Magneto Convective Flow of a Non-Newtonian Fluid through Non-Homogeneous Porous Medium past a Vertical Porous Plate with Variable Suction

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Received 29 December 2015; accepted 19 February 2016; published 22 February 2016

1. Introduction

MHD free convection fluid flows frequently occur in natural world. Fluid passes through porous medium are of great interest nowadays and many researchers attract towards the applications in the fields of science and technology namely in the area of agriculture engineering to know about the ground water resources, in fuel technology to study the moment of natural gas, oil, and water through the oil reservoirs. Chaudhary et al. [1] considered Hall effect on MHD mixed convection flow of a visco-elastic fluid past an infinite vertical porous plate with mass transfer and radiation. Satyanarayana et al. [2] discussed MHD free convective heat and mass transfer past a vertical porous plate with variable temperature. Kesavaiah et al. [3] studied and presented effects of chemical reaction and radiation absorption on unsteady MHD convection heat and mass transfer flow past a semi-infinite vertical permeable moving plate embedded in porous medium with heat source and suction. Reddy et al. [4] considered effects of chemical reaction and radiation absorption on unsteady magnetohydrodynamic double diffusive convective flow of viscous fluid past a semi-infinite porous plate. Raju et al. [5] investigated radiation and mass transfer effects on a free convection flow through porous medium bounded by a vertical surface. Muthukumaraswamy et al. [6] analyzed first order chemical reaction on flow past an impulsively started vertical plate with uniform heat and mass flux. Das et al. [7] studied the effects of mass transfer on a flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction. Kandaswamy et al. [8] focused on the problem of chemical reaction, heat and mass transfer on magnetohydrodynamic flow over a vertical stretching surface with heat source and thermal stratification effects. Effects of chemical reaction and thermophoresis on MHD mixed convective heat and mass transfer flow along an inclined plate in the presence of heat generation/absorption with viscous dissipation and joule heating was considered by Alam et al. [9] . Muthucumaraswamy et al. [10] analyzed the effects of chemical reaction on moving infinite vertical plate with uniform heat flux and variable mass diffusion. Mahapatra et al. [11] investigated the effect of chemical reaction on free convection flow through a porous medium bounded by a vertical surface. Mishra et al. [12] considered the effect of mass and heat transfer on magnetohydrodynamic flow of a visco-elastic fluid through porous medium with oscillatory suction and heat source. Beget et al. [13] had established computational fluid dynamics modeling of bouncy induced visco-elastic flow in a porous medium with magnetic field effect. Soundalgekar et al. [14] investigated effects of mass transfer and natural convection effects on MHD stokes problem for a vertical plate. Kandasamy et al. [15] studied the effects of chemical reaction, heat and mass transfer along a wedge with heat source and concentration in the presence of source or injection. Gurmindersingh et al. [16] analyzed the mass transfer with chemical reaction in MHD mixed convection flow along a vertical stretching sheet. Radiation effects on MHD free convection flow over a vertical plate with heat and mass flux was considered by Sivaiah et al. [17] . Sahim et al. [18] considered Laplace technique on MHD radiating and chemically reacting fluid over an infinite vertical surface. Singh et al. [19] investigated heat and mass transfer in MHD flow of a viscous fluid past a vertical plate under oscillatory suction velocity. Reddy et al. [20] had presented thermal radiation and chemical reaction effects on magnetohydrodynamic mixed convective boundary layer slip flow in a porous medium with heat source and Ohmic heating. Rout et al. [21] studied effect of radiation and chemical reaction on free convective MHD flow through a porous medium with double diffusion. Effects of chemical reaction and radiation absorption on MHD flow of dusty visco-elastic fluid were considered by Prakash et al. [22] . Damala et al. [23] discussed radiation absorption, chemical reaction and magnetic field effects on the free convection and mass transfer flow through porous medium with constant suction and constant heat flux. Raju et al. [24] studied unsteady MHD free convection and chemically reactive flow past an infinite vertical porous plate. Umamaheswar et al. [25] investigated unsteady MHD free convective visco-elastic fluid flow bounded by an infinite inclined porous plate in the presence of heat source, viscous dissipation and Ohmic heating. Reedy et al. [26] considered chemical reaction and radiation effects on unsteady MHD free convection flow near a moving vertical plate. Rajesh et al. [27] studied radiation effects on MHD flow through a porous medium with variable temperature or variable mass diffusion. Kesavaiah et al. [28] investigated radiation and mass transfer effects on moving vertical plate with variable temperature and viscous dissipation. Recently Ravikumar et al. [29] investigated combined effects of heat absorption and MHD on convective Rivlin-Ericksenflow past a semi-infinite vertical porous plate. And Venkateswarlu et al. [30] had presented chemical reaction and radiation absorption effects on the flow and heat transfer of a Nano fluid in a rotating system.

The objective of the present paper is to analyze radiation absorption and chemical reaction on MHD visco-elastic free convection flow through porous medium bounded by a vertical surface with constant heat and mass flux in the presence of homogeneous chemical reaction. The dimensionless equations of continuity, linear momentum, energy and diffusion which governed the flow field were solved using perturbation technique. The behavior of velocity, temperature, concentration and skin friction coefficient was discussed for various parameters involved in the governing equations the applicable criteria that follow.

2. Formulation of the Problem

The unsteady free convection, viscous incompressible electrically conducting flow of a radiation absorption, chemically reacting and visco-elastic (Walters B^{*}) fluid past asemi-infinite vertical porous plate in a porous medium with variable suction as well as permeability in presence of a transverse magnetic field is considered. Let x^{*}-axis be along the plate in the direction of the fluid flow and y^{*}-axis perpendicular to it. It is assumed that, magnetic Reynolds number is much less than unity so that the induced magnetic field is neglected in comparison with the applied transverse magnetic field. The basic flow in the medium is therefore, entirely due to the buoyancy force caused by the temperature difference between the wall and the medium. This assumed that initially, at t^{* }≤ 0,the plate as well as fluids are at the same temperature and concentration. As the concentration of the species is very low so that the Soret and Dofour effects are neglected. When t^{*} > 0, the temperature of the plate is instantaneously raised to and the concentration of the species is set to (see Figure 1).

It is considered that the permeability of the porous medium in the following form

(1)

where is K^{*}_{p} is porosity, ω^{*} is frequency of oscillation, t^{*} is time, ε is a small positive constant.

The suction velocity is assumed to be time varying and it takes the following form

(2)

Here and are positive constants. Under the above assumptions with usual Boussineq’s approximation (Mishra et al. [31] , Raju and Varma [32] [33] ), the governing equations and boundary conditions are given by

(3)

(4)

(5)

Figure 1. Physical configuration and coordinate geometry.

(6)

Introducing the non-dimensional quantities,

(7)

The Equations (3) to (5) are reduced to the following dimensionless equations

(8)

(9)

(10)

(11)

3. Method of Solution

In view of transient suction, temperature and concentration at the plate let us assume the velocity, temperature, concentration in the neighborhood of the plate.

(12)

(13)

(14)

Substituting above Equations (12)-(14) into the Equations (8)-(10) and equating the ^{0} coefficient and coefficient of ^{1}, we get the following equations.

(15)

(16)

(17)

(18)

(19)

(20)

Now the boundary conditions are reduced to the following forms

(21)

The Equations (15) and (16) are not solvable by using the given boundary conditions (21). Hence the perturbation method has been applied using R_{c} (R_{c} < 1), the elastic parameter as the perturbation parameter.

(22)

Substituting Equation (22) into Equations (15) and (16), equating the coefficients of R_{c}^{0} and R_{C}^{1} to zero, we get the following set of equations.

Zeroth order equations

(23)

(24)

First order equations

(25)

(26)

Using the perturbation the boundary conditions are reduced as follows:

(27)

Solving these differential equations by using the boundary conditions we get the following results (Appendix)

The skin friction, Nusselt number and Sherwood number at the plate are defined as follows:

4. Results and Discussion

The present study considers the effects of radiation absorption and chemical reaction effect on transient free convection flow of a non-Newtonian fluid through non-homogeneous porous medium past a vertical porous plate with magnetic field and variable suction. Solutions for velocity, temperature and concentration field are obtained by using perturbation technique. The effects of various parameters like Grashof number for heat and mass transfer Gr and Gc, chemical reaction Kr, Radiation absorption R_{1},Prandtl number Pr, on velocity, temperature and concentration have been studied analytically and effects are executed with the help of Figures 2-15. Also the behavior of skin friction, rate of heat transfer and rate of mass transfer with respect to various parameters have been studied and results were presented in Tables 1-10.

Figure 2. Effect of Kr on concentration.

Figure 3. Effects of Sc on concentration.

Figure 4. Effects of S on temperature.

Figure 5. Effects of Sc on temperature.

Figure 6. Effects of Kr on temperature.

Figure 7. Effects of Pr on temperature.

Figure 8. Effects of Gc on velocity.

Figure 9. Effects of Gr on velocity.

Figure 10. Effects of Kr on velocity.

Figure 11. Effects of M on velocity.

Figure 12. Effects of Pr on velocity.

Figure 13. Effects of R_{1} on velocity.

Figure 14. Effects of S on velocity.

Figure 15. Effects of Sc on velocity.

Table 1. Effect of Sc on skin friction, Sherwood number, Nusselt number, z = 0.01, t = 1, n = 0.1.

Table 2. Effect of Pr on skin friction, Sherwood number, Nusselt number, z = 0.01, t = 1, n = 0.1.

Table 3. Effect of Gr on skin friction, Sherwood number, Nusselt number, z = 0.01, t = 1, n = 0.1.

Table 4. Effect of Gc on skin friction, Sherwood number, Nusselt number, z = 0.01, t = 1, n = 0.1.

Table 5. Effect of M on skin friction, Sherwood number, Nusselt number, z = 0.01, t = 1, n = 0.1.

Table 6. Effect of Kr on skin friction, Sherwood number, Nusselt number, z = 0.01, t = 1, n = 0.1.

Table 7. Effect of Kp on skin friction, Sherwood number, Nusselt number, z = 0.01, t = 1, n = 0.1.

