Analytical expressions of a third grade fluid flow through porous medium using the Homotopy perturbation method

Introduction

Nowadays the majority of the fluids dealt with by Engineers and Scientists, such as air, water and oils can be regarded as Newtonian under most conditions of interest. However, in many cases the assumption of Newtonian behavior is not valid and the rather more complex non-Newtonian response must be modeled. These include non-Newtonian fluid friction reduction, oil-pipeline friction reduction, surfactant applications to large-scale heating and cooling systems, scale-up, and flow tracers. Such situations also arise in the chemical processing industry, plastics processing industry and mining industry. The simulation of non-Newtonian fluid flow phenomena is therefore of importance to industry. The classical Navier -Stokes equations cannot accurately describe the rheological behavior of these complex fluids about these applications. The third grade fluid model has been used to model the flow of non-Newtonian fluids [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] undergoing strongly exothermic chemical reaction due to its ability to explain the trim thickening/thinning properties of these fluids. The flow of a reactive fluid through a channel filled with saturated porous medium was examined by Makinde [7] . There are some methods to obtain approximate solutions of this kind of equations. One of them is Homotopy perturbation method. The method introduced by He in 1999 [16][17][18] has been developed and improved by himself. In this paper, the non linear differential equations are suitable to the presence of exponential temperature dependent internal heat generation occurring within the channel. So the approximate analytical solution will be obtained by Homotopy perturbation method.

Mathematical formulation of the problem

Consider the steady flow of a viscous incompressible third grade fluid between infinite parallel isothermal plates separated by distance 2a. The channel is assumed saturated with porous materials and the geometry of the situation is shown in Figure 1. The channel walls are maintained at temperature Ta. Disregarding the reactant consumption, the equations governing the hydrodynamically and thermally developed flow can be expressed as [1][2][3][4][5][6][7][8][9][10] follows:

Subject to the appropriate boundary conditions

(4) Here u  is the fluid velocity, P  is the fluid pressure, 3  is the material coefficient, k is the thermal conductivity,  is the dynamic viscosity,  is the dimensionless third grade material parameter, T is the fluid temperature, U is the characteristic velocity, a T is the wall temperature, Q is the heat of reaction, A is the rate constant, R is the universal gas constant, E is the activation energy, 0 C is the initial concentration of the reactant species, a is the channel width, y is the distance measured in the normal direction, additional term in the eqns. ( 1) and ( 2) is the third grade material effect defined in [1][2][3][4][5][6][7][8][9] . Now we introduce the following dimensionless parameters:

Then we obtain the following dimensionless eqns. ( 5) and ( 6) (with appropriate boundary conditions):

Here  is the viscous heating parameter,  is the dimensionless temperature, u is the dimensionless velocity, M is the dimensionless pressure gradient, 2  is the porous permeability parameter, Da is the Darey parameter, K is the porous permeability,  is the Frank -Kameneskii parameter.

The physical quantities of this problem are the skin friction parameter

Solution of the non-linear boundary value problem using the Homotopyt perturbation method

Linear and non-linear phenomena are of fundamental importance in various fields of science and engineering. Most models of real -life problems are still very difficult to solve. Therefore, approximate analytical solutions such as Homotopy perturbation method (HPM) [16] were introduced. The HPM is the most effective and convenient method for both linear and non-linear equations. Perturbation method is based on assuming a small parameter. The majority of non-linear problems, especially those having strong non-linearity, have no small parameters at all and the solutions obtained by the perturbation methods are only approximate. In most cases, the solutions are valid only for small values of the small parameter. Generally, the perturbation solutions are uniformly valid as long as a scientific system parameter is small. However, we cannot rely fully on the approximations, because there is no criterion for the existence of a small parameter. Thus, it is essential to check the validity of the approximations numerically and/or experimentally. To overcome these difficulties, HPM have been proposed recently. Recently, many authors have applied the HPM to solve the non-linear problem in physics and engineering sciences. A combination of Homotopy in topology and classic perturbation techniques is useful in solving some of the non-linear problem in physical sciences. The HPM is unique in its applicability, accuracy and efficiency. The HPM uses the imbedding parameters a small parameter, and only a few iterations are needed to search for an asymptotic solution.

The approximate analytical solutions of the eqns. ( 5) and ( 6) obtain by applying the Homotopy perturbation method is given by

The analytical expressions of the eqns. ( 7) and ( 8) obtained by using the eqns. ( 9) and ( 10) is given by

Result and Discussions

Figure 1 versus dimensionless distance y , The curves are plotted using the eqn. ( 9) for various values of  and in some fixed values of the other dimensionless parameters. 12) for various values of  and in some fixed values of the other dimensionless parameters.

Fig 5(h): Nusselt number

Nu versus non-Newtonian material  using the eqn. ( 12) for various values of  and in some fixed values of the other dimensionless parameters.

Fig 5(i): Nusselt number

Nu versus non-Newtonian material  . The curves are plotted using the eqn. ( 12) for various values of  and in some fixed values of the other dimensionless parameters.

Conclusion

In this paper a mathematical analysis of the reaction of third grade fluid flow through the porous channel is done. The analytical expressions for the velocity and temperature profiles are obtained using Homotopy perturbation method. The analytical expressions of skin friction and Nusselt number are also obtained. If non-Newtonian parameter value is increased then the fluid flow decreases within the channel. The Frank-Kameneskii parameter becomes large in the fluid temperature due to internal heat generation as a result of the exothermic chemical reactions. Our analytical results are compared with previous work and a satisfactory agreement is noted. This method can be easily extended to solve nonlinear boundary value problem in fluid mechanics, heat and mass transfer and other fields of engineering.

References

We construct the Homotopy as follows The initial approximations are as follows: