Cmjv04i05p0326 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.4 (5), 326-338 (2013)

Radiation and Diffusion-Thermo Effects on MHD Flow Past An Infinite Vertical Porous Plate in the Presence of A Chemical Reaction K. RAJASEKHAR1, G. V. RAMANA REDDY2 and B. D. C. N. PRASAD3 1

Department of Mathematics, RVR & JC College of Engineering, Guntur, INDIA. 2 Department of Mathematics, KL University, Vaddeswaram, Guntur (Dt), INDIA. 3 Department of Mathematics, PVP Siddhartha Institute of Technology, Vijayawada, INDIA. (Received on: August 23, 2013) ABSTRACT The objective of the present study is to investigate the effect of flow parameters on the free convection and mass transfer of an unsteady magnetohydrodynamic flow of an electrically conducting, viscous, and incompressible fluid past an infinite vertical porous plate under oscillatory suction velocity and thermal radiation. The Dufour (diffusion thermo) and Chemical reaction effects are taken into account. The problem is solved numerically using the perturbation technique for the velocity, the temperature, and the concentration field. The expression for the skin friction, Nusselt number and Sherwood number are obtained. The effects of various thermo-physical parameters on the velocity, temperature and concentration as well as the skin-friction coefficient, Nusselt number and Sherwood number has been computed numerically and discussed qualitatively. Keywords: Radiation, chemical reaction, temperature, porous plate, MHD, mass transfer.

1. INTRODUCTION The range of free convective flows that occur in nature and in engineering practice is very large and has been extensively

considered by many researchers (1980, 1977). When heat and mass transfer occur simultaneously between the fluxes, the driving potentials are of more intricate nature. An energy flux can be generated not

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K. Rajasekhar, et al., J. Comp. & Math. Sci. Vol.4 (5), 326-338 (2013)

only by temperature gradients but by composition gradients as well. The energy flux caused by a composition gradient is called the Dufour or diffusion-thermo effect. Temperature gradients can also create mass fluxes, and this is the Soret or thermaldiffusion effect. Generally, the thermaldiffusion and the diffusion-thermo effects are of smaller order magnitude than the effects prescribed by Fourierâ&#x20AC;&#x2122;s or Fickâ&#x20AC;&#x2122;s laws and are often neglected in heat and mass transfer processes. The study of radiative heat and mass transfer in convective flows is important from many industrial and technological points of view. Mass transfer is one of the most commonly encountered phenomena in chemical industries as well as in physical and biological sciences. When mass transfer takes place in a fluid at rest, the mass is transferred purely by molecular diffusion resulting from concentration gradients. For low concentration of the mass in the fluid and low mass transfer rates, the convective heat and mass transfer process are similar in nature. A number of investigations have already been carried out with combined heat and mass transfer under the assumption of different physical situations. Thermal radiation in free convection has also been studied by many authors because of its applications in many engineering and industrial processes. Examples include nuclear power plant, solar power technology, steel industry, fossil fuel combustion, space sciences applications, etc. In many chemical engineering processes, there does occur the chemical reaction between a foreign mass and the fluid in which the plate is moving. These processes take place in numerous industrial applications, namely, polymer

production, manufacturing of ceramics or glassware, and food procession. In recent years, progress has been considerably made in the study of heat and mass transfer in magnetohydrodynamic (MHD) flows due to its application in many devices, like the MHD power generators and Hall accelerators. Kinyanjui et al. (2001) analyzed simultaneous heat and mass transfer in unsteady free convection flows with radiation absorption past an impulsively started infinite vertical porous plate subjected to a strong magnetic field. Yih (1997) numerically analyzed the effect of the transpiration velocity on the heat and mass transfer characteristics of the mixed convection about a permeable vertical plate embedded in a saturated porous medium under the coupled effects of thermal and mass diffusion. Elbashbeshy (2003) studied the effect of the surface mass flux on the mixed convection along a vertical plate embedded in a porous medium. Chin et al. (2007) obtained numerical results for the steady mixed convection boundary layer flow over a vertical impermeable surface embedded in a porous medium when the viscosity of the fluid varies inversely as a linear function of the temperature. Pal and Talukdar (2009) analyzed the combined effect of the mixed convection with the thermal radiation and chemical reaction on the MHD flow of viscous and electrically conducting fluid past a vertical permeable surface embedded in a porous medium. Mukhopadhyay (2009) performed an analysis to investigate the effects of the thermal radiation on the unsteady mixed convection flow and heat transfer over a porous stretching surface in a porous medium. Hayat et al. (2009) analyzed a

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K. Rajasekhar, et al., J. Comp. & Math. Sci. Vol.4 (5), 326-338 (2013)

mathematical model in order to study the heat and mass transfer characteristics in the mixed convection boundary layer flow about a linearly stretching vertical surface in a porous medium filled with a viscoelastic fluid, by taking into account the diffusion thermo (Dufour) and thermal diffusion (Soret) effects. Li et al. (2006) took an account of the thermal diffusion and diffusion-thermo effects to study the properties of the heat and mass transfer in a strongly endothermic chemical reaction system for a porous medium. Gaikwad et al. (2007) investigated the onset of the double diffusive convection in a two-component couple of the stress fluid layer with the Soret and Dufour effects using both linear and nonlinear stability analyses. Osalusi et al. (2008) investigated the thermo-diffusion and diffusion-thermo effects on combined heat and mass transfer of a steady hydromagnetic convective and slip flow due to a rotating disk in the presence of viscous dissipation and Ohmic heating. Shateyi and Petersen (2008) investigated the thermal radiation and buoyancy effects on heat and mass transfer over a semi-infinite stretching surface with suction and blowing. Ambethkar (2008) studied numerical solutions of heat and mass transfer effects of an unsteady MHD free convective flow past an infinite vertical plate with constant suction. Shanker and Gnaneshwar (2007) investigated the radiation effects on an MHD flow past an impulsively started infinite vertical plate through a porous medium with the variable temperature and mass diffusion. Alam et al. (2006) studied the Dufour and Soret effects on a steady MHD combined free-forced convective and mass transfer flow past a semi-infinite vertical

328

plate. Alam and Rahman (2006) investigated the Dufour and Soret effects on the mixed convection flow past a vertical porous flat plate with variable suction. Abreu et al. (2008) discussed boundary layer flows with the Dufour and Soret effects. Lyubimova et al. (2005) investigated numerical study of high frequency vibration influence on measurement of Soret and diffusion coefficients in low gravity conditions. Postelnicu (2004) discussed influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in a porous media considering the Soret and Dufour effects. The interaction of buoyancy with the thermal radiation has increased greatly during the last decade due to its importance in many practical applications. The thermal radiation effect is important under many isothermal and non-isothermal situations. If the entire system involving the polymer extrusion process is placed in a thermally controlled environment, then the thermal radiation could be important. The knowledge of radiation heat transfer in the system can, perhaps, lead to a desired product with sought characteristics. Motsa (2008) investigated the effect of both the Soret and Dufour effects on the onset of double diffusive convection. Mansour et al. (2008) investigated the effects of chemical reaction, thermal stratification, Soret and Dufour numbers on MHD free convective heat and mass transfer of a viscous, incompressible and electrically conducting fluid on a vertical stretching surface embedded in a saturated porous medium. Srihari et al. (2006) studied the Soret effect on an unsteady MHD free convective mass transfer flow past an infinite vertical porous plate with the oscillatory suction velocity

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and heat sink. Motivated by the above reference works and the numerous possible industrial applications of the problem (like in isotope separation), it is of paramount interest in this study to investigate the effects of thermal radiation, Soret and Dufour on an unsteady MHD free convective mass transfer flow past an infinite vertical porous plate with the oscillatory suction velocity. None of the above investigations simultaneously studied the Soret and Dufour effects on an unsteady MHD free convective mass transfer flow past an infinite vertical porous plate with the oscillatory suction velocity in the presence of the thermal radiation. Hence, the purpose of this paper is to extend the results of Srihari et al. (2006) to study the more general problem which includes the Soret and Dufour effects on an unsteady MHD free convective mass transfer flow past an infinite vertical porous plate with the oscillatory suction velocity in the presence of the thermal radiation. In this study, the effects of different flow parameters encountered in the equations are also studied. The problem is solved numerically using the perturbation technique method, which is more economical from the computational view point. 2. MATHEMATICAL FORMULATION We consider the unsteady hydromagnetic flow of an incompressible, electrically conducting viscous fluid through a porous medium past an infinite vertical plate with the oscillatory suction and the thermal radiation. In the Cartesian

coordinate system, the x′ − axis is assumed to be along a plate in the direction of the

flow and the y ′ − axis normal to it. A

normal magnetic field is assumed to be

′

applied in the y − direction, and the induced magnetic field is negligible in comparison with the applied one which corresponds to a very small magnetic Reynolds number. The surface is maintained at a uniform constant temperature and concentration. The flow has significant thermal radiation, Soret and Dufour effects. Within the above framework, the equations which govern the flow under the usual Boussinesq’s approximation are as follows: ∂v′ =0 ∂y′

(1)

∂u ′ ∂u′ ∂ 2u′ σ B02 ν + v′ =ν + g β T ' − T∞' + g β ∗ C ' − C∞' − u '− u ' ∂t ′ ∂y′ ∂y ′2 k' ρ

(

)

(

)

(2) ∂T ′ ∂T ′ k ∂ T ′ 16σ T ∂ T ′ Dm kT ∂ C ′ + v′ = + + ∂t ′ ∂y′ ρ c p ∂y ′2 3ke ρ c p ∂y ′2 CS c p ∂y ′2 2

3 ∞

(3) ∂C ′ ∂C ′ ∂ 2C ′ + v′ =D − K r′ ( C ′ − C∞′ ) ∂t ′ ∂y ′ ∂y ′2

(4)

where u ′, v ′ are the velocity components in

x′, y ′ directions respectively, t ′ − the time,

ρ − the fluid density, g − the acceleration ∗ due to gravity, β and β − the thermal and concentration expansion coefficients respectively, K ′ − the permeability of the porous medium, T ′ − the temperature of the fluid in the boundary layer, ν − the kinematic viscosity, σ − the electrical

T′ −

∞ conductivity of the fluid, the temperature of the fluid far away from the plate, C′ − the species concentration in the

boundary

layer,

C∞′ −

the

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species

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K. Rajasekhar, et al., J. Comp. & Math. Sci. Vol.4 (5), 326-338 (2013)

concentration in the fluid far away from the cp − plate, B 0 − the magnetic induction, specific heat at constant, D − the coefficient of chemical molecular diffusivity, K ′ − the r

chemical reaction rate constant. The boundary conditions for the velocity, temperature, and concentration fields are given as follows:

u ′ = 0, T ′ = Tw′ + ε (Tw′ − T∞′ ) eη ′t ′ , C ′ = C w′ + ε ( C w′ − C ∞′ ) eη ′t ′ u ′ = 0, T ′ → T∞′ , C ′ → C ∞′

at as

y′ = 0 y′ → ∞

(5)

where, T w′ and C w′ are the temperature and concentration near the plate respectively and

where

From the continuity equation, it can be seen that v′ is either a constant or a function of time. Thus, assuming the suction velocity to be oscillatory about a non-zero constant

η is the frequency of oscillation, 1 is a positive constant. The and ε negative sign indicates that the suction velocity is directed towards the plate. In order to write the governing Equations and the boundary conditions in dimensional following non-dimensional quantities are introduced.

η′ is the constant.

mean, one can write y=

v ′ = − v 0 (1 + ε e iη ′t ′ ) ,

(v0

v0 is the mean suction velocity

> 0 ),

v0 y ′ t ′v 2 T ′ − T∞′ C ′ − C ∞′ K ′v 02 u′ ,u = , t = 0 ,θ = ,C = ,K = , v0 T w′ − T∞′ C w′ − C ∞′ 4ν 4ν ν2

Gr = Sc =

νρ C p g β ν (Tw′ − T∞′ ) g β ∗ν ( C w′ − C ∞′ ) K r′ν , Pr = , K = , G m = , r v 03 k v 02 v 03 ν D

,η =

σ B 02ν 4νη ′ D ( C w′ − C ∞′ ) M , = , Du = 2 2 ρ v0 ν (Tw′ − T∞′ ) v0

The governing equations for the momentum, the energy, and the concentration in a dimensionless form are 1 ∂u ∂u ∂2u  1 − (1+ ε eiηt ) = Grθ + GmC + 2 −  M +  u 4 ∂t ∂y ∂y  K

