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The objective of present work is to study the thermo diffusion effect on an unsteady simultaneous convective heat and mass transfer flow of an incompressible, electrically conducting, heat generating/absorbing fluid along a semi-infinite moving porous plate embedded in a porous medium with the presence of pressure gradient, thermal radiation field and chemical reaction. It is assumed that the permeable plate is embedded in a uniform porous medium and moves with a constant velocity in the flow direction in the presence of a transverse magnetic field. It is also assumed that the free stream consists of a mean velocity, temperature and concentration over which are super imposed an exponentially varying with time. The equations of continuity, momentum, energy and diffusion, which govern the flow field, are solved by using a regular perturbation method. The behavior of the velocity, temperature, concentration, Skin-friction, rate of heat transfer and rate of mass transfer has been discussed for variations in the physical parameters. An increase in both Pr and R results a decrease in thermal boundary layer thickness. However, concentration decreases as Kr, Sc increase but it increases with an increase in both So and δ.

The study of magneto hydrodynamic flows through porous media is of considerable interest because of its abundant applications in several branches of science and technology; such as in astrophysical, geo-physical problem and in developing magnetic generator for obtaining electrical energy at minimum cost. The theory developed by viscous flow through porous media is useful in analyzing the influence of temperature and pressure on the flow of soil water. The unsteady free convection flows over semi-infinite vertical plate was studied by Takhar et al. [

Motivated by the above studies, in this paper we have considered thermo diffusion effect on an unsteady simultaneous convective heat and mass transfer flow of an incompressible, electrically conducting, heat generating/absorbing fluid along a semi-infinite moving porous plate embedded in a porous medium with the presence of pressure gradient, thermal radiation field and chemical reaction. The novelty of this study is the consideration of a double diffusion fluid (mass diffusion and as we as thermal diffusion) along with chemical reaction and heat source of a radiating fluid past or through a porous plate in a conducting filed. In spite of that many authors contributed similar works, still there are many interesting aspects to be concentrated. Hence authors are interested to carry on this investigation.

Consider a two-dimensional unsteady flow of a laminar incompressible electrically conducting and heat generating/absorbing fluid with mass transfer, past a semi-infinite vertical porous plate embedded in a porous medium in the presence of thermal radiation, chemical reaction and thermal diffusion. A uniform magnetic field is applied perpendicular to the plate. There is no applied voltage which implies the absence of the electric field. The transversely applied magnetic field and magnetic Reynolds number are assumed to be small and hence the induced magnetic field is negligible. Since the plate is semi-infinite in length all the flow variables are functions of y and t only. Under the above conditions and the usual Boussinesq’s approximation the governing equations are given as:

Continuity equation:

Momentum equation:

Energy equation:

Species diffusion equation:

By using the Rosseland approximation the radiative heat flux in y direction is given by

where σ_{s} and k_{e} are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively. It should be noted that by using the Rosseland approximation we limit our analysis to optically thick fluids. It is assumed that the temperature differences are small, then Equation (5) can be linearized by expanding T^{*4} into Taylor series about T_{∞}, and neglecting higher order terms to give:

The heating due to viscous dissipation is neglected for small velocities in energy conservation Equation (3) and Boussinesq approximation is used to describe buoyancy force in Equation (2). It is assumed that the free stream velocity, the suction velocity, the plate temperature and the plate concentration follow an exponentially increasing or decreasing small perturbation law.

Under these assumptions, the appropriate boundary conditions for the velocity, temperature and concentration fields are

From the continuity equation, it is clear that the suction velocity normal to the plate is a function of time only and we shall take it in the form

where A is a real positive constant, ε and εA are small less than unity, and V_{0} is scale of suction velocity which is a non-zero positive constant. Outside the boundary layer, Equation (2) gives

Now introduce non-dimensional parameters as follows

After substituting boundary conditions and dimensionless parameters the governing Equations (2)-(5) reduce to

where

The boundary conditions (7) are given by the following non-dimensional form

In order to reduce the above system of partial differential equations to a system of ordinary differential equations in dimensionless form, for

Substituting these Equations (15)-(17) into Equations (11)-(13) and equating the harmonic and non-harmonic terms, also neglecting the coefficient of O(e^{2}), we get the following pairs of equations.

Here primes denote differentiation with respect to y.

The corresponding boundary conditions are

The solutions of Equations (18)-(23) with satisfying boundary conditions (24) are given by

Given the velocity field in the boundary layer, we can now calculate the skin friction at the wall as

We calculate the heat transfer coefficient in terms of Nusselt number as follows

Similarly the mass transfer coefficient in terms of Sherwood number, as follows

In order to assess the accuracy of the numerical results, we have compared our results with accepted data sets for the velocity, temperature and concentration profiles for a stationary vertical porous plate corresponding to the case computed by Kim (2000). i.e., in the absence of the diffusion effects we observed that the effects of all parameters on velocity and temperature profiles are in good agreement with the comparison of Kim (2000).

