# Asymptotic Behavior of Solutions to Free Boundary Problem with Tresca Boundary Conditions

Authors: Abdelkader Saadallah; Nadhir Chougui; Fares Yazid; Mohamed Abdalla; Bahri Belkacem Cherif; Ibrahim Mekawy
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we study the asymptotic behavior of an incompressible Herschel-Bulkley fluid in a thin domain with Tresca boundary conditions. We study the limit when theε tends to zero, we prove the convergence of the unknowns which are the velocity and the pressure of the fluid, and we obtain the limit problem and the specific Reynolds equation.


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## Body

## 1. Introduction

In 1926, the model of Herschel-Bulkley fluid introduced is called a non-Newtonian fluid, whose flow properties differ in any way from those of any Newtonian fluids. There are many phenomena in nature and industry exhibiting the behavior of the Herschel-Bulkley fluid medium and has been used in various publications to describe the flow of metals, plastic solids, and some polymers. The literature concerning this topic is extensive (see e.g., [1–14]). Further, let us mention the works which is realized by many authors in this area, for example, (see [2, 4, 9, 10, 13–21]).This paper is to discuss the asymptotic behavior of steady flow of Herschel-Bulkley fluid in a three-dimensional thin layer with Tresca boundary conditions.The paper is organized as follows. In Section2, we introduce some notations, preliminaries, and the mechanical problem of the steady flow of Herschel-Bulkley fluid in a three-dimensional thin layer. In Section 3, we investigate some estimates and convergence theorem. To this aim, we use the change of variable x3/ε, to transform the initial problem posed in the domain Ωε into a new problem posed on a fixed domain Ω independent of the parameter ε. Finally, the a priori estimate allows us to pass to the limit when ε tends to zero, and we prove the convergence results and limit problem with a specific weak form of the Reynolds equation and two-dimensional constitutive equation of the model flow.


## 2. Problem Statement and Variational Formulation

Letω be fixed region in plan x=x1,x2∈ℝ2. We assume that ω has a Lipschitz boundary and is the bottom of the fluid domain. The upper surface Γ1ε is defined by x3=εhx where 0<ε<1 is a small parameter that will tend to zero and h a smooth bounded function such that 0<h∗<hx<h∗ for all x,0∈ω and ΓLε the lateral surface. We denote by Ωε the domain of the flow:
(1)Ωε=x,x3∈ℝ3:x,0∈ω,0<x3<εhx.The boundary ofΩε is Γε. We have Γε=Γ¯1ε∪Γ¯Lε∪ω¯ where Γ¯Lε is the lateral boundary.
(i)
The law of conservation of momentum is defined by(2)uε∇uε=divσε+fεinΩε,where divσε=σij,jε and fε=fiε1≤i≤3 denote the body forces.(ii)
The stress tensorσε is decomposed as follows
(3)σijε=σ~ijε−pεδij,σ~ε=αεDuεDuε+μDuεr−2DuεifDuε≠0,σ~ε≤αεifDuε=0.where αε≥0 is the yield stress, μ>0 is the constant viscosity, uε is the velocity field, pε is the pressure, δij is the Kronecker symbol, 1<r≤2 and Duε=1/2∇uε+∇uεT. For any tensor D=dij, the notation D represents the matrix norm: D=∑i,jdijdij1/2.(iii)
The incompressibility equation(4)divuε=0inΩε.Our boundary conditions is described as(iv)
At the surfaceΓ1ε∪ΓLε, we assume that
(5)uε=0.(v)
Onω, there is a no-flux condition across ω so that
(6)uε×n=0.(vi)
The tangential velocity onω is unknown and satisfies Tresca boundary conditions:
(7)στε<kεuτε=0στε=kε∃λ≥0,uτε=−λστεinω.Here,kε is the friction yield coefficient and . is the Euclidean norm in ℝ2; n=n1,n2,n3 is the unitoutward normal to Γ1ε, and
(8)unε=uε.n=uiε.ni,uτiε=uiε−unεni,σnε=σ.nn=σijεninj,στiε=σijεnj−σnεni.In order to, we observe that(9)Kε=φ∈W1,rΩε3:φ=0onΓ1ε∪ΓLε,φ.n=0onω,Kdivε=φ∈Kε:divφ=0,L0r′Ωε=q∈Lr′Ωε:∫Ωεqdxdx3=0.A formal application of Green’s formula, using (1)–(6), leads to the following weak formulation:Find a velocity fielduε∈Kdivε andpε∈L0r′Ωε, 1/r+1/r′=1 such that:
(10)auε,φ−uε+Buε,uε,φ−uε−pε,divφ+jφ−juε≥fε,φ−uε,for all φ∈Kε, where
(11)auε,φ−uε=μ∫ΩεDuεr−2DuεDvεdxdx3,Buε,uε,v=∑i=13∫Ωεuiε∂uε∂xivdxdx3,pε,divφ=∫Ωεpεdivφdxdx3,jv=∫ωkεvdx+αε∫ΩεDvdxdx3,fε,v=∫Ωεfεvdxdx3=∑i=13∫Ωεfiεvidxdx3.As in [6, 8], we can show that this variational problem has a unique solution.Now, we state some the following results (see, [15]).
(12)∇uεLrΩε≤CDuεLrΩεKorninequality,(13)uεLrΩε≤εh∗∂uε∂zLrΩεfori=1,2Poincare′inequality,(14)ab≤arr+br′r′,∀a,b∈ℝ2Younginequality.


