Analysis for Flow of an Incompressible Brinkman-Type Fluid in Thin Medium with Friction

Abstract

In this paper, we consider the Brinkman equation in the three-dimensional thin domain . The purpose of this paper is to evaluate the asymptotic convergence of a fluid flow in a stationary regime. Firstly, we expose the variational formulation of the posed problem. Then, we presented the problem in transpose form and prove different inequalities for the solution independently of the parameter . Finally, these estimates allow us to have the limit problem and the Reynolds equation and establish the uniqueness of the solution.

1. Introduction

Darcy’s law is a physical law that describes the flow of a fluid along a porous medium. It is first named after the French researcher Henri Darcy in 1947 who created a pipeline procedure to supply water around a French town [1]. In 1949, another researcher gave an extension to Darcy’s law to which he added the term Brinkman. Thanks to this term, they discovered a new fluid called Darcy-Brinkman [2], which was used to account for transitional flow between boundaries. This model describes a flow in porous media that is fast enough where the drive for flow includes kinetic potential related to fluid velocity, pressure, and gravitational potential. They appear as a mix of Darcy’s law and the Stokes equations and extend Darcy’s law to account for dissipation of kinetic energy by viscous shear as in the Stokes equation. The average fluid flow through an array of sparse, spherical particles can be modeled via the Brinkman equation [2, 3]. In addition, Brinkman’s equations describe transitions between two flows, one slow and the other fast, where the first type occurs in a porous medium subject to Darcy’s law, while the second type occurs in channels subject to Stokes’ equations. His equation is where is the Newtonian fluid viscosity, is the resistance parameter, which is assumed constant (isotropic), is the fluid velocity, is the pressure, and represents the body force density applied on the fluid.

In recent decades, many authors have studied the Brinkman equation. For instance, Durlofsky and Brady in [4] employ Stokesian dynamics to approximate the fundamental solution or Green’s function for flow in random porous media. The authors in [5] investigate a three-dimensional model of flagellar swimming in a Brinkman fluid. In this study, the utility of the Brinkman equation is to model the mean flow rate of the fluid as well as the resistive effects of fibers on the fluid. The fractional Brinkman-type fluid in a channel under the effect of MHD with the Caputo-Fabrizio fractional derivative was studied by Khana et al. in [6]. In the same idea, Liu in [7] has investigated a biobarrier to remove chlorobenzenes from slow-moving ground contaminated water. The goal of the present paper is not only to give existence and uniqueness solution of the boundary value problems governed by the Brinkman fluid but also to obtain rigorously the equation describing such a phenomenon in a thin film flow by way of an asymptotic analysis in which a small parameter is the width of the gap. Several authors are interested in the study of asymptotic convergence of Newtonian and non-Newtonian fluids. In the work of Dilmi et al. [8], the authors studied the asymptotic convergence of a Bingham fluid in a thin domain with the Fourier and Tresca boundary condition on the bottom surface. The asymptotic analysis of an incompressible Herschel-Bulkley fluid with friction law is given by [9]. Many works have focused on mechanics of the fluids in a thin domain in the stationary case which is found in the works of [10, 11]. Other mechanical contact problems similar to incompressible flows in thin medium with friction can be found in [12, 13]. This paper is divided into four sections: in a first step, we discussed the variational formulation of the problem and demonstrate the results of existence and uniqueness of the weak solution; then, we move on to the study of asymptotic analysis. For this, using the change in the variable and unknown news to conduct the study on a domain does not depend on . Then, we prove after an explicit work different inequalities for the solution which the thickness becomes infinitely small in the variational formulation. Finally, these estimates allow us to have the limit problem and the Reynolds equation and establish the uniqueness of the solution.

2. The Problem Statement

In this section, we give an overview of the thin domain. Next, we introduce the problem considered in this domain. Finally, we explore the theorem of existence and uniqueness of the weak solution.

2.1. The Domain

We denote by the vector of whose is the generic vector of and .

Let be a domain of the plane and be a bounded continuous function defined on , with of class such that . The fluid is contained between the lower and the upper surface defined by . Let where is a small parameter that will tend to be zero. The boundary of is , where is the lateral boundary.

Let be the unit outward normal to the boundary . The normal and tangential components of are given by

Similarly, for a regular tensor field , we denote by and the normal and tangential components of given by

We introduce the following functional framework:

2.2. The Model Problem

The boundary value problem describing the stationary flow for the incompressible Brinkman fluid is described by the following:

It is supposed that the law of behavior follows the law of Stokes: where is the Krönecker symbol.

Our problem comes down to finding the solution which satisfies the following equations:

Problem 1. Find the velocity field and the pressure , such that Equation (7) represents the law of conservation of momentum where is the external force. Relation (8) gives the law of behavior of the incompressible fluid. The Dirichlet boundary conditions are given by (9) and (10). The no-flux condition across is given by equation (11). Equation (12) is the condition of the Tresca friction law on the part , with being the friction coefficient (see [13]).

2.3. Weak Variational Formulations

Let be the solution of (7)–(12), multiplying equation (7) by and then integrating over , and using Green’s formula and the conditions (9)–(12), we can show that Problem 1 is equivalent to the following variational problem:

Problem 2. Find the pair , such that where

Theorem 3. If and , then there exists a unique and (to an additive constant) solution to problem (13).

