Paper2

International Journal of Scientific Engineering and Technology Volume No.2, Issue No.2, pg : 56-72

(ISSN : 2277-1581) 1 Feb. 2013

Radiation and Thermo - Diffusion Effects on Mixed Convective Heat and Mass Transfer Flow af a Viscous Dissipated Fluid over a Vertical Surface in the Presence of Chemical Reaction with Heat Source D Chenna Kesavaiah1, P V Satyanarayana2, S Venkataramana3 1 Department of H & BS, Visvesvaraya College of Engineering & Technology, Greater Hyderabad, A.P, India chennakesavaiah@gmail.com 2 Fluid Dynamics Division, School of Advanced Science, VIT University, Vellore - 632 014, T N, India 3 Department of Mathematics, S V University, Tirupati - 517 502, A.P, India ABSTRACT

very light molecular weight  H 2 , He  and the medium

The present work analyzes the influence of chemical reaction on MHD mixed convection heat and mass transfer for a viscous fluid past an infinite vertical plate embedded in a porous medium with radiation and heat generation. The dimensionless governing equations for this investigation are solved analytically using two - term harmonic and non-harmonic functions. The effects of various parameters on the velocity, temperature and concentration fields as well as the skin-friction coefficient, Nusselt number are presented graphically and Sherwood number is presented numerically.

molecular weight

Keywords: Radiation, mass transfer, chemical reaction, heat source and concentration I. INTRODUCTION The phenomenon of heat and mass transfer has been the object of extensive research due to its applications in Science and Technology. Such phenomena are observed in buoyancy induced motions in the atmosphere, in bodies of water, quasisolid bodies such as earth and so on. This assumption is true when the concentration level is very low. Therefore, so ever, exceptions the thermal diffusion effects for instance, has been utilized for isotropic separation and in mixtures between gases with IJSET@2013

 N2

air) the diffusion – thermo

effects was found to be of a magnitude just it cannot be neglected.

The heat and mass transfer simultaneously affecting each other that will cause the cross diffusion effect. The heat transfer caused by concentration gradient is called the diffusion-thermo or Dufour effect. On the other hand, mass transfer caused by temperature gradients is called Soret or thermal diffusion effect. Thus Soret effect is referred to species differentiation developing in an initial homogenous mixture submitted to a thermal gradient and the Dufour effect referred to the heat flux produced by a concentration gradient. The Soret effect, for instance has been utilized for isotope separation, and in mixture between gases with very light molecular weight

 H2 , He

and of medium molecular weight  N2 , air 

Soret effect [thermal-diffusion] refers to mass flux produced by a temperature gradient. These effects are neglected on the basis that they are of a smaller order of magnitude than the effects described by Fourier’s and Fick’s laws. In view of the importance of this diffusion – thermo effect. Similarity equations of the momentum energy and concentration equations are derived by introducing a time dependent length scale. Malsetty et. al [19] have studied the effect of both the Soret coefficient and Dufour coefficient on the double diffusive Page 56

International Journal of Scientific Engineering and Technology Volume No.2, Issue No.2, pg : 56-72 convective with compensating horizontal thermal and solutal gradients. Saritha and Satya Narayana [25] thermal diffusion and chemical reaction effects on unsteady MHD free convection flow past a semi infinite vertical permeable moving plate. Mohamed [21] Double - Diffusive convection - radiation Interaction on Unsteady MHD flow over a vertical moving porous plate with heat generation and Soret effects. Bhupendra Kumar et.al [5] Three-Dimensional mixed convection flow past an infinite vertical plate with constant surface heat flux. Ahmed and Kalita [2] Soret and magnetic field effects on a transient free convection flow through a porous medium bounded by a uniformly moving infinite vertical porous plate in presence of heat source. Gireesh Kumar and Satyanarayana [15] Mass transfer effects on MHD unsteady free convective Walter's memory flow with constant suction and heat sink.

Hydromagnetic flows and heat transfer have become more important in recent years because of its varied applications in agriculture, engineering and petroleum industries. Raptis [24] studied mathematically the case of time varying two-dimensional natural convective flow of an incompressible, electrically conducting fluid along an infinite vertical porous plate embedded in a porous medium. Soundalgekar [28] obtained approximate solutions for two-dimensional flow of an incompressible viscous flow past an infinite porous plate with constant suction velocity, the difference between the temperature of the plate and the free stream is moderately large causing free convection currents. Takhar and Ram [32] studied the MHD free convection heat transfer of water at 4ď Ż C through a porous medium. Elbashbeshy [12] studied heat and mass transfer along a vertical plate under the combined buoyancy effects of thermal and species diffusion, in the presence of magnetic field.

In all these investigations, the radiation effects are neglected. For some industrial applications such as glass production and furnace design and in space technology applications, such as cosmical sight aerodynamics rocket, propulsion systems, plasma physics and IJSET@2013

(ISSN : 2277-1581) 1 Feb. 2013

spacecraft re-entry aerothermodynamics which operate at higher temperatures, radiation effects can be significant. Alagoa et al. [3] studied radiative and free convection effects on MHD flow through porous medium between infinite parallel plates with timedependent suction. Bestman and Adjepong [4] analyzed unsteady hydromagnetic free convection flow with radiative heat transfer in a rotating fluid. Olanrewaju et.al [23] further results on the Effects of Variable Viscosity and Magnetic Field on Flow and Heat Transfer to a Continuous Flat Plate in the Presence of Heat Generation and Radiation with a Convective Boundary Condition.

