Research Article | Open Access
On the Stream Function-Vorticity Finite Element Formulation for Incompressible Flow in Porous Media
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.
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
where is the projection operator defined from onto , such as
By choosing , as test function in the last equation of the problem , we get
Due to the embedding of onto , we have
and also, due to the propriety of the operator , we have
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
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
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.
Proof. For solution of the problem and solution of the problem , we have
Therefore, using the propriety of the operator , we have
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