Homogenization of Bingham Flow in thin porous media

By using dimension reduction and homogenization techniques, we study the steady flow of an incompresible viscoplastic Bingham fluid in a thin porous medium. A main feature of our study is the dependence of the yield stress of the Bingham fluid on the small parameters describing the geometry of the thin porous medium under consideration. Three different problems are obtained in the limit when the small parameter $\varepsilon$ tends to zero, following the ratio between the height $\varepsilon$ of the porous medium and the relative dimension $a_\varepsilon$ of its periodically distributed pores. We conclude with the interpretation of these limit problems, which all preserve the nonlinear character of the flow.


Introduction
We study in this paper the steady incompressible flow of a Bingham fluid in a thin porous medium containing an array of vertical cylindrical obstacles (the pores). The model of thin porous medium of thickness much smaller than the distance between the pores was introduced in [27], where a stationary incompressible Navier-Stokes flow was studied. Recently, the model of thin porous medium under consideration in this paper was introduced in [15], where the flow of an incompressible viscous fluid described by the stationary Navier-Stokes equations was studied by the multiscale expansion method, which is a formal but powerful tool to analyse homogenization problems. These results were rigorously proved in [4] using an adaptation introduced in [3] of the periodic unfolding method from [12]. This adaptation consists of a combination of the unfolding method with a rescaling in the height variable, in order to work with a domain of fixed height, and to use monotonicity arguments to pass to the limit. In [3], in particular, the flow of an incompressible stationary Stokes system with a nonlinear viscosity, being a power law, was studied. For non-stationary incompressible viscous flow in a thin porous medium see [1], where a non-stationary Stokes system is considered, and [2], where a non-stationary non-newtonian Stokes system, where the viscosity obeyed the power law, is studied. For the periodic unfolding method applied to the study of problems stated in other type of thin periodic domains we refer for instance to [18] for crane type structures and to [19], [20] for thin layers with thin beams structures, where elasticity problems are studied.
If Π is a three-dimensional domain with smooth boundary ∂Π and f = (f 1 , f 2 , f 3 ) are external given forces defined on Π, then the velocity u = (u 1 , u 2 , u 3 ) of a fluid and its pressure p satisfy the equations of motion completed with the fluid's incompressibility condition div u = 3 i=1 ∂ xi u i = 0 in Π, and the no-slip boundary condition u = 0 on the boundary ∂Π. What distinguishes different fluids is the expression of the stress tensor σ.
Newtonian fluids are the most encountered ones in real life and as typical examples one can mention the water and the air. For a newtonian fluid, the entries of the stress tensor σ(p, u) are given by (σ(p, u)) ij = −pδ ij + 2µ(D(u)) ij , 1 ≤ i, j ≤ 3 (2) where δ ij is the Kronecker symbol, the real positive µ is the viscosity of the fluid and the entries of the strain tensor are (D(u)) ij = (∂ xj u i + ∂ xi u j )/2. If f belongs to (L 2 (Π)) 3 and the space V is defined by V = {v ∈ (H 1 0 (Π)) 3 | div v = 0}, then u and p satisfying (1) with (2) are such that (see for instance [17]): (Stokes) There is a unique u ∈ V and a unique (up to an additive real constant) p ∈ L 2 (Π) such that (if < ·, · > is the dual pairing between (H −1 (Π)) 3 and (H 1 0 (Π)) 3 ) with a(u, v) = 2µ A fluid whose stress is not defined by relation (2) is called a non-newtonian fluid. There are several classes of non-newtonian fluids, as the power law, Carreau, Cross, Bingham fluids. It is on the study of the last type of fluid that we are interested in this paper. We refer to [13] for a review on non-newtonian fluids. For a Bingham fluid, the nonlinear stress tensor is defined by (see [14]) where |D(u)| 2 = D(u) : D(u) and the positive number g represents the yield stress of the fluid. If g = 0, then (4) becomes (2). Viscoplastic Bingham fluids are quite often encountered in real life. As examples one can mention volcanic lava, fresh concrete, the drilling mud, oils, clays and some paintings. For u g and p g satisfying (1) with (4), according to [14], one has the following result: (Bingham) There is a unique u g ∈ V and a (non-unique) p g ∈ L 2 (Π)/R such that Here a, l, < ·, · > are as before and If the yield stress of the Bingham fluid is of the form g(ε), with ε ∈]0, 1[ and such that g(ε) tends to zero when ε tends to zero, then, according to [[14], Chapter 6, Théorème 5.1.], the following result holds When ε tends to zero, one has for the solution u ε of problem (5) corresponding to g(ε) the following convergence where u is the solution of problem (3).
