# Analysis for Flow of an Incompressible Brinkman-Type Fluid in Thin Medium with Friction

Authors: Soumia Manaa; Salah Boulaaras; Hamid Benseridi; Mourad Dilmi; Sultan Alodhaibi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we consider the Brinkman equation in the three-dimensional thin domainℚε⊂ℝ3. The purpose of this paper is to evaluate the asymptotic convergence of a fluid flow in a stationary regime. Firstly, we expose the variational formulation of the posed problem. Then, we presented the problem in transpose form and prove different inequalities for the solution uε,pε independently of the parameter ε. Finally, these estimates allow us to have the limit problem and the Reynolds equation and establish the uniqueness of the solution.


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## Body

## 1. Introduction

Darcy’s law is a physical law that describes the flow of a fluid along a porous medium. It is first named after the French researcher Henri Darcy in 1947 who created a pipeline procedure to supply water around a French town [1]. In 1949, another researcher gave an extension to Darcy’s law to which he added the term Brinkman. Thanks to this term, they discovered a new fluid called Darcy-Brinkman [2], which was used to account for transitional flow between boundaries. This model describes a flow in porous media that is fast enough where the drive for flow includes kinetic potential related to fluid velocity, pressure, and gravitational potential. They appear as a mix of Darcy’s law and the Stokes equations and extend Darcy’s law to account for dissipation of kinetic energy by viscous shear as in the Stokes equation. The average fluid flow through an array of sparse, spherical particles can be modeled via the Brinkman equation [2, 3]. In addition, Brinkman’s equations describe transitions between two flows, one slow and the other fast, where the first type occurs in a porous medium subject to Darcy’s law, while the second type occurs in channels subject to Stokes’ equations. His equation is
(1)−μ∇2u+∇p+μαε2u=f,where μ is the Newtonian fluid viscosity, αε is the resistance parameter, which is assumed constant (isotropic), u is the fluid velocity, p is the pressure, and f represents the body force density applied on the fluid.In recent decades, many authors have studied the Brinkman equation. For instance, Durlofsky and Brady in [4] employ Stokesian dynamics to approximate the fundamental solution or Green’s function for flow in random porous media. The authors in [5] investigate a three-dimensional model of flagellar swimming in a Brinkman fluid. In this study, the utility of the Brinkman equation is to model the mean flow rate of the fluid as well as the resistive effects of fibers on the fluid. The fractional Brinkman-type fluid in a channel under the effect of MHD with the Caputo-Fabrizio fractional derivative was studied by Khana et al. in [6]. In the same idea, Liu in [7] has investigated a biobarrier to remove chlorobenzenes from slow-moving ground contaminated water. The goal of the present paper is not only to give existence and uniqueness solution of the boundary value problems governed by the Brinkman fluid but also to obtain rigorously the equation describing such a phenomenon in a thin film flow by way of an asymptotic analysis in which a small parameter ε is the width of the gap. Several authors are interested in the study of asymptotic convergence of Newtonian and non-Newtonian fluids. In the work of Dilmi et al. [8], the authors studied the asymptotic convergence of a Bingham fluid in a thin domain with the Fourier and Tresca boundary condition on the bottom surface. The asymptotic analysis of an incompressible Herschel-Bulkley fluid with friction law is given by [9]. Many works have focused on mechanics of the fluids in a thin domain in the stationary case which is found in the works of [10, 11]. Other mechanical contact problems similar to incompressible flows in thin medium with friction can be found in [12, 13]. This paper is divided into four sections: in a first step, we discussed the variational formulation of the problem and demonstrate the results of existence and uniqueness of the weak solution; then, we move on to the study of asymptotic analysis. For this, using the change in the variable x3 and unknown news to conduct the study on a domain ℚ does not depend on ε. Then, we prove after an explicit work different inequalities for the solution uε,pε which the thickness becomes infinitely small in the variational formulation. Finally, these estimates allow us to have the limit problem and the Reynolds equation and establish the uniqueness of the solution.


## 2. The Problem Statement

In this section, we give an overview of the thin domain. Next, we introduce the problem considered in this domain. Finally, we explore the theorem of existence and uniqueness of the weak solution.


