HEAT AND MASS TRANSFER ANALYSIS ON MHD MIXED CONVECTION FLOW OF RADIATIVE CHEMICALLY HEAT GENERATING FLUID WITH VISCOUS DISSIPATION AND THERMO-DIFFUSION EFFECT

In this paper an analysis on heat and mass transfer is made to study magnetohydrodynamic (MHD) mixed convective flow of an incompressible viscous fluid flowing past an inclined plate. A magnetic field of uniform strength is applied to the plate to influence the flow. Due to weak voltage differences caused by the very low polarization charges, the influence of electric field is considered to be neglected. Again large temperature gradient ensures cross diffusion effect like thermo-diffusion (Soret) in the field. The governed set of non-linear partial differential equations is solved by developing a multi-parameter asymptotic perturbation scheme. The influence of various physical parameters such as heat source parameters (Qs), chemical reaction parameter (CR), magnetic field parameter (M), Eckert number (Ec), thermal radiation parameter (R), permeability parameter (K) and plate inclination parameter (ψ) on the velocity, concentration and temperature profiles as well as skin-fraction, Nusselt number and Sherwood number are simulated numerically for the study. It reveals that, an increase in magnetic field parameter (M) decreases the axial velocity field, coefficient of skin-friction and Sherwood number but increases Nusselt number, temperature and concentration profiles. Again an increase in Eckert number (Ec) decreases both the co-efficient of skin-friction and Nusselt number while increases the temperature, Sherwood number and concentrations of the fluid particles.


