On the Bingham Flow in a Thin Y-Like Shaped Structure

We consider the steady Bingham flow in a two-dimensional thin Y-like shaped structure, with no-slip boundary conditions and under the action of given external forces. After passage to the limit with respect to a small parameter related to the thickness of the domain, we obtain three uncoupled problems. Each of these problems describes an anisotropic flow, corresponding to a lower-dimensional “Bingham-like” constitutive law. These results are in accordance with the engineering models.


Introduction
We consider a steady Bingham flow in a two-dimensional thin Y-like shaped structure, with no-slip boundary conditions and under the action of given external forces.
We take the following multi-structure Ω n (see Fig. 1), where h n is a small positive parameter representing the order of the thickness of each branch, α ∈ 0, π 2 [and β ∈] − π 2 , 0 are two angles describing the deviation of the two oblique branches from the vertical direction, and L r cos α , L l cos β , and L b are the main lengths of the three branches (from now on, the exponents r, l and b stand for right, left and below, respectively).
The Bingham fluid is a particular non-newtonian fluid, used as a model for viscoplastic materials as the volcanic lava, the mud for the oil extraction, some paints, etc. (for instance, see [6,16,17]). Its constitutive law is non linear and is characterized by the presence of a given threshold (see (2.3)). The fluid moves like a rigid body as long as a certain function of the stress tensor is below the threshold and obeys a non linear constitutive law beyond the threshold. In this perspective, also the blood can be modelized as a Bingham fluid in the first approximation, since it has a yield stress as the Bingham fluid, but at high shear stress it behaves as a pseudoplastic fluid. Thus, the Bingham fluid in our geometry can also modelize the microcirculation in blood vessels.
According to [19], the steady flow of a Bingham fluid in Ω n , with no-slip boundary conditions, corresponds to the following variational inequality, (1.1) for passing to the limit in each branch, we first get a local reformulation of our problem. Eventually, test functions satisfying the constraint u 1 = cu 2 will be used for passing to the limit and for deducing the variational inequality satisfied by u 2 in each branch. As far as the limit force is concerned, only its projection on the axis of each branch is involved in the limit inequalities (see the first term in the right-hand side of (4.8), (4.9), (4.10), and Remark 4.2). The limit velocity in the vertical branch is classical by now. Indeed, as already observed in [10][11][12], the component in the transverse direction, i.e. u 1 , is zero, while u 2 is the solution to problem (4.10). One of the novelties of this paper appears in the limit velocities in the two oblique branches. On each oblique branch, the velocity has two non-zero components and its direction coincides with the axis of the branch (see Remark 4.2). Let us remark that the case corresponding to α = π 2 and/or β = − π 2 is not included in our study, since in this particular case the initial domain Ω n degenerates, at least one of the branches becoming a horizontal line. We refer [10] for the study of this problem in a T -structure, which is not a particular case of the present work.
Note that problem (1.1) becomes the system modeling the Stokes flow in a thin multi-structure when g = 0. For such problems we refer to [7,31]. For Navier-Stokes flow in a thin tube or thin layer we refer to [13,30,36], while for multi-structures we refer to [23,32,33].
As far as the study of Bingham flow in domains depending on small geometrical parameters is concerned, we recall that the first results for the Bingham flow in periodic domains are obtained in [28]. The justification of the convergence for these results is due to [8,9]. In the recent paper [5], the authors present a model of Bingham flow in a thin periodic domain which contains an array of obstacles modelized as vertical cylinders. Up to our knowledge, the first convergence result on Bingham flow in thin periodic domains is due to [1], followed by [2]. We refer to [18] for Bingham flows in periodic domains of infinite length. We refer to [14] for Bingham flows in a thin domain with periodically oscillating boundary. For other models of non-newtonian fluid in thin multi-structures we refer to [34,35].
For other problems in thin multi-structures we refer to [3,4,[20][21][22][24][25][26][27]. The paper is organized as follows. In Sect. 2 we introduce the domain Ω n , the physical properties of the Bingham fluid, and the variational inequality which modelizes the flow. In Section 3 the problem is rescaled on a fixed domain. Section 4 contains the main result of the paper, the convergence theorem. A priori estimates and convergence results for the velocity and the pressure are obtained in Sect. 5 and in Sect. 6, respectively. These last results will allow us to prove the convergence theorem in Sect. 7. The paper ends with some conclusions, in Sect. 8.

