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A simulation was carried out on an unsteady flow of a viscous, incompressible and electrically conducting fluid past an infinite vertical porous plate. A generic computer program using the Galerkin finite element method is employed to solve the coupled non-linear differential equations for velocity and temperature fields. The diffusion equation, the energy equation, the momentum equations and other relevant parameters are transformed into interpretable postfix codes. Numerical calculations are carried out on the flow fields both in the presence of cooling and heating of the plate by free convection currents. The effects of the dimensionless parameters, namely, the Prandtl number, the Eckert number, the modified Grashof number, the Schmidt number and the time on the temperature and velocity distributions are discussed.

The effect of an applied magnetic field on unsteady free convection flow along a vertical plate has been given considerable interest because of its application in the cooling of nuclear reactors or in the study of the structures of stars and planets which are greatly influenced by the thermal convection processes in their interiors. A number of authors have made important contributions in that area and in the area of fluid flows in general like [

Yusuf et al. [

In this paper, we are making use of a generic computer tool based on the Galerkin finite element method in order to compute the solution of an unsteady boundary layer flow of an incompressible, viscous and electrically conducting fluid past an infinite vertical porous plate.

The geometry and the unsteady flow fields for this problem are described by Bitok [

The boundary conditions are given as follows:

and

where

・ u and w are the components of the dimensionless velocity;

・ θ is the dimensionless temperature;

・ C is the dimensionless species concentration;

・ Gr is the Grashof number;

・ Gc is the modified Grashof number;

・ Pr is the Prandtl number;

・ Ec is the Eckert number;

・ Sc is the Schmidt number;

・ M is the magnetic parameter;

・ m is the Hall parameter;

・ α is the angle of the uniform magnetic field with the y-axis.

The system of Equations (1)-(4) with boundary conditions (5) has been solved numerically by a generic computer program based on the finite element method in steps 1 - 3. In all mathematical formulations,

where

We solve Equation (1) with the help of boundary conditions (5). Constructing the quasi-variational equivalent of Equation (1), we obtain:

Consider an N elements mesh and a two parameter (semi discrete) Galerkin approximation of the form [

Using Equations (6) and (8), Equation (7) reduces to:

where

Using the Q-family of approximation developed by Reddy [

where

The initial value

For t > 0,

We solve Equation (2) with the help of boundary conditions (5). Constructing the quasi-variational equivalent of Equation (2), we obtain:

Consider an N elements mesh and a two parameter (semi discrete) Galerkin approximation of the form [

Using Equations (6) and (16), Equation (15) reduces to:

where

Using the Q-family of approximation developed by Reddy [

where

The initial value

For t > 0,

We solve Equations (3), (4) with the help of boundary conditions (5). Constructing the quasi-variational statement of Equations (3), (4), we obtain:

Consider an N elements mesh and a two parameter (semi discrete) Galerkin approximation of the form [

Using Equations (6), (25) and (26), Equations (23), (24) reduce to:

where

Using the Q-family of approximation developed by Reddy [

where

The initial values

For t > 0,

The numerical values of the temperature and velocity fields have been computed from equations (14), (22), (35) and (36). All input elements such as matrix and vector elements are transformed into interpretable postfix codes.

Numerical calculations have been carried out for the velocity and temperature distributions. The analysis of Bitok [

From

1) The temperature (θ) decreases away from the plate. The decrease is greater for a Newtonian fluid than it is for a non-Newtonian fluid (θ decreases with Pr);

2) There is a rise in temperature profiles (θ) due to an increase in the time (t);

3) There is an insignificant change in the temperature profiles (θ) due to an increase in the Eckert number (Ec).

From

1) The primary velocity (u) decreases due to an increase in the Prandtl number (Pr) and the Schmidt number (Sc);

2) An increase in the modified Grashof number (Gc) and the time (t) leads to a rise in the primary velocity (u);

3) There is an insignificant change in the primary velocity (u) due to an increase in the Eckert number (Ec).

From

1) The primary velocity (u) decreases due to an increase in the Schmidt number (Sc);

2) An increase in the Prandtl number (Pr), the modified Grashof number (Gc) and the time (t) leads to a rise in the primary velocity (u);

3) There is an insignificant change in the primary velocity (u) due to an increase in the Eckert number (Ec).

From

1) The secondary velocity profile (w) increases due to an increase in the time (t);

2) An increase in the Prandtl number (Pr), the Eckert number (Ec), the modified Grashof number (Gc) and the Schmidt number (Sc) leads to an insignificant change in the secondary velocity profile (w).

In this work, a computer simulation was carried out on the unsteady flow of a viscous, incompressible and electrically conducting fluid past an infinite vertical porous plate. The velocity and the temperature fields were computed using a generic software tool based on the Galerkin finite element method. The results obtained reveal that the use of interpretable codes provides a good solution to fluid flow problems.

Naroua, H. and Bachir, M.I. (2016) A Generic Computational Solution of a Natural Convection Flow past an Infinite Vertical Porous Plate. American Journal of Computational Mathematics, 6, 287-297. http://dx.doi.org/10.4236/ajcm.2016.64030