This means that, in a fixed domain, the nonlinear character of the Bingham flow is lost in the limit (when the yield stress tends to zero), as it is expected. A natural question that arises is the following: If the yield stress g(ε) is as before and, moreover, the domain Π itself depends on the small parameter ε, what happens when ε tends to zero? The answer is that, in the limit, the nonlinear character of the flow may be preserved. For instance, if Π ε is a classical rigid porous medium, it was proven in [24] with the asymptotic expansion method that, in a range of parameters, the nonlinear character of the Bingham flow is preserved in the homogenized problem, which is a nonlinear Darcy equation. The convergence corresponding to the above mentioned result was proven in [6] with the two-scale convergence method and then recovered in [8] with the periodic unfolding method. The case of a doubly periodic rigid porous medium was studied in [7], where a more involved nonlinear Darcy equation is derived. Another class of domains for which the nonlinear character of the flow may be preserved in the limit is those of thin domains. The case of a domain Π ε which is thin in one direction was addressed in [10] and [11]. We refer to [9] for the asymptotic analysis of a Bingham fluid in a thin T-like shaped domain. In all these cases, a lower-dimensional Bingham-like law was exhibited in the limit. This law was already encountered in engineering (see [26]), but no rigurous mathematical justification was previously known. For the shallow flow of a viscoplastic fluid we refer the reader to [16], [21], [23] and [22].
In this paper we study the asymptotic behavior of the flow of a viscoplastic Bingham fluid in a thin porous medium. We refer the reader to the very recent paper [5] and the references therein for the application of our study to problems issued from the real life applications. As a first example one can mention the flow of the volcanic lava through dense forests (see [25]). Another important application is the flow of fresh concrete spreading through networks of steel bars.
The paper is organized as follows. In Section 2. we state the problem: we define in (6) the thin porous medium Ω ε (see also Figure 1), of height ε and relative dimension a ε of its periodically distributed pores. In Ω ε we consider the flow of a viscoplastic Bingham fluid with velocity u ε and pressure p ε verifying the nonlinear variational inequality (9). In Section 3. we give some a priori estimates for the velocity and for the pressure obtained after the change of variables (10) and verifying (12), and then for the velocity and for the pressure defined in (21). In Section 4. by passing to the limit ε → 0, we prove the main convergence results of our paper, stated in Theorems 4.4, 4.6 and 4.8, respectively. Up to our knowledge, problems (36), (57) and (78) are new in the mathematical literature. We conclude in Section 5. with the interpretation of these limit problems, which all three preserve the nonlinear character of the flow; both effects of a nonlinear Darcy equation and a lower dimensional Bingham-like law appear. The paper ends with a list of References.

Statement of the problem
A periodic porous medium is defined by a domain ω and an associated microstructure, or periodic cell Y ′ = [−1/2, 1/2] 2 , which is made of two complementary parts: the fluid part Y ′ f , and the solid part . More precisely, we assume that ω is a smooth, bounded, connected set in R 2 , and that Y ′ s is an open connected subset of Y ′ with a smooth boundary ∂Y ′ s , such that Y ′ s is strictly included in Y ′ . The microscale of a porous medium is a small positive number a ε . The domain ω is covered by a regular mesh of size a ε : for k ′ ∈ Z 2 , each cell Y ′ k ′ ,aε = a ε k ′ + a ε Y ′ is divided in a fluid part Y ′ f k ′ ,aε and a solid part Y ′ s k ′ ,aε , i.e. is similar to the unit cell Y ′ rescaled to size a ε . We define Y = Y ′ × (0, 1) ⊂ R 3 , which is divided in a fluid part Y f and a solid part Y s , and consequently Y k ′ ,aε = Y ′ k ′ ,aε × (0, 1) ⊂ R 3 , which is also divided in a fluid part Y f k ′ ,aε and a solid part Y s k ′ ,aε .
