Transmission Problem Between Two Herschel-Bulkley Fluids in a Three Dimensional Thin Layer

. The paper is devoted to the study of steady-state transmission problem between two Herschel-Bulkley ﬂuids in a three dimensional thin layer.


Introduction
The rigid viscoplastic and incompressible fluid of Herschel-Bulkley has been studied and used by many mathematicians, physicists and engineers, to model the flow of metals, plastic solids and a variety of polymers. Due to the existence of the yield limit, the model can capture phenomena connected with the development of discontinuous stresses. A particularity of Herschel-Bulkley fluid lies in the presence of rigid zones located in the interior of the flow and as yield limit increases, the rigid zones become larger and may completely block the flow, this phenomenon is known as the blockage property. The literature concerning this topic is extensive; see e.g. [4,11,12,14,15]. The purpose of this paper is to study the asymptotic behavior of the steady flow of Herschel-Bulkley fluid in a three-dimensional thin layer. The paper is organized as follows. In section 2 we present the mechanical problem of the steady flow of Herschel-Bulkley fluid in a three-dimensional thin layer. We introduce some notations and preliminaries. Moreover, we define some function spaces and we recall the variational formulation.
In Section 3, we are interested in the asymptotic behavior, to this aim we prove some convergence results concerning the velocity and pressure when the thickness tends to zero. Besides, the uniqueness of a limit solution has been also established.

Problem statement
Denoting by ω the fixed region in plan x = (x 1 , x 2 ) ∈ R 2 . Introducing the function h : ω → R such that 0 < h 0 ≤ h(x, y ) ≤ h 1 for all (x, y ) ∈ R 2 , where h 0 and h 1 are constants.
Considering the following domains Ω 1 = (x, y , z) ∈ R 3 / (x, y ) ∈ ω and 0 < z < h(x, y ) , , ∈ Ω i . This permits us to define, for every function ϕ ε i : We consider a mathematical problem modeling the steady flow of a rigid viscoplastic and incompressible Herschel-Bulkley fluid. We suppose that the consistency and yield limit of the fluid are respectively µ i ε p , g i ε where µ i , g i > 0, i = 1, 2 and p represents the power-law index. The first fluid occupies a bounded domain Ω ε 1 ⊂ R 3 with the boundary ∂Ω ε 1 of class C 1 . The second one occupies a bounded domain Ω ε 2 ⊂ R 3 with the boundary ∂Ω ε 2 of class C 1 . We denote by Ω ε the domain Ω ε 1 ∪ Ω ε 2 and we suppose that ∂Ω ε 1 = ω ∪ Γ ε 1 and ∂Ω ε 2 = ω ∪ Γ ε 2 the velocity is known and equal to zero, where ω, Γ ε 1 , Γ ε 2 are measurable domains and meas(Γ ε 1 ), meas(Γ ε 2 ) > 0. The fluids are acted upon by given volume forces of densities f ε 1 , f ε 2 respectively. We denote by S 3 the space of symmetric tensors on R 3 . We define the inner product and the Euclidean norm on R 3 and S 3 , respectively, by Here and below, the indices l and m run from 1 to 3 and the summation convention over repeated indices is used. We denote by σ ε i the deviator of σ ε i given by where p ε i , i = 1, 2 represents the hydrostatic pressure and I 3 denotes the identity matrix of size 2. We consider the rate of deformation operator defined for every v ε i ∈ W 1, p i (Ω ε i ) 3 by We denote by n the unit outward normal vector on the boundary ω oriented to the exterior of Ω ε 1 and to the interior of Ω ε 2 , see the figure below. For every vector field v ε i ∈ W 1, p i (Ω ε i ) 3 we also write v ε i for its trace on ∂Ω ε i , i = 1, 2. The steady-state transmission problem for the Herschel-Bulkley fluids in thin layer is given by the following mechanical problem. Let us define now the following Banach spaces : 14) For the rest of this article, we will denote by c possibly different positive constants depending only on the data of the problem.
The use of Green's formula permits us to derive the following variational formulation of the mechanical problem (P ε ), see [4,13,15].
It is known that this variational problem has a unique solution (u ε , see for more details [11,13,15].

Asymptotic behavior
In this section, we establish some results concerning the asymptotic behavior of the solution when ε tends to zero. We begin by recalling the following lemmas see [1,3,4,8,16].
The main results of this section are stated by the following proposition.
this permits us to obtain, making use of Poincaré's and Korn's inequalities and by passage to variables x , y and z Moreover, we get using the incompressibility condition (2.5), (2.6) and Green's formula, for any We can then extract a subsequences still denoted by ( u ε 1 , u ε 2 ) such that as test function in inequality (2.20), using the incompressibility conditions (2.5) and (2.6) as well as the Green formula and Holder's inequality (3.14) On the other hand, it is easy to check that, after some algebraic manipulations, we find (Ω 2 ) 3 .

(3.16)
Passing to the variables x, y and z in the left hand side of (3.16 ) we find the following estimates Consequently, we can extract a subsequence still denoted by p ε 1 , p ε 2 such that which achieves the proof. This proof permits also to deduce that limit pressure verify p ε 1 (x, y , z) , p ε 2 (x, y , z) = ( p 1 (x, y ), p 2 (x, y )).

Proposition 3.2. The velocity limit given by (3.3) verifies
Proof. We know from incompressibility conditions (2.5) and (2.6) that This implies, using Green's formula Hence, by passage to the variables x, y and z using Geen's formula, we can infer Then, Moreover, the fact that x, y , z)dz is continuous and linear, it is weakly continuous. Thus, by passage to the limit when ε tends to zero, taking into account the boundaries conditions (2.7) ,(2.8) and (2.9) it follows that We derive in the proposition below the strong equation verified by the limit solution = (0, 0) then the limit point ( u 1 , u 2 ) and ( p 1 , p 2 ) given by (3.3) and (3.5) verify the limit problem Proof. Introducing the operator Φ defined as follows It is easy to verify that Φ is monotone and hemi-continuous (see for more details the reference [2,5,15] ). Moreover, we know that the functional is proper and convex. Then, the use of Minty's lemma permits us to affirm that (2.20) is equivalent to the following inequality Our goal now is to pass to the limit when ε tends to zero.To this aim, we use Proposition (3.3) and the weak lower semi-continuity of the convex and continuous functional We find the following limit inequality Consequently, we get by combining these two inequalities Due to the fact that W 1, p [1,6], we can take w 1 = v 1 − u 1 and Changing (w 1 , w 2 ) to (−w 1 , −w 2 ) in (3.27), we obtain Now, utilising the Hahn-Banach theorem, in Ω 2 , Then, Ω ( a 1 + a 2 ).w dxdy dz = Ω 1 ( a 1 + a 2 ).w 1 dxdy dz + Which eventually gives (3.23).
The following proposition shows the uniqueness of the limit solution ( u 1 , p 1 ) and ( u 2 , p 2 ).

Conflicts of Interest:
The authors declare that there are no conflicts of interest regarding the publication of this paper.