### 2.1. The Domain

We denote byy,y3 the vector of ℝ3 whose y=y1,y2 is the generic vector of ℝ2 and y3∈ℝ.LetΓb be a domain of the y plane and h be a bounded continuous function defined on Γb, with h of class C1 such that 0<hm≤hy≤hM,∀y,0∈Γb. The fluid is contained between the lower Γb and the upper surface Γ¯uε defined by y3=εhy. Let
(2)ℚε=y,y3∈ℝ3suchthaty∈Γb,and0<y3ε<hy,where ε∈0,1 is a small parameter that will tend to be zero. The boundary of ℚε is Γε=Γb¯∪Γ¯uε∪Γ¯lε, where Γ¯lε is the lateral boundary.Letν=ν1,ν2,ν3 be the unit outward normal to the boundary Γε. The normal and tangential components of uε are given by
(3)uνε=uε·ν=uiενi,uτiε=uiε−uνενi.Similarly, for a regular tensor fieldσε, we denote by σνε and στε the normal and tangential components of σε given by
(4)σνε=σε·ν·ν=σijενiνj,στiε=σijενj−σνενi.We introduce the following functional framework:(5)Eε=v∈H1ℚε3suchthatv=0onΓuε∪Γlε,v·ν=0onΓb,Edivε=v∈Eεsuchthatdivv=0,L02ℚε=q∈L2ℚεsuchthat∫ℚεqdydy3=0.


### 2.2. The Model Problem

The boundary value problem describing the stationary flow for the incompressible Brinkman fluid is described by the following:It is supposed that the law of behavior follows the law of Stokes:(6)σijε=−pεδij+2μdijuε,dijuε=12∂uiε∂yj+∂ujε∂yi,1≤i,j≤3,where δij is the Krönecker symbol.Our problem comes down to finding the solutionuε,pε which satisfies the following equations:Problem 1.
Find the velocity fielduε:ℚε⟶ℝ3 and the pressure pε:ℚε⟶ℝ, such that
(7)−μΔuε+∇pε+μαε2uε=fεinℚε,(8)divuε=0inℚε,(9)uε=0onΓuε,(10)uε=0onΓlε,(11)uε·ν=0onΓb,(12)στε<kε⇒uτε=0,στε=kε⇒∃β≥suchthatuτε=−βστε,onΓb.
Equation (7) represents the law of conservation of momentum where fε:ℚε⟶ℝ3 is the external force. Relation (8) gives the law of behavior of the incompressible fluid. The Dirichlet boundary conditions are given by (9) and (10). The no-flux condition across Γb is given by equation (11). Equation (12) is the condition of the Tresca friction law on the part Γb, with kε being the friction coefficient (see [13]).


### 2.3. Weak Variational Formulations

Letuε be the solution of (7)–(12), multiplying equation (7) by φ−uε and then integrating over ℚε, and using Green’s formula and the conditions (9)–(12), we can show that Problem 1 is equivalent to the following variational problem:Problem 2.
Find the pairuε,pε∈Edivε×L02ℚε, such that
(13)auε,φ−uε−pε,divφ+jφ−juε≥fε,φ−uε,∀φ∈Eε,where
(14)auε,φ−uε=∫ℚε2μdijuεdijφi−uiεdydy3+μ∫ℚεαε2uiεφi−uiεdydy3,pε,divφ=∫ℚεpεδij∂∂yjφi−uiεdydy3,jφ=∫Γbkεφdy,fε,φ−uε=∫ℚεfiεφi−uiεdydy3.Theorem 3.
Ifkε∈L+∞Γb and fε∈L2ℚε3, then there exists a unique uε∈Edivε and pε∈L02ℚε (to an additive constant) solution to problem (13).Proof.
Since our goal in this work is to prove the asymptotic convergence of the problem posed, we only give the steps followed for the proof of this theorem. Letφ∈Edivε in (13); we have the following:
Finduε∈Edivε, such that
(15)auε,φ−uε+jφ−juε≥fε,φ−uε,∀φ∈Edivε.
Using the Cauchy-Schwartz inequality and∑i,j=13dijuε2≤∇uε2, we obtain that the bilinear from a.,. is continuous:
(16)auε,v=∫ℚε2μdijuεdijvdydy3+∫ℚεμαε2uεvdydy3≤2μ+μαε2uεH1ℚvH1ℚ,∀uε,v∈Edivε2.
By Korn’s inequality, we obtain(17)auε,uε=2μ∑i,j=13dijuεL2ℚε2+μαε2uεL2ℚε2≥min2μCk,μαε2uεH1ℚ2,∀uε∈Edivε,where Ck>0 independent of ε. We deduce that a.,. is coercive on Edivε×Edivε. Moreover, j is convex and continuous on Edivε. This ensures the existence and uniqueness of uε∈Edivε satisfying the variational inequality (15). In addition, using the techniques of [14], we can prove the existence of pε∈L02ℚε for which uε,pε is a solution of (13).☐


