# Asymptotic Behavior of Weak Solutions of Nonisothermal Flow of Herschel–Bulkley Fluid to Free Boundary

Authors: Fares Yazid; Abdelkader Saadallah; Djamel Ouchenane; Nadhir Chougui; Mohamed Abdalla; Ahmed Himadan
Journal: Discrete Dynamics in Nature and Society
 (2022)

---

## Abstract

In this manuscript, the behavior of a Herschel–Bulkley fluid has been discussed in a thin layer inℝ3 associated with a nonlinear stationary, nonisothermal, and incompressible model. Furthermore, the limit problem has been considered, and the studied problem in Ωε is transformed into another problem defined in Ωε without the parameter Ωε (ε is the parameter representing the thickness of the layer tend to zero is studied). We also investigated the convergence of the unknowns which are the velocity, pressure, and the temperature of the fluid. In addition, we established the limit problem and the specific Reynolds equation.


---

## Body

## 1. Introduction

In a recent study of problems for the asymptotic behavior for a problem of continuum mechanics in a thin domainΩε, the problem is transformed into an equivalent problem on a domain Ω independent of the parameter ε. This phenomenon has been presented by many researchers, see, e.g., [1–5]. Specifically, the case of Herschel–Bulkley fluid has been archived in several articles, for instance, [6, 7]. A particularity of Herschel–Bulkley fluid lies in the presence of rigid zones located in the interior of the flow, and as the yield limit increases, the rigid zones become larger and may completely block the flow (see, e.g., [8–10]).This work is to study the asymptotic behavior for weak solutions of a linked system, including of an incompressible Herschel–Bulkley fluid and the equation of the heat energy, in a three-dimensional bounded domain satisfying Tresca-type fluid solid boundary conditions. The boundary of this thin domain consists of three parts: the bottom, the lateral part, and the top surface.The article is organized as follows: in Section2, we present the mechanical problem of the steady-state flow of Herschel–Bulkley fluid in a three-dimensional thin domain. We also introduce some notations, preliminaries, and some function spaces of our coupled problem.In Section3, we use the asymptotic analysis, in which the small parameter ε is the height of the domain. We also discuss some estimates, independent on the parameter ε, for the velocity, the pressure, and the temperature. Moreover, we give some convergence results. The main results concerning the limit problem with a specific weak form of the Reynolds equation are established in Section 4. Finally, in Section 5, we include some remarks and conclusions on the work.