In all these investigations, the viscous dissipation is neglected. The viscous dissipation heat in the natural convective flow is important, when the flow field is of extreme size or at low temperature or in high gravitational field. Gebhart [14] shown the importance of viscous dissipative heat in free convection flow in the case of isothermal and constant heat flux at the plate.

Israel Cookey et al. [17] investigated the influence of viscous dissipation and radiation on unsteady MHD free convection flow past an infinite heated vertical plate in a porous medium with time dependent suction. Sreekantha Reddy et.al [29] Effects of chemical reaction, radiation and thermo-diffusion on convective heat and mass transfer flow of a viscous dissipated fluid in a vertical channel with constant heat flux. Md. Abdul Sattar and Md. Alam [20] Thermal diffusion as well as transportation effect on MHD free convection and Mass Transfer flow past an accelerated vertical porous plate.

Satyanarayana et.al [26] viscous dissipation and thermal radiation effects on an unsteady MHD convection flow past a semi-infinite vertical permeable moving porous plate Sudheer Babu et. al [31] Effects of thermal radiation and chemical reaction on MHD convective flow of a polar fluid past a moving vertical plate with viscous dissipation.

The effects of radiation on MHD flow and heat transfer problem have become more important industrially.

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International Journal of Scientific Engineering and Technology Volume No.2, Issue No.2, pg : 56-72

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Many processes in engineering areas occur at high temperature, and knowledge of radiation heat transfer becomes very important for the design of the pertinent equipment. Nuclear power plants, Gas turbines and various propulsion devices, for aircraft, missiles, satellites and space vehicles are examples of such engineering areas. Shercliff [27] and Ferraro and Plumpton [13]. Hossian and Rees [16] examined the effects of combined buoyancy forces from thermal and mass diffusion by natural convection flow from a vertical wavy surface. Combined heat and mass transfer in MHD free convection from a vertical surface has been studied by Chein-Hsin-Chen [8]. Further, the effect of Hall current on the fluid flow with variable concentration has many applications in MHD power generation, in several astrophysical and meteorological studies as well as in plasma flow through MHD power generators. From the point of application, model studies on the Hall Effect on free and forced convection flows have been made by several investigators. Datta and Jana [10], Acharya et al.[1] and Biswal and Sahoo [6] have studied the Hall effect on the MHD free and forced convection heat and mass transfer over a vertical surface.

Stanford Shateyi et.al [30] The effects of thermal radiation, hall currents, Soret and Dufour on MHD flow by mixed convection over a vertical surface in porous media.

investigated the effect of the first order homogeneous chemical reaction on the process of an unsteady flow past a vertical plate with a constant heat and mass transfer. Chamkha [7] studied the MHD flow of a numerical of uniformly stretched vertical permeable surface in the presence of heat generation/ absorption and a chemical reaction. Muthucumaraswamy [22] presented heat and mass transfer effects on a continuously moving isothermal vertical surface with uniform suction by taking into account the homogeneous chemical reaction of first order.

Kesavaiah Ch et.al [18] Effects of the chemical reaction and radiation absorption on an unsteady MHD convective heat and mass transfer flow past a semiinfinite vertical permeable moving plate embedded in a porous medium with heat source and suction Keeping the above application in view we made attempt in this paper to study the present work analyzes the influence of a first - order homogeneous chemical reaction on MHD mixed convection heat and mass transfer for a viscous fluid past an infinite vertical plate embedded in a porous medium with radiation and heat generation. The dimensionless governing equations for this investigation are solved analytically using two term harmonic and non-harmonic functions

The growing need for chemical reactions in chemical and hydrometallurgical industries requires the study of heat and mass transfer with chemical reaction. There are many transport processes that are governed by the combined action of buoyancy forces due to both thermal and mass diffusion in the presence of the chemical reaction effect. These processes are observed in nuclear reactor safety and combustion systems, solar collectors, as well as metallurgical and chemical engineering. Their other applications include solidification of binary alloys and crystal growth dispersion of dissolved materials or particulate water in flows, drying and dehydration operations in chemical and food processing plants, and combustion of atomized liquid fuels. Dekha et al. [11]

We consider the mixed convection flow of an incompressible, viscous, electrically conducting viscous fluid embedded in a uniform porous medium in the presence of thermal diffusion, chemical reaction, radiation, thermal and concentration buoyancy effects

IJSET@2013

II. FORMULATION OF THE PROBLEM

such that x  -axis is taken along the plate in upwards direction and y  -axis is normal to it. A transverse constant magnetic field is applied i.e. in the direction of

y  - axis. Since the motion is two dimensional and length of the plate is large therefore all the physical variables are independent of x  . Let u  and v  be the components of velocity in x  and y  directions, respectively, taken along and perpendicular to the plate.

