Advances in Numerical Analysis

Volume 2014 (2014), Article ID 459761, 11 pages

http://dx.doi.org/10.1155/2014/459761

Research Article

On the Stream Function-Vorticity Finite Element Formulation for Incompressible Flow in Porous Media

Abdellatif Agouzal,¹ Karam Allali,² and Siham Binna²

¹University Lyon 1, CNRS UMR 5208, Institute Camille Jordan, 69100 Villeurbanne, France

²University Hassan II, FSTM, P.O. Box 146, Mohammedia, Morocco

Received 28 May 2014; Revised 30 August 2014; Accepted 3 September 2014; Published 18 September 2014

Academic Editor: Zhangxin Chen

Copyright © 2014 Abdellatif Agouzal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Stream function-vorticity finite element formulation for incompressible flow in porous media is presented. The model consists of the heat equation, the equation for the concentration, and the equations of motion under the Darcy law. The existence of solution for the discrete problem is established. Optimal a priori error estimates are given.

1. Introduction

The stream function-vorticity formulation is widely used to perform the numerical simulation of an incompressible fluid flow in porous media [1, 2]. The main advantage of the formulation is a reduction of the numerical problem unknowns with the fact that the continuity equation remains satisfied. In this paper, we are interested in studying the finite element of stream function-vorticity formulation for propagating reaction front in porous media. The model considered is a system of reaction-diffusion equations coupled with the hydrodynamic under the Darcy-Boussinesq approximation in the open bounded convex domain [3–5]:

where is the temperature, is the concentration, is the velocity, is the pressure, is the thermal diffusivity, is the diffusion, is the viscosity, is the permeability, is the activation energy, is the universal gas constant, is the Arrhenius preexponential factor, is a mean value of temperature, is the gravity, is the upward unit vector, and is the coefficient of the thermal expansion of the fluid.

The boundary conditions are of Dirichlet-Neumann type for the temperature and the concentration and of impermeability type for the normal component of velocity:

where and are disjoint open parts of such that .

Due to the incompressibility of the fluid, we introduce the stream function :

and by introducing also the vorticity , the problem becomes

where , , and .

Because there is no flow of fluid through the boundary (impermeability boundary), we will have the following zero-flux condition:

The paper is organized as follows. We present the variational formulation of the problem in the next section. We establish the existence result in Section 3. A priori error estimates are given in Section 4. We conclude in the last section.

2. The Variational Formulation

In order to state the problem in the variational form, we first describe the functional framework needed for our study. We set

We introduce now the constant of Friedrichs-Poincaré which depends on the geometry of the domain :

The variational form of the continued problem can be written as the following form:

where

Here, is the dual of the space . We can see clearly that the parameters and the functions and of the problem verify the following conditions:

where is the temperature of the unburned mixture. For convenience, for all and , we introduce the forms defined by

The variational formulation is rewritten as follows:

3. The Existence Result

3.1. The Semidiscrete Problem

In order to give the semidiscrete problem, we will need the following spaces:

where is a strictly positive constant. We assume that the spaces , , and satisfy the following assumptions.

(1)For all , there exists a linear continuous operator from onto such that, for all ,

(2)For all , there exists a linear continuous operator from onto such that, for all ,

Examples of such spaces verifying these conditions are given in [6, 7].

The discretized form of the problem is given as follows:

With the initial conditions,

In the sequel, we assume for simplicity that

The main result of the paper is written as follows.

Theorem 1. The problem admits a unique solution. Moreover, for , , , and solutions of the problem and for , , , and solutions of the problem , one has

under the following conditions:

where

Throughout the paper, we often use the following notation.

For each ; without further specification, the constant is independent of the mesh size and the solutions.

3.2. Existence of Semidiscrete Solutions

In order to prove the existence result of the problem , we need the following lemmas; first for the concentration, we have the following.

Lemma 2. For any local solution of the problem , one has the a priori estimate

Proof. By choosing , as test function in the third equation of the problem , we have

Integrating the last equality and noticing that is positive, we obtain

By using the two last inequalities, we obtain the result of the lemma.

Also, for the temperature, we have the following result.