## 2. Statement of the Problem and Variational Formulation

Here, letϖ be fixed region in plan s=s1,s2∈ℝ2. We assume that ϖ has a Lipschitz boundary and is the bottom of the fluid domain. The upper surface Γ1ε is defined by s3=εhs where 0<ε<1 is a small parameter that will tend to zero and h a smooth bounded function such that 0<h∗≤hs≤h∗ for all s,0∈ϖ and ΓLε the lateral surface. We denote by Ωε the domain of the following:(1)Ωε=s,s3∈ℝ3:s,0∈ϖ,0<s3<εhs.The boundary ofΩε is Γε where Γε=Γ¯1ε∪Γ¯Lε∪ϖ¯ with Γ¯Lε is the lateral boundary. We denote by Σε the deviatoric part and ρε the pressure. The fluid is supposed to be viscoplastic, and the relation between Σε and Dwε is given by(2)Σijε=Σ˜ijε−ρεδij,Σ˜ε=gεTεDwεDIIwε+ΛεTεDwεν−2Dwε,ifDwε≠0,Σ˜ε≤gεTε,ifDwε=0.For any tensorD=dij, the notation D represents the matrix norm: DII=1/2∑i,j=13dijdij1/2. Let n=n1,n2,n3 the unit outward normal vector on the boundary Γε. The normal and the tangential velocity on the boundary Ωε are wnε=wε.n,wτε=wε−wnε.n. Also, Σε is a regular stress tensor field, further let Σnε and Στε are the normal and tangential components of Σε on the boundary ϖ by Σnε=Σε.n.n, Στε=Σε.n−Σnε.n.Problem 1.
Find a velocity fieldwε:Ωε⟶ℝ3, the pressure ρε and a temperature: Ωε⟶ℝ such that(3)divΣε+fε=0inΩε,(4)Σ˜ε=gεTεDwεDIIwε+ΛεTεDwεν−2DwεifDwε≠0,Σ˜ε≤gεTεifDwε=0,(5)divwε=0inΩε,−∇.Kε∇Tε=ΛεTεDwεr+2gεTεDwε−αεTεinΩ,wε=0onΓ1∪ΓL,(6)wε×n=0onϖ,(7)Στε<kε⇒wτε=0Στε=kε⇒∃λ≥0,wτε=−λΣτεonϖ,Tε=0onΓ1∪ΓL,(8)∂Tε∂n=0onϖ.where divΣ=Σij,j and divw=wi,i. The flow is given by the (3) where the density is assumed equal to one. (4) represents the constitutive law of a Herschel–Bulkley fluid whose the consistency Λε and the yield limit gε depend on the temperature, 1<ν<2 is the power law exponent of the material. (5) represents the incompressibility condition. Equation (5) represents the energy conservation where the specific heat is assumed equal to one, Kε>0 is the thermal conductivity and the term −αεTε represents the external heat source with αε>0. (5) gives the velocity on Γ1ε∪ΓLε. As there is no-flux condition across ϖ, then we have equation (6). Condition (7) represents a Tresca thermal friction law on ϖ, where kε is the friction yields coefficient (8) gives the temperature on Γ1ε∪ΓLε. (8) is a homogeneous Neumann boundary condition on ϖ:(9)W1,νΩε=ϑ∈LνΩε3:∂ϑi∂sj∈LνΩεfori,j=1,…,3,VεΩε=ϑ∈W1,νΩε3:ϑ=0onΓ1ε∪ΓLε,ϑ.n=0onϖ,VdivεΩε=ϑ∈Kε:divϑ=0,L0ν′Ωε=ϑ∈Lν′Ωε:∫Ωεϑdsds3=0,and(10)WΓ1ε∪ΓLε1,qΩε=Φ∈W1,qΩε3:Φ=0onΓ1ε∪ΓLε.
A formal application of Green’s formula, using (3)–(8) leads to the weak formulation: Find a velocity field wε∈Vdivε, ρε∈L0ν′Ωε and Tε∈WΓ1ε∪ΓLε1,qΩε, 1<q<3/2 such that(11)awε,ϑ−wε−ρε,divϑ+jTε,ϑ−jTε,wε≥fε,ϑ−wε,∀ϑ∈VεΩε,bTε,Φ=Cwε,Tε,Φ,∀Φ∈WΓ1ε∪ΓLε1,qΩε,where(12)awε,ϑ−wε=∫ΩεΛεTεDwεν−2DwεDvεdsds3,ρε,divϑ=∫Ωερεdivϑdsds3,jTε,v=∫ϖkεvds+2∫ΩεgεTεDvdsds3,fε,v=∫Ωεfεvdsds3=∑i=13∫Ωεfiεvidsds3,bTε,Φ=∫ΩεKε∇Tε∇Φdsds3,Cwε,Tε,Φ=∫ΩεΛεTεDwενΦdsds3+2∫ΩεgεTεDwεΦdsds3+∫ΩεαεTεΦdsds3.
It is known that this variational problem has a unique solution, see for more details [10–12].
We assume that there existΛ∗,Λ∗,g∗,K∗ε,Kε∗,α∗ε,αε∗ in ℝ such that(13)0≤Λ∗≤Λε≤Λ∗,0≤gε≤g∗,fε∈W1,ν′Ωε3,and(14)0≤K∗ε≤Kε≤Kε∗,0≤α∗ε≤αε≤αε∗.
Following some previous results that are useful in the next sections (cf. [13])(15)∇wεLνΩε≤CDwεLνΩε,Cis a positive constant independent ofε,1<ν<2Korn inequality,(16)wiεLνΩε≤εh∗∂wiε∂κLνΩεfori=1,2;h∗=maxhs,1<ν<2Poincaré inequality,(17)ab≤aνν+bν′ν′,∀a,b∈ℝ2,1<ν<2,1ν+1ν′=1Young inequality,(18)a+bρ≤2ρ−1aρ+bρ,∀a,b∈ℝ+∗2,∀ρ>1,a+bρ≤aρ+bρ,∀a,b∈ℝ+∗2,0<ρ<1.