INTRODUCTION
Magneto-hydrodynamic (MHD) is the study of an electromagnetic field interacting with the velocity field of an electrically conducting fluid. It played a vital role in the field of astrophysics, geophysics, nuclear reactor and many electronics devices. Manglesh and Gorla (2012) studied the effects of thermal radiation, chemical reaction and rotation on unsteady MHD visco-elastic slip flow. Ahmed etal (2013) made an exact analysis for MHD free convection mass transfer flow past an oscillating plate embedded in a porous medium with Soret effect. Nasir Uddin et al. (2014) considered the effect of conjugate heat and mass transfer on MHD mixed convective flow past inclined porous plate in porous medium. Heat and mass transfer are kinetic process that may take place jointly or separately, it can be study combined or separately. Heat and mass transfer has a significant role in many industrials and environmental processes particularly in energy utilization, thermal processing, in aerospace, thermal control and food processing. Rajuet al. (2015) analyzedheat and mass transfer in MHD mixed convection flow on a movinginclined porous plate.  investigated free convective chemically absorption fluid past an impulsively accelerated plate with thermal radiation variable wall temperature and concentrations. Sengupta and Karmakar (2016) studied the MHD mixed convection chemically reactive flow in radiative heat generating medium with Soret effect. Prashant et al. (2016) investigated heat transfer in MHD mixed convection viscoelastic fluid flow over a stretching sheet embedded in a porous medium with viscous dissipation and non-uniform heat source/sink. Misra and Adhikary (2016) studied the MHD oscillatory channel flow, heat and masstransfer in a physiological fluid in presence ofchemical reaction. Mondal et al (2017) investigated thermophoresis and soret-dufour on MHD mixed convection mass transfer over an inclined plate with non-uniform heat source/sink and chemical reaction.
Heat generation (heat source) /absorption (heat sink) which minimizes the longevity of many electronics device, it has significant role in industries as well as in nuclear reactors. So due to importance of heat generation/absorption in industrial and engineering field, many researchers worked on its effects. Ravikumar et al. (2012) studied heat and mass transfer effects on MHD flowof viscous fluid through non-homogeneous porousmedium in presence of temperaturedependent heat source. Bhavanaet al. (2013) investigated the Soret effect on free convective unsteady MHD flow over a vertical plate with heat source. Ramandeviet al. (2017) analyzed combined influence of viscous dissipation and non -uniform heat source/sink on MHD non-Newtonian fluid flow with cattaneo-christov heat flux. Recently Venkataramana et al. (2017) investigated the influence of heat generation (absorption) and thermal radiation on MHD laminar boundary layer flow over a moving cylindrical rod.
Chemical reaction has an important role in chemical reaction process such as polymer production food processing. Due to which researchers draw a kin interest in its effect. Many researchers work on effects of chemical reaction. Saleh et al. (2010) studied heat and mass transfer in MHD visco-elastic fluid flow through a porous medium over a stretching sheet with chemical reaction. Pal and Talukdar (2011) investigated the combined effects of joule heating and chemical reaction on unsteady magnetohydrodynamic mixed convection of a viscous dissipating fluid over a vertical plate in porous media with thermal radiation. Pal and Mondal (2012) studied the influence of chemical reaction and thermal radiation on mixed convection heat and mass transfer over a stretching sheet in Darcian porous medium with Soret and Dufour effects. Ibrahim (2014) considered the effects of chemical reaction on dissipative radiative MHD flow through a porous medium over a non-isothermal stretching sheet. Rudraswamy and Gireesha (2014) studied the influence of chemical reaction and thermal radiation on MHD 2 boundary layer flow and heat transfer of a nano fluid over an exponentially stretching sheet. Jonnadula et al. (2015) investigated the influence of thermal radiation and chemical reaction on MHD flow, heat and mass transfer over a stretching surface. Sengupta and Ahmed (2015) studied the effect of chemical reaction interaction on unsteady MHD free convective radiative flow past an oscillating plate embedded in porous media with thermal diffusion. Nayak et al. (2016) studied heat and mass transfer effects on MHD visco-elastic fluid over a stretching sheet through porous medium under the influence of chemical reaction. Jena et al. investigated the chemical reaction effect on MHD viscoelastic fluid flow over a vertical stretching sheet with heat source/sink. Srinivasacharya and Swamy Reddy,(2016) investigated chemical reaction and radiation effects on mixed convection heat and mass transfer over a vertical plate in power-law fluid saturated porous medium. Reddy et al. (2016) studied the influence of chemical reaction, radiation and rotation on MHD nano fluid flow past a permeable flat plate in porous medium.
The process through which the fluids takes energy (kinetic energy) from the motion of the fluid and transform it into internal energy of the fluid is called viscous dissipation. It is partially an irreversible process. Reddyet al. (2015) studied the effects of viscous dissipation and heat source on unsteady MHD flow over a stretching sheet. Of late Ahmed et al. (2017) studied the effects of chemical reaction, heat and mass transfer and viscous dissipation over a MHD flow in a vertical porous wall using perturbation method. Shamshuddin (2017) investigated the influence of heat and mass transfer on the unsteady MHD flow of chemically reacting micropolar fluid with radiation and Joule heating. It is well understood that driving potential for the phenomenon of transport of mass is not only concentration gradients, but also high temperature gradients are also one of the means. The dual process by which mass gets transferred is known as thermo-diffusion or Soret effect. Due to the scientific and industrial importance of Soret effect, many researchers contributed a lot on its effects. Sengupta and Sen (2013) considered the study of thermo-diffusion (Soret) on a free convective heat and mass transfer flow past an oscillating plate with heat generation and thermal radiation. Mamatha et al. (2015) investigated the thermal diffusion effect on MHD mixed convection unsteady flow of a micro polar fluid past a semi-infinite vertical porous plate with radiation and mass transfer.  made an analysis of Soret effect on unsteady heat and mass transfer flow of radiative chemically reactive fluid past an oscillating plate embedded in porous media. Rashidi et al. (2015) investigated heat and mass transfer for MHD viscoelastic fluid flow over a vertical stretching sheet with considering Soret and Dufour effects. Reddy et al. (2016) investigated the Soret and Dufour effects on MHD free convection flow of Rivlin-Ericksen fluid past a semi-infinite vertical plate. Sengupta and Ahmed (2016) studied MHD free convective mass transfer flow of radiative uniform heat generation (absorption) fluid through a wavy permeable channel in presence of Soret and Dufour effects. Ullah et al. (2017) investigated Soret as well as Dufour effects on unsteady mixed convection slip flow of Caisson fluid over anonlinearly stretching sheet withconvective boundary condition. Very recently Sengupta and Karmakar (2018) investigated the induced magnetic field interaction in free convective heat and mass transfer flow of a chemically reactive heat generating fluid with thermo-diffusion and diffusion-thermo effects. Shamshuddin et al. (2018) compute the unsteady MHD mixed convective heat and mass transfer in dissipative reactive micropolar flow considering Soret and Dufour effects.
In spite of these studies, less attention has been given by authors to free stream oscillation flow with combine effect of viscous dissipation, thermo-diffusion (Soret) and thermal radiation on a flow of electrically conducting fluid. The objective of the present study is to make an analysis of combine effect of thermo-diffusion (Soret), viscous dissipation and thermal radiation of electrically conducting and chemically reactive fluid in heat generating, Darcian porous media in presence of magnetic field with an inclined plate condition.

Basic Equations
The vector forms of equations that describe the flow situation are as follows The velocity vector and the magnetic field vectors are given by With electric field E = (0, 0, 0), magnetic field B = (0, 0 B , 0 ) and  , as the electrical conductivity of the medium.

The vector form of energy equation gives
and represents Cauchy's stress tensor. The species continuity equations as: The other symbols are specified in the Nomenclature.

Basic Assumptions
The fundamental assumptions considered for the study are as follows: a) All the fluid properties except possibly the pressure are independent of variations of x*-scale. b) The strength of the applied magnetic field, which is applied to the inclined plate, considered as moderate and uniform. Due to the small value of magnetic Reynold's number, the influence of induced magnetic field in comparison with the applied magnetic field is less prominent and thus can be ignored for the study. c) The Hall as well as magnetic heating effects gets less importance due to moderate strength of the magnetic field. d) As the applied voltage at the ends of the plate is weak as such polarization of charges is neglected, so the electric field is not considered for the study. e) Due to higher temperature gradient, the cross diffusion effects, like thermo-diffusion (Soret) have been considered for the study.
f) The plate is considered to be magnetically insulated. The temperature as well as concentration of fluid particles near the plate surface is supposed to be more than their respective components at the free stream region i.e., Tw>T∞, Cw>C∞.

