Matthieu Bonnivard

Dorin Bucur

The uniform rugosity effect

Keywords: rugosity effect, non-periodic boundaries, uniform decreasing of the velocity

Relying on the effect of microscopic asperities, one can mathematically justify that viscous fluids adhere completely on the boundary of an impermeable domain. The rugosity effect accounts asymptotically for the transformation of complete slip boundary conditions on a rough surface in total adherence boundary conditions, as the amplitude of the rugosities vanishes. The decreasing rate (average velocity divided by the amplitude of the rugosities) computed on close flat layers is definitely influenced by the geometry. Recent results prove that this ratio has a uniform upper bound for certain geometries, like periodical and "almost Lipschitz" boundaries. The purpose of this paper is to prove that such a result holds for arbitrary (non-periodical) crystalline boundaries and general (non-smooth) periodical boundaries.

Introduction

It is commonly accepted that viscous fluids adhere to rough surfaces. One mathematical explanation is based on the so called rugosity effect. For the Navier-Stokes equation, complete slip boundary conditions on a rough surface transform asymptotically in no-slip conditions as the amplitutde of the rugosities vanishes, provided that the energy of the solutions is uniformly bounded and the there is "enough roughness" of the oscillating boundaries. We refer the reader to the pioneering paper of Casado-Díaz, Fernández-Cara and Simon where this result is proved in the case of periodic, self-similar C 2 -boundaries [START_REF] Casado-Díaz | Why viscous fluids adhere to rugose walls: A mathematical explanation[END_REF]. In their proof, the authors prove implicitly that the rugosity effect has a uniform character, in the sense that the decreasing rate of the average velocity on a flat layer close to the boundary can be estimated uniformly.

Recent results obtained in [START_REF] Bucur | On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries[END_REF][START_REF] Bucur | Boundary behavior of viscous fluids: Influence of wall roughness and friction-driven boundary conditions[END_REF][START_REF] Bucur | Influence of wall roughness on the slip behaviour of viscous fluids[END_REF] (see also [START_REF] Jaeger | On the roughness-induced effective boundary conditions for an incompressible viscous flow[END_REF][START_REF] Mohammadi | Rough boundaries and wall laws. Finite elements in fluids[END_REF]) give a quite complete understanding of the rugosity effect for arbitrary boundaries. For equi-Lipschitz domains, if Ω ε is a geometric perturbation of Ω (in the sense that the Hausdorff distance vanishes, d H (Ω ε , Ω) → 0), the solutions u ε of a Stokes equation with complete slip boundary conditions in Ω ε converge to the solution u of the same Stokes equation complemented by the so called friction-driven boundary conditions (see [START_REF] Bucur | Boundary behavior of viscous fluids: Influence of wall roughness and friction-driven boundary conditions[END_REF]): there exists a suitable trio {µ, A, V } such that • µ is a capacitary measure concentrated on ∂Ω • {V (x)} x∈Γ is a family of vector subspaces in R N -1

• A is a positive symmetric matrix function A defined on ∂Ω and formally the boundary conditions read u(x) ∈ V (x) for q.e. x ∈ ∂Ω

D[u] • n + µAu • v = 0 for any v ∈ V (x), x ∈ Γ (1)
The rugosity effect holds provided that above V (x) = {0} for a.e. x ∈ ∂Ω, which implies that u = 0 on ∂Ω. However, assuming that the energies of the solutions u ε H 1 (Ωε) are uniformly bounded, the estimate of the average velocity decay rate is more delicate and relies on finding on an upper bound for lim sup

ε→0 ∂Ωε |u ε | 2 dS d H (Ω ε , Ω) . (2)
For general situations one can not expect this number to be finite. If finite, the rugosity effect is said to be uniform (we refer to [START_REF] Bucur | The incompressible limit of the full Navier-Stokes-Fourier system on domains with rough boundaries[END_REF] for applications). All previous results in the literature providing an upper bound for [START_REF] Amirat | The Stokes equations with Fourier boundary conditions on a wall with asperities[END_REF] were given for periodic self-similar boundaries in the framework of [START_REF] Casado-Díaz | Why viscous fluids adhere to rugose walls: A mathematical explanation[END_REF], under more or less regularity assumptions. We refer to the recent paper of Březina [START_REF] Březina | Asymptotic properties of solutions to the equations of incompressible fluid mechanics Preprint 1[END_REF], where the C 2 -regularity is weakened to almost Lipschitz boundaries (see the precise sense in [START_REF] Březina | Asymptotic properties of solutions to the equations of incompressible fluid mechanics Preprint 1[END_REF]). Nevertheless, at least intuitively, less regularity of the boundary should enforce the physical rugosity effect but, of course, new technical difficulties arise.

The main purpose of the paper is to analyse general geometric perturbations of flat domains for which an upper bound can be found in [START_REF] Amirat | The Stokes equations with Fourier boundary conditions on a wall with asperities[END_REF]. On the one hand, we remove any smoothness hypothesis and prove that the rugosity effect is uniform for general, continuous, periodic self-similar boundaries which are not riblets. This first result is only a technical improvement of previous results of [START_REF] Casado-Díaz | Why viscous fluids adhere to rugose walls: A mathematical explanation[END_REF][START_REF] Bucur | The incompressible limit of the full Navier-Stokes-Fourier system on domains with rough boundaries[END_REF][START_REF] Březina | Asymptotic properties of solutions to the equations of incompressible fluid mechanics Preprint 1[END_REF] and involves a weak interpretation of the riblets associated to continuous boundaries (gradientless formulation). In a second step, we remove the periodicity assumption and prove that for arbitrary crystalline boundaries the rugosity effect is again uniform. This second result relies on a fine use of the Young measures to understand second order pointwise oscillations of the boundaries, i.e. the oscillations of the normal fields of locally rescaled domains. Uniform behaviours are in general difficult to capture for non-periodic structures. In our case, this is possible due to the very specific crystalline structure. We also refer to the paper [START_REF] Basson | Wall laws for fluid flows at a boundary with random roughness[END_REF] for a different result involving uniformity in a random geometric framework.

General settings and main results

Throughout the paper, we fix the dimension of the space N = 3. For simplicity, and without losing generality, we shall assume that Ω is the cube (0, 1) 2 × (-1, 0). Let us denote ϕ ε : (0, 1) 2 → [0, 1] a family of lower semi-continuous functions such that (ϕ ε ) ε converges uniformly to zero as ε → 0. We introduce the geometric perturbations of Ω,

Ω ε = (x 1 , x 2 , x 3 ) | (x 1 , x 2 ) ∈ (0, 1) 2 , -1 < x 3 < ϕ ε (x 1 , x 2 ) , (3)
which are open sets approaching in a certain geometric sense Ω, as ε → 0. In this case, it is convenient to replace in (2) the Hausdorff distance with the amplitude of the rugosities ϕ ε ∞ . Note that discontinuity points on ϕ ε are a priori admissible and correspond to severe "roughness points" of the boundary. Define D = (0, 1) 2 × (-1, 2) so that

Ω ε ⊂ D ∀ε > 0.
The non penetration condition. We denote

H 1 (Ω ε ) the usual Sobolev space H 1 (Ω ε , R 3 ). For a function v ∈ H 1 (Ω ε ), the non-penetration condition v • n ε = 0 on Γ ε = {x ∈ ∂Ω ε : x 3 0},
has to be understood in a weak sense, as soon as the normal vector field n ε is not properly defined on Γ ε . Precisely, if ϕ ε is Lipschitz, the trace of v on Γ ε and the normal vector field n ε are pointwise defined a.e. If ϕ ε is only continuous, we say that v ∈ H 1 (Ω ε ) satisfies the non-penetration condition provided that

∀ψ ∈ C 1 c (D) Ωε [(div v)ψ + v • ∇ψ]dx = 0. (4)
If ϕ ε is lower semi-continuous, the above form of the non-penetration condition is incomplete, and special attention has to be given to discontinuity points. The following form of the nonpenetration condition turns out to be equivalent to (4) as soon as the boundary is represented by a continuous graph, but captures also information at discontinuity points:

∀ψ ∈ H 1 (Ω ε ) Ωε [(div v)ψ + v • ∇ψ]dx = 0. ( 5
)
Rugosity effect and uniform decay rates. The rugosity effect reads: let

v ε ∈ H 1 (Ω ε ) satisfy the non-penetration condition on Γ ε , such that (1 Ωε v ε , 1 Ωε • ∇v ε ) converge weakly in L 2 (D, R 12 ) to (1 Ω v, 1 Ω • ∇v). Then v = 0 on Γ = (0, 1) 2 × {0}.
The rugosity effect above is said to be uniform if the following estimate holds:

∃C > 0, ∃ε 0 > 0, ∀ε 0 > ε > 0, ∀v ∈ H 1 (Ω ε ), v • n ε = 0 on Γ ε (6) Γ |v| 2 dS C ϕ ε ∞ Ωε |∇v| 2 dx. ( 7
)
Notice that in this inequality, one does not require v to vanish on some part of the boundary. As a consequence of the continuity of the trace operator, estimate [START_REF] Bucur | Influence of wall roughness on the slip behaviour of viscous fluids[END_REF] proves that the rugosity effect holds, when ε → 0.

Provided that ϕ ε are equi-Lipschitz, one can replace in ( 6)-( 7) the sum on Γ by the sum on Γ ε Γε

|v| 2 dS C ′ ϕ ε ∞ Ωε |∇v| 2 dx, (8)
which provides an upper bound in [START_REF] Amirat | The Stokes equations with Fourier boundary conditions on a wall with asperities[END_REF]. If ϕ ε were not smooth, the trace on Γ ε is not defined.

In this case, one can expect only estimates of type [START_REF] Bucur | Influence of wall roughness on the slip behaviour of viscous fluids[END_REF].

Main results of the paper. We first consider periodic structures generated by

ϕ ε (x 1 , x 2 ) = εϕ( x 1 ε , x 2 ε ), ( 9
)
where ϕ is a continous function on the two-dimensional torus in

T 2 = [0, 1] 2 | {0,1} extended by periodicity in R 2 .
We give a characterization of the continuous functions ϕ such that (7) holds, removing the "almost Lipschitz" hypothesis considered in [START_REF] Březina | Asymptotic properties of solutions to the equations of incompressible fluid mechanics Preprint 1[END_REF]. Loosely speaking, non-smooth boundaries should provide a stronger rugosity effect than the smooth ones! For technical reasons we restrict our study only to continuous boundaries but in the final section we shall discuss briefly boundaries with singularities generated by lower-semicontinuous functions ϕ.

We say ϕ is a riblet if there exists (c

1 , c 2 ) ∈ R 2 \ {0} such that for every (x 1 , x 2 ) ∈ R 2 and every h ∈ R, ϕ(x 1 + hc 1 , x 2 + hc 2 ) = ϕ(x 1 , x 2 ). (10)
If ϕ were differentiable, this would correspond to ∇ϕ • (c 1 , c 2 ) = 0.

Theorem 2.1 (Characterization of the uniform rugosity effect) Assume that ϕ is a continuous strictly positive function on T 2 . The following statements are equivalent:

(i) there exist k > 0 and ε 0 > 0 such that for every ε 0 > ε > 0 and every v ∈ H 1 (Ω ε ) satisfying the non-penetration condition on Γ ε (in its weak integral form), the following inequality holds:

Γ |v| 2 dS εk ∇v 2 L 2 (Ωε) , (11)
(ii) ϕ is not a riblet.

In the case of lower-semicontinuous boundaries the characterization Theorem 2.1 can be rephrased, but provides a less clear geometric criterion for the uniform rugosity effect.

The second result of the paper is concerned with arbitrary crystalline boundaries (the periodicity assumption is removed). We prove that estimate [START_REF] Bucur | Influence of wall roughness on the slip behaviour of viscous fluids[END_REF] holds under a mild nondegeneracy assumption, similar to the one introduced in [START_REF] Bucur | On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries[END_REF]. Let us consider a finite set K ⊂ R 2 which satisfies the non-degeneracy assumption

∀y 1 , y 2 ∈ K, 0 / ∈ [y 1 , y 2 ]. ( 12
)
This condition simply avoids the creation of riblets on a crystalline structure. A function ϕ ∈ W 1,∞ ((0, 1) 2 ) is admissible provided that for a.e. y ∈ (0, 1) 2 ϕ(y) ∈ [0, 1] and ∇ϕ(y) ∈ K.