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

Here, we apply the technique of scaling inΩε on the coordinate x3. With the variables z=x3/ε, we get
(15)Ω=x,z∈ℝ3:x,0∈ω,0<z<hx.Next, we denote byΓ=Γ¯1∪Γ¯L∪ω¯ its boundary, then, we define the following functions in Ω:
(16)u^iεx,z=uiεx,x3,i=1,2,u^3εx,z=ε−1u3εx,x3,p^εx,z=εrpεx,x3.Now, we assume that(17)f^x,z=εrfεx,x3,α^=εr−1αε,k^=εr−1kε,and we consider the sets
(18)KΩ=φ^∈W1,rΩ3:φ^=0onΓ1∪ΓL;φ^.n=0onω,KdivΩ=φ^∈KΩ:divφ^=0,Vz=φ^∈LrΩ2;∂φ^i∂z∈LrΩ:φ^=0onΓ1∪ΓL,V~z=φ^∈Vz:φ^satisfyD′,where the condition D′ is given by
(19)D′∫ωφ^1∂θ∂x1+φ^2∂θ∂x2dxdz=0,forallφ^∈LrΩ2andθ∈C0∞Ω.By injecting the new data, unknown factors in (10) and after multiplication by εr−1, we deduce that
(20)a0u^ε,φ^−u^ε+B0u^ε,u^ε,φ^−p^ε,divφ^−u^ε+j0φ^−j0u^ε≥f^,φ^−u^ε,∀φ^∈KΩ,where
(21)a0u^ε,φ^−u^ε=∑2i,j=1∫Ωε2μD~u^εr−212∂u^iε∂xj+∂u^jε∂xi∂φ^i−u^iε∂xjdxdz+∑2i=1∫ΩμD~u^εr−212∂u^iε∂z+ε2∂u^3ε∂xi∂φ^i−u^iε∂zdxdz+∫ΩμD~u^εr−2ε2∂u^3ε∂z∂φ^3−u^3ε∂zdxdz+∑2j=1∫Ωε2μD~u^εr−212ε2∂u^3ε∂xj+∂u^jε∂z∂φ^3−u^3ε∂xjdxdz,p^ε,divφ^−u^ε=∫Ωεp^εdivφ^−u^εdxdz,B0u^ε,u^ε,φ^=∑i,j=12∫Ωε2u^iε∂u^jε∂xiφ^dxdz+∑i=12∫Ωε4u^iε∂u^3ε∂xiφ^dxdz+∑j=12∫Ωε2u^3ε∂u^jε∂zφ^jdxdz+∫Ωε4u^3ε∂u^3ε∂zφ^3dxdz,j0φ^=α^∫ΩD~φ^dxdz+∫ωk^φ^dx,f^ε,φ^−u^ε=∑2j=1∫Ωf^iφ^i−u^iεdxdz+∫Ωεf^3φ^3−u^3εdxdz,D~u^ε=14∑2i,j=1ε2∂u^iε∂xj+∂u^jε∂xi2+12∑2i=1∂u^iε∂z+ε2∂u^3ε∂xi2+ε2∂u^3ε∂z21/2.We now establish the estimates for the velocity fieldu^ε and the pressure p^ε in Ω.Theorem 1.
Letu^ε,p^ε∈KdivΩ×L0r′Ω be the solution of variational problem (20), then there exists a constant C>0 independent of ε such that:
(22)∑i,j=12ε∂u^iε∂xjLrΩr+ε∂u^3ε∂zLrΩr+∑i=12∂u^iε∂zLrΩr+ε2∂u^3ε∂xiLrΩr≤C.Proof.
Choosingφ=0 as test function in inequality (10), we get
(23)auε,uε+Buε,uε,uε+αε∫ΩεDuεdxdx3+∫ωkεuεdx≤fε,uε,and because Buε,uε,uε=0, we obtain
(24)auε,uε+αε∫ΩεDuεdxdx3+∫ωkεuεdx≤fε,uε.