## 2.1. The Domain

We denote byy,y3 the vector of ℝ3 whose y=y1,y2 is the generic vector of ℝ2 and y3∈ℝ.LetΓb be a domain of the y plane and h be a bounded continuous function defined on Γb, with h of class C1 such that 0<hm≤hy≤hM,∀y,0∈Γb. The fluid is contained between the lower Γb and the upper surface Γ¯uε defined by y3=εhy. Let
(2)ℚε=y,y3∈ℝ3suchthaty∈Γb,and0<y3ε<hy,where ε∈0,1 is a small parameter that will tend to be zero. The boundary of ℚε is Γε=Γb¯∪Γ¯uε∪Γ¯lε, where Γ¯lε is the lateral boundary.Letν=ν1,ν2,ν3 be the unit outward normal to the boundary Γε. The normal and tangential components of uε are given by
(3)uνε=uε·ν=uiενi,uτiε=uiε−uνενi.Similarly, for a regular tensor fieldσε, we denote by σνε and στε the normal and tangential components of σε given by
(4)σνε=σε·ν·ν=σijενiνj,στiε=σijενj−σνενi.We introduce the following functional framework:(5)Eε=v∈H1ℚε3suchthatv=0onΓuε∪Γlε,v·ν=0onΓb,Edivε=v∈Eεsuchthatdivv=0,L02ℚε=q∈L2ℚεsuchthat∫ℚεqdydy3=0.


## 2.2. The Model Problem

The boundary value problem describing the stationary flow for the incompressible Brinkman fluid is described by the following:It is supposed that the law of behavior follows the law of Stokes:(6)σijε=−pεδij+2μdijuε,dijuε=12∂uiε∂yj+∂ujε∂yi,1≤i,j≤3,where δij is the Krönecker symbol.Our problem comes down to finding the solutionuε,pε which satisfies the following equations:Problem 1.
Find the velocity fielduε:ℚε⟶ℝ3 and the pressure pε:ℚε⟶ℝ, such that
(7)−μΔuε+∇pε+μαε2uε=fεinℚε,(8)divuε=0inℚε,(9)uε=0onΓuε,(10)uε=0onΓlε,(11)uε·ν=0onΓb,(12)στε<kε⇒uτε=0,στε=kε⇒∃β≥suchthatuτε=−βστε,onΓb.
Equation (7) represents the law of conservation of momentum where fε:ℚε⟶ℝ3 is the external force. Relation (8) gives the law of behavior of the incompressible fluid. The Dirichlet boundary conditions are given by (9) and (10). The no-flux condition across Γb is given by equation (11). Equation (12) is the condition of the Tresca friction law on the part Γb, with kε being the friction coefficient (see [13]).


