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

Applied Math Lab, Department of Mathematics, Setif 1 University, 19000, Algeria Laboratory of Pure and Applied Mathematics, University of Laghouat, Algeria Mathematics Department, College of Science, King Khalid University, Abha 61413, Saudi Arabia Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt Preparatory Institute for Engineering Studies in Sfax, Tunisia Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia

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 x 3 /ε, to transform the initial prob-lem posed in the domain Ω ε 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 twodimensional constitutive equation of the model flow.

Problem Statement and Variational Formulation
Let ω be fixed region in plan x = ðx 1 , x 2 Þ ∈ ℝ 2 . We assume that ω has a Lipschitz boundary and is the bottom of the fluid domain. The upper surface Γ ε 1 is defined by x 3 = εhðxÞ where ð0 < ε < 1Þ is a small parameter that will tend to zero and h a smooth bounded function such that 0 < h * < hðxÞ < h * for all ðx, 0Þ ∈ ω and Γ ε L the lateral surface. We denote by Ω ε the domain of the flow: The boundary of Ω ε is Γ ε . We have (i) The law of conservation of momentum is defined by where div ðσ ε Þ = ðσ ε ij,j Þ and f ε = ð f ε i Þ 1≤i≤3 denote the body forces.
(ii) The stress tensor σ ε is decomposed as follows where α ε ≥ 0 is the yield stress, μ > 0 is the constant viscosity, u ε is the velocity field, p ε is the pressure, δ ij is the Kronecker symbol, 1 < r ≤ 2 and Dðu ε Þ = 1 /2ð∇u ε + ð∇u ε Þ T Þ. For any tensor D = ðd ij Þ, the notation jDj represents the matrix norm: jDj = Our boundary conditions is described as (iv) At the surface Γ ε 1 ∪ Γ ε L , we assume that (v) On ω, there is a no-flux condition across ω so that (vi) The tangential velocity on ω is unknown and satisfies Tresca boundary conditions: Here, k ε is the friction yield coefficient and j:j is the Euclidean norm in ℝ 2 ; n = ðn 1 , n 2 , n 3 Þ is the unitoutward normal to Γ ε 1 , 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 u ε ∈ K ε div and p ε ∈ L r ′ 0 ðΩ ε Þ, ð1/r + 1/ r′ = 1Þ such that: As in [6,8], we can show that this variational problem has a unique solution. Now, we state some the following results (see, [15]).
ab ≤ a r r

Change of the Domain and Study of Convergence
Here, we apply the technique of scaling in Ω ε on the coordinate x 3 . With the variables z = x 3 /ε, we get Next, we denote by Γ = Γ 1 ∪ Γ L ∪ ω its boundary, then, we define the following functions in Ω: Now, we assume that and we consider the sets where the condition ðD ′ Þ is given by By injecting the new data, unknown factors in (10) and after multiplication by ε r−1 , we deduce that where We now establish the estimates for the velocity fieldû ε and the pressurep ε in Ω: be the solution of variational problem (20), then there exists a constant C > 0 independent of ε such that: Proof. Choosing φ = 0 as test function in inequality (10), we get 3 Journal of Function Spaces and because Bðu ε , u ε , u ε Þ = 0, we obtain Using now (13) and (14) will yield after some algebra From (24) and (25), we deduce that We multiply (26) by ε r−1 , we get According to Korn's inequality and (28), such that C K independent of ε, we have Using (29), we deduce (22), Theorem 2. Under the conditions in Theorem (1), there exists a constant C ′ > 0 independent of ε such that ∂p ε ∂z Proof. To get the first estimate on the pressure in (31)-(32), we choose in (20) Keeping in mind that jDðû ε + ψÞj ≤ ffiffi ffi 2 p ðjDðû ε Þj + jDð ψÞjÞ, it follows that Using Hölder formula, we get By similar arguments, we choose in (20) b φ =û ε − ψ and ψ ∈ W 1,r 0 ðΩÞ 3 to obtain 4 Journal of Function Spaces We combine now (35) and (36) to see that Next, for i = 1, 2, we choose ψ = ðψ 1 , 0, 0Þ then ψ = ð0, ψ 2 , 0Þ in the inequality (37) and using (22), we find where jΩj = mesðΩÞ. Then, (31) holds for i = 1, 2.
To get (32), we take ψ = ð0, 0, ψ 3 Þ 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 ε tends towards zero and what is the limit problem who should check this limit?

Theorem 3. Under the same assumptions as in Theorem
Proof. By Theorem (1), there exists a constant C independent of ε such that and using Poincare's inequality, we deduce that that is to say,û ε i is bounded in V z , i = 1, 2, this implies the existence ofû * i in V z such thatû ε i converges toû * i in L r ðΩÞ. The same, the inequality (22), we give so ε∂û ε i /∂x j converges to ∂û * i /∂x j and as kû ε i k L r ðΩÞ ≤ C, then ∂û ε i /∂x j converges weakly to ∂û * i /∂x j ; which gives the converges weakly of ∂û ε i /∂x j to 0 in L r ðΩÞ. Well thanks to the inequality: ε 2 k∂û ε 3 /∂x j k L r ðΩÞ ≤ C, we have the convergence ε 2 ∂û ε 3 /∂x j ⟶ ∂û * 3 /∂x j and ε∂ u ε 3 /∂x j ⟶ ∂û * 3 /∂x j : This shows that ∂û ε 3 /∂x j converges weakly to 0 in L r ðΩÞ. Finally, using (31) and (32), we get (45).

Study of the Limit Problem
In this section, we give both the equations satisfied by p * and u * in Ω and the inequalities for the trace of the velocity u * ðx, 0Þ and the stress ∂u * /∂zðx, 0Þ on ω.

Journal of Function Spaces
Theorem 4. With the same assumptions of Theorem (3) the solution ðu * , p * Þ satisfying the following relations where The proof of this theorem is based on the following lemma.

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 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][23][24][25][26][27][28][29][30][31][32][33][34][35]).

Data Availability
No data were used to support the study.

Conflicts of Interest
The authors declare that they have no competing interests.