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International Journal of Scientific Engineering and Technology Volume No.2, Issue No.2, pg : 56-72 The governing equations of continuity, momentum and energy for a flow of an electrically conducting fluid along a hot, non-conducting porous vertical plate in the presence of concentration and radiation is given by

(ISSN : 2277-1581) 1 Feb. 2013

qr   4(T   T ) I  y  



where I   K w 0

(7)

 eb d  , K w  is the absorption T 

v  0 y 

(1)

coefficient at the wall and eb is Planck’s function, I 

v  v0 (Constant)

(2)

is absorption coefficient The boundary conditions are

p   0  p  is independent of y   y



  v 



 C p  v 



T y 

 K



 u   q  r      y  y 

C   2C   D  Kr   C   C  M y  y 2 

T  DT 2 y 2

C (5)



the

temperature of the fluid near the plate, T the free stream temperature, C

y

v0 y 



,

T

T   T Tw  T

 g Tw  T  C   C , Gr  C w  C v03

Ec 

(6)

Here, g is the acceleration due to gravity, T 

u , v0

u





v

(4)

u

2

(8)

Introducing the following non-dimensional quantities

 T    k 2  Q0 T  T  y  2

T   Tw , C  C ; y   0

u*  0, T   T , C   C ; y   

u    2u      g  T   T    2 y  y

  g  *  C   C    B02u  

u   0,

(3)

 v02

 Tw  T 

R

,

4 I   C p v02

Q

C p Q0 Kr  , Kr  , Pr  2 2  C p v0 v0 k

M

 B0 2 2 ,  v0 2

Gm 

S0 

DT Tw  T    Cw  C 

 * g  Cw  C  3 0

v

,

Sc 

(9)

 D

III. SOLUTION OF THE PROBLEM

concentration,  the coefficient of

thermal expansion, k the thermal conductivity, P the

In the equations (4), (5), (6) and (8), we get

pressure, C p the specific heat of constant pressure, B0

 2u u  1    M   u   GrT  Gm C (10) 2 y y  K

the magnetic field coefficient,  viscosity of the fluid,

q  the radiative heat flux,  the density,  the magnetic permeability of fluid V0 constant suction velocity,  the kinematic viscosity and D molecular diffusitivity.  The radiative heat flux qr is given by equation (5) in

the sprit of Cogly et.al [9] IJSET@2013

 2T T  Pr   F  Q  Pr T 2 y y  u    Pr Ec    y 

2

 2C C  2T  Sc  ScKrC   ScS 0 y 2 y y 2

(11)

(12) Page 59

International Journal of Scientific Engineering and Technology Volume No.2, Issue No.2, pg : 56-72 where Gr is Grashof number, Gm is the mass Groshof number, Pr is Prandtl number, M is magnetic parameter, R is Radiation parameter, Sc is Schmidt number, Q is heat source parameter, Ec is the Eckert number, M is magnetic parameter Kr is Chemical reaction parameter and S 0 is Soret number.

The corresponding boundary condition in dimensionless form are reduced to

u  0,

T  1, C  1

y0

u  0, T  0, C  0

y

(13)

The physical variables u, T and C can expand in the power of Eckert number  Ec  (E). This can be possible physically as Ec for the flow of an incompressible fluid is always less than unity. It can be interpreted physically as the due to the Joules dissipation is super imposed on the main flow.

u  y   u0  y   Ec u1  y   O  E

 T  y   T  y   Ec T  y   O  E  (14) u  y   C  y   Ec C  y   O  E  2

2

0

1

2

0

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u0  0,

u1  0, T0  1   y0 T1  0, C0  1, C1  0  u0  0, u1  0, T0  0  y T1  0, C0  1, C1  0 

(21)

Solving equations (15) to (20) with the help of (21), we get

u  y   A1e m 2 y  A2e m 4 y  A3e

m2y



 A4e m6 y

 Ec A5e m 8 y  A6 e 2 m 6 y  A7 e 2 m 2 y  A8e

2m 4 y

 A11e  A13e

 A9 e 2 m2 y  A10 e

m2  m4  y

m2  m6  y

A12 e



 A14 e

m2  m6  y

m4  m6  y

m2  m4  y

 A15e 2 m 2 y

 A16 e m 8 y  A17 e m 8 y  A18e 2 m 6 y  A19 e 2 m 2 y  A20 e 2 m 4 y  A21e 2 m 2 y  A22 e  A24 e

m 2 m 6  y

m 2m 4  y

A23e m 2  m 4  y



 A25e

m 2m 6  y

 A26e

m 4m 6  y

 A27 e 2 m 2 y  A28e m 10 y  A29e m 12 y 

1

Using equation (14) in equations (10)–(12) and equating the coefficient of like powers of E, we have

  y   em y 

Ec

2

B e

2 m6 y

1

 B2e 2 m2 y

1  u0  u0   M   u0   Gr T0  Gm C0 (15) K 

 B3e 2 m4 y  B4 e 2 m2 y  B5e

T0  Pr T0   F  Q  Pr T0  0

(16)

 B9e

C0  ScC0  KrScC0  ScS0T0

(17)

1  u1  u1   M   u1   Gr T1  Gm C1 (18) K 

 B6 e

m2  m4  y

C  y   D1e

 B7 e

m4  m6  y

m2y

 B8e

m2  m6  y

 B10e 2 m2 y  B11e m8 y

 D2e

m4y





 Ec D3e m8 y

 D4 e 2 m6 y  D5e 2 m2 y  D6e 2 m4 y

T1  Pr T1   F  Q  Pr T1   Pr u02

(19)

 D7 e 2 m2 y  D8e

C1 ScC1  KrScC1  ScS0T1

(20)

 D10 e

The corresponding boundary conditions are

m2  m4  y

m2  m6  y

 D12 e

m2  m4  y

m4  m6  y

m2  m6  y

 D9e



D11e

m2  m4  y

m2  m6  y

 D13e 2 m2 y  D14 e m10 y



Skin – friction: The skin-friction coefficient at the plate is given by IJSET@2013

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International Journal of Scientific Engineering and Technology Volume No.2, Issue No.2, pg : 56-72

 u 

Final results are computed for the main physical parameters which are presented by means of graphs. The