## 2.3. Weak Variational Formulations

Letuε be the solution of (7)–(12), multiplying equation (7) by φ−uε and then integrating over ℚε, and using Green’s formula and the conditions (9)–(12), we can show that Problem 1 is equivalent to the following variational problem:Problem 2.
Find the pairuε,pε∈Edivε×L02ℚε, such that
(13)auε,φ−uε−pε,divφ+jφ−juε≥fε,φ−uε,∀φ∈Eε,where
(14)auε,φ−uε=∫ℚε2μdijuεdijφi−uiεdydy3+μ∫ℚεαε2uiεφi−uiεdydy3,pε,divφ=∫ℚεpεδij∂∂yjφi−uiεdydy3,jφ=∫Γbkεφdy,fε,φ−uε=∫ℚεfiεφi−uiεdydy3.Theorem 3.
Ifkε∈L+∞Γb and fε∈L2ℚε3, then there exists a unique uε∈Edivε and pε∈L02ℚε (to an additive constant) solution to problem (13).Proof.
Since our goal in this work is to prove the asymptotic convergence of the problem posed, we only give the steps followed for the proof of this theorem. Letφ∈Edivε in (13); we have the following:
Finduε∈Edivε, such that
(15)auε,φ−uε+jφ−juε≥fε,φ−uε,∀φ∈Edivε.
Using the Cauchy-Schwartz inequality and∑i,j=13dijuε2≤∇uε2, we obtain that the bilinear from a.,. is continuous:
(16)auε,v=∫ℚε2μdijuεdijvdydy3+∫ℚεμαε2uεvdydy3≤2μ+μαε2uεH1ℚvH1ℚ,∀uε,v∈Edivε2.
By Korn’s inequality, we obtain(17)auε,uε=2μ∑i,j=13dijuεL2ℚε2+μαε2uεL2ℚε2≥min2μCk,μαε2uεH1ℚ2,∀uε∈Edivε,where Ck>0 independent of ε. We deduce that a.,. is coercive on Edivε×Edivε. Moreover, j is convex and continuous on Edivε. This ensures the existence and uniqueness of uε∈Edivε satisfying the variational inequality (15). In addition, using the techniques of [14], we can prove the existence of pε∈L02ℚε for which uε,pε is a solution of (13).☐