For every admissible function ϕ we define the crystalline boundary

Γ ϕ = (x 1 , x 2 , ϕ(x 1 , x 2 )) | (x 1 , x 2 ) ∈ (0, 1) 2 ,
and the corresponding domain

Ω ϕ = (x 1 , x 2 , x 3 ) | (x 1 , x 2 ) ∈ (0, 1) 2 , -1 < x 3 < ϕ(x 1 , x 2 ) .
Theorem 2.2 (Crystalline boundaries) There exist k > 0 and ε 0 > 0 such that inequality

Γ |v| 2 dS + Γϕ |v| 2 dS k ϕ ∞ ∇v 2 L 2 (Ωϕ) (13)
holds for every admissible function ϕ verifying ε 0 > ϕ ∞ > 0, and every v ∈ H 1 (Ω ϕ ) satisfying the non-penetration condition on Γ ϕ .

From a technical point of view, the uniformity of the rugosity effect is related to the analysis of the spectral abscissa of an elliptic operator in a family of infinite domains with "wildly" moving boundaries. The rugosity (Young) measures introduced in [START_REF] Bucur | On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries[END_REF] provide a useful tool for dealing with the crystalline case. For our purpose, their use has to be refined in order to understand the local oscillations of the rescaled boundaries. All results of the paper are given in H 1 but they can be extended easily to W 1,p spaces. For instance, inequality (13) becomes

v L q (Γ) + v L q (Γϕ) k ϕ α ∞ ∇v L p (Ωϕ) , (14)
which holds for every for every admissible function ϕ ∈ W 1,∞ ((0, 1) 2 ) such that ε 0 > ϕ ∞ > 0 and every v ∈ W 1,p (Ω ϕ , R 3 ) satisfying the non-penetration condition on Γ ϕ , where

1 < p < 3, 1 < q < 2 3 -p , α = 1 - 3 p + 2 q .
3 Pointwise roughness and Young measures A fine tool allowing to understand the rugosity effect is given by the general theory of Young measures (see for instance [START_REF] Evans | Weak convergence methods for nonlinear partial differential equations[END_REF][START_REF]Parametrized measures and variational principles[END_REF]). For the sake of clarity, we recall the fundamental theorem of Young measures.

Theorem 3.1 (Fundamental theorem of Young measures) Let U ⊂ R n be an open set and K ⊂ R m be a compact set. Consider a sequence (f k ) k∈N ⊂ L ∞ (U, K).
There exists a subsequence (f k j ) and for a.e. y ∈ U a Borel probability measure R y on R m such that for each

F ∈ C(R m ) we have F (f k j ) ⇀ F weakly- * L ∞ (U ),
where

F (y) = R m F (Z)dR y (Z) a.e. y ∈ U.
We call {R y } y∈U a family of Young measures associated with the subsequence (f k j ).

Remark 3.2 We recall that the family of Young measures {R y } y∈U may be not unique for a given sequence (f k ) k∈N and satisfies the following property: spt(R y ) ⊂ K a.e. y ∈ U.

In particular, for the crystalline framework when K is finite, the support of the measures R y is discrete.

Remark 3.3 Let (ϕ k ) k be a bounded sequence in W 1,∞ ((0, 1) 2 , [0, 1]
). We assume, up to renaming the indices, that there exists ϕ ∈ W 1,∞ ((0, 1) 2 , [0, 1]) such that

ϕ k → ϕ uniformly on (0, 1) 2 , ∇ϕ k ⇀ ∇ϕ weakly- * L ∞ ((0, 1) 2 ).
We denote by {R y } y∈(0,1) 2 a family of Young measures associated with a subsequence of (∇ϕ k ) k∈N . Then

R 2 ZdR y (Z) = ∇ϕ(y) a.e. y ∈ (0, 1) 2 .
This is a direct consequence of the fundamental theorem of Young measures, taking for F the identity function.

Ω k = (x 1 , x 2 , x 3 ) | (x 1 , x 2 ) ∈ (0, 1) 2 , -1 < x 3 < ϕ k (x 1 , x 2 ) ,
and consider u k ∈ H 1 (Ω k ) satisfying the non-penetration condition

u k • n k = 0 on Γ k = {x ∈ ∂Ω k : x 3 0} converging weakly in H 1 (D) to c = (c 1 , c 2 , c 3 ) ∈ R 3 .
Then for every g ∈ C(R 2 , R) we have

(c 1 , c 2 ) • R 2 g(Z)ZdR y (Z) = c 3 R 2
g(Z)dR y (Z) a.e. y ∈ (0, 1) 2 .

Proof Let ξ ∈ C(R 2 , R). We prove that for every ψ ∈ C ∞ c ((0, 1) 2 , R),

(0,1) 2 ψ(y) (c 1 , c 2 ) • R 2 √ 1 + Z 2 ξ(Z)ZdR y (Z) -c 3 R 2 √ 1 + Z 2 ξ(Z)dR y (Z) dy = 0.
(15) Taking into account that n k (y, ϕ k (y)) is co-linear with (-∇ϕ k (y), 1), the non penetration condition on Γ k yields

Γ k ψ(y)ξ(∇ϕ k (y))(-∇ϕ k (y), 1) • u k (y, ϕ k (y))dS = 0.
Performing a change of variables from Γ k to Γ, we get

(0,1) 2 ψ(y)ξ(∇ϕ k (y))∇ϕ k (y) • (u 1 k (y, ϕ k (y)), u 2 k (y, ϕ k (y))) 1 + ∇ϕ k (y) 2 dy (16) = (0,1) 2 ψ(y)ξ(∇ϕ k (y))u 3 k (y, ϕ k (y)) 1 + ∇ϕ k (y) 2 dy
Without loss of generality, we may assume that u k belongs to H 1 (D) ∩ C ∞ (D, R 3 ). Then integrating on vertical lines and using the Cauchy-Schwartz inequality, we get

(0,1) 2 |u k (y, ϕ k (y)) -u k (y, ϕ(y))| dy ϕ k -ϕ ∞ ∇u k L 2 (D) → 0 as k → +∞.
Passing to the limit in (16), we get (15). Since ψ is arbitrary, we finish the proof choosing

ξ(Z) = g(Z) √ 1+Z 2 . ✷ Lemma 3.5 Let (ϕ k ) k be a bounded sequence in W 1,∞ ((0, 1) 2 )
. Let (R y ) y∈(0,1) 2 be a family of Young measures associated to a subsequence of (∇ϕ k ) k , and

N ∈ K. If lim inf k→+∞ y ∈ (0, 1) 2 | ∇ϕ k (y) = N > 0. ( 17
) then | y ∈ (0, 1) 2 | N ∈ spt(R y ) | > 0.
Proof Assume for contradiction that

| y ∈ (0, 1) 2 | N ∈ spt(R y ) | = 0.
Since N / ∈ spt(R y ) for a.e. y ∈ (0, 1) 2 , there exists r > 0 such that

R y (B(N, r)) = 0 for a.e. y ∈ G. Let ψ ∈ C ∞ c (R 2 ) such that spt(ψ) ⊂ B(N, r), ψ(N) = 1, ψ(n) = 0 ∀n ∈ K \ {N }.
By definition of the Young measures, up to a subsequence,

ψ(∇ϕ k ) ⇀ 0 weakly * in L ∞ ((0, 1) 2 ), which yields (0,1) 2 ψ(∇ϕ k )dx → 0, that is, lim k→+∞ y ∈ (0, 1) 2 | ∇ φε (y) = N = 0
in contradiction with hypothesis (17). ✷

4 Proof Theorem 2.1

In order to prove Theorem 2.1, we start with two general results involving lower semicontinuous boundaries. First, we recall the following Poincaré Lemma, which does not require smoothness over all the boundary. The proof of this result is standard, summing on vertical lines. Lemma 4.1 (Poincaré Lemma for l.s.c. boundaries) Let ϕ : (0, 1) 2 → [0, 1] be lower semicontinuous. Then there exists a constant K > 0 depending only on ϕ, such that

Ωϕ |u| 2 dx K Ωϕ |∇u| 2 dx + Γ |u| 2 dS ∀u ∈ H 1 (Ω ϕ ) (18)
wher Ω ϕ is the open set defined by

Ω ϕ = (x 1 , x 2 , x 3 ) | (x 1 , x 2 ) ∈ (0, 1) 2 , -1 < x 3 < ϕ(x 1 , x 2 ) .
Lemma 4.2 Assume that ϕ : T 2 → [0, 1] is a continuous function and define ϕ ε by [START_REF] Casado-Díaz | Why viscous fluids adhere to rugose walls: A mathematical explanation[END_REF].

The following statements are equivalent:

(i) the only constant c ∈ R 3 satisfying ∀ψ ∈ C 1 c (D) Ω 1 c • ∇ψdx = 0 (19) is c = 0;
(ii) there exists k, ε 0 > 0 such that for every ε 0 > ε > 0 and every v ∈ H 1 (Ω ε ) satisfying the weak form of the non-penetration condition (4), the following inequality holds:

Γ |v| 2 dS εk ∇v 2 L 2 (Ωε) . (20)
Proof (i)⇒(ii) Considering a subdivision of (0, 1) 2 in squares (S ε i ) i of size ε × ε parallel to the axes, it is enough to prove the following:

S ε i ×{0} |v| 2 dS εk ∇v 2 L 2 (V ε i ) (21)
where

V ε i = {(x 1 , x 2 , x 3 ) | (x 1 , x 2 ) ∈ S ε i , -ε < x 3 < ϕ ε (x 1 , x 2 )} .
Performing a translation if necessary, we may assume that the square S ε i coincides with (0, ε) × (0, ε). Using the scaling function

H ε : (x 1 , x 2 , x 3 ) ∈ R 3 → ( x 1 ε , x 2 ε , x 3 ε ) ∈ R 3 , (21) is equivalent to Γ |v| 2 dS k ∇v 2 L 2 (Ω 1 ) , (22)
where

Ω 1 = H ε (V ε i ) is not depending on ε.
Assume for contradiction that there exists a sequence v k ∈ H 1 (Ω 1 ) satisfying the nonpenetration condition (4) and such that Γ |v k | 2 = 1 and ∇v k L 2 (Ω 1 ) → 0. From Lemma 4.1 we get that v k H 1 (Ω 1 ) is bounded, so there exists v * in H 1 (Ω 1 ) such that up to a subsequence,

v k ⇀ v * weakly in H 1 (Ω 1 ).
As a result, ∇v * L 2 (Ω 1 ) = 0 so there exists c ∈ R 3 such that

v * = c a.e. in Ω 1 .
Using the compactness of the trace operator from H 1 (Ω 1 ) to L 2 (Γ), we get

Γ |v * | 2 dS = 1
which implies that c = 0. This contradicts hypothesis (i) passing to the limit the nonpenetration condition (4) for v k .

(ii)⇒(i) Assume for contradiction that there exists c ∈ R 3 \ {0} such that (19) holds. Then the function v ∈ H 1 (Ω 1 ) defined by v = c satisfies the non-penetration condition in the weak form ( 4), but contradicts inequality [START_REF] Jaeger | On the roughness-induced effective boundary conditions for an incompressible viscous flow[END_REF] for every k > 0. ✷ Proof of Theorem 2.1. (i)⇒(ii) Assume for contradiction that ϕ is a riblet in the sense of [START_REF] Evans | Weak convergence methods for nonlinear partial differential equations[END_REF]. From Lemma 4.2, it is enough to show that a constant vector field of the form

v = (c 1 , c 2 , 0), with (c 1 , c 2 ) ∈ R 2 \ {0}, satisfies (19)
(or ( 4)). This would readily imply that hypothesis (i) does not hold. Since ϕ is a riblet, one may consider a sequence of smooth functions θ n approaching uniformly ϕ on R × {0} × {0} and construct a riblet from θ n in the direction (c 1 , c 2 , 0). Then, the weak form of the non-penetration condition can be written on the domain defined by the smooth riblet associated to θ n

Θ n = {(x 1 , x 2 , x 3 ) | (x 1 , x 2 ) ∈ (0, 1) 2 , -1 < x 3 < θ n (x 1 , x 2 )}. Since 1 Θn → 1 Ω 1 in L 1 (R 3
), the passage to the limit of the non-penetration condition concludes the proof. (ii)⇒(i) We need the following result concerning the uniqueness of riblet structures associated to a non constant, periodic and continuous function.

