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 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., [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 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
from (63) and (64), we deduce that
Finally, to prove the uniqueness of the pressure, we use equation (59) with the two pressures and , we find
Taking and using Poincaré inequality, we obtain . Then, .
5. Conclusion
In this work, the asymptotic behavior of an incompressible Herschel-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 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]).
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare that they have no competing interests.
Acknowledgments
The fourth author extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant (R.G.P.2/1/42).