## 3. Dilatation in the Variabley3

In this section, we use the dilatation in the variabley3 given by y3=zε; then, our problem will be defined on a domain ℚ which does not depend on ε given by
(18)ℚ=y,z∈ℝ3suchthaty∈Γband0<z<hy,and its boundary Γ=Γ¯1∪Γ¯L∪Γb¯.After this change, here are the new functions defined on the fixed domainℚ:
(19)u^iεy,z=uiεy,y3,i=1,2,u^3εy,z=ε−1u3εy,y3,p∧εy,z=ε2pεy,y3.Likewise for new data,(20)f^y,z=ε2fεy,y3,α^=εαε,k^=εkε.We denote by(21)E=v^∈H1ℚ3:v^=0onΓu∪Γl,v^·ν=0onΓb,Ediv=v^∈E:divv^=0,L02ℚ=q∈L2ℚ:∫ℚqdydy3=0,Vz=Φ=Φ1,Φ2∈L2ℚ2:∂Φi∂z∈L2ℚ,Φ=0onΓu∪Γl,the Banach space with the norm
(22)ΦVz=∑i=12ΦiL2ℚ2+∂Φi∂zL2ℚ21/2.According to (19) and (20), then Problem 2 leads to the following:Problem 4.
Findu∧ε,p∧ε∈E×L02ℚ, such that
(23)∫ℚqdivu∧εdydz=0,∀q∈L02ℚ,(24)∑i,j=12∫ℚε2μ∂u^iε∂yj+∂u^jε∂yi−p∧εδij∂∂yjφ^i−u^iεdydz+∑i=12∫ℚμ∂u^iε∂z+ε2∂u^3ε∂yi∂∂zφ^i−u^iεdydz+∑j=12∫ℚμε2∂u^3ε∂yj+∂u^iε∂zε2∂∂yiφ^3−u^3εdydz+∫ℚ2με2∂u^3ε∂z−p∧ε∂∂zφ^3−u^3εdydz+∑i=12μα∧2∫ℚu^iεφ^i−u^iεdydz+ε2μα∧2∫ℚu^3εφ^3−u^3εdydz+∫Γbk^φ^−u∧εdy≥∑i=12∫ℚf^iφ^i−u^iεdydz+ε∫ℚf^3φ^3−u^3εdydz,∀φ∈E.In the next, we will do the estimates of velocityu∧ε and then on the pressure p∧ε solution of our variational problem in the fixed domain.Theorem 5.
Assuming (19) and (20), the following estimate on u∧ε is satisfied:
(25)∑i,j=12ε2∂u^iε∂yjL2ℚ2+∑i=12∂u^iε∂zL2ℚ2+ε2∂u^3ε∂zL2ℚ2+ε2∑i=12∂u^3ε∂yiL2ℚ2+∑i=12u^iεL2ℚ2+ε2u^3εL2ℚ2≤C.Proof.
Letuε be the solution of (15), we choose φ=0, and we obtain
(26)auε,uε≤∫ℚfεuεdydy3,
Now,auε,uε=2μ∫ℚεdijuεdijuεdydy3+μ∫ℚεuiεuiεdydy3. By Korn’s inequality, there exists a constant Ck>0 independent of ε, such that
(27)2μ∫ℚεdijuεdijuεdydy3≥2μCk∇uεL2ℚε2.
We apply the Cauchy-Schwarz inequality and then the Young inequality; we obtain the following:(28)fε,uε≤hM2ε24μCkfεL2ℚε2+CkμhM2ε2uεL2ℚε2.
By the Poincaré inequality, we give(29)μCk∇uεL2ℚε2+μαε2uεL2ℚε≤hM2ε24μCkfεL2ℚε2.
Asε2fεL2ℚε2=ε−1f^L2ℚ2,ε≤1, then we multiply the last equation by ε; we obtain