MATHEMATICAL FORMULATIONS OF THE PROBLEM AND SOLUTIONS
A co-ordinate system (x*, y*, z*) has been introduced, with its x*-axis along the length of the plate in the upward vertical direction, y*-axis taken perpendicular to the plate towards the fluid region and z*-axis considered along the width of the plate. A uniform magnetic field of strength 0 B is applied to the plate. Following Bhuvaneswari et al. 3 (2010) and Barik and Dash (2014), an MHD fluid model has been developed based on conservation principles of mass, momentum, energy and species concentration as well as Boussineque approximations with initio-boundary conditions as: (2)

Fig. 1 A Schematic representation of the flow configuration and geometry
(3) With boundary conditions as: The hydrostatics pressure in (2) can be calculated from Bernoulli's pressure equations as: The radiative heat flux for an optically thick boundary layer flow can also be simplified with the Rosseland approximation model as: (7) Where ϭ1 and k1are the Stefan Boltzmann constant and Rosseland mean absorption coefficient respectively. Assuming that the temperature differences with the flow are sufficiently small, so T *4 may be expressed as: We consider the following dimensionless variables as: On using the dimensionless quantities, equations (1) -(4) can thus be transformed as: (7) and (8) in (3) gives, With non -dimensional boundary conditions as: The equation (7) suggests an asymptotic time dependent solution of v:

METHODOLOGY
To integrate the set of non-liner, coupled system of equations (9) to (12) subject to conditions (13) and (14), we propose a multi -parameter perturbation method. Following this we take 1 st set asymptotic solutions form as: Where f stands for u, Ɵ and φ. On collecting the co-efficient of various powers of fluctuating parameter, we obtain a set of non -perturbed and perturbed system of equations as: With the boundary conditions follow as, We again propose a set of 2nd part of trial solutions as: With the boundary conditions as: Again a set of perturbed equations also formed as: Subject to a set of boundary conditions:  11 11 11 11 11 0, 0, 0, 0, 0, 0, when t>0 , 0, , 0, , 0 when t>0 On using straight forward integration schemes, the closed form solutions for the equations (22) to (27)

Skin friction at the plate:
The real part of the non-dimensional skin-friction coefficient at the plate is obtained as:

Rate of mass transfer coefficient:
The mass transfer coefficient in terms of the Sherwood number as:

x a A A a x a A A a x a A A a x a A A a
Again it is found that, skin-friction and Nusselt number decrease but Sherwood number increases due to increase in Soret number (Sr). It is also clearly observed that, the Nusselt number and Sherwood number increase, while skin-friction decreases due to increase in the values of solutal Grashof number (Gm). The increase in permeability parameter (K) is seen to increase the skin-friction but a reverse trend is observed in case of Nusselt and Sherwood numbers. Table1 also reflects that Nusselt number and Sherwood number increase while skin-friction decreases due to increase in values of Eckert number (Ec).

CONCLUSIONS
A theoretical study on heat and mass transfer analysis on megnetohydrodynamic (MHD) mixed convection flow of an incompressible viscous fluid in presence of thermo-diffusion along with first order chemical reaction flowing past an inclined plate is made. The significant outcome of the study is as follows: (a) The temperature field increases due to increase in values of Eckert number, Magnetic field parameter, heat source parameter and thermal radiation parameter. (b) The concentration field increases due to increase in values of heat source parameter, Eckert number, magnetic field parameter and Soret number, while concentrations decrease due to rise in values of chemical reaction parameter and Schmidt number. (c) The velocity field increases due to increase in values of heat source parameter, permeability parameter, thermal radiation parameter, Soret number and Eckert number, while it decreases due to step up in values of magnetic field parameter, chemical reaction parameter, and plate inclination parameter. (d) The coefficient of skin-friction increases due to increase in values of thermal Grashof number and permeability parameter but the surface friction decreases due to increase in magnetic field parameter, Soret number, solutal Grashof number, Eckert number and plate velocity parameter. (e) Due to increase in magnetic field parameter, Soret number and plate velocity parameter, the Nssult number is found increasing, while an enhancement in thermal as well as solutal Grashof number, the heat transfer rate is found decreasing. (f) The step-up in values of Soret number, Grashof number, permeability parameter, Eckert number and plate velocity parameter accelerates the mass transfer rates, while decrease with rise in magnetic field parameter.

ACKNOWLEDGEMENT
The authors extend their heartfelt thanks to the Reviewers and the Editor for providing valuable comments / suggestions to improve the quality of the paper.