It is noticed that the velocity decreases slightly with the increase of magnetic parameter M, also observed that the velocity increases with thermal Grashoff number Gr. But when the value of Gr increases the effect of magnetic parameter on velocity has high impact. The effect of the porosity parameter K on the velocity profiles has been shown in

From this figure we see that the concentration increases with the increase of both Soret number So and heat generation parameter. We have also shown some Tables 1-3 of the skin-friction, rate of heat transfer and rate of mass transfer by the effects of the Gr, Gm, K, M, Pr, Sc, R, Kr, δ and So. From these tables it is noticed that the

R | δ | Pr | Nu | S_{h} | |
---|---|---|---|---|---|

1 | 0.1 | 0.71 | 4.4530 | 0.0804 | 0.3438 |

1 | 0.1 | 1 | 4.2160 | 0.1392 | 0.3281 |

1 | 0.1 | 5 | 3.3527 | 0.9944 | 0.0766 |

1 | 0.2 | 0.71 | 4.4780 | 0.0708 | 0.3462 |

1 | 0.3 | 0.71 | 4.5065 | 0.0602 | 0.3489 |

10 | 0.1 | 0.71 | 3.9634 | 0.2354 | 0.3010 |

100 | 0.1 | 0.71 | 3.8931 | 0.2719 | 0.2905 |

So | Sc | Kr | S_{h} | |
---|---|---|---|---|

0.5 | 0.6 | 0.2 | 4.4530 | 0.3438 |

1 | 0.6 | 0.2 | 4.5413 | 0.3277 |

1.5 | 0.6 | 0.2 | 4.6296 | 0.3116 |

1 | 0.94 | 0.2 | 4.2269 | 0.5111 |

1 | 1.17 | 0.2 | 4.1269 | 0.6233 |

1 | 0.6 | 0.4 | 4.3389 | 0.4427 |

1 | 0.6 | 0.8 | 4.2059 | 0.5850 |

Gr | Gm | K | M | |
---|---|---|---|---|

2 | 2 | 0.2 | 0.5 | 7.3024 |

5 | 2 | 0.2 | 0.5 | 12.4800 |

10 | 2 | 0.2 | 0.5 | 21.1093 |

2 | 5 | 0.2 | 0.5 | 10.6732 |

2 | 10 | 0.2 | 0.5 | 16.2912 |

2 | 2 | 1 | 0.5 | 4.3351 |

2 | 2 | 2 | 0.5 | 4.3269 |

2 | 2 | 0.2 | 1 | 4.3404 |

2 | 2 | 0.2 | 2 | 4.3291 |

skin-friction at the wall increases as Grashoff number Gr or modified Grashoff number Gm or Soret number So or heat generation/absorption parameter δ increases. And it decreases with increasing Prandtl number Pr or Schmidt number Sc or radiation parameter R or chemical reaction parameter Kr or permeability parameter K or magnetic parameter M. From the analytical results, it can be seen that the rate of heat transfer depends on radiation parameter R, Prandtl number Pr and heat generation/absorption parameter δ and the rate of mass transfer depends on chemical reaction parameter Kr, Schmidt number Sc and Soret number So and the Prandtl, radiation, and heat generation/absorption as well.

The mentioned tables reveal that as Schmidt number or chemical reaction parameter increases, the rate of mass transfer (Sh) increases but it decreases with the increase of Prandtl number Pr or Soret number So or radiation parameter R. Also we observe that the rate of heat transfer (Nu) increases with the increase of Prandtl number or radiation parameter but it decreases with increasing heat generation parameter.

We have considered thermo diffusion effect on an unsteady simultaneous convective heat and mass transfer flow of an incompressible, electrically conducting, heat generating/absorbing fluid along a semi-infinite moving porous plate embedded in a porous medium with the presence of pressure gradient, thermal radiation field and chemical reaction.

1) An increase in K leads to a raise in the velocity but a reverse effect is seen in the case of Kr and M.

2) An increase in both Pr and R results a decrease in thermal boundary layer thickness. However, concentration decreases as Kr, Sc increase but it increases with an increase in both So and δ.

3) The skin-friction at the wall increases as Gr and Gm or M or So or δ increases and it decreases with increase in Pr or Sc or R or Kr or K.

4) As Sc or Kr increase, the rate of mass transfer increases but it decreases with the increase in Pr or So or R. Also we observe that the rate of heat transfer increases with the increase of Pr or R but it decreases with decreasing δ.

5) This work can be extended for other non-Newtonian fluids such as viscoelastic fluid, Rivlin Ericksen fluid, micropolar fluid etc.,

Authors are very much thankful to the anonymous reviewers for giving the constructive suggestion which helped us in improving the quality of the manuscript.

Gurivireddy P.,Raju M. C.,Mamatha B.,Varma S. V. K., (2016) Thermal Diffusion Effect on MHD Heat and Mass Transfer Flow past a Semi Infinite Moving Vertical Porous Plate with Heat Generation and Chemical Reaction. Applied Mathematics,07,638-649. doi: 10.4236/am.2016.77059