Lemma 4.3 Let ϕ ∈ C(T 2 ) be a non-constant function, extended on R 2 by periodicity. Assume there exist (c 1 , c 2 , c 3 ), (c * 1 , c * 2 , c 3 ) in R 3 \ {0} such that for every (x 1 , x 2 ) ∈ (0, 1) 2 and every h ∈ R, ϕ(x 1 + hc 1 , x 2 + hc 2 ) = ϕ(x 1 , x 2 ) + hc 3 , ϕ(x 1 + hc * 1 , x 2 + hc * 2 ) = ϕ(x 1 , x 2 ) + hc 3 . ( 23
)
Then either c 3 = 0 and (c 1 , c 2 ), (c * 1 , c * 2 ) are co-linear, or (c 1 , c 2 ) = (c * 1 , c * 2 ). Proof Assume that (c 1 , c 2 ) = α(c * 1 , c * 2 )
, for some α ∈ R. Then, relations (23) give directly hc 3 = hαc 3 for every h ∈ R. Consequently, either c 3 = 0, or α = 1.

Assume now that (c 1 , c 2 ) and (c * 1 , c * 2 ) are not co-linear. We will prove that ϕ is constant, in contradiction with our hypothesis. Indeed, let (x 1 , x 2 ) ∈ (0, 1) 2 . For every α, β > 0 we define

(d 1 , d 2 ) = α(c 1 , c 2 ) + β(c * 1 , c * 2 ). ( 24
)
By formula (23) we get

ϕ(x 1 + d 1 , x 2 + d 2 ) = ϕ(x 1 , x 2 ) + (α + β)c 3 . ( 25
)
Since ϕ is continuous and periodic, and since (d 1 , d 2 ) is arbitrary, (25) implies that c 3 = 0, i.e. ϕ is constant. ✷ Proof of Theorem 2.1 (continuation). Assume for contradiction that condition (i) in Theorem 2.1 does not hold. If ϕ is constant then (10) holds for every (c 1 , c 2 ) ∈ R 2 \{0}. Assume that ϕ is not constant. From the proof of Lemma 4.2, there exists c = (c 1 , c 2 , c 3 ) ∈ R 3 \{0} such that (19) holds. We prove that for every (x 1 , x 2 ) ∈ (0, 1) 2 and every h ∈ R such that (

x 1 + hc 1 , x 2 + hc 2 ) ∈ (0, 1) 2 , ϕ(x 1 + hc 1 , x 2 + hc 2 ) = ϕ(x 1 , x 2 ) + hc 3 . (26)
Let ρ be the standard mollifier in R 3 . For every ε > 0 we define ρ ε by

ρ ε (x) = 1 ε 3 ρ x ε ∀x ∈ R 3 .
Let η ∈ (0, 1) be a given constant. We introduce the following open subset of D:

D η = {x ∈ D | dist(x, ∂D) > η} .
We will prove that for every ε ∈ (0, η), every x ∈ D η and every h ∈ R such that x + hc ∈ D η , the following equality holds:

1 Ω 1 * ρ ε (x + hc) = 1 Ω 1 * ρ ε (x). ( 27
) Let ψ ∈ C 1 c (D) such that supp(ψ) ⊂ D η , and let ε ∈ (0, η). Applying (19) to ψ * ρ ε ∈ C 1 c (D) we get D 1 Ω 1 c • ∇(ψ * ρ ε )dx = 0 which yields D (1 Ω 1 * ρ ε )c • ∇ψdx = 0.
Integrating by part we get

D ∂(1 Ω 1 * ρ ε ) ∂c ψdx = 0. (28)
Since (28) holds for every ψ ∈ C 1 c (D η ) and since

1 Ω 1 * ρ ε ∈ C ∞ (D η ), we obtain ∂(1 Ω 1 * ρ ε ) ∂c (x) = 0 ∀x ∈ D η .
Consequently (27) is proved. Let us denote Γ 1 the upper part of ∂Ω 1 , namely

Γ 1 = (x 1 , x 2 , x 3 ) | (x 1 , x 2 ) ∈ (0, 1) 2 , x 3 = ϕ(x 1 , x 2 ) .
We need the following result.

Lemma 4.4 Let φ : R → R be a function satisfying the following property:

∃b ∈ R, ∀x ∈ R, ∀y ∈ R φ(x) > y =⇒ ∀h ∈ R φ(x + h) y + hb. (29)
Then φ(h) = φ(0) + bh for every h ∈ R.

Proof On the one hand, for every ε > 0, we have φ(0) > φ(0) -ε, so that

∀h ∈ R, φ(h) φ(0) -ε + hb. Making ε → 0, we get ∀h ∈ R, φ(h) φ(0) + hb.
Assume for contradiction that there exists

x 0 ∈ R such that φ(h 0 ) > φ(0) + h 0 b.
Then, using property (29), we get for a suitable ε > 0

φ(h 0 + h) φ(0) + h 0 b + ε + hb.
Taking h = -h 0 we get a contradiction. ✷

We shall prove in the sequel that for every x ∈ D and for every h ∈ R such that x + hc ∈ D, we have

x ∈ Ω 1 ⇒ x + hc ∈ Ω 1 ∪ Γ 1 ,
or equivalently that property (29) is satisfied by the restriction of ϕ on any line parallel to (c 1 , c 2 ). Let x ∈ Ω 1 and h ∈ R such that x + hc ∈ D. There exists η > 0 such that x, x + hc ∈ D η so (27) holds for every ε ∈ (0, η). Since ϕ is continuous, Ω 1 is open so there exists η 1 > 0 such that B(x, η 1 ) ⊂ Ω 1 . In particular, for every ε ∈ (0, min(η, η 1 )),

1 Ω 1 * ρ ε (x) = 1 Ω 1 (x) = 1.
To prove that x + hc ∈ Ω 1 ∪ Γ 1 we argue by contradiction. Suppose that

x + hc ∈ D \ (Ω 1 ∪ Γ 1 ). Since ϕ is continuous, D \ (Ω 1 ∪ Γ 1
) is open, so by the same argument as above we obtain that there exists η 2 > 0 such that

1 Ω 1 * ρ ε (x + hc) = 0 ∀ε ∈ (0, η 2 ).
5 Proof of Theorem 2.2

Proof of Theorem 2.2. Since Γ ϕ is a W 1,∞ -parametrization of Γ, with bounded norm independent on ϕ, it is enough to prove that there exists ε 0 > 0 such that

Γ |v| 2 dS k ϕ ∞ ∇v 2 L 2 (Ωϕ) (36)
for every admissible function ϕ ∈ W 1,∞ ((0, 1) 2 ) such that ε 0 > ϕ ∞ > 0 and every v ∈ H 1 (Ω ϕ ) satisfying the non-penetration condition on Γ ϕ . We fix M > 0. The value of M depends only on K, and will be fixed later in the proof.

Let Γ M = (0, M ) 2 × {0}, φ(x 1 , x 2 ) = 1 ϕ ∞ ϕ( ϕ ∞ x 1 , ϕ ∞ x 2 ) ∀(x 1 , x 2 ) ∈ (0, M ) 2 .
and

U ϕ = (x 1 , x 2 , x 3 ) | (x 1 , x 2 ) ∈ (0, M ) 2 , -1 < x 3 < φ(x 1 , x 2 ) .
Using the same scaling method as in the proof of Theorem 2.1 (formally for ε = ϕ ∞ ), it is enough to prove the existence of a positive constant k such that

Γ M |v| 2 dS k ∇v 2 L 2 (Uϕ) (37)
for every admissible function ϕ ∈ W 1,∞ ((0, 1) 2 ) with ε 0 > ϕ ∞ > 0 and for every v ∈ H 1 (U ϕ ) satisfying the non-penetration condition

Uϕ [(div v)ψ + v • ∇ψ]dx = 0 ∀ψ ∈ C 1 c ((0, M ) 2 × (-1, 2)). ( 38
)
Assume for contradiction that there exists a sequence of admissible functions ϕ k and a sequence of functions

v k ∈ H 1 (U ϕ k ) satisfying (38) such that Γ M |v k | 2 dS = 1 and ∇v k L 2 (Uϕ k ) → 0. (39)
Since ( φk ) k is uniformly bounded in W 1,∞ ((0, M ) 2 ), there exists a Lipschitz-continuous function φ ∈ C((0, M ) 2 ) such that, up to a subsequence, φk → φ uniformly on (0, M ) 2 and weakly * in W 1,∞ ((0, M ) 2 ). We now define

U := (x 1 , x 2 , x 3 ) | (x 1 , x 2 ) ∈ (0, 1) 2 , -1 < x 3 < φ(x 1 , x 2 ) . Clearly 1 Uϕ k → 1 U in L 1 (R 3 ). ( 40

) 13

As a consequence of a lemma 4.1, there exists a constant C > 0 such that

∀k, v k H 1 (Uϕ k ) C.
Define D M = (0, M ) 2 × (-1, 2). Since the family (U ϕ k ) k is equi-Lipschitz, there exists a family of extension operators

P k : H 1 (U ϕ k ) → H 1 (D M )
and a constant C > 0 such that

P k C ∀k ∈ N. As a consequence, v k (identified to P k (v k )) is uniformly bounded in H 1 (D M ), so there exists v * ∈ H 1 (D M ) such that, up to a subsequence, v k ⇀ v * weakly in H 1 (D M ). (41)
By (39), there exists

c = (c 1 , c 2 , c 3 ) ∈ R 3 such that v * = c a.e. in U ϕ .
Using the compactness of the trace operator from H 1 (U ) to L 2 (Γ M ), we obtain that M 2 c 2 = 1, hence c = 0. At this point, we get from Lemma 3.4 that for every g ∈ C(R 2 , R)

(c 1 , c 2 ) • R 2 g(Z)ZdR y (Z) = c 3 R 2 g(Z)dR y (Z) a.e. y ∈ (0, M ) 2 .
On the other hand, from Remark 3.3 we get R 2

ZdR y (Z) = ∇ φ(y) a.e. y ∈ (0, M ) 2 , while the non-penetration condition gives (in a purely geometric way, or as a consequence of Lemma 3.4 for g ≡ 1)

-(c 1 , c 2 ) • ∇ φ(y) + c 3 = 0, a.e. y ∈ (0, M ) 2 . (42)
Finally, for every g ∈ C(R 2 , R)

(c 1 , c 2 ) • R 2 g(Z)ZdR y (Z) - R 2 ZdR y (Z) R 2 g(Z)dR y (Z) = 0, a.e. y ∈ (0, M ) 2 . ( 43
)
If for some y ∈ (0, M ) 2 the support of R y contains two independent vectors of K, say N 1 and N 2 , then we get (c 1 , c 2 ) = 0. This can be proved by taking test functions g of the form

g(Z) = ag 1 (Z) + bg 2 (Z),
where a, b ∈ R and g i are continuous functions equal to 1 on N i and vanishing on K \ {N i }, i = 1, 2. Consequently, from (42) we also have c 3 = 0, in contradiction with M 2 c 2 = 1. We may assume that for a.e. y ∈ (0, M ) 2 , the support of R y contains only co-linear vectors of K. Recall, that the non degeneracy hypothesis implies that all co-linear vectors in K have the same sense. Since ∇ φ(y) = R 2 ZdR y (Z), there exists α y 1 and N y ∈ K such that ∇ φ(y) = α y N y .

For simplicity we choose N y as the shortest vector in each family of co-linear vectors in K.

If c 3 = 0 then from (42) all N y coincide with some fixed vector v ∈ K, otherwise (c 1 , c 2 ) would vanish. Consequently,

∂ φ ∂v (y) v 2 a.e. y ∈ (0, M ) 2 .
Since K is finite, we can fix M > 0 depending only on K to conclude that φ ∞ > 1 and get a contradiction. Assuming now that c 3 = 0, we get from (42) that -(c 1 , c 2 )α y N y + 1 = 0 for a.e. y ∈ (0, M ) 2 .

This implies that there exists a half plane containing almost every N y . Hence, we can repeat the argument above with a suitable vector v contained in the half plane and such that v • N y is bounded from below by a positive constant depending only on K. ✷

Further remarks on singular rough boundaries

Assume that the function ϕ : T 2 → [0, 1] is only lower semicontinuous and ϕ ε (x 1 , x 2 ) = εϕ( x 1 ε , x 2 ε ). Notice that their sub-graphs are open sets. Due to the lower semicontinous character, singular rugosities may occur around discontinuity points. In this case, the nonpenetration condition has to be weakened, in the sense of [START_REF] Bucur | On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries[END_REF]. Around discontinuity points of ϕ, testing only with C 1 c (D)-functions in (4) is not enough to capture the non-penetration of a vertical obstacle.

