Variable Viscosity and Thermal Conductivity Effect of Soret and Dufour on Inclined Magnetic Field in Non-Darcy Permeable Medium with Dissipation

The analysis of thermal-diffusion (Soret) and diffusion-thermo (Dufour) effects on variable thermal conductivity and viscosity in a dissipative


Introduction
The study of the consequence of variable viscosity of the fluid and thermal conductivity with an inclined magnetic field in an incompressible flow stimulated by the instantaneous actions of buoyancy forces consequential from non-Darcy porous medium on heat and mass transfer is important from the practical and theoretical point of view due to it applications in planetary atmosphere research and others.During many mechanical forming processes, heat generation is essential in the aspect of chemical reaction.Presently, improvement has been significantly achieved in the analysis of MHD heat and mass transfer flow as a result of it usefulness in several devices, such as Hall accelerator, power engineering, MHD power generator and underground spreading of chemical wastes are the few areas where the combined diffusion-thermo and thermal-diffusion influences are observed.
There are numerous engineering cases where joint heat and mass transfer take place concurrently such as desert coolers, chemical reactors, humidifiers, dehumidifiers etc.In few of these, [1][2][3] carried out analysis on radiative mixed convection MHD flow in a permeable medium with heat and mass transfer near a vertical surface while Singh & Makinde [4] analyzed computational dynamics of Newtonian heating magnetohydrodynamic flow of volumetric heat generation past an inclined surface.[5][6] studied flow of heat and mass transfer for hydrodynamic radiative fluid through a porous moving plate.It was reported that the interface of the magnetic field is counter prolific in improving the concentration and velocity profiles favorable in achieving superior temperature inside the fluid flow field.I-Chung [7] reported on heat and mass transfer over a stretching sheet in magnetohydrodynamic flow.It was observed that the temperature at unchanging position raised with an increase in the magnetic field and heat generation terms but reduced with an increase in the Prandtl number.The magnetic field term affects the velocity profile and as well accelerates the temperature profile indirectly.
Due to its several applications, an analytical study was carried out involving permeable plates with an inclined magnetic poiseuille fluid flow by Manyonge et al. [8].Inclined magnetic field with chemical reaction effects on semi infinite porous surface through a permeable media was examined by Sugunamma et al. [9].It was noticed that the velocity decreased as the inclined magnetic field and Hartmann number increases.
The above cited authors studied with the assumption that the physical characteristics of the ambient fluid were constants.However, the physical characteristics of the flow fluid can vary considerably in the presence of temperature, particularly for fluid viscosity.In other to forecast the flow and the rate of heat transfer, it is important to consider the temperature dependent viscosity of the fluid.[10][11] studied the effects of magnetohydrodynamic fluid, variable thermal conductivity and viscosity in a convective boundary conditions past a permeable sheet.[12][13][14][15] carried out analysis on the influences of variable thermal conductivity and viscosity on hydromagnetic flow over a moving permeable surface with heat source.It was reported that an increase in parameter value of r θ caused the velocity profiles to increase.Devi and Gururaj [16] examined the flow of heat transfer of power-law velocity and nonlinear radiation along with variable viscosity on magnetohydrodynamic past a moving surface.
The above studies continued their discussion by assuming the magnetic field to be at right angle to the flow, Soret and Dufour effects were also taken to be insignificant.Nevertheless, it was believed that these bodily characteristics changes considerably whenever the effect of variable thermal conductivity and viscosity are regarded.An incompressible flow fluid possessions are appreciably varied contrast to constant physical properties.The present study focused on the fluid chattels which are temperature dependent.Consequently, the main objective of the study is to examine the effects of Soret and Dufour on variable thermal conductivity and viscosity with viscous dissipation and inclined magnetic field in a permeable medium.