(30)minμCk,μα∧2ε∇uεL2ℚε2+uεL2ℚε2≤hM24μCkf^L2ℚ2.
Writing this equation in terms ofu∧ε, we deduce (25).☐Theorem 6.
Assuming (19), the following estimates on pε are satisfied:
(31)∂p∧ε∂zH−1ℚ≤εC1,(32)∂p∧ε∂yiH−1ℚ≤C2,i=1,2,where C1 and C2 denote the constants independent of ε.Proof.
Letψ∈H01ℚ; putting in (24) φ^i=u^iε (for i=1,2) and φ^3=u^3ε±ψ, we deduce
(33)∑i=12∫ℚε2με2∂u^3ε∂yj+∂u^iε∂z∂ψ∂yidydz+∫ℚ2ε2μ∂u^3ε∂z−p∧ε∂ψ∂zdydz+ε2μα∧2∫ℚu^3εψdydz=ε∫ℚf^3ψdydz.
Therefore,(34)∫ℚp∧ε∂ψ∂zdydz=∑i=12∫ℚε2με2∂u^3ε∂yj+∂u^iε∂z∂ψ∂yidydz+∫ℚ2ε2μ∂u^3ε∂z∂ψ∂zdydz+ε2μα∧2∫ℚu^3εψdydz−ε∫ℚf^3ψdydz.
Using Green’s formula and the Cauchy-Schwarz inequality, we have(35)∫ℚ∂p∧ε∂zψdxdz≤ε4μ∑i=12∂u∧3ε∂xjL2ℚ21/2+ε2μ∑i=12∂u∧jε∂zL2ℚ21/2+2ε2μ∂u^3ε∂zL2ℚ+ε2μα∧2u^3εL2ℚ+εf^3L2ℚψH01ℚ.
From Theorem5, we deduce the inequality (31).
Taking in (24) φ^1ε=u^1ε±ψ, ψ in H01ℚ; φ^i=u^iε,i=2,3; we have
(36)∫ℚp∧ε∂ψ∂y1dydz=∫ℚμ∂u^1ε∂z+ε2∂u^3ε∂y1∂ψ∂zdydz+∫ℚ2ε2μ∂u^1ε∂y1∂ψ∂y1dydz+∫ℚε2μ∂u^1ε∂y2+∂u^2ε∂y1∂ψ∂y2dydz+μα∧2∫ℚu^1εψdydz−∫ℚf^1ψdydz.
In the same way, by the chosenφ^1=u^1ε, φ^3=u^3ε, and φ^2=u^2ε±ψ, ψ in H01ℚ, we have
(37)∫ℚp∧ε∂ψ∂y2dydz=∫ℚμ∂u^2ε∂z+ε2∂u^3ε∂y2∂ψ∂zdydz+∫ℚ2ε2μ∂u^2ε∂y2∂ψ∂y2dydz+∫ℚε2μ∂u^1ε∂y2+∂u^2ε∂y1∂ψ∂y1dydz+μα∧2∫ℚu^2εψdydz−∫ℚf^2ψdydz.
We use the same technical in (36) and (37) to obtain inequality (32).☐Thanks to the estimations (25)–(32), we have the following convergence result:Theorem 7.
Suppose that the estimations (25)–(32) hold. There exists u∗,p∗=u1∗,u2∗,p∗ in Vz×L02ℚ such that we have the following:
(38)u^iε⇀ui∗inVz,1≤i≤2,(39)ε∂u^iε∂yj⇀0inL2ℚ,1≤i,j≤2,(40)ε∂u^3ε∂z⇀0inL2ℚ,(41)ε2∂u^3ε∂yi⇀0inL2ℚ,1≤i≤2,(42)εu^3ε⇀0inL2ℚ,(43)p∧ε⇀p∗inL02ℚ.Proof.
Using the estimate (25) and Poincaré’s inequality in the domain ℚ:
(44)∂u^iε∂zL2ℚ2+u^iεL2ℚ2≤Ci=1,2,we deduce (38). Also, equations (39)–(42) follow from (25) and divu∧ε=0. Finally, the weakly convergences (43) follow from (31) and (32) and [15].☐