If ϕ is lower semicontinuous, a suitable version of Lemma 4.2, in which the non-penetration condition (4) is replaced by ( 5) can be interpreted as a characterisation of the uniform rugosity effect. The main difficulty is to give a correct geometric interpretation of a riblet, similar to the one obtained in Theorem 2.1.

The following geometry, which is not a riblet in the sense of [START_REF] Evans | Weak convergence methods for nonlinear partial differential equations[END_REF], does not produce a uniform rugosity effect:

ϕ : T 2 → R, ϕ(x 1 , x 2 ) = 1 -x 1 1 2 δ {x 2 = 1 2 } .
In fact, around the discontinuity points of ϕ, vertical surfaces of positive capacity have a crucial influence on the rugosity effect. This kind of situation falls out from [START_REF] Evans | Weak convergence methods for nonlinear partial differential equations[END_REF], and can not be captured, from a technical point of view, by the proof of Theorem 2.1. Indeed, the convolution method used in Theorem 2.1 ignores sets of zero Lebesgue measure, even though their capacity is strictly positive.

If the uniform rugosity effect does not hold, one we can prove, using the convolution technique of Theorem 2.1, that if x ∈ Ω 1 , there exists ε > 0 and c = (c 1 , c 2 , 0) = 0 such that for every h ∈ R |Ω c 1 ∩ B(x + hc, ε)| = 0. If for every x 0 ∈ T 2 , the value ϕ(x 0 ) coincides with its approximate lower limit at x 0 ϕ(x 0 ) = ap lim y→x 0 ϕ(y), then Lemma 4.4 can be used and ϕ is a ribblet. So Theorem 2.1 can be extended for lower and approximate lower semicontinuous functions. An example of such a function is ϕ : T 2 → R, generated by ∀(x 1 , x 2 ) ∈ [0, 1) 2 ϕ(x 1 , x 2 ) = x 1 .

Lemma 3 . 4

 34 With the notations of Remark 3.3, let us denote

This is a contradiction with (27) and (4).

Applying Lemma 4.4 to the function

we get (26).

In order to prove that c 3 = 0, we reproduce the previous argument for p periods of ϕ. Let us fix p ∈ N * . We define the following subsets of R 3 :

As above, there exists

We first prove that c ′ = ±c. By (30) we get that for every (x 1 , x 2 ) ∈ (0, p) 2 and every h ∈ R such that (

Since |c ′ | = |c| it is enough to prove that c ′ and c are colinear. Recall the following relations:

which hold for every (x 1 , x 2 ) ∈ (0, 1) 2 and every h, h * ∈ R such that (x 1 +hc 1 , x 2 +hc 2 ), (

in (34), we get that for every (x 1 , x 2 ) ∈ (0, 1) 2 and for h small enough,

Lemma 4.3 and relations (33)-(35) imply that (

). Denoting h * = λh in (34) and using (33) we get c 3 = λc ′ 3 . Thus c = λc ′ , with λ = ±1.

Finally, for every p ∈ N * relation (32) holds with the same constant vector c. Since ϕ is bounded, we get that c 3 = 0 hence ϕ is a riblet. ✷


Matthieu Bonnivard

The Drag in a Navier-Stokes Flow with Friction-Driven Boundary Conditions

. These boundary conditions account for the asymptotic effect of rough boundaries on the solutions to the Navier-Stokes equations. Consequently, they appear as a natural control variable in order to reduce the drag using the effect of micro-rugosities. In this paper, we prove the existence of a drag associated to friction-driven boundary conditions, we give a uniqueness criterion and, in the particular case of Navier's friction law, we prove that the drag is differentiable with respect to the friction coefficient, and compute its gradient.

Introduction

In this paper, we are interested in the definition and the mathematical properties of the drag of an obstacle in a Navier-Stokes flow, associated to the so-called friction-driven boundary conditions on the solid-fluid interface [START_REF] Bucur | Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions[END_REF].

Let N ∈ {2, 3} be the space dimension. We define the obstacle S as a compact subset of R N , with a Lipschitz boundary. The friction-driven boundary conditions are determined by a triplet {µ, A, V}, where:

• µ is a capacitary measure concentrated on ∂S,

• A is a positive symmetric matrix function,

• V = {V(x)} x∈∂S is a collection of vector spaces, where for q.e. x ∈ ∂S, V(x) is a subspace of the tangent hyperplane to S at x.

Let u ∞ ∈ R N be the velocity of the fluid at infinity. To define the drag of the solid S in the direction u ∞ , we (artificially) introduce an open subset Ω ⊂ R N such that S ⊂ Ω, and define the fluid domain F = Ω \ S.

We note L 2 0 (F) the subspace of L 2 (F) consisting of functions with mean value equal to zero. We consider the following problem: find (u, p) ∈ H 1 (F, R N ) × L 2 0 (F) such that -div(σ(u, p)) + (u • ∇)u = 0 in F, (1) div u = 0 in F,

u = u ∞ on ∂Ω, (2)
u(x) ∈ V(x) for q.e. x ∈ ∂S, (4) [2νD(u)n + µAu] • φ = 0 for any φ ∈ V(x), x ∈ ∂S, [START_REF] Březina | Asymptotic properties of solutions to the equations of incompressible fluid mechanics[END_REF] where σ(u, p) is the stress tensor defined by σ(u, p) = 2νD(u) -pId, ν > 0 being the viscosity of the fluid and D(u) the symmetric part of ∇u, defined by

D(u) = 1 2 (∇u) + (∇u) T .
In condition [START_REF] Březina | Asymptotic properties of solutions to the equations of incompressible fluid mechanics[END_REF], n stands for the normal vector field to ∂S oriented towards the solid. Condition (4) expresses that the flow is driven by the family of linear spaces {V(x)} x∈∂S . Above, [2νD(u)n + µAu] • φ = 0 for φ ∈ V(x) is a formal pointwise relation, which has to be understood globally on ∂S. Precisely, condition [START_REF] Březina | Asymptotic properties of solutions to the equations of incompressible fluid mechanics[END_REF]

holds provided that 2ν ∂S D(u)n • φ dx + ∂S Au • φ dµ = 0
for every φ ∈ H 1 (F, R N ) such that φ(x) ∈ V(x) for q.e. x ∈ ∂S. In particular, condition (5) is a friction condition, characterized by the measure µ and the matrix A.

Friction-driven boundary conditions have been obtained by Bucur, Feireisl and Nečasová in [START_REF] Bucur | Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions[END_REF], as the general expression of the boundary conditions resulting from the rugosity effect on perfectly slippery walls. The rugosity effect has been investigated in [START_REF] Mohammadi | Rough boundaries and wall laws[END_REF][START_REF] Jäger | On the roughness-induced effective boundary conditions for an incompressible viscous flow[END_REF][START_REF] Casado-Díaz | Why viscous fluids adhere to rugose walls: a mathematical explanation[END_REF][START_REF] Amirat | Effect of rugosity on a flow governed by stationary Navier-Stokes equations[END_REF] and in the more recent papers [START_REF] Bucur | On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries[END_REF][START_REF] Bucur | Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions[END_REF][START_REF] Březina | Asymptotic properties of solutions to the equations of incompressible fluid mechanics[END_REF]. It consists in approximating a given domain U by a sequence of geometric perturbations (U ε ) ε>0 , and studying the asymptotic behaviour of a family of vector fields u ε ∈ H 1 (U ε , R N ) in the neighborhood of the boundary of the limit domain U . In [START_REF] Bucur | Boundary behavior of viscous fluids: influence of wall roughness and friction-driven boundary conditions[END_REF], the authors have considered a sequence (u ε ) ε>0 of weak solutions to the Navier-Stokes equations with perfect slip boundary conditions in equi-Lipschitz domains (U ε ) ε>0 . They have proved the existence of a triplet {µ, A, V}, such that, up to a subsequence, (u ε ) ε>0 converges weakly to a function u solution to the Navier-Stokes equations with friction-driven boundary conditions. This asymptotic process is an indirect proof of the existence of a solution to the Navier-Stokes equations with friction-driven boundary conditions, for admissible triplets {µ, A, V}, that is, triplets that can be effectively obtained by a sequence of geometric perturbations of the domain. However, since this result relies on abstract Γ-convergence tools, the class of admissible triplets cannot be explicitely described, at least for very general perturbations. The purpose of this paper is to prove that, in fact, problem (1)-( 5) admits at least one weak solution (in a sense that will be precised below), for every ν > 0, u ∞ ∈ R N and every triplet {µ, A, V}. This result is stated in Theorem 1. Moreover, we prove that for a suitable choice of ν, u ∞ and V, which does not depend on {µ, A}, every weak solution to System (1)-( 5) is unique (see Theorem 2).

In the second part of the paper, we study the differentiability of the drag function with respect to the friction part of the boundary condition. Let ν > 0 and u ∞ ∈ R. We can associate to every triplet {µ, A, V} and every weak solution u to System (1)-( 5), a drag T (µ, A, V, u) defined by

T (µ, A, V, u) = 2ν F |D(u)| 2 dx + ∂S Au • u dµ.
In the case where u is unique, the drag depends only on the choice of the triplet {µ, A, V}, which appears as a natural control variable in drag minimization problems. As a first step in the study of the regularity of the drag with respect to {µ, A, V}, we consider Navier's boundary conditions on ∂S, which are given by the non penetration condition

u • n = 0 on ∂S (6)
and by the friction law 2ν [D(u)n] tan + βu = 0 on ∂S.

Here, n is the normal vector field to ∂S oriented towards the solid and β ∈ L 2 (∂S) is a friction coefficient. For any vector field v, defined on ∂S, we note [v] tan its tangential part, defined by

[v] tan = v -(v • n)n.
Boundary conditions ( 6)-( 7) correspond to the case where V(x) is equal to the tangent hyperplane to ∂S on q.e. x on the boundary, A ≡ Id and µ is absolutely continuous with respect to the surface measure on ∂S with a density β ∈ L 2 (∂S). We prove that the drag is differentiable with respect to β and compute the gradient of the drag in the L 2 (∂S) topology (see Theorem 3 and Corollary 4). The structure of the paper is the following. In Section 2, we introduce the functional framework associated with System (1)-( 3) and boundary conditions (4)-( 5) and ( 6)- [START_REF] Bucur | On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries[END_REF], and state our main results. In Section 3, we recall some abstract material from [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF] which we use in the proof of Theorems 1 and 2. Section 4 is devoted to the proof of Theorem 3 and Corollary 4.

Statement of the Main Results

We note

H 1 (F) = H 1 (F, R N ).
To define the weak solutions to System (1)-( 5), we introduce the following subsets of H 1 (F):

K = {v ∈ H 1 (F) | div v = 0 in L 2 (F), v(x) ∈ V(x) for q.e. x ∈ ∂S, v = u ∞ on ∂Ω} and V = {φ ∈ H 1 (F) | div φ = 0 in L 2 (F), v(x) ∈ V(x) for q.e. x ∈ ∂S, φ = 0 on ∂Ω}.
Let us fix the family of linear spaces V such that V is non empty. In particular, V is a separable Hilbert space for the H 1 scalar product. We note V ′ its dual space for the H 1 norm. Let µ be a capacitary measure concentrated on ∂S and A a positive symmetric matrix function.

We say that u ∈ H 1 (F) is a weak solution to system (1)-( 5) if u ∈ K and for every

φ ∈ V , 2ν F D(u) : D(φ) dx + F [(u • ∇)u] • φ dx + ∂S Au • φ dµ = 0. ( 8
)
The following results hold:

Theorem 1 Given ν > 0 and u ∞ ∈ R N , there exists at least one weak solution to System (1)-( 5). Moreover, there exists a constant M > 0 depending only on ν, u ∞ , Ω and S, such that every weak solution u ∈ K satisfies the following estimate:

u H 1 (F ) M. (9)
Theorem 2 There exists a constant C = C(Ω, S) > 0 such that, for all u ∞ ∈ R N , there exists ν 0 > 0 satisfying

ν 0 C(|u ∞ | + |u ∞ | 2 ) 1/2 (10)
and such that for all ν > ν 0 , System (1)-( 5) has a unique weak solution, which satisfies the following estimate:

u H 1 (F ) C ′ (ν + |u ∞ |), (11)
where C ′ is a positive constant depending only on Ω, S.