## 3. Change of the Domain and Study of Convergence

In this section, we will use the technique of scaling inΩε on the coordinate s3, by introducing the change of the variables κ=s3/ε. We obtain a fixed domain Ω which is independent of ε: Ω=s,κ∈ℝ3:s,0∈ϖ,0<κ<hs.We denote its boundary byΓ=Γ¯1∪Γ¯L∪ϖ¯, also we have(19)w^iεs,κ=wiεs,s3,i=1,2,w^3εs,κ=ε−1w3εs,s3,ρ^εs,κ=ενρεs,s3.Assume that(20)K^s,κ=εβ+ν−2Kεs,s3,α^s,κ=εβ+ναεs,s3,Λ^=Λε,f^s,κ=ενfεs,s3,g^=εν−1gε,k^=εν−1kε,,with(21)β=32−ν3−ν.Let(22)VΩ=ϑ^∈W1,νΩ3:ϑ^=0onΓ1∪ΓL;ϑ^.n=0oϖ,VdivΩ=ϑ^∈KΩ:divϑ^=0,Vκ=ϑ^∈LrΩ2;∂ϑ^i∂κ∈LνΩ:ϑ^=0onΓ1∪ΓL,V˜κ=ϑ^∈Vκ:ϑ^satisfyD′,Πκ=ϑ^∈LqΩ2;∂ϑ^i∂κ∈LqΩ,where the condition D′ is given by(23)∫ϖϑ^1∂ϑ∂s1+ϑ^2∂ϑ∂s2dsdκ=0,for allϑ^∈LνΩ2,ϑ∈C0∞Ω.D.By injecting the new data and unknown factors in (19) and (20), we prove that w^ε,ρ^ε,T^ε is a solution of the following problem:(24)a0T^ε,w^ε,ϑ^−w^ε−ρ^ε,divϑ^−w^ε+j0T^ε,ϑ^−j0T^ε,w^ε≥f^,ϑ^−w^ε,∀ϑ^∈VΩ,b0T^ε,Φ^=C0w^ε,T^ε,Φ^,∀Φ^∈WΓ1ε∪ΓLε1,qΩ,where(25)a0T^ε,w^ε,ϑ^−w^ε=∑i,j=12∫Ωε2Λ^T^εD˜w^εν−212∂w^iε∂sj+∂w^jε∂si∂ϑ^i−w^iε∂sjdsdκ+∑i=12∫ΩΛ^T^εD˜w^εν−212∂w^iε∂κ+ε2∂w^3ε∂si∂ϑ^i−w^iε∂κdsdκ+∫ΩΛ^T^εD˜w^εν−2ε2∂w^3ε∂κ∂ϑ^3−w^3ε∂κdsdκ++∑j=12∫Ωε2Λ^T^εD˜w^εν−212ε2∂w^3ε∂sj+∂w^jε∂κ∂ϑ^3−w^3ε∂sjdsdκ,ρ^ε,divϑ^−w^ε=∫Ωερ^εdivϑ^−w^εdsdκ,j0T^ε,ϑ^=2∫Ωg^T^εD˜ϑ^dsdκ+∫ϖk^|ϑ^|ds,f^ε,ϑ^−w^ε=∑j=12∫Ωf^iϑ^i−w^iεdsdκ+∫Ωεf^3ϑ^3−w^3εdsdκ,b0T^ε,Φ^=∫Ωε2∇εT^ε∇εΦ^dsdκ=∑i=12∫Ωε2K^∂T^ε∂si∂Φ^∂sidsdκ+∫ΩK^∂T^ε∂κ∂Φ^∂κdsdκ,C0w^ε,T^ε,Φ^=∫ΩεβΛ^T^εD˜w^ενΦ^dsdκ+2∫Ωεβg^T^εD˜w^εΦ^dsdκ−∫Ωα^T^εΦ^dsdκ,D˜w^ε=14∑i,j=12ε2∂w^iε∂sj+∂w^jε∂si2+12∑i=12∂w^iε∂κ+ε2∂w^3ε∂si2+ε2∂w^3ε∂κ21/2.


### 3.1. A Priori Estimates on the Velocity and the Pressure

Theorem 1.
For all1<ν<2 and under assumptions (13) and (14) and (20), there exists a constant C>0 independent of ε such that(26)∑i,j=12ε∂w^iε∂sjLνΩν+∑i=12∂w^iε∂κLνΩν+ε2∂w^3ε∂siLνΩν+ε∂w^3ε∂κLνΩν≤C,∂ρ^ε∂siW−1,ν′Ω≤Cfori=1,2,∂ρ^ε∂κW−1,ν′Ω≤εC.Proof.
Choosingϑ=0 as test function in inequality (11), we get(27)awε,wε+2∫ΩεgεTεDwεdsds3+∫ϖkεwεds≤fε,wε,from (16) and (17) we have(28)fε,wε≤εh∗∇wεLνΩfεLν′Ωε≤12Λ∗Ck∇wεLνΩεν+εh∗ν′ν′1/2Λ∗νCkν′/νfεLν′Ωεν′.
From (27) and (28), we deduce(29)awε,wε+2∫ΩεgεTεDwεdsds3+∫ϖkεwεds≤12Λ∗Ck∇wεLνΩεν+εh∗ν′ν′1/2Λ∗νCkν′/νfεLν′Ωεν′.
We multiply (29) by εν−1, we get(30)εν−1awε,wε+2∫Ωεg^T^D˜w^εdsdκ+∫ϖk^w^εds≤12Λ∗Ckεν−1∇wεLνΩεν+εν−1εh∗ν′ν′1/2Λ∗νCkν′/νfεLν′Ωεν′.
Asεν′fεLν′Ωεν′=ε1−νf^Lν′Ων′, we have(31)εν−1awε,wε+2∫Ωg^T^εD˜w^εdsdκ+∫ϖk^w^εds≤12Λ∗Ckεν−1∇wεLνΩεν+h∗ν′ν′1/2Λ∗νCkν′/νf^Lν′Ων′.
From Korn’s inequality and (15), there exists a constant Ck independent of ε, such that(32)12Λ∗Ckεν−1∇wεLνΩεν+2∫Ωg^T^D˜w^εdsdκ+∫ϖk^w^εds≤h∗ν′ν′1/2Λ∗νCkν′/νf^Lν′Ων′
From (32), we deduce (26), with C=1/2Λ∗CK−1h∗ν′/ν′1/2Λ∗νCkν′/νf^Lν′Ων′, and(33)εν−1∇wεLνΩεν=∇w^εLνΩν=∑i,j=12ε∂w^iε∂sjLνΩν+ε∂w^3ε∂κLνΩν+∑i=12∂w^iε∂κLνΩν+ε2∂w^3ε∂siLνΩν.
We prove (26) and (26) as in [14].


