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

Abstract

In this manuscript, the behavior of a Herschel–Bulkley fluid has been discussed in a thin layer in associated with a nonlinear stationary, nonisothermal, and incompressible model. Furthermore, the limit problem has been considered, and the studied problem in is transformed into another problem defined in without the parameter ( 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.

1. Introduction

In a recent study of problems for the asymptotic behavior for a problem of continuum mechanics in a thin domain , the problem is transformed into an equivalent problem on a domain independent of the parameter . 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 Specifically, the case of Herschel–Bulkley fluid has been archived in several articles, for instance, [6, 7].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 Section 2, 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 Section 3, we use the asymptotic analysis, in which the small parameter is the height of the domain. We also discuss some estimates, independent on the parameter , 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, 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 following:

The boundary of is where with is the lateral boundary. We denote by the deviatoric part and the pressure. The fluid is supposed to be viscoplastic, and the relation between and is given by

For any tensor , the notation represents the matrix norm: . Let the unit outward normal vector on the boundary . The normal and the tangential velocity on the boundary are . Also, is a regular stress tensor field, further let and are the normal and tangential components of on the boundary by , .

Problem 1. Find a velocity field , the pressure and a temperature: such thatwhere and . 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 and the yield limit depend on the temperature, 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, is the thermal conductivity and the term represents the external heat source with . (5) gives the velocity on . As there is no-flux condition across , then we have equation (6). Condition (7) represents a Tresca thermal friction law on , where is the friction yields coefficient (8) gives the temperature on . (8) is a homogeneous Neumann boundary condition on :andA formal application of Green’s formula, using (3)–(8) leads to the weak formulation: Find a velocity field , and , such thatwhereIt is known that this variational problem has a unique solution, see for more details [10–12].
We assume that there exist in such thatandFollowing some previous results that are useful in the next sections (cf. [13])

3. Change of the Domain and Study of Convergence

In this section, we will use the technique of scaling in on the coordinate , by introducing the change of the variables . We obtain a fixed domain which is independent of : .

We denote its boundary by , also we have

Assume thatwith

Letwhere the condition is given by

By injecting the new data and unknown factors in (19) and (20), we prove that is a solution of the following problem:where

3.1. A Priori Estimates on the Velocity and the Pressure

Theorem 1. For all and under assumptions (13) and (14) and (20), there exists a constant independent of such that

Proof. Choosing as test function in inequality (11), we getfrom (16) and (17) we haveFrom (27) and (28), we deduceWe multiply (29) by , we getAs , we haveFrom Korn’s inequality and (15), there exists a constant independent of , such thatFrom (32), we deduce (26), with , andWe 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 temperature , for this we need to establish the following result:

Theorem 2. Assume that the assumptions of Theorem 1 are satisfied. Moreover, assume that there exist , , such that

Then, there exists a positive constant independent of , such that

Proof. Choosing in (24), where is defined byWe obtainOn the other hand,where depends only on .
As and then , so using this inequality and (26), we deduceUsing Holder’s inequality with the exponents and , for , we obtainusing (39), we getwhere .
By (18) and (18), we findNow using the Poincaré-Sobolev inequality, we haveOn the other hand, for all and , we have the implication:Hence from (43) and (44) and the fact that , we deducewhereandAs , then . So for , we obtainFrom (42) and (45), we getas and , we obtainwherewhere is a constant independent of .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 Theorem 1 and Theorem 2, there exist , and such that

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 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 satisfies the following relations:andwhereand

The proof of this theorem is based on the following lemma.

Lemma 1 (Minty). Let be a Banach spaces, a monotone and hemicontinuous operator, a proper and convex functional. Let and . Then the following assertions are equivalent:

Proof. By using Minty’s Lemma 1 and the fact that in , then (24) is equivalent toFrom (57), we have almost everywhere. As is continuous function on , thenUsing Theorem 4 and the fact is convex and lower semicontinuous, , we findand as , because independent of , we getUsing again Minty’s lemma for the second time, thus (61) is equivalent to (59). Now, we can choose and respectively in (59), we find (60). For (61), we choose for all . Passing to the limit on tend to in (24) and using (52)–(54),(57)–(58) we getby Green’s formula, we obtain

Theorem 5. Let us setthenwhere and .

Proof. If , from (73)we get . For all , choosing , then in (61), we obtainwhereNow, utilising the Hanh-Banach theorem, then, , with , such thatIn particular, from (60) and (76), we getAlso, from (76) and (77), we haveNext using (78), we haveAs , we deduceHence, if , by (73), we obtainIn this case, ; therefore, we can writefor every and from (79), there exist such thatUsing (83) and (84) becomesand choosing in (85), we find (74).

The convergence of our problem towards the Reynolds equation given by the following result: Theorem 5.for all where

Proof. To prove (86), we integrate twice (74) from 0 to , then taking , we obtain the requested result.

The uniqueness of the limit velocity and pression are given in the following theorem:

Theorem 6. The solution in of equality (86) is unique.

Proof. Let and be two solutions of (59)–(63) and (86); then and solve (62)–(63), so satisfies the problemso , thus . Taking and respectively, as test function in (59) we getObserve that for every we obtainwhere , Using Hölder’s inequality, we deduceFrom (91) and (92), we obtainusing Poincare’s inequality, we deduceFinally, to prove the uniqueness of the pressure, we use (86) with the two pressures and , we findTaking , and by Poincare’s inequality, we deduce . So .

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].

Data Availability

No data were used to support the study.

Conflicts of Interest

This work does not have any conflicts of interest.

Acknowledgments

The fifth-named author extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the Research Groups Program [grant number R.G.P.2/11/43].