Now we consider System (1)-( 3) with Navier's boundary conditions ( 6)-( 7) on ∂S. To define weak solutions to this system, we introduce

H τ = {v ∈ H 1 (F) | div v = 0 in L 2 (F), v • n = 0 on ∂S},
and define, as above,

K τ = {v ∈ H τ | v = u ∞ on ∂Ω} and V τ = {φ ∈ H τ | φ = 0 on ∂Ω} We say that u ∈ H 1 (F) is a weak solution to system (1)-(3) with boundary conditions (6)-(7) if u ∈ K τ and for every φ ∈ V τ , 2ν F D(u) : D(φ) dx + F [(u • ∇)u] • φ dx + ∂S βu • φ dH N -1 = 0. ( 12
)
Let u ∞ ∈ R N and ν 0 > 0 as in Theorem 2. We fix ν > ν 0 , so that for every β ∈ L 2 (∂S), any weak solution to system (1)-( 3), ( 6)- [START_REF] Bucur | On the asymptotic limit of the Navier-Stokes system on domains with rough boundaries[END_REF], is unique. To each friction coefficient β ∈ L 2 (∂S), we associate a drag T (β) defined by

T (β) = 2ν F |D(u)| 2 dx + ∂S β|u| 2 dH N -1 ,
where u ∈ K τ satisfies (12).

To address the differentiability of the drag with respect to β, we introduce the following open subset of L 2 (∂S)

O = {β ∈ L 2 (∂S) | β > 0 q.e. on ∂S},
and define the mapping

R : O → H τ β → u,
where u ∈ K τ satisfies (12). The following result holds:

Theorem 3 The mapping β ∈ O → R(β) ∈ H τ
is differentiable, and for every β ∈ O, its derivative

DR(β) ∈ L(L 2 (∂S), H τ ) at point β is defined by DR(β)h = w ∀h ∈ L 2 (∂S),
where w is the unique solution to the system:

           -div(σ(w, p)) + (w • ∇)u + (u • ∇)w = 0 in F, div w = 0 in F, w = 0 on ∂Ω, w • n = 0 on ∂S, 2ν [D(w)n] tan + βw = -hu on ∂S, (13)
in the following sense: w ∈ V τ and for every φ ∈ V τ ,

2ν F D(w) : D(φ) dx+ F [(w•∇)u+(u•∇)w]•φ dx+ ∂S βw•φ dH N -1 = - ∂S hu•φ dH N -1 . (14)
The following corollary has interesting applications to the drag optimization problem.

Corollary 4 The mapping β ∈ O → T (β) ∈ R
is differentiable, and for every β ∈ O, its derivative DT (β) ∈ L(L 2 (∂S), R) is defined by

DT (β)h = 2 2ν F D(u) : D(w) dx + ∂S βu • w + ∂S h|u| 2 dH N -1 ∀h ∈ L 2 (∂S)
where u = R(β) and w = DR(β)h is the solution to system (13). For every β, the gradient of T at β with respect to the L 2 (∂S) topology is given by the following formula:

∇T (β) = (u + ψ) • u q.e. on ∂S, (15)
where u = R(β) and ψ is the unique solution to

           -div(σ(ψ, p)) + (∇u) T ψ -(u • ∇)ψ = 2(u • ∇)u in F, div ψ = 0 in F, ψ = 0 on ∂Ω, ψ • n = 0 on ∂S, 2ν [D(ψ)n] tan + βψ = 0 on ∂S, (16)
in the sense that ψ ∈ V τ and for every φ ∈ V τ ,

2ν F D(ψ) : D(φ) dx+ F (∇u) T ψ -(u • ∇)ψ •φ dx+ ∂S βψ•φ dH N -1 = 2 F [(u•∇)u]•φ dx. ( 17
)
3 Proof of Theorems 1 and 2

In order to prove Theorems 1 and 2, we follow the standard steps from the theory of stationary Navier-Stokes equations, which we adapt to the framework of friction-driven boundary conditions. These arguments are developed in [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF], for instance.

We define the bilinear form a 0 :

H 1 (F) × H 1 (F) → R by a 0 (v 1 , v 2 ) = 2ν F D(v 1 ) : D(v 2 ) dx + ∂S Av 1 • v 2 dµ ∀(v 1 , v 2 ) ∈ H 1 (F) × H 1 (F), (18)
and the trilinear form a 1 :

H 1 (F) × H 1 (F) × H 1 (F) → R by a 1 (v, v 1 , v 2 ) = F [(v • ∇)v 1 ] • v 2 dx ∀(v, v 1 , v 2 ) ∈ H 1 (F) × H 1 (F) × H 1 (F). (19) Let u 0 ∈ K such that u 0 H 1 (F ) C 0 |u ∞ | (20)
where C 0 > 0 depends only on F. The existence of C 0 and u 0 ∈ K satisfying (20) was proved by Bogovskiȋ [START_REF] Bogovskiȋ | Solution of the first boundary value problem for an equation of continuity of an incompressible medium[END_REF]. Setting u = uu 0 , we get that a function u ∈ K satisfies (12) if and only if u ∈ V and satisfies the following identity:

∀φ ∈ V a 0 ( u, φ)+a 1 (u 0 , u, φ) + a 1 ( u, u 0 , φ) + a 1 ( u, u, φ) = -a 0 (u 0 , φ) -a 1 (u 0 , u 0 , φ). (21)
We define the mapping a :

H 1 (F) × H 1 (F) × H 1 (F) → R by a(v, v 1 , v 2 ) = a 0 (v 1 , v 2 ) + a 1 (u 0 , v 1 , v 2 )+a 1 (v 1 , u 0 , v 2 ) + a 1 (v, v 1 , v 2 ) ∀(v, v 1 , v 2 ) ∈ H 1 (F) × H 1 (F) × H 1 (F), ( 22
)
and the linear form

f = f (u 0 ) ∈ V ′ by f , φ V ′ ×V = -a 0 (u 0 , φ) -a 1 (u 0 , u 0 , φ) ∀φ ∈ V.
The problem we consider is equivalent to the following : find u 0 ∈ K and u ∈ V such that

a( u, u, φ) = f (u 0 ), φ V ′ ×V ∀φ ∈ V. ( 23
)
We recall the following results from [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF].

Theorem 5 Assume that the following hypotheses hold:

(i) for all v ∈ H 1 (F), the mapping (v 1 , v 2 ) → a(v, v 1 , v 2 ) is continuous on H 1 (F) × H 1 (F);
(ii) there exists a constant α > 0 such that a(φ, φ, φ) α φ 2

H 1 (F ) ∀φ ∈ V ; ( 24
)
(iii) the space V is separable and, for all φ ∈ V , the mapping

v → a(v, v, φ)
is sequentially weakly continuous on V , i.e., weak lim

m→∞ v m = v in V implies lim m→∞ a(v m , v m , φ) = a(v, v, φ) ∀φ ∈ V. (25)
Then, problem (23) admits at least one solution.

Theorem 6 Moreover, assume that (iv) the bilinear form a(v, •, •) is uniformly V -elliptic with respect to v, i.e., there exists a constant α > 0 such that

a(v, φ, φ) α φ 2 H 1 (F ) ∀(v, φ) ∈ V × V ; (26)
(v) there exists a continuous and monotonically increasing function L : R + → R + such that for all η > 0

|a(v 1 , v 1 , v 2 ) -a(v 2 , v 1 , v 2 )| L(η) v 1 H 1 (F ) v 2 H 1 (F ) v 1 -v 2 H 1 (F ) ∀v 1 , v 2 ∈ V, ∀v 1 , v 2 ∈ S η = {v ∈ V | v H 1 (F ) η}. ( 27
)
Then, under the condition

[ f V ′ /(α 2 )]L( f V ′ /α) < 1, (28)
the solution u ∈ V to problem (23) is unique, and satisfies the following estimate:

u H 1 (F ) f V ′ /α. (29)
Proof of Theorem 1. Let us fix u ∞ ∈ R N and ν > 0. We need to prove that the mapping a defined by ( 22) satisfies the assumptions of Theorem 5. By Sobolev imbedding theorem (see the standard monograph by Adams [START_REF] Adams | Sobolev spaces[END_REF]), H 1 (F) is continuously imbedded in L 4 (F, R N ) for N = 2, 3. Therefore, by Hölder inequality, the mapping a 1 defined by ( 19) is continuous. Since a 0 defined by ( 18) is continuous, we obtain the continuity of a, and consequently, condition (i) from Theorem 5 holds.

To prove (ii), we prove the stronger condition (iv). We use the following Korn inequality, which is proved for instance in [4] (see the proof of Lemma VII.3.5, p. 329):

Lemma 7 There exists C 1 > 0 such that for all φ ∈ V , ∇φ L 2 (F ,R N ×N ) C 1 D(φ) L 2 (F ,R N ×N ) .
Consequently, by Poincaré inequality, there exists C 2 > 0 such that for all φ ∈ V ,

1 C 2 φ 2 H 1 (F ) 2 D(φ) 2 L 2 (F ,R N ×N ) . (30)
Let v, φ ∈ V . We observe that

a(v, φ, φ) = a 0 (φ, φ) + a 1 (φ, u 0 , φ).
Indeed, a 1 (u 0 , φ, φ) = 0 and a 1 (v, φ, φ) = 0, which comes from the following computation:

a 1 (u 0 , φ, φ) = 1 2 F (u 0 • ∇)(|φ| 2 ) dx = 1 2 ∂S |φ| 2 u 0 • n dH N -1 + ∂Ω |φ| 2 u 0 • n dH N -1 . (31)
Since φ ∈ V , φ = 0 on ∂Ω, and since u 0 ∈ K, u 0 • n = 0 on ∂S. Therefore, last boundary integrals vanish. Replacing u 0 by v we obtain the desired conclusion.

For every u 0 ∈ K satisfying (20), we note ρ(u 0 ) the best positive constant such that

|a 1 (φ, u 0 , φ)| ρ(u 0 ) φ 2 H 1 (F ) ∀φ ∈ V, (32)
We prove the following result:

Lemma 8 For every δ > 0, there exists u 0 ∈ K satisfying (20) and such that ρ(u 0 ) δ.

Proof. We consider a bounded extension operator

P : H 1 (F) → H 1 0 (Ω, R N ).
Let δ > 0. By the proof of Lemma 2.3 in [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF], chapter IV, there exists

u 0 ∈ H 1 (Ω, R N ) such that    div u 0 = 0 in Ω, u 0 = u ∞ on ∂Ω, u 0 = 0 in a neighborhood of S,
and such that for every ϕ ∈ H

1 0 (Ω, R N ), Ω [(ϕ • ∇) u 0 ] • ϕ dx δ P ϕ H 1 (Ω,R N ) . (33)
Let φ ∈ V . By inequality (33), we obtain

Ω [((P φ) • ∇) u 0 ] • (P φ) dx δ P P φ H 1 (Ω,R N ) δ φ H 1 (F ) .
Since u 0 vanishes on S, observing that

Ω [((P φ) • ∇) u 0 ] • (P φ) dx = F [(φ • ∇) u 0 ] • φ dx,
we obtain the desired result with u 0 = ( u 0 ) |F . ✷ Let u 0 ∈ K such that ρ(u 0 ) < ν/C 2 , where C 2 is defined by (30). By conditions (30) and (32), for every v, φ ∈ V ,

a(v, φ, φ) ν C 2 -ρ(u 0 ) φ 2 H 1 (F ) .
Consequently, condition (iv) holds with α = ν/C 2ρ(u 0 ). To prove condition (25), we only need to consider the nonlinear term a 1 (v m , v m , φ). Integrating by part and using the incompressibility condition, we obtain

a 1 (v m , v m , φ) = - F [(v m • ∇)φ] • v dx + ∂S∪∂Ω (v m • φ)(v m • n) dH N -1 .
Since φ, v m ∈ V , we conclude as above that

a 1 (v m , v m , φ) = - F [(v m • ∇)φ] • v m dx. Since H 1 (F) is continuously imbedded in L 4 (F, R N ), we obtain lim m→∞ a 1 (v m , v m , φ) = a 1 (v, v, φ).

