The Dufour and Soret effects in the combined heat and mass transfer processes, due to the thermal energy flux resulting from concentration gradients and the thermal diffusion flux resulting from the temperature gradients, may be significant in the areas of geosciences and chemical engineering, Eckert and Drake [1]. Such physical effects were explored in the papers by Kafoussias and Williams [2], by Anghel et al. [3] and by Lin et al. [4], amongst others. Tai and Char [5] examined the Soret and Dufour effects on free convection flow of non-Newtonian fluids along a vertical plate embedded in a porous medium with thermal radiation. Venkateswarlu et al. [6-9] presented heat and mass transfer characteristics on MHD flows with chemical reaction and thermal radiation.

We consider the unsteady laminar slip flow of an incompressible, viscous and electrically conducting fluid through a channel with non-uniform wall temperature bounded by two parallel plates separated by a distancea . The channel is assumed to be filled with a saturated porous medium. A uniform magnetic field of strength oBis applied perpendicular to the plates. The above plate is heated at constant temperature and thermal radiation effect is also taken in to account. It is assumed that there exist a homogeneous chemical reaction of first order with constant rate rKbetween the diffusing species and the fluid. Geometry of the problem is presented in Figure. 1. We choose a Cartesian coordinate system (,)xy where xlies along the centre of the channel, y is the distance measured in the normal section such that ya is the channel‘s width as shown in the figure below. Under the usual Boussinesq‘s approximation, the equations of conservation of mass, momentum, energy and concentration governing the natural convective nonlinear boundary layer flow over a laminar porous plate in porous medium can be expressed as:

Continuity equation:

 (1)

Momentum equation:

6

 (2)

Energy equation:

222

 

 (3)

Diffusion equation;

2

 (4)

where u fluid velocity in xdirection, v fluid velocity along ydirection, pfluid pressure, g acceleration due to gravity,  fluid density, T coefficient of thermal expansion, C coefficient of concentration volume expansion, t time, K permeability of porous medium, oB magnetic induction, T fluid temperature, oT temperature at the cold wall, TK thermal diffusivity of the fluid,  dimensional radiation parameter, C species concentration in the fluid, oC concentration at the cold wall, e fluid electrical conductivity,pc specific heat at constant pressure, mD chemical molecular diffusivity,sc concentration susceptibility, kinematic viscosity of the fluid and rK dimensional chemical reaction parameter respectively. Assuming that slipping occurs between the plate and fluid, the corresponding initial and boundary conditions of the system of partial differential equations for the fluid flow problem are given below

 

100 211

11

,,at0 ,expint, expintat

  

 (5)

where 1Tfluid temperature at the heated plate, 1Cspecies concentration at the heated plate, 1cold wall dimensional slip parameter, 2heated wall dimensional slip parameter, n frequency of oscillation and1 is a very small positive constant.

2122

2

211

,,,,,,

  

 (6)

Equations (2), (3) and (4) are reduced to the following non-dimensional form

2

21UdPUGrGmMUdDa

 (7)

22

221

PrDuH

 (8)

2



 (9)

Here 312TogTThGrv is the thermal buoyancy force,3102CgCChGmv is the concentration buoyancy force, 22eoBhMv is the magnetic parameter,2KDah is the Darcy parameter,PrpTcK is the Prandtl number, 224phNc is the thermal radiation parameter, 11mTospoDKCCDuccTT is the Dufour number,mScD is the Schmidt number and 2rhKrK is the chemical reaction parameter respectively. Corresponding initial and boundary conditions are given by



,0,0at0 ,1exp,1expat1

 

 (10)

Following Adesanya and Makinde [12], for purely an oscillatory flow we take the pressure gradient of the form

01exp()dPitd

(11)

where 0and1are constants and is the frequency of oscillation.