### 3.2. A Priori Estimates on the Temperature

In this subsection, we look for a priori estimates on the temperatureT^ε, for this we need to establish the following result:Theorem 2.
Assume that the assumptions of Theorem1 are satisfied. Moreover, assume that there exist K∗, K∗, such that(34)0<K∗≤K^≤K∗.Then, there exists a positive constantC1 independent of ε, such that(35)∂T^ε∂κW1,qΩ≤C1,∑i=12ε∂T^ε∂siW1,qΩ≤C1.Proof.
ChoosingΦ=ϑTε in (24), where ϑ is defined by(36)ϑt=ζsignt∫0tdτ1+tζ+1=signt1−11+tζ,
We obtain(37)ζK∗∫Ω∇T^ε21+T^εζ+1≤Λ∗εβ−2∫ΩD˜w^ενdsdκ+εβ−22g∗∫ΩD˜w^εdsdκ.
On the other hand,(38)∫ΩD˜w^ενdsdκ≤Cν∑i,j=12ε∂w^iε∂sjLνΩν+∑i=12∂w^iε∂κLνΩν+ε2∂w^3ε∂siLνΩν2+ε∂w^3ε∂κLνΩν,where Cν>0 depends only on ν.
Asν>1 and 0<ε<1 then εν−1≤1, so using this inequality and (26), we deduce(39)∫Ω∇εT^ε21+T^εζ+1≤1ζK∗Λ∗CνC+2g∗Cεβ−2.
Using Holder’s inequality with the exponents2/q and 2/2−q, for q<3/2, we obtain(40)∫Ω∇εT^εqdsdκ≤∫Ω∇εT^ε21+T^εζ+1q/2∫Ω1+T^εζ+1q/2−q2−q/2,using (39), we get(41)∫Ω∇εT^εqdsdκ≤1ζK∗Λ∗CνC+2g∗Cεβ−2q/2∫Ω1+T^εq∗2−q/2,where q∗=3q/3−q≥ζ+1q/2−q.
By (18) and (18), we find(42)∫Ω∇εT^εqdsdκ≤1ζK∗Λ∗CνC+2g∗Cεβ−2q/22q∗−12−q/2Ω2−q/2+∫ΩT^εq∗2−q/2.
Now using the Poincaré-Sobolev inequality, we have(43)∫ΩT^εq∗dsdκ1/q∗≤C′∇εT^εLqΩ≤1ζK∗Λ∗CνC+2g∗C1/2εβ/2−12q∗−12−q/2q×C′Ω2−q/2q+∫ΩT^εq∗2−q/2q.
On the other hand, for alla>0,b>0,c>0 and 0<s<t, we have the implication:(44)Ifat≤b+casthena≤max1,b+c1t−s.
Hence from (43) and (44) and the fact that 2−q/2<1, we deduce(45)∫ΩT^εq∗dsdκ≤ξε,where(46)ξ=max1,γεβ/2−11/q∗−2−q/2q−12−q/2=max1,γε3β/2−12−q,and(47)γ=1ζK∗Λ∗CνC+2g∗C1/22q∗−12−q/2qC′Ω2−q/2q+11/q∗−2−q/2q−1=1ζK∗Λ∗CνC+2g∗C1/22−q/2q2q∗−12−q/2qC′Ω2−q/2q+16.
Asβ=32−q/3−q, then 32−qβ/2−1<0. So for ε≤β−32−qβ/2−1−1, we obtain(48)ξ=γε3β/2−12−q≥1.
From (42) and (45), we get(49)εq∫Ω∇εT^εqdsdκ≤1ζK∗Λ∗CνC+2g∗Cq/22q∗−12−q/2Ω2−q/2εβ/2q+γε3β/2−12−q+β/2q,as 3β/2−12−q+β/2q=0 and β/2q>0, we obtain(50)εq∫Ω∇εT^εqdsdκ≤C1,where(51)C1=1ζK∗Λ∗CνC+2g∗Cq/22q∗−12−q/2Ω2−q/2+γ.where C1 is a constant independent of ε.Thus, we obtain (35) and (35)The following theorem states some immediate estimates of the limit of our initial problem.Theorem 3.
Under the same assumptions as in Theorem1 and Theorem 2, there exist w⋆=w1⋆,w2⋆∈V˜κ, ρ⋆∈L0r′Ω and T∗∈Πκ such that(52)w^iε⇀wi⋆,i=1,2weakly inV˜κ,(53)ε∂w^iε∂sj⇀0,i,j=1,2weakly inLνΩ,ε∂w^3ε∂κ⇀0,weakly inLνΩ,(54)ε2∂w^3ε∂si⇀0,i=1,2weakly inLνΩ,(55)εw^3ε⇀0,weaklyinLνΩ,(56)ρ^ε⇀ρ⋆,weakly inLν′Ω,ρ⋆depend only ofs,(57)T^ε⇀T∗weakly inΠκ,(58)∂T^ε∂si⇀0,i=1,2weaklyinLqΩ.Proof.
The convergence of (52) to (53) is a direct result of inequality (26). Using (26) and (26), we get (56), while (57) and (58) follow from (35).