This proves condition (iii).

Now let u ∈ K be a weak solution to System (1)-( 5). We define u = uu 0 . Applying relation (23) to φ = u and using inequality (26), we obtain

u H 1 (F ) 1 α f (u 0 ) V ′ .
By the triangular inequality, estimate (9) holds with M = u 0 H 1 (F ) + 1/α f (u 0 ) V ′ . This concludes the proof of Theorem 1. ✷ with the estimate

ξ H 1 (F ) C h V ′ τ . (42)
Fix (β, v) ∈ O × V 0 . We will prove that the bilinear form associated to D v F (β, v) satisfies the assumptions of Lax-Milgram theorem. The continuity results from the continuity of a 0 and a 1 , and the coercivity is a consequence of the definition of V 0 . Indeed, by computation (31), for all φ ∈ V τ ,

D v F (β, v)φ, φ V ′ τ ×Vτ = a 0 (φ, φ) + a 1 (φ, u 0 , φ) + a 1 (φ, v ,
φ). Consequently, using property (30) and definitions (32) and (34), we obtain for all φ ∈ V τ ,

D v F (β, v)φ, φ V ′ τ ×Vτ ν C 2 -ρ(u 0 ) -N v H 1 (F ) φ 2 H 1 (F ) . Since v ∈ V 0 , ν C 2 -ρ(u 0 ) -N v H 1 (F ) > 0.
This proves that the mapping

β ∈ O → u ∈ V τ
where u satisfies (39), is differentiable. Consequently, since for all

β ∈ O, R(β) = u 0 + u, R is differentiable on O. Moreover, for every β ∈ O, DR(β) ∈ L(L 2 (∂S), H) is defined by DR(β) = -[D v F (β, R(β) -u 0 )] -1 D β F (β, R(β) -u 0 ). Since for every (β, v) ∈ O × V 0 , D β F (β, v) ∈ L(L 2 (∂S), V ′ τ ) is defined by ∀h ∈ L 2 (∂S), ∀φ ∈ V τ D β F (β, v), h = ∂S h(v + u 0 ) • φ dH N -1 ,
defining w = DR(β)h for h ∈ L 2 (∂S), we obtain that w ∈ V τ satisfies the following identity:

∀φ ∈ V τ a 0 (w, φ) + a 1 (u 0 , w, φ) + a 1 (w, u 0 , φ) + a 1 (w, R(β)u 0 , φ)

+ a 1 (R(β) -u 0 , w, φ) = - ∂S hR(β) • φ dH N -1 .
Using the trilinearity of a 1 , we obtain (14). ✷ This proves (15). ✷

Remark 9

The coercivity of the bilinear form associated to the weak formulation (43) results from the following estimate (see the proof of Theorems 1 and 2):

ρ(u) ρ(u 0 ) + ρ( u) ρ(u 0 ) + N u H 1 (F ) < ν C 2 .

Proof of Corollary 4 .F[ 1 = 2 F 2 FFDFD 1 = 2 F 2 F

 4122122 Noticing that for every φ ∈ V τ ,[(∇u) T ψ] • φ = [(φ • ∇)u] • ψ and -(u • ∇)ψ] • φ dx = F [(u • ∇)φ] • ψ dx,(17)can be written in the following form:∀φ ∈ V τ 2ν F D(ψ) : D(φ) dx + F ψ • [(φ • ∇)u + (u • ∇)φ] dx + ∂S βψ • φ dH N -[(u • ∇)u] • φ dx. (43)Let us fix β ∈ O and h ∈ L 2 (∂S). We note u = R(β) and w = DR(β)h. Taking φ = w in (12), we express DT (β)h as follows:DT (β)h = -[(u • ∇)u] • w dx + ∂S h|u| 2 dH N -1 .(44)Besides, taking φ = ψ in (14) and φ = w in (43), we obtain 2ν (w) :D(ψ) dx + F [(w • ∇)u + (u • ∇)w] • ψ dx + ∂S βw • ψ dH N -1 = -∂S hu • ψ dH N -1 , 2ν (ψ) : D(w) dx + F ψ • [(w • ∇)u + (u • ∇)w] dx + ∂S βψ • w dH N -[(u • ∇)u] • w dx.Substracting these equalities, we obtain -[(u • ∇)u] • w dx = ∂S hu • ψ dH N -1and finally, by (44),DT (β)h = ∂S [(u + ψ) • u]h dH N -1 .

Proof of Theorem 2. Let us fix u ∞ ∈ R N .

To prove (v), we observe that for v, v, v 1 , v 2 ∈ V ,

Introducing

we obtain that (27) holds with L(η) = N for all η > 0. We define

Estimate [START_REF] Jäger | On the roughness-induced effective boundary conditions for an incompressible viscous flow[END_REF] results from the existence of a constant C 3 > 0, such that for every φ ∈ V ,

As a result, by (20), there exists C 4 > 0 such that

and consequently, inequality (35) implies that there exists a constant C 5 > 0 such that

This proves [START_REF] Jäger | On the roughness-induced effective boundary conditions for an incompressible viscous flow[END_REF]. Let ν > ν 0 . By definition of the infimum, there exists u 0 ∈ K satisfying (20) and such that

We fix α = ν/C 2ρ(u 0 ). By (35), α > 0 and by the same argument as above, α satisfies (26). Now let u ∈ K be a weak solution to System (1)-( 5) and let u = uu 0 . Condition (28) reads

By (35), last estimate holds. By Theorem 6, u ∈ V is the unique solution to (23). Using [START_REF] Jäger | On the roughness-induced effective boundary conditions for an incompressible viscous flow[END_REF] and (36), we obtain the following estimate:

Consequently, be (20) and the triangular inequality,

This proves [START_REF] Mohammadi | Rough boundaries and wall laws[END_REF]. By taking the infimum over all admissible functions u 0 , for ν > ν 0 , we obtain the uniqueness of the function u ∈ K satisfying (21). Therefore, System (1)-( 5) admits one unique weak solution u 0 + u ∈ H 1 (F).

✷ 10 4 Proof of Theorem 3

In this section, we assume that V(x) is equal to the tangent hyperplane to ∂S on q.e. x on the boundary, A ≡ Id and µ is absolutely continuous with respect to the surface measure on ∂S with a density β ∈ L 2 (∂S).

Let u ∞ ∈ R N . Let ν 0 > 0 as in Theorem 2, and ν > ν 0 . Let u 0 ∈ K satisfying (20) and such that

We use the mappings introduced in Section 3. We define g = g(u

The problem we consider is equivalent to the following: find

Proof of Theorem 3. The proof is based on the implicit function theorem. We will prove that the mapping

where u satisfies (21), is differentiable. We introduce the open subset of V τ :

and define the mapping

The mapping F is well defined and F is of class C ∞ . By the proof of Theorems 1 and 2, for every β ∈ O there exists one unique u ∈ V 0 satisfying (21). We need to prove that for every

τ ×Vτ = a 0 (ξ, φ) + a 1 (u 0 , ξ, φ) + a 1 (ξ, u 0 , φ) + a 1 (ξ, v, φ) + a 1 (v, ξ, φ).

We need to prove that for every (β, v) ∈ O × V 0 , there exists a constant C > 0 such that for every h ∈ V ′ τ , there exists one unique ξ ∈ V τ such that


Nicola C Hunt

Liam M Grover
email: l.m.grover@bham.ac.uk

Á L M Grover

Cell encapsulation using biopolymer gels for regenerative medicine

Keywords: Tissue engineering, Regenerative medicine, Hydrogels, Cell encapsulation, Biomaterials

There has been a consistent increase in the mean life expectancy of the population of the developed world over the past century. Healthy life expectancy, however, has not increased concurrently. As a result we are living a larger proportion of our lives in poor health and there is a growing demand for the replacement of diseased and damaged tissues. While traditionally tissue grafts have functioned well for this purpose, the demand for tissue grafts now exceeds the supply. For this reason, research in regenerative medicine is rapidly expanding to cope with this new demand. There is now a trend towards supplying cells with a material in order to expedite the tissue healing process. Hydrogel encapsulation provides cells with a three dimensional environment similar to that experienced in vivo and therefore may allow the maintenance of normal cellular function in order to produce tissues similar to those found in the body. In this review we discuss biopolymeric gels that have been used for the encapsulation of mammalian cells for tissue engineering applications as well as a brief overview of cell encapsulation for therapeutic protein production. This review focuses on agarose, alginate, collagen, fibrin, hyaluronic acid and gelatin since they are widely used for cell encapsulation. The literature on the regeneration of cartilage, bone, ligament, tendon, skin, blood vessels and neural tissues using these materials has been summarised.

Introduction

As the mean life expectancy of the developed world has increased, there has been an ever increasing demand for the development of new strategies for the repair of diseased and damaged tissues [START_REF] Crimmins | Trends in healthy life expectancy in the United States, 1970-1990: gender, racial, and educational differences[END_REF]. Ceramic, metallic and polymeric materials have been investigated widely for the direct replacement of tissues with good success [START_REF] Zhang | Nanotechnology and nanomaterials: promises for improved tissue regeneration[END_REF]. While the majority of licensed treatments consist of a synthetic material alone, there is a current trend towards the delivery of cells within the material matrix in order to expedite healing [START_REF] Kretlow | Injectable biomaterials for regenerating complex craniofacial tissues[END_REF]. Much of this work has involved the use of sponge-like scaffold materials exhibiting interconnected porosity [START_REF] Rosa | Macroporous scaffolds associated with cells to construct a hybrid biomaterial for bone tissue engineering[END_REF]. Although within such structures, the cells are arranged spatially in three dimensions with respect to one another, they still attach to a two dimensional surface and as such do not exhibit a phenotype that would be expected in native tissue. Since cell phenotype is critical to correct healing of the damaged tissue, a number of workers have attempted to deliver cells within a matrix that is more akin to the extracellular matrix (ECM) than sponge-like polymers or ceramics. The structure of a hydrogel is morphologically similar to that of the ECM and when used as an encapsulation medium [START_REF] Bokharia | The enhancement of osteoblast growth and differentiation in vitro on a peptide hydrogel-polyHIPE polymer hybrid material[END_REF], it enables the cell population to exhibit phenotypes more similar to those in vivo than when the cells are grown in monolayer culture [START_REF] Abbott | Biology's new dimension[END_REF]. A vast range of different hydrogel based materials are available with contrasting chemistries which may or may not allow cell attachment [START_REF] Lee | Hydrogels for tissue engineering[END_REF]. In this review, we discuss approaches that have been taken to repair diseased or damaged tissues using biopolymeric hydrogels for cell encapsulation. We give a basic overview of the most frequently used biopolymer gel materials (Table 1) and also discuss their application in the repair of a number of tissue types.

Biopolymer gels

Alginate

Alginate is a polysaccharide isolated from brown algae which has been used with great success as a wound dressing [START_REF] Augst | Alginate hydrogels as biomaterials[END_REF]) and as a food additive. On dissolution in an aqueous medium, alginate forms a hydrocolloid, which gels ionotropically following the addition of multivalent cations. Alginate is formed from polysaccharides derived from a range of seaweeds found world-wide and consists of a mixture of b-D-mannuronic acid (M) and a-L-guluronic acid residues [START_REF] Lee | Hydrogels for tissue engineering[END_REF].

The ratio of M to G blocks can vary significantly depending upon the source of the raw materials used in alginate manufacture. Both cell adhesion and hydrogel stiffness can be influenced by M to G ratio [START_REF] Wang | Evaluation of sodium alginate for bone marrow cell tissue engineering[END_REF]. Since hydrogel formation occurs following electrostatic interaction between the carboxylic moieties on the G blocks of alginate and multivalent cations, the higher the ratio of G:M, the stiffer the resulting gel. Cell adhesion to alginate gels can be increased by covalently modifying the polymer with molecules such as RGD [START_REF] Rowley | Alginate type and RGD density control myoblast phenotype[END_REF].

Fibrin

Fibrin gels occur widely in the human body and are important in haemostasis following injury [START_REF] Janmey | Fibrin gels and their clinical and bioengineering applications[END_REF]. Fibrin gels form following the cleavage of fibrinogen by thrombin to expose regions on the fibrin molecules that interact allowing self assembly of protofibrils which aggregate and lengthen. At certain points in the fibrin network, branching occurs, which increases the volume of the gel. The presence of RGD motifs within the fibrin network allow cell adhesion and binding of a range of important growth factors. In the presence of mammalian cells, however, fibrin can degrade rapidly due to the localised secretion of proteolytic enzymes (Ye et al. 2000).