We denote by : the full contraction of two matrices; for A = (a i,j ) 1≤i,j≤3 and B = (b i,j ) 1≤i,j≤3 , we have In order to apply the unfolding method, we will need the following notation. For k ′ ∈ Z 2 , we define κ : Remark that κ is well defined up to a set of zero measure in R 2 (the set ∪ k ′ ∈Z 2 ∂Y ′ k ′ ,1 ). Moreover, for every a ε > 0, we have We denote by C a generic positive constant which can change from line to line.
The points x ∈ R 3 will be decomposed as We also use the notation x ′ to denote a generic vector of R 2 .
In Ω ε we consider the stationary flow of an incompressible Bingham fluid. As already seen in the Introduction, following Duvaut and Lions [14], the problem is formulated in terms of a variational inequality.

For a vectorial function
We introduce the following spaces where the yield stress g(ε) will be made precise in Section 3.1. Let f ∈ (L 2 (Ω)) 3 be given such that The model of the flow is described by the following variational inequality: From Duvaut and Lions [14], we know that there exists a unique u ε ∈ V (Ω ε ) solution of problem (8). Moreover, from Bourgeat and Mikelić [6], we know that if p ε is the pressure of the fluid in Ω ε , then problem (8) is equivalent to the following one: Find u ε ∈ V (Ω ε ) and p ε ∈ L 2 0 (Ω ε ) such that Problem (9) admits a unique solution u ε ∈ V (Ω ε ) and a (non) unique solution p ε ∈ L 2 0 (Ω ε ), where L 2 0 (Ω ε ) denotes the space of functions belonging to L 2 (Ω ε ) and of mean value zero.
Our aim is to study the asymptotic behavior of u ε and p ε when ε tends to zero. For this purpose, we first use the dilatation of the domain Ω ε in the variable x 3 , namely in order to have the functions defined in an open set with fixed height, denoted Ω ε .
Let us introduce some notation which will be useful in the following. For a vectorial function v = (v ′ , v 3 ) and a scalar function w, we will denote Moreover, associated to the change of variables (10), we introduce the operators: D ε , D ε , div ε and ∇ ε , defined by We introduce the following spaces Using the transformation (10), the variational inequality (8) can be rewritten as: and (9) can be rewritten as: Our goal now is to describe the asymptotic behavior of this new sequence (ũ ε ,p ε ).

A Priori Estimates
We start by obtaining some a priori estimates forũ ε .

and taking into account that
For the cases a ε ≈ ε or a ε ≪ ε, taking into account Remark 4.3(i) in [3], we obtain the second estimate in (13), and, consequently, from classical Korn's inequality we obtain the last estimate in (13). Now, from the second estimate in (13) and Remark 4.3(i) in [3], we deduce the first estimate in (13). For the case a ε ≫ ε, proceeding similarly with Remark 4.3(ii) in [3], we obtain the desired result.

3.1
The extension of (ũ ε ,p ε ) to the whole domain Ω We extend the velocityũ ε by zero to the Ω\ Ω ε and denote the extension by the same symbol. Obviously, estimates (13)- (14) remain valid and the extension is divergence free too.
We study in the sequel the following cases for the value of yield stress g(ε): ii) if a ε ≫ ε, then g(ε) = g ε.
These choices are the most challenging ones and they answer to the question adressed in the paper, namely they all preserve in the limit the nonlinear character of the flow.
In order to extend the pressure to the whole domain Ω, the mapping R ε (defined in Lemma 4.5 in [3] as R ε 2 ) allows us to extend the pressure p ε to Q ε by introducing F ε in (H −1 (Q ε )) 3 : Setting Moreover, if div w = 0 then F ε , w Qε = 0, and the DeRham Theorem gives the existence of P ε in L 2 0 (Q ε ) with F ε = ∇P ε .