     A1m2  A2 m4  A3m2  A4 m6  y  y 0

influence of the thermal Grashof number Gr  , solutal

 Ec  A5 m8  2 A6 m6  2 A7 m2

Grashof number  Gm  , the magnetic field parameter

 2 A8 m4  2 A9 m2   m2  m6  A10

M  ,

  m2  m4  A11   m2  m4  A12   m2  m6  A13 

 m4  m6  A14

 2 A15 m2  A16 m8  A17 m8

heat source parameter

Q ,

thermal radiation

 F  , Prandtl number  Pr  , Porosity parameter  K  chemical reaction parameter  Kr  , Soret number  S0  and Schmidt number  Sc  on the parameter

Heat

 2 A18 m6  2 A19 m2  2 A20 m4

 m2  m6  A22  A23  m2  m4    m2  m4  A24   m2  m6  A25   m4  m6  A26  2 A27 m2  2 A28 m10  A29 m22   2 A21m2

(ISSN : 2277-1581) 1 Feb. 2013



velocity, temperature and concentration profiles can be analyzed from Figures (1) - (26). The values of Gr & Gm are taken to be both positive as this value represent respectively cooling and the Eckert number

 Ec  is taken to be 0.001 .

Transfer: The rate of heat transfer in terms of Nusselt number at the plate is given by

 T  Nu     m2  Ec 2 B1m6  2m2 B2  y  y 0  2m4 B3  2m2 B4   m2  m6  B5   m2  m4  B6   m2  m4  B7   m2  m6  B8   m4  m6  B9  2m2 B10



m8 B11

Sherwood number

 C  Sh     m 2 D1  m 4 D2  Ec m8 D3  y  y 0  2m6 D4  2m2 D5  2m4 D6

 m2  m6  D8   m2  m4  D9   m2  m4  D10   m2  m6  D11   m4  m6  D12  2m2 D13  m10 D14   2m2 D7 

IV. RESULTS AND DISCUSSION IJSET@2013

Velocity Profiles The velocity profiles for different values of the Grashof number Gr  and the modified Grashof numbers Gc  are defined in Figures (1) and (2) respectively. It can be observed that an increase in Gr or Gc leads to the rise in the values of velocity, Here, the positive values of Gr correspond to a cooling of the surface by natural convection. In addition, the curves show that the peak value of the velocity increases rapidly near the wall of the porous plate as Gr or Gc increases, and then decays to the free stream velocity. The effect of thermal radiation parameter on the velocity field has been illustrated in Figure (3). It is seen that as the thermal radiation parameter increases the velocity field decreases. For various values of the permeability parameter  K  , the profiles of the velocity across the boundary layer are shown in Figure (4). The velocity increases for increasing values of the permeability parameter K . Figure (5) illustrate the variation in velocity distributions across the boundary layer for various values of the chemical reaction parameter  Kr  .

It can be seen that the velocity decreases in the Page 61

International Journal of Scientific Engineering and Technology Volume No.2, Issue No.2, pg : 56-72

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destructive reaction  Kr  0 . The MHD and the

profiles for different values of the Prandtl number  Pr 

concentration boundary layer become thin as the reaction parameters. For different values of the magnetic field

is shown in Figure (12). The results show that an increase of the Prandtl number results in a decrease in the thermal boundary layer thickness and a more uniform temperature distribution across the boundary layer. The reason is that the smaller values of Pr are equivalent to increasing the thermal conductivities, and therefore heat is able to diffuse away from the heated surface more rapidly than for the larger values of Pr .

Hence, the thicker the boundary layer is the shower the rate of heat transfer. At high Prandtl fluid has low velocity, which in turn also implies that at lower fluid velocity the specie diffusion is comparatively lower and hence higher specie concentration is observed at high Prandtl number. Figures (13) present the decreasing result of temperature when heat source parameter is increasing. Figures (14) - (16) shows the temperature profiles for different values of Gm, Gr and Sc . It is

parameter  M  , the velocity profile is plotted in Figure (6). It is obvious that the effect of increasing values of the parameter M results in a decreasing velocity distribution across the boundary layer. Figure (7) present the velocity profiles for different values of the Prandtl number  Pr  . The results show that the effect of increasing values of Pr results in a decrease of the velocity that physically is true because the increase in the Prandtl number is due to the increase in the viscosities of the fluid which makes the fluid thick and hence causes a decrease in the velocity of fluid. The effects of internal heat generation parameter  Q  on the velocity are displayed in Figure (8). It is clear that as the parameter  Q  increases, the velocity profiles lead to a fall. Figure (9) depicts the effect of Soret number  Sr  on the fluid velocity and we observed an increase in the fluid velocity as Soret number

 Sr 

observed that an increasing values of Gm, Gr and

Sc the results decreases the thermal boundary layer thickness across the plate.

increases. This is

because an increase in the volumetric rate of generation connotes increase in buoyancy force thereby increasing fluid velocity. For different values of the Schmidt number  Sc  , the velocity profiles are plotted in Figure (10). It is obvious that the effect of increasing values of Sc results in a decreasing velocity distribution.

Concentration Profiles The radiation  F  effects shows in figure (17). It is observed that and increasing values of F the concentration profiles increases. Figures (18) and (19) depict the effect of various values of the mass and thermal

Temperature profiles Figure (11) show the effect of radiation parameter on the temperature profile. A rise in F causes a significant fall in the temperature values from the highest value at the wall

 y  0

across the boundary layer to the free

stream. Thus, greater value of F corresponds to smaller radiation flux and the minimum temperature is observed.