## 4. Study of the Limit Problem

To reach the desired goal, we need in the rest of this paragraph the results of previous convergences.We go to the limit in (24), and using (23) (ε⟶0), we find
(45)∑i=12∫ℚμ∂ui∗∂z∂∂zφ^i−ui∗dydz−∫ℚp∗∂φ^1∂y1+∂φ^2∂y2dydz+∑i=12μα^∫ℚui∗φ^i−ui∗dydz+k^∫Γbφ^−u∗dy≥f^,φ^−u∗.Moreover, if(46)∫ℚφ^1∂θ∂y1y+φ^2∂θ∂y2ydydz=0∀θ∈C01Γb,then
(47)∑i=12∫ℚμ∂ui∗∂z∂∂zφ^i−ui∗dydz+∑i=12μα∧2∫ℚui∗φ^i−ui∗dydz+k^∫Γbφ^−u∗dy≥∑i=12∫ℚf^iφ^i−ui∗dydz,∀φ^∈ΠE,where
(48)ΠE=ψ=ψ^1,ψ^2∈H1ℚ2:∃ψ^3suchthatψ=ψ^1,ψ^2,ψ^3∈E.Lemma 8.
If the estimations (25)–(32) hold, then u∗,p∗ satisfy
(49)μ∫ℚ∂u∗∂z2dydz+μα∧2∫ℚu∗2dydz+k^∫Γbu∗dy−∫ℚf^u∗dydz=0,(50)μ∫ℚ∂u∗∂z∂φ^∂zdydz+μα∧2∫ℚu∗φ^dydz+k^∫Γbφ^dy≥∫ℚf^φ^dydz,∀φ^∈ΣE,where ΣE=φ¯∈ΠE:φ¯satisfiescondition32}.Proof.
As in [16], by choosing φ^=0 in (47), we find that the second member of this inequality is an upper bound of its first member. Then, for φ^=2u∗ in the same inequality, we get the reverse, which therefore gives equality (49). Now, for φ^=u∗+φ^ in (47), we deduce directly the inequality (50). It is shown reciprocally that (49) and (50) imply (47). For this, we consider the mapping:
(51)Λ:ΣE⟶L1Γb2,φ~⟶Λφ^=k~φ~.
Let us defineF as
(52)Fk^φ^=μ∫ℚ∂u∗∂z∂φ^∂zdydz+μα∧2∫ℚu∗φ^dydz−∫ℚf^φ^dydz.
For allψ^∈ΣE, we choose φ^=−ψ^ in (49):
(53)Fk^ψ^≤∫Γbk^ψ^dy.
By (53), F is a continuous linear function on the subspace of L1Γb2 which is the image of ΣV by Λ. Then, by the Han-Banach theorem, there exists χ∈L∞Γb2, with χ≤1, such that
(54)Fk^ψ^=−∫Γbχk^ψ^dy.
In particular, from (50) and (54), we get
(55)∫Γbk^u∗dy=∫Γbχk^u∗dy.
Also,(56)μ∫ℚ∂u∗∂z∂ψ^∂zdydz+μα∧2∫ℚu∗ψ^dydz−∫ℚf^ψ^dydz+∫Γbχk^ψ^dy=0.
Then, from (49) and (56), we have
(57)μ∫ℚ∂u∗∂z∂ψ^∂z−∂u∗∂zdydz+μα∧2∫ℚu∗ψ^−u∗dydz−∫ℚf^ψ^−u∗dydz+∫Γbk^ψ^−u∗dy≥0,
which gives (47).☐Theorem 9.
With the same assumptions as Theorems5 and 6, the pair u∗,p∗ satisfies
(58)p∗∈H1Γb,(59)−μ∂2ui∗∂z2+∂p∗∂yi+μα∧2ui∗=fi^inL2Γb.Proof.
We choose in (24) φ^3=u^3ε±ψ with ψ in H01ℚ and φ^i=u^iε (for i=1,2); we give (34). Using the convergence limit ((38) and (40)–(42)) in (34), we find
(60)∫ℚp∗∂ψ∂zdydz=0,∀ψ∈H01ℚ,then,
(61)∂p∗∂z=0inH−1ℚ.
Choosingφ^i=u^iε±ψ (for i=1,2) with ψi in H01ℚ and φ^3=u^3ε leads to
(62)∑i,j=12∫ℚε2μ∂u^iε∂yj+∂u^jε∂yi−p∧εδij∂ψ∂yjdydz+∑i=12∫ℚμ∂u^iε∂z+ε2∂u^3ε∂yi∂ψ∂zdydz+∑i=12∫ℚμα∧2u^iεψidydz=∑i=12∫ℚf^iψdydz.
Using (38), (39), (41), and (43) and choosing ψ1=0 and ψ2∈H01ℚ and then with ψ2=0 and ψ1 in H01ℚ, we find
(63)∑i=12∫ℚ−p∗∂ψi∂yidydz+∑i=12∫ℚμ∂ui∗∂z∂ψi∂zdydz+∑i=12∫ℚμα∧2ui∗ψidydz=∑i=12∫ℚf^iψidydz.
By Green’s formula, we deduce(64)∫ℚ−μ∂2ui∗∂z2ψidydz+∫ℚ∂p∗∂yiψidydz+μα∧2∫ℚui∗ψidydz=∫ℚf^iψidydz,∀ψi∈H01ℚ.
Then,(65)−μ∂2ui∗∂z2+∂p∗∂yi+μα∧2ui∗=f^i,inH−1ℚi=1,2.
To provep∗∈H1Γb, we see from Theorem 6 that p∗ does not depend on z. Then, as in [16], we choose ψiy,z=zz−hyγy in (63); with γ∈H01Γb, we obtain
(66)∫Γb∫0hy−p∗∂∂yizz−hyγydydz+∫Γb∫0hyμ∂ui∗∂z2z−hyγydydz+∫Γb∫0hyμα∧2ui∗zz−hyγydydz=∫Γb∫0hyf^y,zzz−hyγydydz.
Using the Green formula, we deduce(67)16∫Γbp∗∂h3γy∂yidy−2μ∫Γb∫0hyui∗y,zγydydz+∫Γb∫0hyμα∧2ui∗zz−hyγydydz=∫Γb∫0hyf^y,zzz−hyγydydz.
Then,(68)16∫Γbp∗∂h3γ∂yidy−2μ∫Γbhyu~i∗γydy+∫Γbμα∧2u^i∗γydy=∫Γbf.~γydy,where