(7) 1 ∂θ ∂θ 1  4 ∂ θ ∂C − (1 + ε eiηt ) = 1 +  2 + Du 2 4 ∂t ∂y Pr  3N  ∂y ∂y (8) 2

1 ∂C ∂C 1 ∂ 2C − (1 + ε eiηt ) = − Kr C 4 ∂t ∂y Sc ∂y 2

(6)

where M , G r , G m , P r, K r , S c , K , D u , N are the magnetic field parameter, Grashof number, modified Grashof number, Prandtl number, Chemical reaction number, Schmidt number, permeability parameter, Dufour number and thermal radiation parameter. The relevant corresponding boundary conditions for t > 0 are transformed to: u = 0, θ = 1 + ε e iη t , C = 1 + ε e iη t u → 0,

θ → 0,

C →0

y=0

as y → ∞

(9)

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(10)

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K. Rajasekhar, et al., J. Comp. & Math. Sci. Vol.4 (5), 326-338 (2013)

3. SOLUTION OF THE PROBLEM

u1′′ + u1′ − A17 u1 = − G rθ 1 − G mC 1 − u 0′

(13)

Equations (12) – (14) are coupled, non – linear partial differential Equations and these cannot be solved in closed form. However, these Equations can be reduced to a set of ordinary differential Equations, which can be solved analytically. This can be done by representing the velocity, temperature and concentration of the fluid in the neighborhood of the fluid in the neighborhood of the plate as

A3θ0′′ + θ0′ = − Du C0′′

(14)

u ( y , t ) = u0 ( y ) + ε eiηt u1 ( y ) + o ( ε 2 ) + ............

θ ( y , t ) = θ 0 ( y ) + ε e θ1 ( y ) + o ( ε iη t

C ( y, t ) = C0 ( y ) + ε eiη t

) + ............ C ( y ) + o (ε ) + ............ 2

(11) Substituting (11) in Equations (7) – (9) and equating the harmonic and non – harmonic terms, and neglecting the higher order terms of

( ) , we obtain

o ε2

1   u 0′′ + u 0′ −  M +  u 0 = − G rθ 0 − G m C 0 K  

(12)

A 3θ 1′′ + θ 1′ −

iη θ 1 = − θ 0′ − D u C 1′′ 4

(15)

C 0′′ + ScC 0′ − S cK r C 0 = 0

iη  C 1′′ + ScC1′ − Sc  K r + 4 

(16)   C1 = − C 0′ 

(17)

where prime denotes ordinary differentiation with respect to y. The corresponding boundary conditions can be written as u0 = 0, u1 = 0,θ0 = 1,θ1 = 1, C0 = 1, C1 = 1

at y = 0

u0 → 0, u1 → 0, θ0 → 0, θ1 → 0, C0 → 0, C1 → 0 as y → ∞

(18) Solving Equations (12) – (17) under the boundary condition (18) we obtain the velocity, temperature and concentration distribution in the boundary layer as

u ( y, t ) = A16 exp(-m4 y ) + A29 exp(-m1 y ) + A14 exp(- A4 y )  A28 exp(-m5 y ) + A18 exp(-m3 y ) + A30 exp(- A4 y )  + ε exp(int )    + A31 exp(-m1 y ) + A32 exp(-m2 y )   A exp(-m3 y ) + A7 exp(- A4 y)  θ ( y, t ) = A5 exp(-m1 y ) + A6 exp(- A4 y) + ε exp(int )  11   + A12 exp(-m1 y ) − A9 exp(-m2 y ) 

C ( y, t ) = exp(-m1 y) + ε exp(int ) [ A1 exp(-m1 y) + A2 exp(-m2 y)]

where

A1 =

m1 3N + 4 1 1 iη , A2 = 1 − A1 , A3 = , A4 = , A17 = M + + iη  3 NPr A3 K 4  m12 − Scm1 −  Kr +  4  Journal of Computer and Mathematical Sciences Vol. 4, Issue 5, 31 October, 2013 Pages (322-402)

K. Rajasekhar, et al., J. Comp. & Math. Sci. Vol.4 (5), 326-338 (2013)

332

1 1  iη  Sc + Sc 2 + 4 KSc  , m2 =  Sc + Sc 2 + 4 Sc  + K   ,  2 2  4   1 m3 = 1 + 1 + iη A3  2 1 1 1   m4 = 1 + 1 + 4  M +   , m5 = 1 + 1 + 4 A17  , 2  K   2  m1 =

