (5.5), for i = 1, 2, the approximate average slip length in direction e i is then defined by V ,∞ app e i • e i . Finally, we introduce the solid fraction φ s , which is defined by the relative area of the no-slip zone T in the elementary square of size (or equivalently, by the area of the rescaled no-slip domain -1 T ). Using definitions (2.4)-(2.5), φ s is given by the following expressions: Notice that system (5.2) is completely determined by φ s and by the domain T (in the case of patches) or the union of intervals I (in the case of riblets).

Computation of the average slip length, in the case of patches. In the case of patches, we have plotted V ,∞ app e 1 • e 1 against 1/ φ s , considering circular and squared patches (see Figure 4). We observe that the dependency is affine, and a linear regression gives the relation V ,∞ app e 1 • e 1 ≈ α/ φ s + β, with α = 0.322, β = -0.429 in the case of the disk, and α = 0.311, β = -0.422 in the case of the square. Note that these coefficients are very close to the ones obtained by Ybert et al. in Ref. [START_REF] Ybert | Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries[END_REF]. Consequently, since lim →0 φ s = 0,

V ,∞ app e 1 • e 1 ∼ α φ s as → 0. (5.9)
To compare this numerical result with the theoretical result given by Theorem 2.1, let us consider the critical case a / 2 → C 0 > 0. In that case, there exists a two by two matrix M 0 , depending on the pattern T , such that lim →0 V ,∞ = 1 C0 M -1 0 . For circular or squared patterns centered in the unit square, as observed above, the matrices V ,∞ , and consequently the matrix M 0 , are diagonal. Moreover, since these patterns are invariant by a rotation of angle π/2, one can easily see that the corresponding matrix V ,∞ satisfies V ,∞ e 1 • e 1 = V ,∞ e 2 • e 2 . Consequently, there exists λ 0 > 0 such that M 0 = λ0 0 0 λ0 , and the following relation holds:

lim →0 V ,∞ e 1 • e 1 = 1 C 0 λ 0 .
Besides, using the definition of φ s in the case of patches, the asymptotic relation (5.9) yields lim

→0 V ,∞ app e 1 • e 1 = α C 0 |T | .
Thus, the numerical value of the slip length α/(C 0 |T |), that can be deduced from the asymptotic behavior (5.9) in the critical case, is consistent with Theorem 2.1. The coefficient of the matrix M 0 can be approximated by λ 0 ≈ |T |/α. We notice that the results concerning the sub-critical and super-critical cases can also be retrieved, at least formally, from relation (5.9). Indeed, since / φ s = 2 /(a |T |), we obtain in the sub-critical case: lim →0 V ,∞ app e 1 • e 1 = +∞, which corresponds formally to an infinite slip length in the e 1 direction, that is, a perfect slip condition. In the same manner, in the super-critical case, we obtain lim →0 V ,∞ app e 1 • e 1 = 0, which corresponds to adherence in the e 1 direction.

Computation of the effective slip of rough hydrophobic surfaces via homogenization 21

Computation of the average slip length, in the case of riblets. In that case, exact computations are available in the literature, that give the average slip lengths in the e 1 and e 2 direction as a function of the solid fraction φ s (see for instance Ref. 17):

V ,∞ e 1 •e 1 = -ln cos π 2 (1 -φ s ) /π, V ,∞ e 2 •e 2 = -ln cos π 2 (1 -φ s ) /(2π).
(5.10) We have plotted in Figure 5 the computed value of the average slip lengths V ,∞ app e 1 •e 1 and V ,∞ app e 2 • e 2 , against φ s , as well as the exact values defined by formulas (5.10). We observe that the numerical values are close to the expected ones.

Once again, formulas (5.10) and the numerical behavior of the average slip length shown in Figure 5, are consistent with the theoretical results of Theorem 2.2. Indeed, in the critical case lim →0 -ln(a ) = C 0 > 0, using the expression φ s = (a |I|)/ , one obtains by a straightforward computation that ln cos π 2 (1 -φ s ) → -C 0 as →. Consequently, the slip length in the directions e 1 and e 2 are respectively given by lim

→0 V ,∞ e 1 • e 1 = C 0 π , lim →0 V ,∞ e 2 • e 2 = C 0 2π .
Influence of the shape of the no-slip area: comparative results. In order to provide a comparison between the efficiency of patches and riblets in terms of slip length, we consider the slip length in the direction of the constant pressure gradient f = 2e i , with i = 1 or i = 2. For circular or squared patterns, the average slip length is given by V ,∞ e 1 • e 1 . In the case of riblets, we consider two configurations of physical interest:

• riblets parallel to the flow: f = 2e 1 , the average slip length is defined by V ,∞ e 1 • e 1 ; • riblets orthogonal to the flow: f = 2e 2 , the average slip length is V ,∞ e 2 •e 2 .