Thermal radiation thereby reduces the rate of energy transport to the fluid. The variation of the temperature IJSET@2013

Grashof

numbers

Gm, Gr 

on

the

concentration boundary layer thickness. It is interesting to note that increase in this parameters bring a small increase across the plate. The effect of permeability parameter K on the concentration profile is illustrated in Figure (20). These results show that the with increasing permeability parameter concentration profiles decreases. Figure (21) show the effect of the chemical reaction parameter on concentration profiles respectively. It is noticed that species concentration and thermal boundary layer are decreasing, as the values of chemical reaction are increasing. Figure (22) shows the Page 62

International Journal of Scientific Engineering and Technology Volume No.2, Issue No.2, pg : 56-72 effects of the magnetic parameter M . It is observed that the concentration decreases with increasing values of M . The influence of Prandtl number and heat source parameter are on the concentration field is seen in Figure (23) and (24). It is noticed that the increase in the Prandtl number and Sherwood number  Sh  heat source Sc Sh Sh F parameter 0.22 0.5 -0.8783 though 0.4078 increases the 0.30 1.0 -0.8533 concentration of 0.4985 the fluid. The 0.38 1.5 -0.8302 influence of S 0 0.5842 on the 0.46 2.0 -0.8085 concentration of 0.6664 the fluid medium is seen in Figure (25). In general it is noted that increase in Soret number contributes to increase in concentration of the medium. Further the effect is found to be diminishing as we move away from the plate.

Figure (26) reflects that with increase in Schmidt number  Sc  the fluid concentration decreases. Figure (27) and (28) shows the effects of Prandtl number and magnetic field on skin friction and Nusselt number versus thermal Grashof number. It is observed that increasing values of

 Pr, M 

the results in both the

case behaviour are decreases.

Table (I) shows that the effects of Schmidt number and radiation parameter  Sc, F  on Sherwood number. It is observed that increasing values of Schmidt number the Sherwood number decreasing but reverse in radiation parameter.

V. CONCLUSIONS The present study provides the effects of thermodiffusion on MHD mixed convective heat and mass transfer flow of a viscous fluid through a porous medium in a vertical surface with radiation and heat generation, IJSET@2013

(ISSN : 2277-1581) 1 Feb. 2013

chemical reaction and radiation. In the light of present investigation, it is found that the concentration of the fluid increased during a generative chemical reaction and decreased during a destructive chemical reaction.

Further, the momentum boundary layer thickness decreases while thermal boundary layer thickness and concentration profiles decreases with increasing permeability parameter.

Increasing Soret number reduce the thermal boundary layer thickness, while reverse trend is seen in concentration profiles.

Table: (1)

REFERENCES

[1] Acharya M, Dash G C and Singh L P (2001): Indian J.of Physics B, 75 B (1), p. 168 [2] Ahmed N and Kalita H (2012): Soret and magnetic field effects on a transient free convection flow through a porous medium bounded by a uniformly moving infinite vertical porous plate in presence of heat source, Int. J. of Appl. Math and Mech. 8(16): pp. 1-21.

[3] Alagoa K D, Tay G and Abbey T M (1999): Radiative and free convection effects of a MHD flow through porous medium between infinite parallel plates with time dependent suction, Astrophysics Space. Sci., Vol. 260, pp.455-468.

[4] Bestman A R and Adjepong S K (1998): Unsteady hydromagnetic free convection flow with radiative transfer in a rotating fluid, Astrophysics. Space Sci., Vol. 143, pp.73-80.

[5] Bhupendra Kumar Sharma, Tara Chand and Chaudhary R C (2011): Three-Dimensional Mixed Convection Flow past an Infinite Vertical Plate with Constant Surface Heat Flux,

Page 63

International Journal of Scientific Engineering and Technology Volume No.2, Issue No.2, pg : 56-72 American Journal of Computational and Applied Mathematics, 1(1): pp. 27-32 [6] Biswal S and Sahoo P K(1994): Proc. Nat. Acad. Sci., 69A, p. 46.

[7]Chamkha A J (2003): MHD flow of a numerical of uniformly stretched vertical permeable surface in the presence heat generation\absorption and a chemical reaction, Int. Commun. Heat Mass Transfer, Vol. 30,413422.

[8] Chein-Hsin-Chen, Int.J.Eng.Science, 42,699-713, 2004.

[9] Cogley A C, Vincenty W G and Gilles S E (1968): AIAAJ, 6, p. 551.

[10]Datta N and Jana R N (1976): J. Phys. Soc. Japan, 40, pp. 1469-1475 [11] Dekha R, Das U N and Soundalgekar V M (1994): Effects on mass transfer on flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction, Forschungim Ingenieurwesen, Vol. 60, 284-209.

[12] Elbashbeshy E M A (1997): Int. Eng. Sc, 34, pp. 515-522.

[13]Ferraro V C A and Plumption C (1996): An Introduction to Magneto Fluid Mechanics, Clarandon Press, Oxford.

[14]Gebhart B (1962): Effects of viscous dissipation in natural convection, J. Fluid Mech., Vol. 14, pp.225-232.

[15]Gireesh Kumar J and Satyanarayana P V (2011): Mass transfer effects on MHD unsteady free convective Walter's memory flow with constant suction and heat sink, Int. J. of Appl. Math and Mech., 7, (19): pp. 97 – 109.