Du m12 A4 A6 A5 m1 , A6 = 1 − A5 , A7 = , A8 = 2 iη iη A3m1 − m1 A3 A42 − A4 − A3m12 − m1 − 4 4 2 2 Du A2 m2 Du A1m1 A9 = , A10 = , A11 = 1 − A7 − A8 + A9 + A10 , i η iη A3m22 − m2 − A3 m12 − m1 − 4 4 −GrA5 −GrA6 −Gm A13 = , A14 = , A15 = , 1 1 1    2 2 2 m1 − m1 −  M +  A4 − A4 −  M +  m1 − m1 −  M +  K K K    −GrA11 −GrA7 −GrA12 A16 = − ( A13 + A14 + A15 ) , A18 = 2 , A19 = 2 , A20 = 2 , m3 − m3 − A17 A4 − A4 − A17 m1 − m1 − A17 A5 =

A21 =

GrA9 A m −GmA1 −GmA2 , A22 = 2 , A23 = 2 , A24 = 2 16 4 , m2 − m2 − A17 m4 − m4 − A17 m − m2 − A17 m1 − m1 − A17

A25 =

A13 m1 A m A A , A26 = 2 14 4 , A27 = 2 15 1 , m − m1 − A17 A4 − A4 − A17 m1 − m1 − A17

2 2

2 1

A28 = − ( A18 + A19 + A20 + A21 + A22 + A23 + A24 + A25 + A26 + A27 ) A29 = A13 + A15 , A30 = A19 + A26 , A31 = A20 + A22 + A25 + A27 , A32 = A21 + A23 ; 4. RESULTS AND DISCUSSION In order to get a physical insight of the problem, the above physical quantities are computed numerically for different values of the governing parameters viz., thermal Grashof number Gr the modified number Gm, magnetic parameter M,

permeability parameter K , Prandtl number Pr , Schmidt number Sc , Dufour number Du , the radiation parameter N and chemical reaction parameter K r . Fig.1. exhibits the variation of velocity profiles for different values of Grashof number Gr, in case of

Journal of Computer and Mathematical Sciences Vol. 4, Issue 5, 31 October, 2013 Pages (322-402)

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K. Rajasekhar, et al., J. Comp. & Math. Sci. Vol.4 (5), 326-338 (2013)

corresponding to cooling of the plate (Gr >0). It is observed that the velocity profiles increases with an increases of Gr. The rise in the values of velocity is due to enhancement in buoyancy force. In addition, it is noticed that the velocity increases rapidly near the wall of the porous plate as Grashof number increases and then decays to the free steam velocity. From Fig.2 it can be seen that, increasing the values of modified Grashof number Gm is to increases the velocity profiles. Fig.3 shows that the effects of magnetic field parameter M on the velocity profiles. It is obvious that an existence of the magnetic field decreases the velocity field. The radiation effects on velocity profiles are shown in Fig.4. It is observed that the velocity profiles increases with an increases of Radiation parameter N. The Fig.5 is shown for the velocity profiles for the different values of permeable parameter K. t is observed that the increasing values of K the velocity profiles is also increases. Fig.6 shows the effects of Dufour number on the velocity profiles. It is found that the velocity increases as Dufour number Du increases in case of cooling of the plate (Gr > 0) for both water and air and a reverse effect is indentified in case of heating of the plate (Gr < 0). Fig.7 shows the effects of chemical reaction parameter on velocity profiles. It is observed that as the chemical reaction parameter increases the velocity profiles decreases.Fig.8 shows the effects of Schmidt Number on velocity profiles. It is noticed that as the Schmidt number increases the velocity profiles decreases. In Fig.9 it can be seen that the effect of Prandtl number Pr on velocity profiles. As Prandtl number increases the velocity profiles decreases.