The results are plotted in Figure 6. As stated in Remark 2.2, page 6, these numerical results confirm that the riblets parallel to the flow are not necessarily optimal. Indeed, if the solid fraction φ s is small enough, say φ s < 0.1, the circular or squared patches produce a superior slip length.

To estimate the influence of the shape of the pattern on the slip length, we have considered families of rectangles of fixed area φ s , that are centered in the unit square. For φ s = 0.01, 0.04, 0.09 we have computed the average slip length V ,∞ app e 1 • e 1 , in the direction e 1 , associated to each of these rectangular patterns. The results are plotted in figure 7, against the dimension L of each rectangular pattern, in the e 1 direction. For each solid fraction φ s , the extremal values associated to L = φ s and L = 1, correspond respectively to a riblet orthogonal to the flow, and parallel to the flow.

We notice that, for each family of rectangular patterns of fixed area, the riblet orthogonal to the flow provides always the smallest average slip length. As already mentionned, the riblet parallel to the flow is not optimal, especially for small values of the solid fraction φ s = 0.01, φ s = 0.04. In that cases, the curves present a unique maximum, and the associated optimal size L of the rectangle is slightly superior to the size φ s of the square of same area. For these values of the solid fraction, the optimal rectangular pattern will present a certain anisotropy in the direction of the flow. Multiplying last inequality by r 2 (r ) 2 sin θ and integrating on r ∈ (0, a α), r ∈ (a α, √

2 ), φ ∈ (0, 2π), θ ∈ (0, π/2), we obtain the inequality I ≤ 2J + 2K (5.13) where the integrals I , J , K are respectively defined by

I = a α r =0 √ 3

Figure 1 .

 1 Figure 1. Patch configuration. For every k= (k 1 , k 2 ) ∈ [|0, . . . , -1 |] 2 , the intersection of the no-slip zone T with the cell [ k 1 , (k 1 + 1)) × [ k 2 , (k 2 + 1)) is defined by k + T = k + a T .

Figure 2 .

 2 Figure 2. Riblet configuration. For k = (k 1 , k 2 ), the intersection of the no-slip zone T with the cell [ k 1, (k 1 + 1)) × [ k 2 , (k 2 + 1)) is defined by k + T = k + ( T) × ( 2 + a I).

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February 19 ,Figure 3 .

 193 Figure 3. Each cube P k is decomposed into a union of subdomains C k , D k and K k , which are separated by spheres of radius /4 and /2 centered in the cube.

February 19 ,

 19 2015 18:39 WSPC/INSTRUCTION FILE BDGV-rev2 12 Bonnivard, Dalibard, Gérard-Varet where C ,± k = C k ∩ R 3 ± . In all sums, k ranges over [|0, -1 |] 2 . Using the estimates (3.13), we infer that

Lemma 3 . 3 (

 33 see Lemma 2.3.8 in Ref. 2). Let α and β be two positive functions in H -1 (Ω) such that 0 ≤ α ≤ β .

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•

  in the case of patches, φ s = a 2 |T |, where |T | stands for the area of the domain T ; • in the case of riblets, φ s = a |I|, where |I| stands for the length of the interval I.

FebruaryFigure 4 .

 4 Figure 4. Numerical value of the average slip length V ,∞ app e 1 •e 1 plotted against 1/ φ s , for circular patches and squared patches.

Figure 5 .

 5 Figure 5. Numerical values of the average slip lengths V ,∞ app e 1 • e 1 and V ,∞ app e 2 • e 2 , plotted against φ s , in the case of riblets. The dashed lines represent the exact value of the average slip lengths, defined by formulas (5.10).

February 19 ,Figure 6 .Dimension of the rectangle along e 1

 1961 Figure 6. Average slip length in the direction of the flow, in the case of circular patches, squared patches and riblets parallel and orthogonal to the flow.

Figure 7 .