## 3.1. A Priori Estimates on the Velocity and the Pressure

Theorem 1.
For all1<ν<2 and under assumptions (13) and (14) and (20), there exists a constant C>0 independent of ε such that(26)∑i,j=12ε∂w^iε∂sjLνΩν+∑i=12∂w^iε∂κLνΩν+ε2∂w^3ε∂siLνΩν+ε∂w^3ε∂κLνΩν≤C,∂ρ^ε∂siW−1,ν′Ω≤Cfori=1,2,∂ρ^ε∂κW−1,ν′Ω≤εC.Proof.
Choosingϑ=0 as test function in inequality (11), we get(27)awε,wε+2∫ΩεgεTεDwεdsds3+∫ϖkεwεds≤fε,wε,from (16) and (17) we have(28)fε,wε≤εh∗∇wεLνΩfεLν′Ωε≤12Λ∗Ck∇wεLνΩεν+εh∗ν′ν′1/2Λ∗νCkν′/νfεLν′Ωεν′.
From (27) and (28), we deduce(29)awε,wε+2∫ΩεgεTεDwεdsds3+∫ϖkεwεds≤12Λ∗Ck∇wεLνΩεν+εh∗ν′ν′1/2Λ∗νCkν′/νfεLν′Ωεν′.
We multiply (29) by εν−1, we get(30)εν−1awε,wε+2∫Ωεg^T^D˜w^εdsdκ+∫ϖk^w^εds≤12Λ∗Ckεν−1∇wεLνΩεν+εν−1εh∗ν′ν′1/2Λ∗νCkν′/νfεLν′Ωεν′.
Asεν′fεLν′Ωεν′=ε1−νf^Lν′Ων′, we have(31)εν−1awε,wε+2∫Ωg^T^εD˜w^εdsdκ+∫ϖk^w^εds≤12Λ∗Ckεν−1∇wεLνΩεν+h∗ν′ν′1/2Λ∗νCkν′/νf^Lν′Ων′.
From Korn’s inequality and (15), there exists a constant Ck independent of ε, such that(32)12Λ∗Ckεν−1∇wεLνΩεν+2∫Ωg^T^D˜w^εdsdκ+∫ϖk^w^εds≤h∗ν′ν′1/2Λ∗νCkν′/νf^Lν′Ων′
From (32), we deduce (26), with C=1/2Λ∗CK−1h∗ν′/ν′1/2Λ∗νCkν′/νf^Lν′Ων′, and(33)εν−1∇wεLνΩεν=∇w^εLνΩν=∑i,j=12ε∂w^iε∂sjLνΩν+ε∂w^3ε∂κLνΩν+∑i=12∂w^iε∂κLνΩν+ε2∂w^3ε∂siLνΩν.
We prove (26) and (26) as in [14].