Using now (13) and (14) will yield after some algebra
(25)fε,uε≤εh∗∇uεLrΩfεLr′Ωε≤12μCk∇uεLrΩεr+εh∗r′r′1/2μrCkr′/rfεLr′Ωεr′.
From (24) and (25), we deduce that
(26)auε,uε+αε∫ΩεDuεdxdx3+∫ωkεuε−sdx≤12μCk∇uεLrΩεr+εh∗r′r′1/2μrCkr′/rfεLr′Ωεr′.
We multiply (26) by εr−1, we get
(27)εr−1auε,uε+α^∫ΩεD~u^εdxdz+∫ωk^u^εdx≤12μCkεr−1∇uεLrΩεr+εr−1εh∗r′r′1/2μrCkr′/rfεLr′Ωεr′.
Now, sinceεr′fεLr′Ωεr′=ε1−rf^Lr′Ωr′, it follows that
(28)εr−1auε,uε+α^∫ΩD~u^εdxdz+∫ωk^u^εdx≤12μCKεr−1∇uεLrΩεr+h∗r′r′1/2μrCkr′/rf^Lr′Ωr′.
According to Korn’s inequality and (28), such that CK independent of ε, we have
(29)12μCKεr−1∇uεLrΩεr+α^∫ΩD~u^εdxdz+∫ωk^u^εdx≤h∗r′r′1/2μrCkr′/rf^Lr′Ωr′.
Using (29), we deduce (22), with C=1/2μCK−1h∗r′/r′1/2μrCkr′/rf^Lr′Ωr′, and
(30)εr−1∇uεLrΩεr=∇u^εLrΩr=∑i,j=12ε∂u^iε∂xjLrΩr+ε∂u^3ε∂zLrΩr+∑i=12∂u^iε∂zLrΩr+ε2∂u^3ε∂xiLrΩr.Theorem 2.
Under the conditions in Theorem (1), there exists a constant C′>0 independent of ε such that
(31)∂p^ε∂xiW−1,r′Ω≤C′fori=1,2,(32)∂p^ε∂zW−1,r′Ω≤εC′.Proof.
To get the first estimate on the pressure in (31)–(32), we choose in (20), φ^=u^ε+ψ,ψ∈W01,rΩ3, to obtain
(33)a0u^ε,ψ+B0u^ε,u^ε,ψ−p^ε,divψ+α^∫ΩD~u^ε+ψdxdz−α^∫ΩD~u^εdxdz≥f^ε,ψ,p^ε,divψ≤a0u^ε,ψ+B0u^ε,u^ε,ψ+α^∫ΩD~u^ε+ψdxdz−α^∫ΩD~u^εdxdz−f^ε,ψ.
Keeping in mind thatD~u^ε+ψ≤2D~u^ε+D~ψ, it follows that
(34)p^ε,divψ≤a0u^ε,ψ+B0u^ε,u^ε,ψ+α^2∫ΩD~ψdxdz+2−1α^∫ΩD~u^εdxdz−∫Ωf^ψdxdz.
Using Hölder formula, we get(35)p^ε,divψ≤μDu^εLrΩr/r′ψW1,rΩ3+u^εW1,rΩ32ψW1,rΩ3+α^2Ω1/r′ψW1,rΩ3+2−1α^D~u^εLrΩ3+f^Lr′Ω3ψW1,rΩ3.
By similar arguments, we choose in (20) φ^=u^ε−ψ and ψ∈W01,rΩ3 to obtain
(36)−p^ε,divψ≤μDu^εLrΩr/r′ψW1,rΩ3+Cu^εW1,rΩ32ψW1,rΩ3+α^2Ω1/r′ψW1,rΩ3+2−1α^D~u^εLrΩ3+f^Lr′Ω3ψW1,rΩ3.
We combine now (35) and (36) to see that
(37)p^ε,divψ≤μDu^εLrΩr/r′ψW1,rΩ3+u^εW1,rΩ32ψW1,rΩ3+α^2Ω1/r′ψW1,rΩ3+2−1α^D~u^εLrΩ3+f^Lr′Ω3ψW1,rΩ3.