Proof. Since our goal in this work is to prove the asymptotic convergence of the problem posed, we only give the steps followed for the proof of this theorem. Let in (13); we have the following:
Find , such that Using the Cauchy-Schwartz inequality and , we obtain that the bilinear from is continuous: By Korn’s inequality, we obtain where independent of . We deduce that is coercive on . Moreover, is convex and continuous on . This ensures the existence and uniqueness of satisfying the variational inequality (15). In addition, using the techniques of [14], we can prove the existence of for which is a solution of (13).☐

3. Dilatation in the Variable

In this section, we use the dilatation in the variable given by ; then, our problem will be defined on a domain which does not depend on given by and its boundary .

After this change, here are the new functions defined on the fixed domain :

Likewise for new data,

We denote by the Banach space with the norm

According to (19) and (20), then Problem 2 leads to the following:

Problem 4. Find , such that

In the next, we will do the estimates of velocity and then on the pressure solution of our variational problem in the fixed domain.

Theorem 5. Assuming (19) and (20), the following estimate on is satisfied:

Proof. Let be the solution of (15), we choose , and we obtain Now, . By Korn’s inequality, there exists a constant independent of , such that We apply the Cauchy-Schwarz inequality and then the Young inequality; we obtain the following: By the Poincaré inequality, we give As , then we multiply the last equation by ; we obtain Writing this equation in terms of , we deduce (25).☐

Theorem 6. Assuming (19), the following estimates on are satisfied: where and denote the constants independent of .

Proof. Let ; putting in (24) (for ) and , we deduce Therefore, Using Green’s formula and the Cauchy-Schwarz inequality, we have From Theorem 5, we deduce the inequality (31).
Taking in (24) , in ; ; we have In the same way, by the chosen , , and , in , we have We use the same technical in (36) and (37) to obtain inequality (32).☐

Thanks to the estimations (25)–(32), we have the following convergence result:

Theorem 7. Suppose that the estimations (25)–(32) hold. There exists in such that we have the following:

Proof. Using the estimate (25) and Poincaré’s inequality in the domain : we deduce (38). Also, equations (39)–(42) follow from (25) and . Finally, the weakly convergences (43) follow from (31) and (32) and [15].☐

4. Study of the Limit Problem

To reach the desired goal, we need in the rest of this paragraph the results of previous convergences.

We go to the limit in (24), and using (23) (), we find

Moreover, if then where

Lemma 8. If the estimations (25)–(32) hold, then satisfy where .

Proof. As in [16], by choosing in (47), we find that the second member of this inequality is an upper bound of its first member. Then, for in the same inequality, we get the reverse, which therefore gives equality (49). Now, for in (47), we deduce directly the inequality (50). It is shown reciprocally that (49) and (50) imply (47). For this, we consider the mapping: Let us define as For all , we choose in (49): By (53), is a continuous linear function on the subspace of which is the image of by . Then, by the Han-Banach theorem, there exists , with , such that In particular, from (50) and (54), we get Also, Then, from (49) and (56), we have which gives (47).☐

Theorem 9. With the same assumptions as Theorems 5 and 6, the pair satisfies

Proof. We choose in (24) with in and (for ); we give (34). Using the convergence limit ((38) and (40)–(42)) in (34), we find then, Choosing (for ) with in and leads to Using (38), (39), (41), and (43) and choosing and and then with and in , we find By Green’s formula, we deduce Then, To prove , we see from Theorem 6 that does not depend on . Then, as in [16], we choose in (63); with , we obtain Using the Green formula, we deduce Then, where whence As and then in , therefore , , and are in . In addition, from (70), we get in . So as belongs to from (61), we have , whence (59) holds, and we also have in ☐

Theorem 10. Suppose that the assumptions of the previous theorem hold; then, the solution satisfies the weak generalized equation of Reynolds: where

Proof. By integrating expression (59) from to , it becomes In particular for , we obtain Integrating (73) from to , we get As , we have From (74)–(76), we deduce (71), which ends the proof requested.☐

Theorem 11. Suppose that the assumptions of the previous theorem hold; then, the solution of the limit problem (45) is unique in .

Proof. Suppose that problem (47) admits two solutions that we denote by and . Using classical techniques, we choose in (47) and then as test functions; then, by summing the two inequalities obtained, we find Then, Consequently, this last inequality ensures the uniqueness of the solution in .
Similarly, we take in the Reynolds equation (71) the pressure value and then , respectively, at the end by subtracting the equations obtained; it becomes Finally, by performing the change of variable and then by the Poincaré inequality, we obtain We deduce the desired result.☐

5. Conclusions

The purpose of this paper is to study the asymptotic convergence of an incompressible Brinkman-type fluid in thin medium with the Tresca friction on the bottom surface. One of the objectives of this study is to obtain a two-dimensional equation that allows a reasonable description of the phenomenon occurring in the three-dimensional domain by passing the limit to 0 on the small thickness of the domain (3D). We show the existence and uniqueness results of the weak solution. We proceed to the study of the convergence analysis. To do this, we use the scale change following the third component and new unknowns to conduct the study on a fixed domain. Then, we prove different inequalities for the solution . Finally, these estimates allow us to have the limit problem and establish the uniqueness of the solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.