Proof. Let us estimate ∇ εPε in the cases a ε ≈ ε or a ε ≪ ε. We estimate the right-hand side of (18). Using Cauchy-Schwarz's inequality and from the second estimate in (13) we have .
Using the assumption made on the function f , we obtain and by Cauchy-Schwarz's inequality and taking into account that | Ω ε | ≤ |Ω|, we obtain Then, from (18), we deduce Taking into account the third point in Lemma 4.6 in [3], we have If a ε ≈ ε we take into account that a ε ≪ 1, and if a ε ≪ ε we take into account that a ε /ε ≪ 1 and a ε ≪ 1, and we see that there exists a positive constant C such that and consequently ∇ εPε (H −1 (Ω)) 3 ≤ C. It follows that (see for instance Girault and Raviart [17], Chapter I, Corollary 2.1) there exists a representative ofP ε ∈ L 2 0 (Ω) such that Finally, let us estimate ∇ εPε in the case a ε ≫ ε. Similarly to the previous case, we estimate the right side of (18) by using Cauchy-Schwarz's inequality and from the second estimate in (14), and we have Taking into account the proof in Lemma 4.5 in [3], the change of variables (10) and that a ε ≫ ε, we can deduce and using that a ε ≪ 1, we see that there exists a positive constant C such that and reasing as the previous case, we have the estimate (19).
According to these extensions, problem (12) can be written as: 2µ for everyṽ that is the extension by zero to the whole Ω of a function in (H 1 0 ( Ω ε )) 3 .

Adaptation of the Unfolding Method
The change of variable (10) does not provide the information we need about the behavior ofũ ε in the microstructure associated to Ω ε . To solve this difficulty, we use an adaptation introduced in [3] of the unfolding method from [12].
Remark 4.1. For k ′ ∈ K ε , the restriction of (û ε ,P ε ) to Y ′ k ′ ,aε × Y does not depend on x ′ , whereas as a function of y it is obtained from (ũ ε ,P ε ) by using the change of variables We are now in position to obtain estimates for the sequences (û ε ,P ε ), as in the proof of Lemma 4.9 in [3].
Lemma 4.2. There exists a constant C independent of ε, such that the couple (û ε ,P ε ) defined by (21) satisfies i) if a ε ≈ ε, with a ε /ε → λ, 0 < λ < +∞, or a ε ≪ ε, and, moreover, in every cases, P ε When ε tends to zero, we obtain for problem (20) different behaviors, depending on the magnitude of a ε with respect to ε. We will analyze them in the next sections.
Proof. We refer the reader to Lemmas 5.2, 5.3 and 5.4 in [3] for the proof of (22)- (25). Here, we prove thatP does not depend on the microscopic variable y. To do this, we choose as test functionṽ(x ′ , y) ∈ D(ω; (20) (we recall that g(ε) = g a ε )) and using that div εũε = 0, we have By the change of variables given in Remark 4.1 and by Lemma 4.2, we get for the first term in relation (26) and for the second term in relation (26) Moreover, applying the change of variables given in Remark 4.1 to the fourth term in relation (26), we have Therefore, applying the change of variables given in Remark 4.1 to relation (26), we obtain According with (23), the first term in relation (30) can be written by the following way In order to pass to the limit in the first nonlinear term, we have Now, in order to pass the limit in the second nonlinear term, we are taking into account that and using (23) and the fact that the function E(ϕ) = |ϕ| is proper convex continuous, we can deduce that Moreover, using (23) the two first terms in the right hand side of (30) can be written by We consider now the terms which involve the pressure. Taking into account the convergence of the pressure (23), passing to the limit when ε tends to zero, we have ω×YP div λṽ dx ′ dy.