Next, fori=1,2, we choose ψ=ψ1,0,0 then ψ=0,ψ2,0 in the inequality (37) and using (22), we find
(38)∫Ω∂p^ε∂xiψdxdz≤C1+α^2Ω1/r′+f^iLr′Ω3ψW1,rΩ3+2−1α^C,where Ω=mesΩ. Then, (31) holds for i=1,2.
To get (32), we take ψ=0,0,ψ3 in the inequality (37) to see that
(39)1ε∫Ω∂p^ε∂zψdxdz≤C1+α^2Ω1/r′+f^3Lr′ΩψW1,rΩ3+2−1α^C.
The question which naturally arises is to know what will be the asymptotic behavior of the fluid when the thickness of the thin film is very small. Mathematically, it is about knowing: do the speed field and the pressure admit a limit whenε tends towards zero and what is the limit problem who should check this limit?
The answer to the first question is given in Theorem (3). However, the answer to the second question will be dealt with in Theorems (4), (7), and (8).Theorem 3.
Under the same assumptions as in Theorem (1) and Theorem (2), there exist u⋆=u1⋆,u2⋆∈V~z and p⋆∈L0r′Ω such that:
(40)u^iε⇀ui⋆,i=1,2weaklyinV~z,(41)ε∂u^iε∂xj⇀0,i,j=1,2weaklyinLrΩ,(42)ε∂u^3ε∂z⇀0,weaklyinLrΩ,(43)ε2∂u^3ε∂xi⇀0,i=1,2weaklyinLrΩ,(44)εu^3ε⇀0,weaklyinLrΩ,(45)p^ε⇀p⋆,weaklyinLr′Ω,p⋆dependonlyofx.Proof.
By Theorem (1), there exists a constant C independent of ε such that
(46)∂u^iε∂zLrΩ≤C,i=1,2,and using Poincare’s inequality, we deduce that
(47)u^iεLrΩ≤h∗∂u^iε∂zLrΩ,i=1,2.that is to say, u^iε is bounded in Vz, i=1,2, this implies the existence of u^i∗ in Vz such that u^iεconverges to u^i∗ in LrΩ. The same, the inequality (22), we give
(48)ε∂u^iε∂xjLrΩ≤C,so ε∂u^iε/∂xj converges to ∂u^i∗/∂xj and as u^iεLrΩ≤C, then ∂u^iε/∂xj converges weakly to ∂u^i∗/∂xj; which gives the converges weakly of ∂u^iε/∂xj to 0 in LrΩ.
Well thanks to the inequality:ε2∂u^3ε/∂xjLrΩ≤C, we have the convergence ε2∂u^3ε/∂xj⟶∂u^3∗/∂xj and ε∂u^3ε/∂xj⟶∂u^3∗/∂xj. This shows that ∂u^3ε/∂xj converges weakly to 0 in LrΩ. Finally, using (31) and (32), we get (45).