The Setting of the Problem
In what follows, a generic element of R 2 is denoted by (X 1 , x 2 ) or (x 1 x 2 ).
Let α ∈ [0, π 2 [ and β ∈] − π 2 , 0], L r , L l , L b ∈]0, +∞[, for every n ∈ N let h n ∈]0, 1[ be a small parameter, and let Ω n be the thin two-dimensional Y-like shaped domain defined by In Ω n we consider the non linear flow of a Bingham fluid. If u n = (u n1 , u n2 ) and p n denote its velocity and pressure, respectively, the corresponding stress tensor is defined by where δ ij is the Kronecker symbol, g is a strictly positive constant related to the yield stress of the fluid, μ is a strictly positive constant related to the viscosity of the fluid, and D T (u n ) the transposed of D(u n ). Moreover, we set We remark that (2.2), i.e., the constitutive law of the Bingham fluid, is valid only if e II (u n ) = 0. In [19] it is shown that this constitutive law is equivalent to the following one We point out that this is a threshold law: as long as the shear stress σ II (u n ) is below gh n , the fluid behaves as a rigid solid. When the value of the shear stress σ II (u n ) exceeds gh n , then the fluid flows obeying a non linear law. We also suppose that the fluid is incompressible, that is Moreover, we apply to the fluid a given external force f n = (f n1 , f n2 ) belonging to L 2 (Ω n ) 2 , and then we have the following relations Furthermore, we assume the no-slip condition to the boundary of the domain, which reads In [19] it is shown that the velocity u n satisfying (2.2), (2.4), (2.5) and (2.6) solves the following variational inequality For each n, this inequality admits a unique solution u n . According to [19], problem (2.7) is equivalent to the following one which admits a solution (u n , p n ), such that u n is unique, but p n is not unique. The aim of this paper is to study the asymptotic behavior, as n diverges, of problem (2.8), under suitable assumption on the given data {f n } n∈N and on the assumption lim n h n = 0. (2.9) We note that the pressure p n is not unique. If p n is a solution of (2.8), then p n + c is also a solution of (2.8), for every c ∈ R. In particular, the constant c can be choosen such that the average of p n + c is zero on a subset of Ω n .

The Rescaled Problem
In order to pass to the limit in (2.8), the first step consists in rewriting problems (2.7) and (2.8) on the domain Ω (which is independent of n). This is done as usual, by a domain dilatation technique (see [15]), through the three different maps More precisely, for every n ∈ N we set and If u n solves (2.7) (or equivalently (u n , p n ) solves problem (2.8)), then u n (or equivalently (u n , p n )) defined by or equivalently Conversely, if u n solves (3.7) (or equivalently (u n , p n ) solves (3.8)), then u n (or equivalently (u n , p n )) defined by solves (2.7) (or equivalently (2.8)). Therefore, the goal of this paper becomes to study the asymptotic behavior, as n diverges, of problem (3.8). To this aim, we assume (3.9)

The Main Results
In order to give the main result of our paper, according to [11] we introduce the applications respectively, and we set

