^{1}

^{1}

^{*}

^{2}

Radiation absorption and chemical reaction effects on unsteady MHD free convective flow of a viscoelastic fluid past a vertical porous plate in the presence of variable suction and heat source is considered. A uniform magnetic field is assumed to be applied in the transverse direction of the flow. The set of non-linear partial differential equations is transformed into a set of ordinary differential equations by super imposing a solution with steady and unsteady part. The set of ordinary differential equations is solved by using regular perturbation scheme. The expressions for velocity, temperature and species concentration fields are obtained and the expressions for Skin friction, Nusselt number and Sherwood number are also derived. The effects of numerous physical parameters on the above flow quantities are studied with the help of graphs and tables.

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. [

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.

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^{* }β€ 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

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

where ^{*}_{p} is porosity, Ο^{*} is frequency of oscillation, t^{*} is time, Ξ΅ is a small positive constant.

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

Here

(4)

Introducing the non-dimensional quantities,

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

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

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

Now the boundary conditions are reduced to the following forms

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.

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

First order equations

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:

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.

Pr | Gr | Gc | M | Sc | Kr | R1 | Kp | Rc | S | π | Sh | Nu |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.71 | 20 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 1.7939 | 0.5513 | 0.3811 |

0.71 | 20 | 10 | 1 | 0.66 | 0.1 | 0.1 | 100 | 0.1 | 1 | 3.9012 | 1.0714 | 0.4061 |

0.71 | 20 | 10 | 1 | 0.78 | 0.1 | 0.1 | 100 | 0.1 | 1 | 5.5524 | 1.2913 | 0.4054 |

Pr | Gr | Gc | M | Sc | Kr | R1 | Kp | Rc | S | π | Sh | Nu |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.025 | 20 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 1.1192 | 0.2999 | 0.0194 |

0.71 | 20 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 9.8535 | 0.2999 | 0.3518 |

7 | 20 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 13.9820 | 0.2999 | 5.1590 |

Pr | Gr | Gc | M | Sc | Kr | R1 | Kp | Rc | S | π | Sh | Nu |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 6.5856 | 0.2999 | 0.3518 |

0.71 | 15 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 8.2196 | 0.2999 | 0.3518 |

0.71 | 20 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 9.8535 | 0.2999 | 0.3518 |

Pr | Gr | Gc | M | Sc | Kr | R1 | Kp | Rc | S | π | Sh | Nu |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.71 | 10 | 10 | 1 | 0.60 | 0.1 | 0.1 | 100 | 0.1 | 1 | 6.5856 | 0.2999 | 0.3518 |

0.71 | 10 | 15 | 1 | 0.60 | 0.1 | 0.1 | 100 | 0.1 | 1 | 8.2444 | 0.2999 | 0.3518 |

0.71 | 10 | 20 | 1 | 0.60 | 0.1 | 0.1 | 100 | 0.1 | 1 | 9.9033 | 0.2999 | 0.3518 |

Pr | Gr | Gc | M | Sc | Kr | R1 | Kp | Rc | S | π | Sh | Nu |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 6.5856 | 0.2999 | 0.3518 |

0.71 | 10 | 10 | 2 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 2.9872 | 0.2999 | 0.3518 |

0.71 | 10 | 10 | 5 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 2.7055 | 0.2999 | 0.3518 |

Pr | Gr | Gc | M | Sc | Kr | R1 | Kp | Rc | S | π | Sh | Nu |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 6.5856 | 0.2999 | 0.3518 |

0.71 | 10 | 10 | 1 | 0.22 | 0.3 | 0.1 | 100 | 0.1 | 1 | 8.5864 | 0.3955 | 0.3634 |

0.71 | 10 | 10 | 1 | 0.22 | 0.5 | 0.1 | 100 | 0.1 | 1 | 10.0082 | 0.4662 | 0.3718 |

Pr | Gr | Gc | M | Sc | Kr | R1 | Kp | Rc | S | π | Sh | Nu |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.1 | 90 | 0.1 | 1 | 6.5832 | 0.2999 | 0.3518 |

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.1 | 95 | 0.1 | 1 | 6.5844 | 0.2999 | 0.3518 |

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 6.5856 | 0.2999 | 0.3518 |

Pr | Gr | Gc | M | Sc | Kr | R1 | Kp | Rc | S | π | Sh | Nu |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 6.5856 | 0.2999 | 0.3518 |

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 3 | 6.5808 | 0.2999 | 0.3568 |

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 5 | 6.5799 | 0.2999 | 0.3577 |

Pr | Gr | Gc | M | Sc | Kr | R1 | Kp | Rc | S | π | Sh | Nu |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0 | 1 | 17.7144 | 0.2999 | 0.3518 |

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.05 | 1 | 3.2928 | 0.2999 | 0.3518 |

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.2 | 1 | 1.3171 | 0.2999 | 0.3518 |

Pr | Gr | Gc | M | Sc | Kr | R1 | Kp | Rc | S | π | Sh | Nu |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.1 | 100 | 0.1 | 1 | 6.5856 | 0.2999 | 0.3518 |

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.3 | 100 | 0.1 | 1 | 6.5993 | 0.2999 | 0.3375 |

0.71 | 10 | 10 | 1 | 0.22 | 0.1 | 0.5 | 100 | 0.1 | 1 | 6.6130 | 0.2999 | 0.3232 |

_{1} results a decrease in the velocity profile. _{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}.

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.

S. HarinathReddy,M. C.Raju,E. KeshavaReddy, (2016) Magneto Convective Flow of a Non-Newtonian Fluid through Non-Homogeneous Porous Medium past a Vertical Porous Plate with Variable Suction. Journal of Applied Mathematics and Physics,04,233-248. doi: 10.4236/jamp.2016.42031

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 Ο: Electrical conductivity

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

Kp: Permeability/porosity parameter t: Time

k: Thermal diffusivity Ξ²: Volumetric coefficient of expansion for heat transfer

M: Magnetic parameter Ξ²^{*}: Volumetric coefficient of expansion with species concentration

N: Nusselt number

Pr: Prandtl number Ξ΅: a small positive constant

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

Sc: Schmidt number

Sh: Sherwood number