## 4. Study of the Limit Problem

In this section, we give both the equations satisfied byp∗ and u∗ in Ω and the inequalities for the trace of the velocity u∗x,0 and the stress ∂u∗/∂zx,0 on ω.Theorem 4.
With the same assumptions of Theorem (3) the solution u∗,p∗ satisfying the following relations
(49)μ∑2i=1∫Ω1212∑2i=1∂ui∗∂z2r−2/2∂ui∗∂z∂φ^i−ui∗∂zdxdz−∫Ωp∗x∂φ^1∂x1+∂φ^2∂x2dxdz+α^22∫Ω∂φ^∂z−∂u∗∂zdxdz+∫ωk^∣φ^∣−∣u∗∣dx≥∑2i=1∫Ωf^iφ^i−ui∗dxdz,∀φ^∈WΓ1∪ΓL,where
(50)WΓ1∪ΓL=φ^=φ^1,φ^2∈W1,rΩ2,φ^=0onΓ1∪ΓL.
The proof of this theorem is based on the following lemma.Lemma 5 (Minty).
LetE be a Banach spaces, T:E⟶E′ a monotone and hemicontinuous operator, and J:E⟶−∞,+∞ a proper and convex functional. Let u∈E and f∈E′. Then, the followings assertions are equivalent:
(51)Tu;v−uE′×E+Jv−Ju≥f;v−uE′×E′,∀v∈E,Tv;v−uE′×E+Jv−Ju≥f;v−uE′×E,∀v∈E.Proof.
By usingMinty’s Lemma (5) and the fact that divu^ε=0 in Ω, then (20) is equivalent to
(52)a0φ^,φ^−u^ε+B0φ^,φ^,φ^−u^ε−∑i=12p^ε,∂φ^i∂xi−p^ε,∂φ^3∂z+j0φ^−j0u^ε≥∑i=12∫Ωf^iφ^i−u^iεdxdz+∫Ωεf^3φ^3−u^3εdxdz.
Using Theorem (3) and the fact that j0 is convex and lower semicontinuous, liminfj0u^ε≥j0u∗, we obtain
(53)μ∑i=12∫Ω1212∑2i=1∂φ^i∂z2r−2/2∂φ^i∂z∂φ^i−ui∗∂zdxdz−∫Ωp∗∂φ^1∂x1+∂φ^2∂x2dxdz−∫Ωp∗∂φ^3∂zdxdz+j0φ^−j0u∗≥∑j=12∫Ωf^iφ^i−ui∗dxdz,and as ∫Ωp∗∂φ^3/∂zdxdz=0, because p∗ independent of z, we deduce that
(54)μ∑i=12∫Ω1212∑i=12∂φ^i∂z2r−2/2∂φ^i∂z∂φ^i−ui∗∂zdxdz−∫Ωp∗∂φ^1∂x1+∂φ^2∂x2dxdz+j0φ^−j0u∗≥∑j=12∫Ωf^iφ^i−ui∗dxdz.
Using againMinty’s Lemma for the second time, thus, (54) is equivalent to (49).Theorem 6.
The variational inequality (49) is equivalent the following system
(55)μ∫Ω12r/2∂u∗∂zrdxdz+α^∫Ω∂u∗∂zdxdz+∫ωk^u∗dx=∫Ωf^u∗dxdz,μ∫Ω12r/2∂u∗∂zr−2∂u∗∂z∂ψ^∂zdxdz+α^∫Ω∂ψ^∂zdxdz+∫ωk^ψ^dx≥∫Ωf^ψ^dxdz,∀ψ^∈ΣK,where
(56)ΣK=ψ^=ψ^1,ψ^2∈H1Ω2:ψ^satisfyD′.Theorem 7.
Let us set(57)σ∗=σ~∗−∇p∗andσ~∗=12r/2μ∂u∗∂zr−2∂u∗∂z+α^π,then
(58)−∂∂z12μ12∑i=12∂ui∗∂z2r−2/2∂u∗∂z+22α^∂u∗/∂z∂u∗/∂z=f^−∇p∗,in W−1,r′Ω2, where π∈L∞Ω2 and π∞,Ω≤1.Proof.
For the proof of this theorem, we follow the same steps as in [13] (Theorem (9)).Theorem 8.
Under the assumptions of preceding theorems,u⋆ and p⋆ satisfy the following inequality
(59)∫ωh312∇p∗+F~+μ∫0h∫0yA∗x,ζ∂u∗x,ξ∂ξdξdy+α^22∫0h∫0y∂u∗∂zx,ξdξdy−hμ2∫0hA∗x,ζ∂u∗x,ξ∂ξdξ−α^2h4∫0h∂u∗∂zx,ξdξ.∇φxdx=0,
for allφ∈W1,rω, where
(60)F~x=∫0hFx,ydy−h2Fx,h,Fx,y=∫0h∫0ξf^x,tdtdξ,A∗x,ξ=1212∑i=12∂u∗∂zx,ξ2r−2/2.Proof.
The proof can be found in [13].