To overcome this problem fibrin degradation inhibitors and fibrin stabilisers, such as aprotinin, factor XIII and e-amino-n-caproic acid, have been added to the gels to maintain the structure over longer time periods [START_REF] Mol | Fibrin as a cell carrier in cardiovascular tissue engineering applications[END_REF][START_REF] Park | Tissueengineered cartilage using fibrin/hyaluronan composite gel and its in vivo implantation[END_REF]Ye et al. 2000). Furthermore, the importance of sufficient thrombin and calcium to prevent excessive degradation has been identified [START_REF] Eyrich | Long-term stable fibrin gels for cartilage engineering[END_REF].

Collagen

Collagen is the most abundant form of protein in humans, constituting 30% of all protein found in the body. There are 29 different forms of collagen in the body, the most ubiquitous which is type I collagen, which comprises triple a-helices, which in the correct environmental conditions self assemble to form a fibrillar structure [START_REF] Pachence | Collagen-based devices for soft tissue repair[END_REF]. The selfassembling tendencies of type I collagen have led to it being used as a hydrogel for use in tissue engineering. Rat tail collagen, for example, may be dispersed in acid medium and when neutralised in culture forms gels which have been widely used since they allow cell adhesion. The two major limitations of collagen-based scaffolds are their weak nature and their extensive contraction by encapsulated cells. One group has sought to address both of these drawbacks by plastic compression of the collagen gel immediately after its formation [START_REF] Brown | Ultrarapid engineering of biomimetic materials and tissues: fabrication of nano-and microstructures by plastic compression[END_REF]).

Gelatin

Gelatin is formed from the hydrolysation of collagen. Two different forms of gelatin with different isoelectric points can be formed depending on the hydrolysation protocol. Gelatin dissolves in water at 60°C then gels as the solution cools to room temperature [START_REF] Young | Gelatin as a delivery vehicle for the controlled release of bioactive molecules[END_REF]. Gelatin is widely used in pharmaceutical and medical applications due to its biodegradable nature [START_REF] Ikada | Gelatin hydrogel as a matrix to release protein drugs[END_REF][START_REF] Miyoshi | Effects of bFGF incorporated into a gelatin sheet on wound healing[END_REF].

Hyaluronic acid

Hyaluronan is found throughout the body in a number of tissues, including skin and cartilage. Colloids of hyaluronic acid (HA) can be gelled by prior chemical modification of hyaluronan with thiols, methacrylates or tyramines, or can be crosslinked in situ using formaldehyde or divinyl sulphone. One of the key advantages of using HA gels for tissue engineering is that their degradation can be mediated by hyaluronidase, an enzyme secreted by a multitude of mammalian cell types [START_REF] Peppas | Hydrogels in biology and medicine: from molecular principles to bionanotechnology[END_REF].

Agarose

Agarose can be formed into a gel that once heated to 90°C forms a polymer solution. When the temperature of this solution is lowered to room temperature gelation will occur. Agarose is used widely in molecular biology and since it is well accepted following implantation it has been evaluated for immunoisolation purposes (Lahooti and Sefton 2000a, b, c). The non-degradable nature of the gel, however, means that it has not been widely used in tissue engineering since scaffold materials used in tissue engineering applications should degrade over time to allow space for accumulation of new tissue.

The application of biopolymer gel encapsulation in regenerative medicine

Cell encapsulation in biopolymer hydrogels was initially investigated for immunoisolation of cells producing therapeutic proteins for treatment of diseases. Some of these studies are summarised in Table 2. More recently, encapsulation of mammalian cells has been used in the regeneration of an array of different tissues. Table 3 summarises a broad cross section of the literature covering the majority of the body tissues.

The remainder of the review, however, will focus on the use of biopolymer gels for the encapsulation of cells for use in musculoskeletal, neural, skin, hepatic and cardiovascular tissue engineering.

Musculoskeletal tissue engineering

During embryogenesis musculoskeletal tissue develops from the mesoderm. Mesenchymal stem cells (MSCs) differentiate to form the individual musculoskeletal tissues, which include muscle, cartilage, bone, tendons and ligaments. The musculoskeletal system enables locomotion and with the ageing population, novel approaches to regenerate parts of the musculoskeletal system and therefore restore patient mobility are gaining increasing attention.

Muscle

In musculogenesis MSCs differentiate to myoblasts, the muscle cell precursors, which further differentiate and fuse to form multinucleated myotubes. These myotubes mature into myofibres which bundle together to form the skeletal muscle [START_REF] Berendse | Myoblast structure affects subsequent skeletal myotube morphology and sarcomere assembly[END_REF]. Encapsulated skeletal myoblasts have been shown to remain viable and differentiate to form myotubules in collagen gel when mechanical conditioning is used after 7 day's culture in vitro. Cell alignment was shown to occur uniaxially in the direction of applied tension [START_REF] Cheema | 3-D in vitro model of early skeletal muscle development[END_REF].

Cartilage

Cartilage is found in joints and prevents bone to bone contact. The tissue consists of mainly chondrocyctes and an ECM containing collagens type I and II and aggregan [START_REF] Sterodimas | Tissue engineering and auricular reconstruction: a review[END_REF]. A large amount of research has focussed on tissue engineering cartilage by the encapsulation of chondrocyctes in a variety of hydrogels including fibrin, collagen, chitosan and alginate. Chondrocyctes encapsulated in fibrin gels have been maintained for 5 weeks in vitro [START_REF] Park | Tissueengineered cartilage using fibrin/hyaluronan composite gel and its in vivo implantation[END_REF]. [START_REF] Park | Tissueengineered cartilage using fibrin/hyaluronan composite gel and its in vivo implantation[END_REF] also showed that addition of HA gel to the fibrin gel may be beneficial since after 4 weeks in vivo culture the degree of contraction was reduced and ECM production was higher in fibrin/HA gels when compared with pure fibrin gels. Chondrocyctes encapsulated in pure HA gels have been evaluated both in vivo up to 12 weeks and in vitro up to 2 weeks post-encapsulation. The constructs were shown to maintain or increase in size and encapsulated chondrocyctes were shown to deposit ECM both in vivo and in vitro. The amount of ECM produced was also shown to be enhanced under mechanical loading [START_REF] Chung | Differential behavior of auricular and articular chondrocytes in hyaluronic acid hydrogels[END_REF].

Chondrocyctes encapsulated in collagen and gelatin gels were shown to produce cartilage specific matrix over 21 days in vitro culture. The collagen gel was shown to contract after 1 day and an increase in cell number was observed throughout the first 7 days. The viable cell number however, fell gradually over the subsequent 14 days culture. In contrast, the viability of chondrocyctes encapsulated in gelatin hydrogels was maintained, with no significant change in cell number [START_REF] Hoshikawa | Encapsulation of chondrocytes in photopolymerizable styrenated gelatin for cartilage tissue engineering[END_REF]. Maintaining the viability of encapsulated chondrocytes can be problematic due to insufficient perfusion of scaffolds with a large volume. The addition of microchannels to polymeric gels has been proven to maintain the viability and function of alginate encapsulated chondrocytes [START_REF] Choi | Microfluidic scaffolds for tissue engineering[END_REF], and this may prove effective for other polymeric gels.

Ligament and tendon

Tendons and fibroblasts are rich in fibroblasts which secrete ECM containing collagen type I/III, elastin and proteoglycans [START_REF] Carvalho | Structure and proteoglycan composition of specialized regions of the elastic tendon of the chicken wing[END_REF][START_REF] Chan | Scaffolding in tissue engineering: general approaches and tissue-specific considerations[END_REF][START_REF] Cheema | Fabricating tissues: analysis of farming versus engineering strategies[END_REF][START_REF] Cleary | Changes in chemical composition during development of bovine nuchal ligament[END_REF]. Collagen and fibrin gel encapsulation of fibroblasts have been shown to have promise for tissue engineering of tendons and ligaments. [START_REF] Marenzana | The origins and regulation of tissue tension: identification of collagen tension-fixation process in vitro[END_REF] showed that tendon fibroblasts encapsulated in collagen gels attached, spread and were orientated in the parallel to the long axis of the collagen fibrils.

The fibroblasts produced tensile forces and remodelled the matrix [START_REF] Marenzana | The origins and regulation of tissue tension: identification of collagen tension-fixation process in vitro[END_REF]. Ligament fibroblasts encapsulated in collagen gels proliferated and secreted further collagen up to day 14, then maintained the collagen level and cell number for a further 7 days culture. Gel contraction was seen to occur over the 21 days culture [START_REF] Murray | The effect of thrombin on ACL fibroblast interactions with collagen hydrogels[END_REF].

Similarly, ligament fibroblasts encapsulated in fibrin gels have been shown to proliferate, contract the gel and secrete ECM [START_REF] Chun | Cultures of ligament fibroblasts in fibrin matrix gel[END_REF].

Bone

Bone tissue consists mainly of osteoblast secreted mineralised ECM containing collagen type 1, collagen type IV, fibronectin and heparan sulphate (Narayanan [START_REF] Williams | Fibrin gel as a three dimensional matrix in cardiovascular tissue engineering[END_REF]. Fibrin gel encapsulation has been investigated as an approach for bone tissue engineering. Where MSCs have been used, their osteogenic differentiation is measured by expression of ALP, osteopontin, bone sialoprotein (BSP), and osteocalcin. MSCs encapsulated in fibrin gel have been shown to proliferate, but at a slower rate than in monolayer culture. Encapsulation was seen to enhance osteogenic differentiation and ECM production, compared with monolayer culture. Mineralised tissue was seen to accumulate in the pores that formed in the fibrin gel, which were surrounded by numerous cells [START_REF] Hou | In vitro evaluation of a fibrin gel antibiotic delivery system containing mesenchymal stem cells and vancomycin alginate beads for treating bone infections and facilitating bone formation[END_REF]. [START_REF] Catelas | Human mesenchymal stem cell proliferation and osteogenic differentiation in fibrin gels in vitro[END_REF] also showed that MSCs encapsulated in fibrin gels proliferate, show osteogenic differentiation, and secrete mineralised tissue. They, however, observed that the MSCs did not fully differentiate to mature osteoblasts within the 28 days of in vitro culture.

Skin

Skin is composed of two anatomically distinct layers known as the epidermis and the dermis. It comprises loose connective tissue rich in collagen type I that contains blood capillaries, smooth muscle fibres, sweat glands and sebaceous glands and their ducts, hair follicles and sensory nerve endings. The epidermis consists mainly of tightly packed stratified keratinocytes upon a basement membrane [START_REF] Fuchs | Skin stem cells: rising to the surface[END_REF][START_REF] Macneil | Progress and opportunities for tissue-engineered skin[END_REF][START_REF] Williams | Fibrin gel as a three dimensional matrix in cardiovascular tissue engineering[END_REF]. For tissue engineering skin fibrin has been used to encapsulate both fibroblasts, to develop a dermal analogue [START_REF] Cox | Behavior of human dermal fibroblasts in three-dimensional fibrin clots: dependence on fibrinogen and thrombin concentration[END_REF][START_REF] Meana | Large surface of cultured human epithelium obtained on a dermal matrix based on live fibroblast-containing fibrin gels[END_REF]) and keratinocytes to create an epidermal analogue [START_REF] Bannasch | Treatment of chronic wounds with cultivated autologous keratinocytes as suspension in fibrin glue[END_REF][START_REF] Bannasch | Cultured keratinocytes in fibrin with decellularised dermis close porcine full-thickness wounds in a single step[END_REF]. Encapsulated fibroblasts were shown to proliferate within the fibrin matrix, but also allow for the stratification of the co-cultured keratinocytes on the surface of the gel both in vivo and in vitro [START_REF] Meana | Large surface of cultured human epithelium obtained on a dermal matrix based on live fibroblast-containing fibrin gels[END_REF]. Fibrinencapsulated keratinocytes applied to the surface of Alloderm produce a continuous epithelium with a cornified layer and basement membrane after 4 weeks in vivo [START_REF] Bannasch | Cultured keratinocytes in fibrin with decellularised dermis close porcine full-thickness wounds in a single step[END_REF]. [START_REF] Brown | Ultrarapid engineering of biomimetic materials and tissues: fabrication of nano-and microstructures by plastic compression[END_REF] and co-workers have shown that plastic compressed collagen gel can be used to encapsulate fibroblasts for tissue engineering the dermis. The compression was only seen to reduce cell viability of encapsulated human dermal fibroblasts by 10% as long as the gel did not become desiccated. A concern when increasing the collagen content of the hydrogel is that diffusion of nutrients and waste products through the scaffold would not be sufficient to maintain cell viability over extended periods of time. To address this problem [START_REF] Nazhat | Controlled microchannelling in dense collagen scaffolds by soluble phosphate glass fibers[END_REF] incorporated micro-channels into the gels using soluble phosphate glass fibres and after 24 h the cell viability of encapsulated fibroblasts was seen to be greater than 80%.