[16]Hossain M A and Rees D A S (1999): Acta Mech., 136, pp. 133-141.

[17]Israel-Cookey C, Ogulu A and Omubo-Pepple V B (2003): Influence of viscous dissipation on unsteady MHD free-convection flow past an infinite heated vertical plate in porous medium with time-dependent suction, Int. J. Heat Mass Transfer, Vol. 46, pp. 2305-2311.

[18]Kesavaiah D Ch, Satyanarayana P V and Venkataramana S (2011): Effects of the chemical reaction and radiation absorption on an unsteady MHD convective heat and mass transfer flow past a semi-infinite vertical permeable moving plate embedded in a porous medium with heat source and suction, Int. J. of Appl. Math and Mech. 7 (1), pp. 52-69.

[19]Malasetty M S and Gaikwad S N (2002): Effect of cross diffusion on double diffusive convection in the presence of horizontal gradient, Int. J. Eng. Science., 40, p. 773.

IJSET@2013

(ISSN : 2277-1581) 1 Feb. 2013

[20]Md. Abdul Sattar, Md. Alam (1995): Thermal diffusion as well as transportation effect on MHD free convection and Mass Transfer flow past an accelerated vertical porous plate, Ind. J. of Pure and App. Maths., 24, p. 679.

[21] Mohamed R A (2009): Double-Diffusive ConvectionRadiation Interaction on Unsteady MHD Flow over a Vertical Moving Porous Plate with Heat Generation and Soret Effects, Applied Mathematical Sciences, Vol. 3, no.

13, pp. 629 – 651.

[22]Muthucumaraswamy R (2002): Effects of a chemical reaction on a moving isothermal surface with suction, Acta Mechanica, Vol. 155, pp. 65-72.

[23]Olanrewaju P O, Anake T, Arulogun O T and Ajadi D A (2012): Further Results on the Effects of Variable Viscosity and Magnetic Field on Flow and Heat Transfer to a Continuous Flat Plate in the Presence of Heat Generation and Radiation with a Convective Boundary Condition, American Journal of Computational and Applied Mathematics, 2(2), pp. 42-48 [24]Raptis A (1986): Flow through a porous medium in the presence of magnetic field, Int. J. Energy Res., Vol.10, pp.

97-101.

[25]Saritha S and Satya Narayana P V (2012): thermal diffusion and chemical reaction effects on unsteady MHD free convection flow past a semi infinite vertical permeable moving plate, Asian Journal of Current Engineering and Maths, 1: 3, pp. 131 – 138.

[26]Satyanarayana P V, Kesavaiah D Ch and Venkataramana S (2011). Viscous dissipation and thermal radiation effects on an unsteady MHD convection flow past a semi-infinite vertical permeable moving porous plate International Journal of Mathematical Archive, 2(4), pp. 476 - 487.

[27]Shercliff J A (1965): A text book of Magnetohydrodynamics, Pergamon Press, London.

[28]Soundalgekar V M, Gupta S K and Birajdar N S (1979): Effects of mass transfer and free convection effects on MHD Stokes problem for a vertical plate, Nucl., Eng. Design, Vol. 53, pp. 309-346.

[29]Sreekantha Reddy M V, Srinivasa Kumar C and Prasada Rao D R V (2012): Effects of chemical reaction, radiation and thermo-diffusion on convective heat and mass transfer flow of a viscous dissipated fluid in a vertical channel with constant heat flux, Adv. Appl. Sci. Res., 3(5), pp. 30323044

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(ISSN : 2277-1581) 1 Feb. 2013

[30] Stanford Shateyi, Sandile Sydney Motsa, and Precious Sibanda (2010): The effects of thermal radiation, hall currents, Soret, and Dufour on MHD flow by mixed convection over a vertical Surface in porous media, Mathematical Problems in Engineering.

[31] Sudheer Babu M, Satyanarayana P V, Sankar Reddy T and Umamaheswara Reddy D (2011): Effects of thermal radiation and chemical reaction on MHD convective flow of a polar fluid past a moving vertical plate with viscous dissipation, Elixir Appl. Math. 40: pp. 51685173.

[32] Takhar H S and Ram P C (1994): Magneto hydrodynamic free convection flow of water at 4ยบC through a porous medium, Int. Comm. Heat Mass Transfer, Vol. 21, pp.

371-376.

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APPENDIX

 Pr  Pr 2  4 Pr  F  Q    m2      2    Sc  Sc 2  4 KrSc  m4       2    1 1 4N  m6     2    Pr  Pr 2  4 N  1  m8      2   1  N1  Pr  F  Q  , N   M   K   Sc  Sc 2  4 KrSc  m10       2    2  2  4N m12    2 

  

ScS0 m22 D1   2 m2  Scm2  KrSc 1  D2  1  D1 , N   M   K  ScS0 m82 B11 D3   2 m8  Scm8  KrSc

