The effect of roughness on the boundary condition on porous wall

The effects of roughness on the Darcy boundary condition for the Stokes system are studied using rigorous asymptotic analysis and homogenization techniques. Starting from the Stokes system in domain with porous part of the boundary and assuming that the porous boundary is periodically oscillating, we determine the effective permeability as a function of roughness.


Introduction
Different boundary conditions, depending on the physical situation under consideration, can be imposed on the Navier-Stokes system. The most frequent is the Dirichlet condition for the velocity, where the value of the fluid velocity is prescribed on the boundary of the domain. That is a kinematic type of the boundary condition and in the weak formulation it is incorporated in the functional space that we are working with. Another possibility is to prescribe the pressure or the stress on the boundary. That is a dynamic kind of the boundary condition and it appears explicitly as the right-hand side in the weak formulation of the problem. However, it rarely makes sense to impose such condition on the fixed, rigid boundary. Recently, in [14] (see also [15,17]), the Darcy type of boundary condition on porous wall has been justified in case of flat porous boundary. It has the form where K is the permeability tensor of the boundary, positive and symmetric, computed from an auxiliary boundary-layer problem. The vector s is the fluid stress on the boundary s = [−2μ e(u) + p I] n (2) and H = P 0 n is the (prescribed) exterior stress caused by the atmospheric pressure. The above boundary condition can be connected with the classical Beavers-Joseph condition [3], which is imposed on an interface between the porous and non-porous domains filled by viscous fluid. The physical meaning of the Darcy condition is that the velocity of the fluid on the porous boundary is proportional to the difference of the inner and the outer stresses. Even though it involves the velocity, it is actually a dynamic boundary condition, appearing explicitly as the right-hand side in the weak formulation of the problem. In case of isotropic geometry of the flat porous boundary, the permeability tensor is diagonal. The permeability of the boundary depends on the form and distribution of the holes and its size (see [15] and [4]). Many natural materials (concrete, stone, skin, textile, etc.) are permeable and also rough. The goal of the present paper is to study the dependence of the permeability on the boundary roughness. In last 40 years, homogenization techniques have been developed by different authors for asymptotic analysis of different equations in domains with periodic rough boundaries, and we employ those techniques here. To mention some, we can start by [18] and [7], where the heat-conduction equation in the domain with oscillating boundary was analyzed. There is a very long list of papers where different aspects of the boundary roughness were studied for equations of fluid mechanics, with different boundary conditions. The reader can consult, for instance, [5,6,9,12,13,16], and [11]. The novelty of our paper is that we use those techniques in order to study the roughness-induced effect on our, new type of boundary condition, aiming to deduce how it affects the effective boundary permeability. The Darcy-type boundary condition (2) can be seen as a kind of non-homogeneous Fourier boundary condition for the Navier-Stokes system. Homogeneous Fourier condition on the rough boundary was studied in [1] and [2], where it was seen as a version of a slip boundary condition. The result was the same type of boundary condition, but with different (homogenized) friction coefficient. That coefficient is resulting from the fact that the rugosities on the boundary are almost periodic. Since the exterior force was absent, in case of purely periodic ruffles, there would be no effects of roughness. In our case, that effect comes from the exterior pressure. Assuming periodicity on the lateral boundary, they derive error estimate for the velocity, the pressure and the drag force. We prove the strong convergence, but not the error estimate.

Description of the problem
The idea is to start with the Stokes system in a domain with rough porous boundary that has periodically distributed rugosities. We continue with the precise description of the geometry.

The domain's geometry
Let Ω ⊂ R 2 be a smooth bounded domain and denote by Γ ⊂ ∂Ω the flat (porous) part of its boundary which we refer to as the bottom (see Fig. 1). We choose the coordinate system with standard basis (i, j) such that the bottom satisfies the equation x 2 = 0 and assume that the rest of the domain lies in the half-space x 2 > 0. More precisely, We denote The effect of roughness on the boundary... Now, we formally describe the periodic geometry of the rough boundary. Let F : R → [0, 1] be a smooth periodic function with period 1, and let F (0) = F (1) = 0. We denote Now, Ω δ ⊃ Ω is the domain bounded by Σ and Γ δ (see Fig. 2). We impose the Dirichlet condition on Σ and the Darcy-type boundary condition on Γ δ .

Remark 1.
We have assumed that the domain is two-dimensional, to simplify its description and our presentation. We could extend our results by taking where ω = 0, 1 2 ⊂ R 2 . Assuming that the shape of the ruffles is given by F ∈ C 1 0 (ω), extended by periodicity to R 2 , we just take Otherwise, the dimension of the domain is not essentially used in the paper, and all the results hold for three-dimensional domain with easily adapted proofs.