Lemma 3. For any local solution of the problem , one has the a priori estimate

Proof. By choosing , as test function in the second equation of the problem , we have

Via integration of the last inequality, it follows that

Thus, using inequality (23), we get

It leads to

From (28) and (29), it follows that

For the vorticity, we have the following.

Lemma 4. For any local solution of the problem , one has the estimate

Proof. By choosing , as test function in the first equation of the problem , we have

Then, by using triangular inequality, we get

While integrating the last inequality, we obtain

However, using Lemma 3, we have

It leads to

From (35) and (36), we obtain

Finally, for the stream function, we have the following.

Lemma 5. For any local solution of the problem , one has the estimate

Proof. Let be the solution of the problem . From [8, 9], we have

However,

where is the projection operator defined from onto , such as

By choosing , as test function in the last equation of the problem , we get

Therefore,

Due to the embedding of onto , we have

and also, due to the propriety of the operator , we have

so

We conclude that

Then, we get

Using Lemma 4, we conclude that

Now, we are able to prove the main theorem of this section more precisely.

Theorem 6. The problem admits at least a solution

Proof. Indeed, it is obvious that the problem admits a local solution in the interval . For which is rather small, Lemmas 2, 3, 4, and 5 show that this solution can be defined on the interval for .

4. A Priori Error Estimations

In this section, we prove some error estimates on the stream function, on the vorticity, on the temperature, and on the concentration. Subsequently, we assume that there exists , such as

Lemma 7. For any and solutions of the problems and , respectively, one has

Proof. For and , respectively, solutions of the continuous and the semidiscrete problem, we have

where is an element of the uniform meshes family . Let be the approximation of in defined by

Hence, for all and , we have the following result:

where is the diameter of the element .

By using equality (53) and setting , we obtain

Therefore, we have

Then,

However, by using the triangular inequality, we can write

It follows that

However, for all and , we have

While multiplying (60) by and using (55) and (61), we obtain

Finally, by integrating and using Jensen’s inequality, we get

However,

We conclude that

Lemma 8. For any and solution of the problems and , respectively, one has

Proof. Let and be the solution of the problems and , respectively. From [8, 9], we have

However,

where is the projection operator verifying (41). Moreover, we have

Then, we get

By choosing as test function in the last equation of the problem , we obtain

It leads to

On the other hand, due to the embedding of onto , we have

In addition,

so

From inequalities (67) and (72), we have

We conclude that

Also, we have the following error estimate for temperature.

Lemma 9. If we assume that the hypothesis is verified, then one has

Proof. For solution of the problem and solution of the problem , we have

Therefore, using the propriety of the operator , we have

So,

By setting , we obtain

Then, we get

We have also

From (83) and (84), we get

Then, we deduce

By multiplying the last inequality by , we obtain

Via integration of the last inequality, we obtain

Therefore,

However,

Then, we have

Inequalities (89) and (91) allow us to obtain the following estimate:

Finally, if we set

we obtain the following estimate:

In addition, we have the following error estimate on concentration.

Lemma 10. For and , respectively, solution of the problems and , one has

Proof. By choosing as test function, we have

However,

Then, we have

Using (96) and (98), we obtain

By using Young’s inequality, we obtain

While multiplying the last inequality by and via integration, we obtain

So, we get

Finally, by the triangular inequality, we obtain

Lemma 11. For and solution of the problems and , respectively, one has

Proof. By using Lemmas 9 and 10, we have

If we set

then we get

Lemma 12. One assumes that the hypotheses , , and are verified, so

Proof. First of all, by using Lemmas 7, 8, and 11, we obtain

If we set

assuming that

for example, if , we obtain

and then we obtain

Finally, using Lemmas 8, 10, 11, and 12, we obtain the a priori error estimate at the same time on the temperature, the concentration, the stream function, and the vorticity, so the main Theorem 1 is proved.

5. Conclusion

The propagation of reaction front in porous media is modelled by a system of equations, coupling hydrodynamic equations and the reaction-diffusion equations. We have taken into account Darcy-Boussinesq approximation. We have adopted the stream function-vorticity formulation of the Darcy equation and we have chosen the appropriate functional framework for our variational problem. We have proved the existence result for the semidiscrete solution. Furthermore, we have established an optimal a priori estimate on the temperature, the concentration, the stream function, and the vorticity.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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