We have shown that fibroblasts encapsulated in alginate hydrogel serve as a dermal analogue allowing for a stratified epithelium to form on the surface of the alginate from the surface seeded keratinocyte coculture (Hunt et al. 2009a). Furthermore, we have shown that these cells remain mitotically inhibited up to 33 days encapsulation (Hunt et al. 2009b) and that the cells remain viable for at least 150 days encapsulation. The hydrogel was shown to degrade by acellular mechanisms in vitro to release the encapsulated fibroblasts, which were seen to subsequently secrete ECM (Hunt et al. 2009a). We have also observed that the alginate is not contracted by the encapsulated fibroblasts, and this may result in reduced detrimental scar contraction which is often seen in large skin wounds [START_REF] Harrison | The mechanism of skin graft contraction: an update on current research and potential future therapies[END_REF].

Neural tissue

Chronic neurological diseases and physical injuries can result in loss of neuronal cell bodies, axons and associated glia support. Since the central nervous system has limited or no capacity to replace the lost neurons, there is a significant interest in the engineering of neural tissue [START_REF] Nisbet | Neural tissue engineering of the CNS using hydrogels[END_REF]). One of the major challenges facing researchers in this area is stimulation of guided axonal extension [START_REF] Norman | Guiding axons in the central nervous system: a tissue engineering approach[END_REF]. To this end, [START_REF] Horn | Influence of cross-linked hyaluronic acid hydrogels on neurite outgrowth and recovery from spinal cord injury[END_REF] encapsulated chick dorsal ganglia in HA and fibrin gels which were evaluated both in vitro. Both gels were shown to support neutrites within the first 60 h, but after 192 h a 50% increase in neutrite length was seen in HA gels when compared with the fibrin gels. Despite this, when the neurons encapsulated in the HA gels were implanted, there was no restoration of spinal cord function. [START_REF] Herbert | Effects of fibrin micromorphology on neurite growth from dorsal root ganglia cultured in three-dimensional fibrin gels[END_REF] also investigated the use of fibrin gels for neural tissue engineering. Encapsulated chick dorsal root ganglia were successfully maintained in the hydrogels and displayed neutrite outgrowth. [START_REF] Bellamkonda | Hydrogel-based 3-dimensional matrix for neural cells[END_REF] showed that PC12 neural progenitor cells and chick dorsal ganglia encapsulated in agarose also produced neutrites of 900 lm in length after 4 days in culture.

Another approach to neural regeneration is the localised and sustained release of nerve growth factor (NGF) to stimulate neural differentiation of precursor cells. For example, [START_REF] Zielinski | Chitosan as a matrix for mammalian-cell encapsulation[END_REF] have shown that a genetically modified fibroblast cell line provided sustained release of nerve growth factor (NGF) during encapsulation in chitosan hydrogels. The amount of NGF secreted was sufficient to induce the differentiation of co-cultured neural progenitor PC12 cells. The limitation of this approach, however, was that fibroblasts were seen to aggregate and after 2 weeks which resulted in necrosis in the centre of the aggregates.

Liver

Hepatic tissue is formed of hepatocytes which are organized into a polarized epithelium with distinct apical and basal domains [START_REF] Dunn | Hepatocyte function and extracellular-matrix geometrylong-term culture in a sandwich configuration[END_REF]. Hepatocytes have a high metabolic activity and produce many liver specific molecules such as urea, amino acids, glycogen and bile. The successful tissue engineering of liver is often assessed by measuring the secretion of these liver specific products.

Hepatocytes encapsulated in fibrin showed maintained viability over the first 3 days culture in vivo, but thereafter showed significant loss despite the maintenance of an even distribution of cells. Neotissue formation, however, was observed with good integration into the host tissue after 7 days in vivo as well as maintained liver specific function and phenotype in viable cells [START_REF] Bruns | Injectable liver: a novel approach using fibrin gel as a matrix for culture and intrahepatic transplantation of hepatocytes[END_REF].In order to facilitate the diffusion of nutrients within hydrogels encapsulating hepatocytes to maintain cell viability and function, various approaches have been taken. For example, the incorporation of micro-channels into gelatin hydrogels was shown to maintain 90% viability and metabolic activity of the encapsulated hepatocytes after 45 days in vitro culture [START_REF] Wang | Generation of three-dimensional hepatocyte/gelatin structures with rapid prototyping system[END_REF]. Similarly, [START_REF] Khattak | Enhancing oxygen tension and cellular function in alginate cell encapsulation devices through the use of perfluorocarbons[END_REF] have shown that perfluorocarbon incorporation can improve metabolic activity and viability of encapsulated hepatocytes for 2 weeks in vitro culture.

Cardiovascular tissue

The cardiovascular network is essential to maintaining the viability and functions of all living tissues. The network can become obstructed of damaged due to disease or injury and thus efforts towards the tissue engineering of blood vessels and cardiac tissue are being made [START_REF] Stegemann | Review: advances in vascular tissue engineering using proteinbased biomaterials[END_REF]. In order to engineer vascular tissue, myofibroblasts have been encapsulated in fibrin gels [START_REF] Mol | Fibrin as a cell carrier in cardiovascular tissue engineering applications[END_REF]Ye et al. 2000) and viability maintained for up to 6 weeks in vitro. The cells were also seen to proliferate and secrete collagen. Smooth muscle cells have also been encapsulated fibrin gels in order to produce an arterial media equivalent. After 3 weeks in vitro culture the arterial equivalent was seen to have a 95% reduction in volume and was shown to accumulate collagen, which was aligned in the circumferential direction of the tubular structure. The accumulation of collagen and the gel contraction was associated with an increase in ultimate tensile strength of the construct [START_REF] Grassl | A fibrin-based arterial media equivalent[END_REF].

Cardiac tissue has successfully been engineered in vivo by encapsulation of neonatal cardiac myocytes in fibrin gel within a silicon chamber implanted near an artery to maintain cell viability. After 3 weeks the implant was found to be viable with large amounts of muscle tissue which contracted in response to electrical stimulation and had neovascularisation throughout [START_REF] Birla | Myocardial engineering in vivo: formation and characterization of contractile, vascularized three-dimensional cardiac tissue[END_REF].

Conclusion

Biopolymer gel encapsulation of mammalian cells is finding increasing application in engineering a variety of different tissues. This review has summarised the most widely used biopolymeric hydrogels in tissue regeneration as well as the response of a range of cell types to encapsulation. Researchers have studied a variety of factors including cell viability, growth, matrix production and differentiation. Due the variation in culture conditions and material formulation, it is often difficult to make conclusions on the effectiveness of a particular polymeric gel for tissue regeneration applications and there is a need for an increase in systematic research in this field. What is clear is that alginate, collagen, fibrin, hyaluronic acid, and gelatin all show promise as materials for the encapsulation of cells for tissue engineering of cartilage, bone, ligament, tendon, skin, blood vessels and neural tissues and there is no doubt that polymeric cell encapsulation has great potential in the future of regenerative medicine. The potential importance, however, of tailoring degradation, ensuring sufficient diffusion through the biopolymer scaffolds and mechanical conditioning to ensure success of tissue engineering by encapsulation of cells in biopolymer gels have been highlighted in this review.

Table 1

 1 Biopolymer gels used for cell encapsulation and tissue regeneration and their key features

	Biopolymer Protein/polysaccharide Source	Gelation method	Mechanism of degradation Cell adhesivity
	Agarose	Polysaccharide	Seaweed	Thermal	Non-degradable	Low
	Alginate			Ionotropic cross-linking Ion exchange
	Collagen	Protein	Animal/human Neutralisation	Enzymatic cleavage	High
	Fibrin			Proteolytic cleavage
	Gelatin			Thermal
	HA	Polysaccharide		Chemical cross-linking
	HA hyaluronic acid

Table 2

 2 Biopolymer gel cell encapsulation for treatment of diseases with non-autologous cells Biopolymer gel Encapsulated cell type Secreted therapeutic protein Disease treatment References

	Agarose	Islet cells	Insulin	Diabetes	Gazda et al. (2007)
		Kidney cells	Hepatic lipase	Hyperlipidemia	Lahooti and Sefton (2000a)
		Fibroblasts	Alkaline phosphatase	Osteoporosis	Lahooti and Sefton (2000b)
	Alginate	Fibroblasts	VEGF	Cardiovascular disease Keshaw et al. (2005)
		Kidneys cells	Endostatin	Brain tumour	Joki et al. (2001),
					Read et al. (2001a, b)
		Myoblasts	Human factor IX	Haemophilia	Hortelano et al. (1996)
		Fibroblasts	Human factor IX	Haemophilia	Liu et al. (1993)
		Fibroblasts	Human growth hormone	Dwarfism	Chang et al. (1994)
		Islet cells	Insulin	Diabetes	Calafiore et al. (2006),
					Soon-Shiong (1999),
					Zimmermann et al. (2007)
	Collagen	Fibroblasts	Alkaline phosphatase	Osteoporosis	Lahooti and Sefton (2000c)

Table 3

 3 A summary of key references from the current literature describing cell encapsulation for the regeneration of a number of tissues

	Biopolymer gel	Encapsulated cell type	Engineered tissue	References
	Agarose	MSCs	Cartilage	Pelaez et al. (2009)
	Alginate	Dorsal root ganglia	Neural	Bellamkonda et al. (1995)
		Fibroblasts	Dermis	Hunt et al. (2009a)
		Chondrocyctes	Cartilage	Choi et al. (2006), Hong et al. (2007)
		MSCs	Bone	Smith et al. (2007), Wang et al. (2003)
		Osteoblasts	Bone	Kong et al. (2003)
		Hepatocytes	Liver	Khattak et al. (2007)
		Neural progenitor cells	Neural	Zielinski and Aebischer (1994)
	Chitosan	Chondrocyctes	Cartilage	Hong et al. (2007)
		Fibroblasts	Neural	Zielinski and Aebischer (1994)
	Collagen	Myofibroblasts	Skeletal muscle	Cheema et al. (2003)
		Fibroblasts	Dermis	Brown et al. (2005), Nazhat et al. (2006),
				Marenzana et al. (2006)
		Fibroblasts	Ligament	Murray et al. (2006)
		Fibroblasts	Tendon	Marenzana et al. (2006)
		Chondrocyctes	Cartilage	Hoshikawa et al. (2006)
		Osteosarcoma cells	Bone	Bitar et al. (2007)
	Fibrin	Myofibroblasts	Cardiac muscle	Ye et al. (2000), Birla et al. (2005)
		Myofibroblasts	Blood vessel	Mol et al. (2005)
		Smooth muscle cells	Blood vessel	Grassl et al. (2003)
		Fibroblasts	Dermis	Tuan et al. (1996), Brown et al. (1993),
				Meana et al. (1998), Cox et al. (2004)
		Fibroblasts	Ligament	Chun et al. (2003)
		Keratinocytes	Epidermis	Bannasch et al. (2000, 2008)
		Chondrocytes	Cartilage	Park et al. (2005), Eyrich et al. (2007),
				Mesa et al. (2006)
		MSCs	Bone	Catelas et al. (2006), Hou et al. (2008)
		Hepatocytes	Liver	Bruns et al. (2005)
		Dorsal root ganglia	Neural	Herbert et al. (1998)
		Urethral cells	Urethra	Bach et al. (2001)
		Keratocytes	Corneal stroma	Alaminos et al. (2006)
	Gelatin	Chondrocyctes	Cartilage	Hoshikawa et al. (2006)
		Hepatocytes	Liver	Wang et al. (2006)
	Hylauronic acid	Chondrocytes	Cartilage	Chung et al. (2008)
		Dorsal root ganglia	Neural	Horn et al. (2007)

BiotechnolLett (2010) 32:733-742