## 3.2. A Priori Estimates on the Temperature

In this subsection, we look for a priori estimates on the temperatureT^ε, for this we need to establish the following result:Theorem 2.
Assume that the assumptions of Theorem1 are satisfied. Moreover, assume that there exist K∗, K∗, such that(34)0<K∗≤K^≤K∗.Then, there exists a positive constantC1 independent of ε, such that(35)∂T^ε∂κW1,qΩ≤C1,∑i=12ε∂T^ε∂siW1,qΩ≤C1.Proof.
ChoosingΦ=ϑTε in (24), where ϑ is defined by(36)ϑt=ζsignt∫0tdτ1+tζ+1=signt1−11+tζ,
We obtain(37)ζK∗∫Ω∇T^ε21+T^εζ+1≤Λ∗εβ−2∫ΩD˜w^ενdsdκ+εβ−22g∗∫ΩD˜w^εdsdκ.
On the other hand,(38)∫ΩD˜w^ενdsdκ≤Cν∑i,j=12ε∂w^iε∂sjLνΩν+∑i=12∂w^iε∂κLνΩν+ε2∂w^3ε∂siLνΩν2+ε∂w^3ε∂κLνΩν,where Cν>0 depends only on ν.
Asν>1 and 0<ε<1 then εν−1≤1, so using this inequality and (26), we deduce(39)∫Ω∇εT^ε21+T^εζ+1≤1ζK∗Λ∗CνC+2g∗Cεβ−2.
Using Holder’s inequality with the exponents2/q and 2/2−q, for q<3/2, we obtain(40)∫Ω∇εT^εqdsdκ≤∫Ω∇εT^ε21+T^εζ+1q/2∫Ω1+T^εζ+1q/2−q2−q/2,using (39), we get(41)∫Ω∇εT^εqdsdκ≤1ζK∗Λ∗CνC+2g∗Cεβ−2q/2∫Ω1+T^εq∗2−q/2,where q∗=3q/3−q≥ζ+1q/2−q.
By (18) and (18), we find(42)∫Ω∇εT^εqdsdκ≤1ζK∗Λ∗CνC+2g∗Cεβ−2q/22q∗−12−q/2Ω2−q/2+∫ΩT^εq∗2−q/2.
Now using the Poincaré-Sobolev inequality, we have(43)∫ΩT^εq∗dsdκ1/q∗≤C′∇εT^εLqΩ≤1ζK∗Λ∗CνC+2g∗C1/2εβ/2−12q∗−12−q/2q×C′Ω2−q/2q+∫ΩT^εq∗2−q/2q.
On the other hand, for alla>0,b>0,c>0 and 0<s<t, we have the implication:(44)Ifat≤b+casthena≤max1,b+c1t−s.
Hence from (43) and (44) and the fact that 2−q/2<1, we deduce(45)∫ΩT^εq∗dsdκ≤ξε,where(46)ξ=max1,γεβ/2−11/q∗−2−q/2q−12−q/2=max1,γε3β/2−12−q,and(47)γ=1ζK∗Λ∗CνC+2g∗C1/22q∗−12−q/2qC′Ω2−q/2q+11/q∗−2−q/2q−1=1ζK∗Λ∗CνC+2g∗C1/22−q/2q2q∗−12−q/2qC′Ω2−q/2q+16.
Asβ=32−q/3−q, then 32−qβ/2−1<0. So for ε≤β−32−qβ/2−1−1, we obtain(48)ξ=γε3β/2−12−q≥1.
From (42) and (45), we get(49)εq∫Ω∇εT^εqdsdκ≤1ζK∗Λ∗CνC+2g∗Cq/22q∗−12−q/2Ω2−q/2εβ/2q+γε3β/2−12−q+β/2q,as 3β/2−12−q+β/2q=0 and β/2q>0, we obtain(50)εq∫Ω∇εT^εqdsdκ≤C1,where(51)C1=1ζK∗Λ∗CνC+2g∗Cq/22q∗−12−q/2Ω2−q/2+γ.where C1 is a constant independent of ε.Thus, we obtain (35) and (35)The following theorem states some immediate estimates of the limit of our initial problem.Theorem 3.
Under the same assumptions as in Theorem1 and Theorem 2, there exist w⋆=w1⋆,w2⋆∈V˜κ, ρ⋆∈L0r′Ω and T∗∈Πκ such that(52)w^iε⇀wi⋆,i=1,2weakly inV˜κ,(53)ε∂w^iε∂sj⇀0,i,j=1,2weakly inLνΩ,ε∂w^3ε∂κ⇀0,weakly inLνΩ,(54)ε2∂w^3ε∂si⇀0,i=1,2weakly inLνΩ,(55)εw^3ε⇀0,weaklyinLνΩ,(56)ρ^ε⇀ρ⋆,weakly inLν′Ω,ρ⋆depend only ofs,(57)T^ε⇀T∗weakly inΠκ,(58)∂T^ε∂si⇀0,i=1,2weaklyinLqΩ.Proof.
The convergence of (52) to (53) is a direct result of inequality (26). Using (26) and (26), we get (56), while (57) and (58) follow from (35).