Problem Formulation
Consider an incompressible, laminar flow fluid with variable thermal conductivity and viscosity through permeable sheet.The flow is driven by of buoyancy forces past a porous medium.The flow fluid is in xdirection with y -axis normal to it.The fluid viscosity is assumed to vary as a reciprocal of a linear function of temperature.An inclined magnetic field 0 with the boundary conditions: where u and v are the Darcian velocity component in x and y direction, T and C are the temperature and species concentration of the fluid.w u is the fluid velocity at the wall.The physical quantities σ The viscosity is taken to be differ as a reciprocal temperature function Lai and Kulacki [17].
where the velocity components, temperature and concentration respectively become Using equations ( 6)-( 9) in the governing equations, the continuity equation is satisfied while equations ( 2 The corresponding boundary conditions becomes is the Eckert number, with w τ , w q and m q are respectively taking to be Thus, the skin friction, Nusselt and Sherwood numbers becomes where is the Reynolds number.

Results and Discussion
The coupled non-linear equations along with the boundary conditions are solved numerically.The computational analysis are examined for different values of the terms.The following parameter values are adopted for the computation: Table 1 represents the computational results, this show the influence of some parameters on the heat transfer rate at the wall in an existing studied comparing with the present study.Table 2 shows the numerical results, this depict the influence of some bodily parameters on flow.It is observed that a rise in the value of the parameters a H , α and a D reduces the skin friction and causes a rise in the energy and mass gradient at the wall while a rise in the values of c E and r θ causes an increase in the skin friction and the temperature gradient at the wall but decreases the concentration gradient at the wall.Also, as the values of a D and m increases there is reduction in the skin friction and increase in the concentration gradient while a rise in the value of a D enhances the temperature gradient at the wall and variational rise in the values of m decreases the heat gradient.retarded the flow velocity and cause it to be heater as it moves beside the sheet which bring about decrease in the velocity profile due to the present of Lorentz force that drag the flow rate .
Fig. 3 shows the effect of the inclined magnetic field on the velocity.It is observed that an increase in the inclination of the magnetic field influence the buoyancy force which accordingly decreases the driving force to the flow fluid and thereby decreases the flow velocity.

Conclusion
The influences of variable thermal conductivity and viscosity dissipative heat and mass transfer on inclined magnetic field in a Darcy-forrcheimer media are investigated.From the numerical results, it can be deduced that, an increase in the values of

BFig. 1 .
Fig. 1.The geometry of the model Both r T and s are constants which as to do with fluid thermal property and the reference state, where 0 is linear, taken to vary as a function of temperature Chiam[18].

θ
in the free stream viscosity value ∞ µ , takes place at the plate surface when is positive for gases and negative for liquids.From the expansion, as of attention for this flow are the skin friction f C , the Nusselt u N as well as sherwood numbers Sh which are respectively defined as:

Fig. 2
Fig. 2 depicts the influence of magnetic field term a H on the fluid flow.An increase in the values of a H

Fig. 2 .
Fig. 2. Velocity profiles for different values of a H Fig. 4 illustrates the effect of variation in the mass transfer boundary layers with Soret number.It is seen that the mass transfer boundary layers thickness increases as the Soret number rises thereby causes a rise in the concentration fields since mass is unable to transfer away from the system due to thickness in the mass boundary layer.The influence of viscosity on the flow, energy and mass transfer are represented in Figs. 5, 6 and 7. A rise in the values of r θ retarded the fluid velocity near the plate surface at 1 ≤ η

Fig. 8
Fig. 8 represents the effect of thermal boundary layers with the Dufour number u D .It is noticed that the thermal boundary layers thickness increases with a rise in the Dufour number and thereby enhance the heat within the system that turn to increase the temperature profile.

Fig. 10
Fig. 10 represents the effect of m on the temperature.It is noticed that an increase in the values of m increases the temperature field.Heat moves from the plate surface to the ambient medium when 0 f m , otherwise it moves away from the ambient medium to the stretching sheet.

aH
, α , r θ and a D retarded the movement of the flow by causes decrease in the flow velocity while a rise in the Soret parameter values manifested as a rise in the flow velocity and concentration distributions.α , δ and u D enhances the temperature boundary layer thickness and 0 Q are the fluid electric conductivity, kinematics viscosity, dynamic viscosity, free stream density, permeability of the medium, specific heat at constant pressure, Forchheimer inertia coefficient, thermal conductivity, mass diffusion coefficient, reaction rate coefficient and internal heat generation respectively.g is the gravitational acceleration, T β and C β are the coefficients of thermal and concentration expansion, A , B , m , n , b are prescribed constants, F is the forchheimer parameter of the medium.

Table 2 .
Effect of a