The influence of various flow parameters on the fluid temperature are illustrated in figures 10-11. Fig.10 depicts that the effects of Dufour number on the fluid temperature. It is seen that the diffusion thermal effects slightly affect the fluid temperature. As the values of Dufour number increase, the fluid temperature is decreases and goes linearly of the boundary layer. The effect of thermal radiation parameter N on the temperature field is illustrated in Fig.11. It is obvious that the radiation parameter increases an increasing of the fluid temperature. The concentration profiles for different values of chemical reaction parameter and Schmidt number are presented through Fig. 12 and13 respectively. From these figures it is seen that the concentration decreases with an increases in chemical reaction parameter or Schmidt number. The combined effect of radiation parameter and the magnetic influence on the skin friction is illustrated in Fig.14. It is observed that the magnetic intensity influences the skin friction and is seen that as the radiation parameter increases, the skin friction increases. The consolidated influence of magnetic field with respect to permeability parameter over skin friction is seen in Fig.15. It is seen that in general, increase in M contributes to decrease of skin friction and is found to be independent of permeability parameter. The consolidated influence of magnetic field with respect to chemical reaction parameter over skin friction is seen in Fig.16. It is seen that in general, increase in M contributes to decrease of skin friction and is found to be independent of chemical reaction parameter.

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Fig.17 reveals that the rate of heat transfer coefficient in terms of Nusselt number for different values of Dufour number against to the Prandtl number. It is observed that Nusselt number increases with an increase values of Dufour number. Fig.18 reveals that

the rate of mass transfer coefficient in terms of Sherwood number for different values of Schmidt number against to the chemical reaction parameter. It is observed that Sherwood number increases with an increase values of Schmidt number.

3.5

7 M=0.5;Gm=5;Sc=0.22;Pr=0.71;Kr=0.5; Du=0.03; ε=0.005;η=1;t=π/2;N=0.5;K=1;

2.5

M=0.5;Gr=5;Sc=0.22;Pr=0.71;Kr=0.5; Du=0.03;ε=0.005; η=1;t= π/2;N=0.5;K=1;

Gr=5,10,15,20

Velocity

Gm=5, 10, 15, 20 2

1.5

0.5

1.5

2.5 y

3.5

4.5

Fig.1. Effects of Grashof number on velocity profiles.

0.5

1.5

2.5 y

3.5

4.5

Fig.2. Effects of modified Grashof number on velocity profiles. 2

2.5

1.8

Gr=10;Gm=5;Sc=0.22;Pr=0.71;Kr=0.5; Du=0.03; ε=0.005;η=1;t=π/2;N=0.5;K=1;

N=0.1, 0.2 ,0.3,0.4

1.6 1.4

M= 0.5, 1.0, 1.5, 2.0

1.2

Velocity

1.5

1 0.8

0.6 0.4

0.5

M=0.5;Gr=5;Gm=5;Sc=0.22;Pr=0.71;Kr=0.5; Du=0.03; ε=0.005; η=1;t=0.1;K=1;

0.2 0

0.5

1.5

2.5 y

3.5

4.5

0.5

1.5

2.5 y

3.5

4.5

Fig.3. Effects of magnetic parameter on velocity profiles. Fig.4. Effects of radiation parameter on velocity profiles. 3

2 M=0.5;Gr=5;Gm=5;Sc=0.22;Pr=0.71;Kr=0.5; Du=0.03;ε=0.005;\eta=1;t=0.1;N=0.5;

2.5

1.8 Du=0.1, 0.5, 0.8, 1.0 1.6

K=0.5, 1.0, 1.5, 2.0

M=0.5;Gr=5;Gm=5;Sc=0.22; Pr=0.71;Kr=0.5;K=1;ε=0.005; ω=1;t=0.1;N=0.5;

1.4

Velocity

1.2 1.5

1 0.8

0.6 0.4

0.5

0.2 0

0.5

1.5

2.5 y

3.5

4.5

0.5

1.5

2.5 y

3.5

4.5

Fig.5. Effects of permeability parameter on velocity profiles. Fig.6. Effects of Dufour number on velocity profiles. Journal of Computer and Mathematical Sciences Vol. 4, Issue 5, 31 October, 2013 Pages (322-402)

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K. Rajasekhar, et al., J. Comp. & Math. Sci. Vol.4 (5), 326-338 (2013) 2

1.8 1.6

1.6 M=0.5;Gr=5;Gm=5;Sc=0.22; Pr=0.71;Du=0.003;K=1; ε=0.005; η=1;t=π/2;N=0.5;

1.4

1.2 Velocity

Velocity

Sc=0.22, 0.30, 0.60, 0.78

1.4

1.2 1

0.8

0.6

0.4

0.2 0

M=0.5;Gr=5;Gm=5;Pr=0.71;Du=0.003; K=1;ε=0.005; η=1;t=π/2;N=0.5;Kr=0.5;

1.8

Kr=0.5, 1.0, 1.5, 2.0

0.2

0.5

1.5

2.5 y

3.5

4.5

0.5

Fig.7. Effects of chemical reaction parameter on velocity profiles.