Due to the selected form of pressure gradient we assume the solution of equations (7) to (9) of the form

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(12)

201(,)()exp()io (13) 201(,)()expio (14) Substituting equations (12) to (14) into equations (7) to (9), then equating the harmonic and non–harmonic terms and neglecting the higher order terms of2o, we obtain 000001UMUGrGmDa (15) 111111UMiUGrGmDa (16) 000PrPrNDu (17) 111PrPrNiDu (18) 000ScKr (19) 110ScKri (20) where the prime denotes the ordinary differentiation with respect to. Initial and boundary conditions in equation (10), can be written as

1111 1111

,,0,0,0,0at0 ,,1,1,1,1at1

 

 (21)

We obtained the analytical solutions for the fluid velocity, temperature and concentration and are presented in the following form









225215 6371831 396386 2342422542

expexp ,sinh()sinh() sinh()sinh() expexp expsinh()sinh() sinh()sinh()







 

 (22)

 

31

5442

42

sinh()sinh() sinh()sinh()expsinh()sinh()

 

 (23)

12

12

sinh()sinh(),expsinh()sinh()

 (24)

Here the constants are not given under the brevity

In order to investigate the influence of various physical parameters such as thermal Grashof number Gr, solutal Grashof number Gm, Darcy parameter Da, pressure gradient , magnetic parameterM, cold wall slip parameter, heated wall slip parameter , Prandtl number Pr, chemical reaction parameter Kr, mass diffusion parameterScand Dufour effect Du on the flow-field, the fluid velocity U, temperature and concentration  have been studied analytically and computed results of the analytical solutions from equations (22) to (24) are displayed graphically from Figs.2 to 20 for various values of these physical parameters. In the present study following default parameter values are adopted for computations: 4,Du

5,10,5,3,1,2,Air(Pr0.71), Water(Pr7.0),Hydrogen(0.22),Ammonia(0.78)

5,0.5,0.1,0.1,0,0.005.



  



Therefore all the graphs are corresponding to these values unless specifically indicated on the appropriate graph. Figure 2: Velocity Uagainst for varying Grand M .

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Figure 3: Velocity Uagainst for varying Grand Pr . Figure 4: Velocity Uagainst for varying Grand Sc. Figure 5: Velocity Uagainst for varying Gmand M . Figure 6: Velocity Uagainst for varying Gmand Pr. Figure 7: Velocity Uagainst for varying Gmand Sc. Figure 8: Velocity Uagainst for varying Daand M . Figure 9: Velocity Uagainst for varying Daand Pr.

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Figure 10: Velocity Uagainst for varying Daand Sc . Figure 11: Velocity Uagainst for varying DuandM. Figure 12: Velocity Uagainst for varying DuandPr. DuandSc. Figure 14: Velocity Uagainst for varying NandM. Figure 15: Velocity Uagainst for varying NandPr. Figure 16: Velocity Uagainst for varying NandSc.

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Figure 17: Velocity Uagainst for varying KrandM. Figure 18: Velocity Uagainst for varying KrandPr. Figure 19: Velocity Uagainst for varying KrandSc. Figure 21: Velocity Uagainst for varying andM. The effect of thermal Grashof number Gr on the velocity U of the flow field is presented in Figs. 2 to 4. Physically, thermal Grashof number Gr signifies the relative strength of thermal buoyancy force to viscous hydrodynamic force in the boundary layer. A study of the curves shows that thermal Grashof numberGraccelerates the velocity of the flow field at all points. This is due to the reason that there is an enhancement in thermal buoyancy force. The effect of solutal Grashof numberGm on the velocity U of the flow field is presented in Figs. 5 to7. Physically, Solutal Grashof numberGmsignifies the relative strength of species buoyancy force to viscous hydrodynamic force in the boundary layer. A study of the curves shows that solutal Grashof numberGmaccelerates the velocity of the flow field at all points. This is due to the reason that there is an enhancement in concentration buoyancy force. Figs. 8 to 10 show the variation of fluid velocity Uwith the Darcy parameterDa. The graph shows that an increase in the Darcy parameter increases the fluid flow except at the flow reversal point at the heated wall. The influence of Dufour number Du on velocityU is plotted in Figs. 11 to 13. The Dufour number signifies the contribution of the concentration gradients to the thermal energy flux in the flow. It is found that an increase in Dufour number causes a rise in the velocity.It is observed that from Figs.14 to 16, the fluid velocity U decreases on increasing the radiation parameterN. Figs. 17 to 19 demonstrate the effects of chemical reaction parameter Kr on the velocity. It is observed that, velocity U increases on increasing the chemical reaction parameterKr. Figs. 20 and 21 shows the fluid velocity profile variations with the cold wall slip parameter  and the heated wall slip parameter. It is observed that, the fluid velocity U increases on increasing the cold wall