## 4. Study of the Limit Problem

In this section, we give both the equations satisfied byρ∗ and w∗ in Ω and the inequalities for the trace of the velocity w∗s,0 and the stress ∂w∗/∂κs,0 on ϖ.Theorem 4.
With the same assumptions of Theorem3, the solution w∗,ρ∗,T∗ satisfies the following relations:(59)∑i=12∫ΩΛ^T∗12ν2∑i=12∂wi∗∂κ2v−2/2∂wi∗∂κ∂ϑ^i−wi∗∂κdsdκ−∫Ωρ∗s∂ϑ^1∂s1+∂ϑ^2∂s2dsdκ+∫Ωg^T∗∂ϑ^∂κ−∂w∗∂κdsdκ+∫ϖk^ϑ^−w∗ds≥∑i=12∫Ωf^iϑ^i−wi∗dsdκ,∀ϑ^∈WΓ1∪ΓL,(60)∫ΩΛ^T∗12v/2∂w∗∂κνdsdκ+∫Ωg^T∗∂w∗∂κdsdκ+∫ϖk^w∗ds=∫Ωf^w∗dsdκ,(61)∫ΩΛ^T∗12ν2∂w∗∂κν−2∂w∗∂κ∂Φ^∂κdsdκ+∫Ωg^T∗∂Φ^∂κdsdκ+∫ϖk^Φ^ds≥∫Ωf^Φ^dsdκ,∀Φ^∈ΣK,and(62)−∂∂κK∂T∗∂κ=−α^T∗inLqΩ,(63)T∗=0inΓ1∪ΓL,∂T∗∂n=0inϖ,where(64)WΓ1∪ΓL=ϑ^=ϑ^1,ϑ^2∈W1,νΩ2,ϑ^=0onΓ1∪ΓL,and(65)Σ˜K=Φ^=Φ^1,Φ^2∈W1,νΩ2:Φ^satisfyD′.The proof of this theorem is based on the following lemma.Lemma 1 (Minty).
LetE be a Banach spaces, T:E⟶E′ a monotone and hemicontinuous operator, J:E⟶−∞,+∞ a proper and convex functional. Let u∈E and f∈E′. Then the following assertions are equivalent:(66)Tw;v−wE′×E+Jv−Jw≥f;v−wE′×E′,∀v∈E,Tv;v−wE′×E+Jv−Jw≥f;v−wE′×E,∀v∈E.Proof.
By using Minty’s Lemma1 and the fact that divw^ε=0 in Ω, then (24) is equivalent to(67)a0T^ε,ϑ^,ϑ^−w^ε−∑i=12ρ^ε,∂ϑ^i∂si−ρ^ε,∂ϑ^3∂κ+j0T^ε,ϑ^−j0T^ε,w^ε≥∑i=12∫Ωf^iϑ^i−w^iεdsdκ+∫Ωεf^3ϑ^3−w^3εdsdκ.
From (57), we have T^ε⟶T∗ almost everywhere. As Λ^ is continuous function on ℝ, then(68)Λ^T^ε⟶Λ^T∗.
Using Theorem4 and the fact j0 is convex and lower semicontinuous, liminfj0T^ε,w^ε≥j0T∗,w∗, we find(69)∑i=12∫Ω12Λ^T∗12∑2i=1∂ϑ^i∂κ2v−2/2∂ϑ^i∂κ∂ϑ^i−wi∗∂κdsdκ−∫Ωρ∗∂ϑ^1∂s1+∂ϑ^2∂s2dsdκ−∫Ωρ∗∂ϑ^3∂κdsdκ+j0T∗,ϑ^−j0T∗,w∗≥∑j=12∫Ωf^iϑ^i−wi∗dsdκ,and as ∫Ωρ∗∂ϑ^3/∂κdsdκ=0, because ρ∗ independent of κ, we get(70)∑i=12∫Ω12Λ^T∗12∑i=12∂ϑ^i∂κ2v−2/2∂ϑ^i∂κ∂ϑ^i−wi∗∂κdsdκ−∫Ωρ∗∂ϑ^1∂s1+∂ϑ^2∂s2dsdκ+j0T∗,ϑ^−j0T∗,w∗≥∑j=12∫Ωf^iϑ^i−wi∗dsdκ.
Using again Minty’s lemma for the second time, thus (61) is equivalent to (59). Now, we can choose ϑ^=2w∗ and ϑ^=0 respectively in (59), we find (60). For (61), we choose Φ^=ϑ^−w∗ for all Φ^∈ΣK. Passing to the limit on ε tend to 0 in (24) and using (52)–(54),(57)–(58) we get(71)∫ΩK^∂T∗∂κ∂Φ^∂κdsdκ=−∫Ωα^T∗Φ^dsdκ,∀Φ^∈WΓ1∪ΓL1,qΩ,by Green’s formula, we obtain(72)−∂∂κK^∂T∗∂κ=α^T∗inW−1,q′Ω.Theorem 5.
Let us set(73)∑∗=∑˜∗−∇ρ∗,∑˜∗=12ν2Λ^T∗∂w∗∂κν−2∂w∗∂κ+g^T∗π,then(74)−∂∂κ12Λ^T∗12∑i=12∂wi∗∂κ2v−2/2∂w∗∂κ+g^T∗∂w∗/∂κ∂w∗/∂κ=f^−∇ρ∗,inW−1,ν′Ω2.where π∈L∞Ω2 and πΩ,∞≤1.Proof.
If∂w∗/∂κ=0, from (73)we get Σ˜∗<g^T∗. For all Φ^∈∑K, choosing Φ^=Φ^, then Φ^=−Φ^ in (61), we obtain(75)Fk^Φ^,∂Φ^∂κ≤∫ϖk^Φ^ds+∫Ωg^T∗∂Φ^∂κdsdκ,where(76)Fk^Φ^,∂Φ^∂κ=∫ΩΛ^T∗12v/2∂w∗∂κν−2∂w∗∂κ∂Φ^∂κdsdκ−∫Ωf^Φ^dsdκ,
Now, utilising the Hanh-Banach theorem, then,∃χ,π∈L∞ϖ2×L∞Ω2, with χϖ,∞≤1πΩ,∞≤1, such that(77)Fk^Φ^,∂Φ^∂κ=−∫ϖχk^Φ^ds−∫Ωπg^T∗∂Φ^∂κdsdκ.
In particular, from (60) and (76), we get(78)∫ϖk^w∗ds+∫Ωg^T∗∂w∗∂κdsdκ=∫ϖχk^w∗ds+∫Ωπg^T∗∂w∗∂κdsdκ.
Also, from (76) and (77), we have(79)∫ΩΛ^T∗12v/2∂w∗∂κν−2∂w∗∂κ∂Φ^∂κdsdκ+∫ϖχk^Φ^ds+∫Ωπg^T∗∂Φ^∂κdsdκ−∫Ωf^Φ^dsdκ=0.
