Journal of Function Spaces

Journal of Function Spaces / 2021 / Article
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Unique and Non-Unique Fixed Points and their Applications

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Research Article | Open Access

Volume 2021 |Article ID 9983950 |

Abdelkader Saadallah, Nadhir Chougui, Fares Yazid, Mohamed Abdalla, Bahri Belkacem Cherif, Ibrahim Mekawy, "Asymptotic Behavior of Solutions to Free Boundary Problem with Tresca Boundary Conditions", Journal of Function Spaces, vol. 2021, Article ID 9983950, 9 pages, 2021.

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

Academic Editor: Liliana Guran
Received24 Mar 2021
Revised06 Apr 2021
Accepted12 Apr 2021
Published24 Apr 2021


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 the 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.

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., [114]). Further, let us mention the works which is realized by many authors in this area, for example, (see [2, 4, 9, 10, 1321]).

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 Section 2, 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 , to transform the initial problem posed in the domain into a new problem posed on a fixed domain independent of the parameter . Finally, the a priori estimate allows us to pass to the limit when 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

Let be fixed region in plan . We assume that has a Lipschitz boundary and is the bottom of the fluid domain. The upper surface is defined by where is a small parameter that will tend to zero and a smooth bounded function such that for all and the lateral surface. We denote by the domain of the flow:

The boundary of is . We have where is the lateral boundary. (i)The law of conservation of momentum is defined by where and denote the body forces.(ii)The stress tensor is decomposed as follows where is the yield stress, is the constant viscosity, is the velocity field, is the pressure, is the Kronecker symbol, and . For any tensor , the notation represents the matrix norm: .(iii)The incompressibility equation

Our boundary conditions is described as (iv)At the surface , we assume that (v)On , there is a no-flux condition across so that (vi)The tangential velocity on is unknown and satisfies Tresca boundary conditions:

Here, is the friction yield coefficient and is the Euclidean norm in ; is the unitoutward normal to , and

In order to, we observe that

A formal application of Green’s formula, using (1)–(6), leads to the following weak formulation:

Find a velocity field and, such that: for all , where

As in [6, 8], we can show that this variational problem has a unique solution.

Now, we state some the following results (see, [15]).

3. Change of the Domain and Study of Convergence

Here, we apply the technique of scaling in on the coordinate . With the variables we get

Next, we denote by its boundary, then, we define the following functions in :

Now, we assume that and we consider the sets where the condition is given by

By injecting the new data, unknown factors in (10) and after multiplication by , we deduce that where

We now establish the estimates for the velocity field and the pressure in

Theorem 1. Let be the solution of variational problem (20), then there exists a constant independent of such that:

Proof. Choosing as test function in inequality (10), we get and because , we obtain Using now (13) and (14) will yield after some algebra From (24) and (25), we deduce that We multiply (26) by , we get Now, since , it follows that According to Korn’s inequality and (28), such that independent of , we have Using (29), we deduce (22), with , and

Theorem 2. Under the conditions in Theorem (1), there exists a constant independent of such that

Proof. To get the first estimate on the pressure in (31)–(32), we choose in (20), , to obtain Keeping in mind that , it follows that Using Hölder formula, we get By similar arguments, we choose in (20) and to obtain We combine now (35) and (36) to see that Next, for , we choose then in the inequality (37) and using (22), we find where . Then, (31) holds for .
To get (32), we take in the inequality (37) to see that 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 when 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 and such that:

Proof. By Theorem (1), there exists a constant independent of such that and using Poincare’s inequality, we deduce that that is to say, is bounded in , , this implies the existence of in such that converges to in . The same, the inequality (22), we give so converges to and as , then converges weakly to ; which gives the converges weakly of to in .
Well thanks to the inequality: , we have the convergence and This shows that converges weakly to in . Finally, using (31) and (32), we get (45).

4. Study of the Limit Problem

In this section, we give both the equations satisfied by and in and the inequalities for the trace of the velocity and the stress on .

Theorem 4. With the same assumptions of Theorem (3) the solution satisfying the following relations where The proof of this theorem is based on the following lemma.

Lemma 5 (Minty). Let be a Banach spaces, a monotone and hemicontinuous operator, and a proper and convex functional. Let and . Then, the followings assertions are equivalent:

Proof. By using Minty’s Lemma (5) and the fact that in then (20) is equivalent to Using Theorem (3) and the fact that is convex and lower semicontinuous, , we obtain and as , because independent of , we deduce that Using again Minty’s Lemma for the second time, thus, (54) is equivalent to (49).

Theorem 6. The variational inequality (49) is equivalent the following system where

Theorem 7. Let us set then in , where and

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, and satisfy the following inequality for all where

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 solution in of inequality (49) is unique.

Proof. Let and be two solutions of (49); taking and , respectively, as test function in (59), we get Keeping in mind that for every we obtain where , .
Using Hölder’s inequality, we deduce