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cold wall slip parameter did not cause any appreciable effect on the heated wall. An increase in the heated wall slip parameterdecreases the fluid velocity minimally at the cold wall and increasing the heated wall slip parameter causes a flow reversal towards the heated wall. It is observed that0 corresponds to the pulsatile case with no slip condition at the heated wall in Fig 21.

In this paper we have studied the Dufour Effect on Radiative MHD Flow of a Viscous Fluid in a Parallel Porous Plate Channel under the Influence of Slip Condition. From the present investigation the following conclusions can be drawn: An increase in the magnetic parameterMdecreases the fluid velocityUdue to the resistive action of the Lorenz forces except at the heated wall where the reversed flow induced by wall slip caused an increase in the fluid velocity. This implies that magnetic field tends to decelerate fluid flow. Velocity profiles are increases on increasing the Prandtl number. Influence of thermal buoyancy force decreases on velocity on increasing the Schmidt number. An increase in the heated wall slip parameterdecreases the fluid velocity minimally at the cold wall and increasing in the heated wall slip parameter causes a flow reversal towards the heated wall. The cold wall slip parameter did not cause any appreciable effect on the heated wall.

E. R. G. Eckert and R. M. Drake, Analysis of Heat and Mass Transfer, McGraw-Hill, New York, 1972. N.G. Kafoussias and E.W. Williams, Thermal-diffusion and diffusion-thermo effects on mixed free-forced convective and mass transfer boundary layer flow with temperature dependent viscosity, Int. J. Eng. Sci. vol. 33, no. 9, pp. 1369-1384, 1995. M. Anghel, H. S. Takhar and I. Pop: Dufour and Soret effects on free-convection boundary layer over a vertical surface embedded in a porous medium, Studia Universitatis Babes Bolyai, Mathematica, vol. XLV, pp. 11-21, 2000. H. C. Lin, M. I. Char and W. J. Chang, Soret effects on non-Fourier heat and non- Fickianmass diffusion transfer in a slab, Numer. Heat Transfer Part A, vol. 55, pp. 1096-1115, 2009. Bo-C. Tai and M.-I. Char Soret and Dufour effects on free convection flow of non- Newtonian fluids along a vertical plate embedded in a porous medium with thermal radiation, Int. Commun.

2010.

M. Venkateswarlu, G. V. Ramana Reddy and D. V. Lakshmi, Thermal diffusion and radiation effects on unsteady MHD free convection heat and mass transfer flow past a linearly accelerated vertical porous plate with variable temperature and mass diffusion, J. Korean Soc. Ind. Appl. Math., vol. 18, pp. 257-268, 2014. M. Venkateswarlu, G. V. Ramana Reddy and D. V. Lakshmi, Radiation effects on MHD boundary layer flow of liquid metal over a porous stretching surface in porous medium with heat generation, J. Korean Soc. Ind. Appl. Math., vol. 19, pp.83-102, 2015. M. Venkateswarlu and P. Padma, Unsteady MHD free convective heat and mass transfer in a boundary layer flow past a vertical permeable plate with thermal radiation and chemical reaction, Procedia Engineering, vol. 127, pp. 791-799, 2015. M. Venkateswarlu, D. Venkata Lakshmi and K. Nagamalleswara Rao: Soret, hall current, rotation, chemical reaction and thermal radiation effects on unsteady MHD heat and mass transfer natural convection flow past an accelerated vertical plate, J. Korean Soc. Ind. Appl. Math., vol. 20, no. 3, pp. 203 -224, 2016. S. O. Adesanya and O. D. Makinde: MHD oscillatory slip flow and heat transfer in a channel filled with porous media, U.P.B. Sci. Bull. Series A, vol.76, pp. 197-204, 2014.

Department of Mathematics, V. R. Siddhartha Engineering College, Krishna (Dist), Andhra Pradesh, India