 7 Figure 7. Numerical values of the average slip length V ,∞ app e 1 •e 1 , produced by rectangular patterns of given area φ s ∈ {0.01, 0.04, 0.09}, and plotted against the dimension L of the rectangle in the e 1 direction. The extremal value L = φ s (resp. L = 1) corresponds to the riblet orthogonal to the flow (resp. parallel to the flow).

Appendix: proof of inequality ( 3 . 2 ) 12 ) 2 . 3 2 3 2 ,

 32122332 To obtain inequality(3.2), it is enough to prove that for every k ∈ [|0,-1 |] 2 , S k ×{0} |u | 2 ≤ η( ) B ,+ k |∇u | 2 ∀0 < < 0 . (5.11) A summation over k ∈ [|0, -1 |] 2 then leads to inequality (3.2). Let k ∈ [|0, -1 |] 2. By rescaling the trace inequality in the half cube [0, 1] 2 ×[0,1 2 ], we obtain the existence of a constant C > 0 such thatS k ×{0} |u | 2 ≤ CTo estimate the L 2 norm of u by the L 2 norm of its gradient, we adapt Lemma 3.4.1 in Ref. 3 to our bidimensional array of holes. We denote by B k the ball circumscribing the cube P k . Of course, the upper half-cube P ,+ k is contained in the upper half-ball B ,+ k . Moreover, since the model no-slip zone T contains a disk of radius α centered at the origin, each elementary no-slip pattern k + T contains a disk of radius a α, centered in the square S k . Let B k be the 3d ball of same center and radius, and B ,+ k be the corresponding half ball. With this notation, we can write To estimate the contribution of the exterior part B ,+ k \ B ,+ k , we use spherical coordinates (ρ, φ, θ) centered at point k + ( 2 , 2 , 0). The radius of B k being equal to √ , integrating along rays, we get for every r , r such that 0 < r < a α < r < √ u (r, φ, θ) = u (r , φ, θ) + r r ∂ ρ u (ρ, φ, θ)dρ, which yields |u (r, φ, θ)| 2 ≤ 2|u (r , φ, θ)| 2 + 2 r r ∂ ρ u (ρ, φ, θ)dρ 2 .

2 ∂ 2 r 2

 222 r=a α θ φ |u(r, φ, θ)| 2 r 2 (r ) 2 sin θ dθ dφ dr dr , |u(r , φ, θ)| 2 r 2 (r ) 2 sin θ dθ dφ dr dr , ρ u (ρ, φ, θ)dρ (r ) 2 sin θ dθ dφ dr dr . |u(r, φ, θ)| 2 r 2 sin θ dθ dφ dr = a 3 α 3 3 B ,+ k \ B ,+ k |u | 2 ,

  [START_REF] Jäger | On the roughness-induced effective boundary conditions for an incompressible viscous flow[END_REF][START_REF] Amirat | Influence de la rugosité en hydrodynamique laminaire[END_REF].

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	Computation of the effective slip of rough hydrophobic surfaces via homogenization 7
	Remark 2.2.
	Remark 2.3. Our results are consistent with those of Ref. 20: indeed, in the case of
	patches, it is shown heuristically there that the slip length is proportional to 2 /a :
	in other words, if a	2 , perfect slip is achieved, if a	2 , a no-slip condition is
	retrieved in the limit, and in the critical case, the slip length is positive and finite.
	Also, explicit calculations (see Ref. 17) recalled in Ref. 20 show that the slip
	length for riblets is equal to -/π ln(a / ) for riblets parallel to the flow, and to
	-/(2π) ln(a / ) for riblets perpendicular to the flow. Once again, this is consistent
	with Theorem 2.2.

  Computation of the effective slip of rough hydrophobic surfaces via homogenization 15 the velocity can be extended though the planar slip zones into a Stokes solution satisfying Dirichlet conditions at the remaining part of the boundary. In this way, the problem becomes very similar to the one raised by Allaire in Section 4 of Ref.3 on fluid flows through porous grids. In this respect, it is different from article 4

	.22)
	where the matrix M 0 ∈ M 2 (R) is defined by (3.19). Since relation (3.22) holds for
	every test function φ, this proves that u = u C0M0 .

Remark 3.1. Theorem 1 expresses that the homogenized boundary condition depends strongly on the ratio between slip and no-slip areas. By simple symmetry,

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Computation of the effective slip of rough hydrophobic surfaces via homogenization 19

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Computation of the effective slip of rough hydrophobic surfaces via homogenization 27 fluids 19 (2007), 123601.