Next using (78), we have(80)∫ϖk^w∗−χw∗ds+∫∂w∗∂κ≠0g^T∗∂w∗∂κ−π∂w∗∂κdsdκ=0.
Asχϖ,∞≤1,πΩ,∞≤1, we deduce(81)∂w∗∂κ=π∂w∗∂κ,w∗−χw∗.
Hence, if∂w∗/∂κ≠0, by (73), we obtain(82)Σ˜∗=12ν2Λ^T∗∂w∗∂κν−2∂w∗∂κ+g^T∗∂w∗/∂κ∂w∗/∂κ.
In this case,Σ˜∗=1/2ν/2Λ^T∗∂w∗/∂κν−1+g^T∗>g^T∗; therefore, we can write(83)12v/2Λ^T∗∂w∗∂κν−2∂w∗∂κ=0,ifΣ˜∗≤α^,Σ˜∗−g^T∗∂w∗/∂κ∂w∗/∂κ,ifΣ˜∗>α^,for every Φ^∈ΣK and from (79), there exist p∗∈Lr′Ω2 such that(84)∫ΩΛ^T∗12v/2∂w∗∂κν−2∂w∗∂κ∂Φ^∂κdsdκ+∫ϖχk^Φ^ds+α^∫Ωπg^T∗∂Φ^∂κdsdκ−∫Ωf^Φ^dsdκ=−∫Ω∇ρ∗Φ^dsdκ.
Using (83) and (84) becomes(85)∫ΩΣ˜∗∂Φ^∂κdsdκ+∫ϖχk^Φ^ds=∫Ωf^Φ^dsdκ−∫Ω∇ρ∗Φ^dsdκ,and choosing Φ^∈W01,νΩ2 in (85), we find (74).The convergence of our problem towards the Reynolds equation given by the following result: Theorem5.(86)∫ϖh312∇ρ∗+F˜+∫0h∫0yΛ^T∗s,ζA∗s,ζ∂w∗s,ξ∂ξdξdy−h2∫0hg^T∗s,ζ∂u∗∂κs,ξdξ.∇ϑsds=0,for all ϑ∈W1,νϖ where(87)F˜s=∫0hFs,ydy−h2Fs,h,Fs,y=∫0h∫0ξf^s,tdtdξ,A∗s,ξ=1212∑2i=1∂w∗∂κs,ξ2ν−2/2.Proof.
To prove (86), we integrate twice (74) from 0 to κ, then taking κ=h, we obtain the requested result.The uniqueness of the limit velocity and pression are given in the following theorem:Theorem 6.
The solutionw∗,T∗,ρ∗ in Vκ×W−1,qΩ×L0ν′ϖ of equality (86) is unique.Proof.
Letw∗,1,T∗,1,ρ∗,1 and w∗,2,T∗,2,ρ∗,2 be two solutions of (59)–(63) and (86); then T∗,1 and T∗,2 solve (62)–(63), so T=T∗,1−T∗,2 satisfies the problem(88)−∂∂κK∂T∂κ=−α^T,T=0inΓ1∪ΓL,∂T∂κ=0onϖ.so T=0, thus T∗,1=T∗,2. Taking ϑ=u∗,2 and ϑ=w∗,1 respectively, as test function in (59) we get(89)∑2i=1∫ΩΛ^T∗12v/2∑2i=1∂wi∗,1∂κ2v−2/2∂wi∗,1∂κ∂∂κwi∗,1−wi∗,2dsdκ≤∑2i=1∫ΩΛ^T∗12v/2∑2i=1∂wi∗,2∂κ2v−2/2∂wi∗,2∂κ∂∂κwi∗,1−wi∗,2dsdκ.
Observe that for everys,y∈ℝn(90)sν−2s−yν−2y,s−y≥ν−1s+yν−2s−y2,∀1<ν≤2,we obtain(91)∫Ω∂w∗,1∂κ+∂w∗,2∂κν−2∂w∗,1∂κ−∂w∗,2∂κ2dsdκ=0,where ∂w∗,j/∂κ=∑i=12∂wi∗,j/∂κ21/2, j=1,2 Using Hölder’s inequality, we deduce(92)∫Ω∂∂κw∗,1−w∗,2νdsdκ≤C∫Ω∂w∗,1∂κ+∂w∗,2∂κν−2∂w∗,1∂κ−∂w∗,2∂κ2dsdκv/2×∫Ω∂w∗,1∂κ+∂w∗,2∂κνdsdκ2−v/2.
From (91) and (92), we obtain(93)∂∂κw∗,1−w∗,2LνΩ=0,using Poincare’s inequality, we deduce(94)w∗,1−w∗,2Vκ=0.
Finally, to prove the uniqueness of the pressure, we use (86) with the two pressures ρ∗,1 and ρ∗,2, we find(95)∫ϖh3/12∇ρ∗,1−ρ∗,2∇ϑds=0.
Takingϑ=ρ∗,1−ρ∗,2, and by Poincare’s inequality, we deduce ρ∗,1−ρ∗,2Lν′ϖ=0. So ρ∗,1=ρ∗,2.


## 5. Conclusions

This work studies the asymptotic analysis of an incompressible Herschel–Bulkley fluid in a thin domain with Tresca boundary conditions. The yield stress and the constant viscosity are assumed to vary with respect to the thin layer parameter. Firstly, the problem statement and variational formulation are formulated. We then obtained the estimates for the velocity field and the pressure independently of the parameter. Finally, we gave a specific Reynolds equation associated with variational inequalities and proved the uniqueness [15–23].



---

*Source: 5610938-2022-05-20.xml*