Theorem 4.4 (Critical case). If a ε ≈ ε, with a ε /ε → λ, 0 < λ < +∞, then (û ε /a 2 ε ,P ε ) converges to (û,P ) in , which satisfies the following variational inequality where D λ [·] = D y ′ [·] + λ∂ y3 [·] and for everyṽ ∈ L 2 (ω; . We first multiply (20) by a −2 ε and we use that div εũε = 0. Then, we take as test function a 2 εṽ ε = a 2 ε (ṽ ′ (x ′ , x ′ /a ε , y 3 ), λε/a ε v 3 (x ′ , x ′ /a ε , y 3 )), withṽ(x ′ , y) = 0 in ω × Y s and satisfying the incompressibility conditions (25), that is, div λṽ = 0 in ω × Y and Yṽ ′ (x ′ , y)dy · n = 0 on ∂ω, and we have 2µ By the change of variables given in Remark 4.1 and by Lemma 4.2, we have (27) for the first term in relation (37), and for the second term in relation (37) we obtain Moreover, applying the change of variables given in Remark 4.1 to the fourth term in relation (37), we have (29). Therefore, applying the change of variables given in Remark 4.1 to relation (37), we obtain According with (23), the first term in relation (39) can be written and, taking into account that λ ε/a ε → 1, this term tends to the following limit The second term in relation (39) writes and, taking into account that the function B(ϕ) = |ϕ| is proper convex continuous and λ ε/a ε → 1, we get that the lim inf ε→0 of this second is greater or equal than In order to pass to the limit in the first nonlinear term, we have and we can deduce that the first nonlinear term tends to the following limit Now, in order to pass the limit in the second nonlinear term, we are taking into account that and using (23) and the fact that the function E(ϕ) = |ϕ| is proper convex continuous, we can deduce that Moreover, using (23) the two first terms in the right hand side of (39) tend to the following limit We consider now the terms which involve the pressure. Taking into account the convergence of the pressure (23) the first term of the pressure tends to the following limit ω×YP div x ′ṽ ′ dx ′ dy, and using (25) and taking into account thatP does not depend on y, we have Finally, using that div λṽ = 0, we have 1 a ε ω×YP ε div y ′ṽ ′ dx ′ dy + λ a ε ω×YP ε ∂ y3ṽ3 dx ′ dy = 0.
Lemma 4.5 (Subcritical case). For a subsequence of ε still denoted by ε, there existũ ∈ (L 2 (Ω)) 3 , wherẽ Proof. See Lemmas 5.2, 5.3 and 5.4 in [3] for the proof of (47)-(50). In order to prove thatP does not depend on y ′ we argue as in the proof of Lemma 4.3 using that a ε ≪ ε, and we obtain ω×YP div y ′ṽ ′ dx ′ dy = 0, which shows thatP does not depend on y ′ . Now, in order to prove thatP does not depend on y 3 , setting εṽ = ε(0,ṽ 3 (x ′ , x ′ /a ε , y 3 )) in (20) (we recall that g(ε) = g a ε )) and using that div εũε = 0, we have Applying the change of variables given in Remark 4.1 to relation (51) and taking into account (27)-(29), we obtain According with (48) and using that a ε ≪ ε, the first term in relation (52) can be written by the following way In order to pass to the limit in the first nonlinear term, we have In order to pass to the limit in the second nonlinear term, we proceed as in Lemma 4.3. Moreover, using (48) the first term in the right hand side of (52) can be written by We consider now the term which involves the pressure. Taking into account the convergence of the pressure (48), passing to the limit when ε tends to zero, we have Therefore, taking into account (33) and (53)-(56), when we pass to the limit in (52) when ε tends to zero, we have 0 ≥ ω×YP ∂ y3ṽ3 dx ′ dy. Now, if we choose as test function −εṽ = −ε(0,ṽ 3 (x ′ , x ′ /a ε , y 3 )) in (20) and we argue similarly, we can deduce thatP does not depend on y 3 , soP does not depend on y.
Using (48) the two first terms in the right hand side of (59) tend to the following limit We consider now the terms which involve the pressure. Taking into account the convergence of the pressure (48) the first term of the pressure tends to the following limit ω×YP div x ′ṽ ′ dx ′ dy, and using (50) and taking into account thatP does not depend on y, we have (45). Finally, using that div y ′ṽ ′ = 0, we have 1 a ε ω×YP ε div y ′ṽ ′ dx ′ dy = 0.
It is straightforward to obtain thatû 3 = 0 and therefore we get (57).