D3  

ScS0 m82 B11 m82  Scm8  KrSc

D4  

4ScS0 m62 B1 4m62  2Scm6  KrSc

D5  

2 2

4ScS0 m B2 4m  2Scm2  KrSc 2 2

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D6  

4ScS0 m42 B3 4m22  2Scm2  KrSc

D7  

4ScS0 m22 B4 4m22  2Scm2  KrSc ScS0  m2  m6  B5

D8   D9  

 m2  m6 

2

 Sc  m2  m6   KrSc

ScS0  m2  m4  B6

 m2  m4 

D10   D11   D12  

2

 Sc  m2  m4   KrSc

ScS0  m2  m4  B7

 m2  m4 

2

 Sc  m2  m4   KrSc

ScS0  m2  m6  B8

 m2  m6 

2

 Sc  m2  m6   KrSc

ScS0  m4  m6  B7

 m4  m6 

2

 Sc  m4  m6   KrSc

4ScS0 m22 B10 D13   2 4m2  2Scm2  KrSc D14    D3  D4  D5  D6  D7  D8  D9  D10  D11  D12  D13  GmD2 Gr , A2   2 m  m2  N m4  m4  N GmD1 A3   2 , A4    A1  A2  A3  m2  m2  N

A1  

2 2

GrB11 GrB1 , A6   2 m  m8  N 4m6  2m6  N GrB2 A7   2 4m2  2m2  N A5  

A8  

2 8

GrB3 4m  2m4  N 2 4

GrB4 4m  2m2  N GrB5 A10   2  m2  m6   Pr  m2  m6   N A9  

2 2

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International Journal of Scientific Engineering and Technology Volume No.2, Issue No.2, pg : 56-72

A11   A12   A13   A14  

 m2  m4 

GrB6

 Pr  m2  m4   N

2

 m2  m4   m2  m6 

GrB7

2

2

A29    A5  A6  A7  A8  A9  A10  A11  A12  A13  A14  A15  A16  A17   A18  A19  A20  A21  A22

 A23  A24  A25  A26  A27  A28 

GrB8

Pr m62 A42 Pr m22 A12 B1   2 B2   2 4m6  2 Pr m6  N1 4m2  2 Pr m2  N1

GrB9

Pr m42 A22 B3   2 4m4  2 Pr m4  N1

 Pr  m2  m6   N

2

 m4  m6 

 Pr  m2  m4   N

 Pr  m4  m6   N

A16  

GrB11 m8  m8  N

B4  

A15  

GrB10 GmD3 A17   2 m8  m8  N 4m2  2m2  N

B5  

2

2

GmD5 GmD4 A19   2 2 4m6  m6  N 4m2  m2  N GmD6 A20   4m4 2  m4  N

A18  

GmD7 A21   4m2 2  m2  N GmD8 A22   2  m2  m6    m2  m6   N

A23   A24   A25   A26   A27  

(ISSN : 2277-1581) 1 Feb. 2013

GmD9

 m2  m4    m2  m4   N 2

GmD10

 m2  m4    m2  m4   N 2

GmD11

 m2  m6    m2  m6   N 2

B6   B7   B8   B9  

Pr m22 A32 4m22  2 Pr m2  N1 2 Pr m6 m4 A1 A4

 m2  m6 

2

 Pr  m2  m6   N1

2 Pr m2 m6 A1 A2

 m2  m4 

2

 Pr  m2  m4   N1

2 Pr m2 m4 A3 A2

 m2  m4 

2

 Pr  m2  m4   N1

2 Pr m2 m6 A3 A4

 m2  m6 

2

2 Pr m4 m6 A2 A4

 m4  m6 

B10  

 Pr  m2  m6   N1

2

 Pr  m4  m6   N1

2 Pr m22 A1 A3 4m22  2 Pr m2  N1

B11    B1  B2  B3  B4  B5  B6  B7  B8  B9  B10 

GmD12

 m4  m6    m4  m6   N 2

GmD13 GmD14 A28   2 4m  2m2  N m10  m10  N 2 2

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2

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1.2 sc=0.65, S 0=1.0,Ec=0.001,Pr=0.025,Gr=5.0

1.5

1

Sc=0.65,S0=1.0,Ec=0.001,Pr=0.75,Gr=5.0

Gm=1.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0

Gm=5.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0

0.8 1

Gm=1.0,2.0,3.0,4.0

u

u

0.6

0.5

0.4

0

0.2 0

-0.5 0

2

4

6

8

10

-0.2 0

y Figure (1): Velocity profiles for different values of Gm

Kr=0.5,1.0,1.5,2.0

1

2

3

4

5

6

y Figure (5): Velocity profiles for different values of Kr

1 2

Sc=0.65,S =1.0,Ec=0.001,Pr=0.025,Gr=5.0 0

0.8

Gm=1.0,F=3.0,Q=1.0, Kr=0.5,K=1.0, M=2.0 1.5

Sc=0.65,S 0 =1.0,Ec=0.001,Pr=0.025,Gr=5.0 Gm=1.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0