The uniqueness of the limit velocity and pression are given by the following theorem.Theorem 9.
The solutionu∗,p∗ in Vz×L0r′ω of inequality (49) is unique.Proof.
Letu∗,1,p∗,1 and u∗,2,p∗,2 be two solutions of (49); taking φ=u∗,2 and φ=u∗,1, respectively, as test function in (59), we get
(61)μ∑i=12∫Ω12r/2∑i=12∂ui∗,1∂z2r−2/2∂ui∗,1∂z∂∂zui∗,1−ui∗,2dxdz≤μ∑i=12∫Ω12r/2∑i=12∂ui∗,2∂z2r−2/2∂ui∗,2∂z∂∂zui∗,1−ui∗,2dxdz.
Keeping in mind that for everyx,y∈ℝn(62)xr−2x−yr−2y,x−y≥r−1x+yr−2x−y2,∀1<r≤2,we obtain
(63)∫Ω∂u∗,1∂z+∂u∗,2∂zr−2∂u∗,1∂z−∂u∗,2∂z2dxdz=0,where ∂u∗,j/∂z=∑2i=1∂ui∗,j/∂z21/2, j=1,2.
UsingHölder’s inequality, we deduce
(64)∫Ω∂∂zu∗,1−u∗,2rdxdz≤C∫Ω∂u∗,1∂z+∂u∗,2∂zr−2∂u∗,1∂z−∂u∗,2∂z2dxdzr/2×∫Ω∂u∗,1∂z+∂u∗,2∂zrdxdz2−r/2,from (63) and (64), we deduce that u∗,1−u∗,2Vz=0.
Finally, to prove the uniqueness of the pressure, we use equation (59) with the two pressures p∗,1 and p∗,2, we find
(65)∫ωh312∇p∗,1−p∗,2∇φdx=0.
Takingφ=p∗,1−p∗,2 and using Poincaré inequality, we obtain p∗,1−p∗,2Lr′ω=0. Then, p∗,1=p∗,2.


## 5. Conclusion

In this work, the asymptotic behavior of an incompressibleHerschel-Bulkley fluid in a thin domain with Tresca boundary conditions is considered, where we prove the convergence of the unknowns which are the velocity and the pressure of the fluid when the ε tends to zero. In addition, the limit problem and the specific Reynolds equation are studied. The aim of our next study is to complement and improve our current results, which is to weaken the hypotheses of fixed point theory by using the following concepts: weak contractual applications, applications that verify some characteristics, normal global operating system, and closed graph applications. We will state and give conclusions on the fixed point theory using the concepts mentioned in recent references. On the other hand, we will study the uniform convergent behavior of a series of designations, of Banach space towards itself, with fixed points, or nearly fixed points in order to show some results in the fixed point theory applied on the problem studied in this paper. With the help of these results, we will introduce some applications and provide some examples and some notes regarding weak contraction mappings. In addition, we will mention and give some results of the fixed point theory of weak contraction mappings by using the studied algorithm in ([15, 22–35]).



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