(69)u~i∗x=1hy∫0hyui∗y,zdz,u^i∗=∫0hyui∗y,zzz−hydz,f~i=∫0hyf^izz−hydz,
whence(70)−h36∂p∗∂yi−2μhyu~i∗+μα∧2u^i∗=f~iinH−1Γb.
Asfi∈L2ℚ and ui∗∈Vz then in L2ℚ, therefore f~i, u~i∗, and u^i∗ are in L2Γb. In addition, from (70), we get p∗ in H1Γb. So as fi belongs to L2ℚ from (61), we have ∂2ui∗/∂z2∈L2ℚ, whence (59) holds, and we also have ∂ui∗/∂z in Vz.☐Theorem 10.
Suppose that the assumptions of the previous theorem hold; then, the solutionu∗,p∗ satisfies the weak generalized equation of Reynolds:
(71)∫Γbh312μ∇p∗+hui~∗y−h2s∗y+α^h2U∗y,h−∫0hU∗y,tdt+F~y∇φydy=0,∀φ∈H1ℚ.where
(72)F~y=1μ∫0hyFy,tdt−h2μFy,h,U∗y,t=∫0t∫0ζui∗y,θdθdζ,Fy,t=∫0t∫0ζf^iy,θdθdζ,s∗y=u∗y,0andτ∗y=∂u∗∂zy,0.Proof.
By integrating expression (59) from 0 to z, it becomes
(73)−μui∗y,z+μui∗y,0+μz∂ui∗∂zy,0+z22∂p∗∂yi+μα∧2∫0z∫0ζui∗y,tdtdζ=∫0z∫0tf^idtdζ.
In particular forz=h, we obtain
(74)μs∗y+μhτ∗y+h22∂p∗∂yi+μα∧2∫0h∫0ζui∗y,tdtdζ=∫0h∫0ζf^iy,tdtdζ.
Integrating (73) from 0 to h, we get
(75)−μ∫0hui∗y,tdt+μhs∗y+h2μ2τ∗y+h36∂p∗∂yi+μα∧2∫0h∫0ζ∫0tui∗y,θdθdtdζ=∫0h∫0ζ∫0tf^iy,θdθdtdζ.
Asu~i∗y=1/h∫0hyui∗y,tdt, we have
(76)−μhu~i∗y+μhs∗y+μh22τ∗y+h36∂p∗∂yi+μα∧2∫0h∫0ζ∫0tui∗y,θdθdtdζ=∫0hyFy,tdt.
From (74)–(76), we deduce (71), which ends the proof requested.☐Theorem 11.
Suppose that the assumptions of the previous theorem hold; then, the solutionu∗,p∗ of the limit problem (45) is unique in Vz×L02ℚ.Proof.
Suppose that problem (47) admits two solutions that we denote by u∗,1,p∗,1 and u∗,1,p∗,2. Using classical techniques, we choose in (47) φ^=u∗,1 and then φ^=u∗,2 as test functions; then, by summing the two inequalities obtained, we find
(77)∑i=12μ∫ℚ∂∂zui∗,1−ui∗,22dydz+∑i=12μα∧2∫ℚui∗,1−ui∗,22dydz≤0.
Then,(78)∂∂zu∗,1−u∗,2L2ℚ2+u∗,1−u∗,2L2ℚ2≤0.
Consequently, this last inequality ensures the uniqueness of the solutionu∗ in Vz.
Similarly, we take in the Reynolds equation (71) the pressure value p∗=p∗,1 and then p∗=p∗,2, respectively, at the end by subtracting the equations obtained; it becomes
(79)∫Γbh312μ∇p∗,1−p∗,2∇φdy=0.
Finally, by performing the change of variableφ=p∗,1−p∗,2 and then by the Poincaré inequality, we obtain
(80)p∗,1−p∗,2H1Γb=0.
We deduce the desired result.☐


## 5. Conclusions

The purpose of this paper is to study the asymptotic convergence of an incompressible Brinkman-type fluid in thin medium with the Tresca friction on the bottom surface. One of the objectives of this study is to obtain a two-dimensional equation that allows a reasonable description of the phenomenon occurring in the three-dimensional domain by passing the limit to 0 on the small thickness of the domain (3D). We show the existence and uniqueness results of the weak solution. We proceed to the study of the convergence analysis. To do this, we use the scale change following the third component and new unknowns to conduct the study on a fixed domain. Then, we prove different inequalities for the solutionuε,pε. Finally, these estimates allow us to have the limit problem and establish the uniqueness of the solution.



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*Source: 5112840-2021-09-22.xml*