Table 8. Effect of S on skin friction, Sherwood number, Nusselt number, z = 0.01, t = 1, n = 0.1.

Table 9. Effect of Rc on skin friction, Sherwood number, Nusselt number, z = 0.01, t = 1, n = 0.1.

Table 10. Effect of R_{1} on skin friction, Sherwood number, Nusselt number, z = 0.01, t = 1, n = 0.1.

Figure 2 exhibits the effect of chemical reaction (Kr) on concentration, it is noticed that the concentration of the fluid decreases as chemical reaction parameter increases. Figure 3 shows the effect of Schmidt number Sc on concentration when the values of Sc increases, the concentration value decreases. From Figure 4 we have noticed that temperature of the fluid increases as source or sink increases. Figure 5 shows the temperature profile for the different values of Sc, from this figure we have noticed that increase in the value of Sc results in increase in the temperature profile. Figure 6 depicts the effect of Kr on temperature, when the values of Kr increases the temperature value increases. Figure 7 shows the effect of Prandtl number on temperature, when Pr value increases, the temperature decreases, similar type of results are noticed with Satyanarayana et al. [2] . Figure 8 depicts the effect of Gc on velocity, the velocity of the fluid increases when Gc increases. Figure 9 exhibits that the effect of Gr on velocity, from this figure we observed that, the velocity of the fluid increases when Gr increases. Figure 10 shows the effect of Kr on velocity, the velocity of the fluid decreases in the increase of Kr. Figure 11 illustrates velocity profiles for different values of M, from this figure we have observed that velocity of the fluid decreases when an increase in the values of M. Figure 12 shows the velocity profile for different values of Pr. It is observed that increase in the value of Pr results in decrease in the velocity profile. Figure 13 shows the velocity profile for different values of radiation absorption, from this figure it is noticed that an increase in the value of R_{1} results a decrease in the velocity profile. Figure 14 illustrates the effect of source/sink on velocity, from this figure it is noticed that velocity of the fluid increases for decreasing values of source/sink. Figure 15 shows the effect of Sc on velocity, from this figure it is noticed that when Sc values increases the velocity of the fluid decreases. On the other hand, Tables 1-4 show, the effect of Sc, Pr, Gr and Gc on the parameters skin friction, Sherwood number, Nusselt number. It can be observed that skin friction coefficient increased with the increase in Sc, Pr, Gr, Gc. It can be clearly observed that rate of heat transfer of the fluid increases for increase in Sc and it is not shown any effect in case of Pr, Gr, Gc. The Nusselt number increased as increase in the Sc and Pr, but it is constant in the case of Gr and Gc. Further, Tables 5-10 show the effect of M, Kr, Kp, S, Rc and R on the parameters skin friction, Sherwood number, Nusselt number. It can be observed that skin friction coefficient increased with an increase in Kr, Kp, Rc, and R_{1 }whereas decreased in the increase of M and S. The rate of the heat transfer of the fluid increases with an increase in Kr, but it not shown any effect in case of M, Kp, S, Rc and R_{1}. The Nusselt number increased with an increase in Kr, S and it is decreased with an increase in R_{1}.

5. Conclusions

The present study is carried out to investigate the magneto convective flow of a non-Newtonian fluid through non-homogeneous porous medium past a vertical porous plate with variable suction. The dimensionless governing equations are solved by using the perturbation technique. The results for velocity and temperature are obtained and plotted graphically. The numerical results for skin friction and Nusselt number are computed in tables. The main conclusions of this study are as follows:

1. Velocity of the fluid increases with an increasing values of S, Gc, Gr. And it decreases in the case of Kr, Sc, Pr, M and R_{1}.

2. Temperature of the fluid increases with an increasing values of Kr, Sc and S, whereas decreased in the case of Pr.

3. Kr and Sc show negative impact on the concentration of the fluid.

4. Coeffecient of skin friction receives positive impact in case of Sc, Pr, Gr, Gc, Kr, Kp, Rc, while negative effect in the case of M and S. Sherwood number increases for increasing values of Sc and Kr. Coefficient of rate of heat transfer increases with an increase in Sc, Pr, Kr and S.

Appendix

Nomenclature

C0: Species concentration R_{1}: Radiation absorption.

C: Non-dimensional species concentration Kr: Chemical reaction

D: Molecular diffusivity

Gc: Grashof number for mass transfer M: Magnetic parameter

Gr: Grashof number for heat transfer B_{0}: Magnetic field of uniform strength

g: Acceleration due to gravity σ: Electrical conductivity

K0: Permeability of the medium ρ: Density of the fluid

Kp: Permeability/porosity parameter t: Time

k: Thermal diffusivity β: Volumetric coefficient of expansion for heat transfer

M: Magnetic parameter β^{*}: Volumetric coefficient of expansion with species concentration

N: Nusselt number

Pr: Prandtl number ε: a small positive constant

S: Heat source parameter R_{c}: Elastic parameter.

Sc: Schmidt number : Kinematic coefficient of viscosity.

Sh: Sherwood number : Constant suction velocity.

References

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