0ũ (x ′ , y 3 )dy 3 with Yû 3 dy = 0 andû 3 independent of y 3 , and P ∈ L 2 0 (ω × Y ), independent of y, such that Proof. See Lemmas 5.2, 5.3 and 5.4 in [3] for the proof of (61)-(64). Here, we prove thatP does not depend on the microscopic variable y. To do this, we choose as test functionṽ(x ′ , y) ∈ D(ω; . In order to prove thatP does not depend on y 3 , we set εṽ(x ′ , x ′ /a ε , y 3 ) in (20) (we recall that g(ε) = g ε))and using that div εũε = 0, we have Applying the change of variables given in Remark 4.1 to relation (65) and taking into account (27)-(29), we obtain According with (62) and using that a ε ≫ ε, one has for the first term in relation (66) We pass to the limit in the first nonlinear term and we have In order to pass the limit in the second nonlinear term, we taking into account that and using (62), with a ε ≫ ε, and the fact that the function E(ϕ) = |ϕ| is proper convex continuous, we can deduce that Moreover, using (62) the two first terms in the right hand side of (66) can be written by We consider now the terms which involve the pressure. Taking into account the convergence of the pressure (62) and a ε ≫ ε, passing to the limit when ε tends to zero, we have Therefore, taking into account (67)-(71), when we pass to the limit in (66) when ε tends to zero, we have 0 ≥ ω×YP ∂ y3ṽ3 dx ′ dy. Now, if we choose as test function −εṽ(x ′ , x ′ /a ε , y 3 ) in (20) and we argue similarly, we can deduce thatP does not depend on y 3 . Now, in order to prove thatP does not depend on y ′ , we set a εṽ = a ε (ṽ ′ (x ′ , x ′ /a ε , y 3 ), 0) in (20) and using that div εũε = 0, we have Applying the change of variables given in Remark 4.1 to relation (72) and taking into account (27)-(29), we obtain According with (62) and using that a ε ≫ ε, the first term in relation (73) can be written by the following way In order to pass to the limit in the first nonlinear term, we have Moreover, using (62) the two first terms in the right hand side of (73) can be written by We consider now the terms which involve the pressure. Taking into account the convergence of the pressure (62), passing to the limit when ε tends to zero, we have ω×YP div y ′ṽ ′ dx ′ dy.

Conclusions
By using dimension reduction and homogenization techniques, we studied the limiting behavior of the velocity and of the pressure for a nonlinear viscoplastic Bingham flow with small yield stress, in a thin porous medium of small height ε and for which the relative dimension of the pores is a ε . Three cases are studied following the value of λ = lim ε→0 a ε /ε and, at the limit, they all preserve the nonlinear character of the flow. More precisely, according to [24], each of the limit problems (36), (57) and (78), is written as a nonlinear Darcy equation: The velocity of filtrationŨ ( We remark that in all three cases, the vertical componentŨ 3 of the velocity of filtration equals zero and this result is in accordance with the previous mathematical studies of the flow in this thin porous medium, for newtonian fluids (Stokes and Navier-Stokes equations) and for power law fluids (see [15], [1], [2], [3], [4]). Moreover, despite the fact that the limit pressure is not unique, the velocity of filtration is uniquely determined (see Section 4.3 in [24]). In (85), the function K λ : R 2 −→ R 2 is nonlinear and its expression can not be made explicit for the Bingham flow (see [24]). Nevertheless, in each case, for a given ξ ∈ R 2 , one has K λ (ξ) = Y χ ξ λ (y)dy, with χ ξ λ solution of a local problem stated in the cell Y . If 0 < λ < +∞, the local problem is a 3-D Bingham problem. If λ = 0, the local problem is a 2-D Bingham problem (defined for each y 3 ∈]0, 1[), while if λ = +∞ the 1-D local problem (defined for each y ′ ∈ Y ′ ) corresponds to a lower-dimensional Bingham-like law (see [11]).
We end with the remark that if in the initial problem (9) we take g = 0, then the problem under study becomes the Stokes problem. We refer to [3] (case p = 2) for the asymptotic analysis of the Stokes problem. If we set g = 0 in the limit problems (36), (57) and (78), they become exactly the ones in [3], Theorem 6.1 (case p = 2), corresponding to the Stokes case.