(4.2)
Theorem 4.1. Assume that (2.9) and (3.9) hold true. For every n ∈ N let (u n , p n ), (u n , q n ), and (u n , s n ) be three solutions to (3.8) such that and set Let W r 0 (Ω r ), W l 0 (Ω l ), and W b 0 (Ω b ) be defined in (4.2). Then, there exist u r in W r 0 (Ω r ), u l in W l 0 (Ω l ), as n diverges. Furthermore, (u r , p), (u l , q), respectively, where (f r 1 , f r 2 ), (f l 1 , f l 2 ), and f b 2 are given by (3.9). Furthermore, u r , u l , and u b are unique.
Remark 4.2. The limit velocity in the right branch is u r (tan α, 1) (see (4.3)). We note that α describes the deviation of the right branch from the x 2 -axis, then the direction (tan α, 1) of the limit velocity is exactly along the axis of the right branch. Moreover, as far as the limit force (f r 1 , f r 2 ) is concerned, only its projection on the axis of the right branch is involved in problem (4.8). Indeed, where P | (tan α,1) denotes the projection operator on the axis x = y tan α.
If α = 0, then the domain Ω r is a vertical rectangle, tan α = 0, and the first limit in (4.3) is (0, u r ). This is in accordance with the results obtained in [10][11][12]. A similar remark holds true for problem (4.9) stated in the left branch, and for problem (4.10) stated in the vertical branch (which corresponds to the previous cases with α = 0).

A Priori Estimates and Convergence Result for the Velocity
In order to use compactness results for passing to the limit in the terms involving the velocity, we first derive a priori estimates for this function.
Proposition 5.1. Assume (2.9) and (3.9) hold true. For every n ∈ N let u n be the solution to (3.7). Then there exists a positive constant c independent on n such that Proof.
Choosing v = 0 and v = 2u n as test functions in (3.7), and comparing the variational inequalities which are obtained give that On the other hand, the homogeneous boundary conditions of u n provide Applying the Hölder inequality to the right-hand side of (5.3) and using (5.4) imply (5.5) Eventually, (5.1) follows from (5.5), (3.9), and the following inequality  Proof. Statements in (5.6) and in (5.7) follow from Proposition 5.1. The first equality in (5.8) is obtained by passing to the limit in div r n (u n ) = 0 in Ω r , ∀n ∈ N, and taking into account the last two convergences in (5.6), and (5.7). Similarly, one can prove the last two equalities in (5.8).
As far as the proof of (5.9) is concerned, equality div r n (u n ) = 0 in Ω r provides (5.10) On the other side, the boundary conditions of u n give where ν 1 is the first component of the exterior unit normal on ∂Ω r . Taking into account again the boundary conditions of u n and combining (5.10) with (5.11) provide which implies T r (u n2 ) = 0 a.e. in ]0, L r [, ∀n ∈ N, (5.12) since T r (u n2 ) (1) = 0. Eventually, combining the first convergence in (5.6) with (5.12) provides the first equality in (5.9), since T r is weakly continuous, being strongly continuous. Similarly, one can prove the last two equalities in (5.9).

A Priori Estimates and Convergence Result for the Pressure
In what follows, we set (6.1) Proposition 6.1. Assume that (2.9) and (3.9) hold true. For every n ∈ N let (u n , p n ) be a solution to (3.8). Then there exists a positive constant c independent on n such that 2) In (3.7) choosing v = 0 first, then v = 2u n , and comparing provide Ω r h 2 n μ|D r n u n | 2 + h n g|D r n u n | dx 1 dx 2 + Ω l h 2 n μ|D l n u n | 2 + h n g|D l n u n | dx 1 dx 2 (7.1) Combining (3.8) and (7.1), and recalling that div r n (u n ) = 0 in Ω r , div l n (u n ) = 0 in Ω l , div b n (u n ) = 0 in Ω b , give Ω r h 2 n μD r n u n D r n v + h n g|D r n v| dx 1 dx 2 + Ω l h 2 n μD l n u n D l n v + h n g|D l n v| dx 1 dx 2 with w in H 1 0 (Ω r ), as test function in (7.2) gets Ω r h 2 n μD r n u n D r n (tan αw, w) + h n g|D r n (tan αw, w)| dx 1 dx 2 ≥ Ω r (f r n1 tan α + f r n2 ) w + tan 2 α + 1 p n ∂ να w dx 1 dx 2 ∀w ∈ H 1 0 (Ω r ).