The governing system
As indicated in Introduction, we assume that the fluid flow is governed by the Stokes system. We add the corresponding boundary conditions to obtain: To stress the dependence of the solution on the small parameter δ, we use the superscript δ for all the fields involved.

The weak formulation and the a priori estimates
The stationary Stokes system with Darcy boundary condition is well-posed (see [14]). An appropriate functional space is We continue by lifting the non-homogeneous Dirichlet data g. We assume that g ∈ H 1 0 (Σ) 2 . In general, Σ g · n = 0 so that we cannot choose the lift function to be zero on Γ δ . Let ϕ ∈ C ∞ 0 (0, 1) be such that 0 ≤ ϕ(x) ≤ 2 and 1 0 ϕ(x)dx = 1. We construct the auxiliary function Clearly (see, e.g., [8]), there exists G ∈ H 1 (Ω δ ) 2 such that: Now, we look for the solution of (5)- (7) in the form where v δ satisfies the following weak formulation: where In addition, we look for a scalar function p δ ∈ L 2 (Ω δ ), such that It should be noticed that, in order to write the weak formulation, we have used the following formulation of the Darcy law , and let K be strictly positive symmetric matrix, whose smallest eigenvalue is k 1 > 0. Then: (a) The problem (12)- (14) has a unique solution

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Proof. Since where k 1 > 0 is the smaller eigenvalue of the permeability K, and C G > 0 is the constant from (10), the existence of such v δ follows from the Lax-Milgram theorem. Furthermore, using v δ as the test function in (12), and estimating the right-hand side as we arrive at Finally, The existence of the pressure p δ is a consequence of the DeRham theorem, and it is obtained in the same way as in case of the standard Dirichlet boundary condition (see, e.g., [8]). Furthermore, clearly (14) implies that

Convergence of the homogenization procedure
From the a priori bounds (15) and (16), we deduce that there exists some u ∈ H 1 (Ω) 2 , such that the sequence of restrictions of u δ and p δ on Ω have convergent subsequences (denoted for simplicity by the same symbols) and that u δ → u weakly in H 1 (Ω) and strongly in L 2 (Ω), It remains to identify the boundary-value problem satisfied by the limit functions (u, p).
We choose a test function Obviously, the restriction of w on Ω 0 = Ω is an element of Testing the equation (5) by w ∈ V δ , we get Passing to the limit as δ → 0, we deduce that

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The boundary integrals on Γ δ are the most interesting ones. We treat them in the following way: To treat the first integral, we use the compactness of the trace operator γ : due to the periodicity, we have For the second integral, due to that fact that the domain of integration is tiny, namely |A δ | = O(δ), we obtain Finally, E. Marušić-Paloka and I. Pažanin ZAMP Finally, due to the continuity of P 0 , using a similar approach as in the above boundary integral, we get Putting together (32), (31) and (36) implies that the limit functions u and p satisfy the boundary-value problem: Observe that is the same type of the problem as the original one (5)- (7), so that the same existence and uniqueness theorem applies. Thus, the accumulation points u and p are unique and the whole sequences u δ and p δ converge in (24), (25), and not just their subsequences. Returning to (33) and using u δ as a test function, repeating the same procedure, we get Following the same steps as in the proof of (35), we get Since the set A δ , defined by (34), is vanishing, we have The effect of roughness on the boundary...

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proving the strong convergence of u δ in H 1 (Ω). To prove the strong convergence of the pressure, we pick v δ ∈ H 1 0 (Ω) 2 such that div v δ = p δ in Ω.
We assume here that i.e., that p δ ∈ L 2 0 (Ω). That implies p ∈ L 2 0 (Ω). Obviously, v δ v weakly in H 1 (Ω) , Using v δ as a test function in (33), and extending it by zero to A δ , gives giving the strong convergence of the pressure.

Remark 2.
It is worth noticing that the homogenized boundary condition on the porous boundary is still of the Darcy-type, but with permeability 1 σ K instead of K. It is obvious from the definition that σ > 1 so that the effective permeability is smaller than the original microscopic permeability. The total flux through the porous boundary did not change, since On the other hand, Γ δ KH · n = Γ δ P 0 Kn · n = = 1 0 P 0 (x, −δF (x/δ)) k 11 F (x/δ) 2 + k 22 + 2k 12 F (x/δ) Thus, for the normal stress exerted by the fluid on the porous boundary s δ , we have