1.5

2.5 y

3.5

4.5

Fig.8. Effects of Schmidt number on velocity profiles.

1 M=0.5;Gr=10;Gm=5;Sc=0.22;Du=0.003; K=1; ε=0.005; η=1;t= π/2;N=0.5;Kr=0.5;

0.9 M=0.5;Gr=10;Gm=5;Sc=0.22;Du=0.003; K=1; ε=0.005;Pr=0.71; η=1;t=π/2;Kr=0.5;

0.8 0.7 Temperature

4 Velocity

0.6 0.5 0.4

N=0.1, 0.3, 0.5, 0.8

0.3 0.2

Pr=0.71, 1.00,3.00,7.00 0.1

0.5

1.5

2.5 y

3.5

4.5

Fig.9. Effects of Prandtl number on velocity profiles.

0.5

1.5

2.5 y

3.5

4.5

Fig.10. Effects of radiation parameter on temperature profiles.

1.2

0.9 1

M=0.5;Gr=5;Sc=0.22;Pr=0.71;Kr=0.5; Gm=5; ε=0.005;η=1;t=π/2;N=0.5;K=1;

M=0.5;Gr=5;Gm=5;Sc=0.22;Pr=0.71; Du=0.003;K=1; ε=0.005;η=1;t=π/2;N=0.5;

0.8

0.7 Concentration

Temperature

0.6

0.4

Du=0.1, 0.3, 0.5, 0.8

0.6 Kr=0.5, 1.0, 1.5, 2.0 0.5 0.4 0.3

0.2

0.2 0 0.1 -0.2

0.5

1.5

2.5 y

3.5

4.5

Fig.11. Effects of Dufour number on temperature profiles.

0.5

1.5

2.5 y

3.5

4.5

Fig.12. Effects of chemical reaction parameter on concentration profiles.

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1 0.9 M=0.5;Gr=5;Gm=5;Pr=0.71;Du=0.003; K=1;ε=0.005;η=1;t=π/2;N=0.5;Kr=0.5;

0.8

0.6

Skin-friction

Concentration

0.7

Sc=0.22, 0.30, 0.60, 0.78 0.5 0.4

N=0.5, 1.0, 1.5, 2.0 10

8 0.3 0.2

0.1 0

0.5

1.5

2.5 y

3.5

4.5

Fig.13. Effects of Schmidt number on concentration profiles.

0.5

1.5

2.5 M

3.5

4.5

Fig.14. Effects of radiation parameter on skin-friction.

7.5

10 M=1, 2, 3, 4

Skin-friction

9 8 7

6.5

6 5.5

M=1, 2, 3, 4

4 3

0.5

1.5

2.5 K

3.5

4.5

Fig.15. Effects of magnetic parameter on skin-friction.

0.5

1.5

2.5 Kr

3.5

4.5

Fig.16. Effects of magnetic parameter on skin-friction.

2.5

Sherwood number

20 Nusselt number

1.5

Du=1, 2, 3, 4

Sc=0.22, 0.30, 0.60, 0.78 0.5

0.5

1.5

2.5 Pr

3.5

4.5

Fig.17. Effects of Dufour number on Nusselt number.

5. REFERRENCES 1. Abreu, C. R. A., Alfradique, M. F., and Telles, A. S. Boundary layer flows with Dufour and Soret effects, I: forced and natural convection. Chemical Engineering Science 61,4282–4289, (2006).

0.5

1.5

2.5 Kr

3.5

4.5

Fig.18. Effects of Schmidt number on Sherwood number.

2. Alam, M. S. and Rahman, M. M. Dufour and Soret effects on mixed convection flow past a vertical porous flat plate with variable suction. Nonlinear Analysis: Modelling and Control 11(1), 3–12, (2006).

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