0.6

u

u

1

Gr=1.0,2.0,3.0,4.0

0.4

0.5

0.2 0

M=0.5,1.0,1.5,2.0

0 -0.5 0

-0.2 0

5

10 y Figure (2): Velcoity profiles for different values of Gr

5

10

15

y Figure (6): Velocity profiles for different values of M

15

1.2

1.4 1

1.2

Sc=0.65, S 0=1.0;Ec=0.001,Pr=0.025,Gr=5.0

Gm=1.0,F=3.0,Q=1.0, Kr=0.5, K=1.0,M=2.0

0.8

Gm=2.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0

1

0.6

u

u

0.8 0.6

0.4

0.4

0.2

0.2

0 Pr=0.025,0.50,0.075,0.1

F=1.0,2.0,3.0,4.0 0 -0.2 0

Sc=0.65,S0=1.0,Ec=0.001,Pr=0.025,Gr=5.0

-0.2 0

5 10 y Figure (3): Velocity profiles for different values of F

15

5

10 y Figure (7): Velocity profiels for different values of Pr

15

1.2

2

1 Sc=0.65,S0=1.0,Ec=0.001,Pr=0.025,Gr=5.0

Gm=2.0,F=3.0,Q=1.0,Kr=0.5, K=1.0, M=2.0

Gm=2.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0

1.5

Sc=0.65,S0=1.0,Ec=0.001,Pr=0.025,Gr=5.0

0.8

u

0.6 1

K=1.0,2.0,3.0,4.0

u

0.4

0.5

0.2 0

0

-0.2 0

-0.5 0

5

10

y Figure (4): Velocity profiles for different values of K

IJSET@2013

15

Q=2.0,4.0,6.0,8.0

5

10 y Figure (8): Velocity profiles for different values of Q

15

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1

Sc=0.65, S 0=1.0,Ec=0.001,Pr=0.75,Gr=5.0

0.6

Ec=0.001,Pr=0.025,Q=1.0Gr=5.0,Gm=2.0 0.8

Gm=1.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0 0.5

0.6 

u

0.4 0.3

S0=1.0,2.0,3.0,4.0

0.4

F=1.0,2.0,3.0,4.0

0.2 0.2

0.1 0 0

1

2

3

4 5 6 7 y Figure (9): Velocity profiles for different values of S 0

0 0

8

5 10 15 y Figure (11): Temperature profiles for different values of F

20

1 1.2

0.8

Ec=0.001,F=2.0,Q=1.0,Gr=5.0,Gm=2.0

Sc=0.65,S0=1.0,Ec=0.001,Pr=0.025,Gr=5.0

1

Gm=1.0, F=3.0, Q=1.0,Kr=0.5,K=1.0,M=2.0

0.6 

0.8

Pr=0.025,0.050,0.075,0.10

0.6 u

0.4

0.4

0.2 0.2 Sc=0.22,0.30,0.60,0.78 0 -0.2 0

0 0 5 10 y Figure (10): Velocity profiles for different values of Sc

15

2 4 6 8 10 12 14 16 y Figure (12): Temperature profiles for different values of Pr

18

1 Ec=0.001,Pr=0.025,F=2.0,Gr=5.0,Gm=2.0

0.8



0.6

0.4

Q=1.0,2.0,3.0,4.0

0.2

0 0

2 4 6 8 10 12 14 16 y Figure (13): Temperature profiles for different values of Q

18

1.2 1 Ec=0.001,Pr=0.71,F=2.0,Q=5.0,Gr=5.0 0.8



0.6 0.4

Gm=1.0,2.0,3.0,4.0

0.2 0 -0.2 0

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0.5

1 1.5 2 y Figure (14): Temperature profeiles for different values of Gm

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1 Sc=0.65,

0.8

Ec=0.001,Pr=0.71,F=3.0,Q=1.0,Gm=10.0

0.8

0.6

C



Gr=5.0:5.0:20.0

Pr=0.75

Gr=5.0,10.0,15.0,20.0

0.2

0.5 1 1.5 2 2.5 y Figure (15): Temperature profiles for different values of Gr

0 0

3

1.4

1 2 3 4 5 y Figure (19): Concentration profiles for different values of Gr

1

Ec=0.001,Pr=0.75,F=1.0,Q=4.0,Gr=4.0 Gm=10.0, M=1.0,K=0.5, Kr=0.5,S 0=3.0

0.8

1

Sc=0.65,S0=1.0,Ec=0.001,Pr=0.025,Gr=50.0 F=2.0, Q=50.0, Kr=0.5, M=2.0, Gm=100.0

0.6

C



0.8 0.6

Ec=0.001,

0.4

0.2

1.2

S 0=0.5,

F=2.0,Q=5.0,Kr=0.5,K=0.5,M=2.0,Gm=50.0

0.6

0.4

0 0

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Sc=0.22,0.08,0.38,0.46

K=0.5,1.0,1.5,2.0 0.4

0.4 0.2

0.2 0 0

0.5

1 1.5 y Figure (16): Temperature profiles for different values of Sc

0 0

2

1

1 Sc=0.65,S0=2.0, Ec=0.001, Pr=0.025

0.8

1 2 3 4 5 6 7 8 y Figure (20): Concentration profiles for different values of K

Q=1.0,Kr=0.5,K=0.5,M=2.0,Gm=2.0

0.8

0.6

Sc=0.65,S0=2.0,Ec=0.001,Pr=0.025,Gr=5.0 F=1.0,Q=1.0, Kr=0.5,K=0.5,M=2.0,Gm=2.0

C

C

0.6

0.4

F=1.0,2.0,3.0,4.0

0.4

0.2

0 0

Kr=0.5,1.0,1.5,2.0

0.2

2 4 6 8 10 y Figure (17): Concentration profiles for different values of F

0 0

5

10 15 y Figure (21): Concentration profiles for different values of Kr

1 Sc=0.65, 0.8

S 0=0.5,

Ec=0.001,

1

Pr=0.75

Gr=50.0,F=2.0,Q=5.0,Kr=0.5,K=0.5,M=2.0

Sc=0.65,S =0.5,Ec=0.001,Pr=0.75,K=0.5 0

0.8 Gm=5.0,10.0,15.0,20.0

0.6

C

0.6

C

0.4

Gr=60.0, F=2.0, Q=5.0, Kr=0.5, Gm=20.0

0.4 0.2

0.2 0 0

1 2 3 4 5 y Figure (18): Concentration profiles for different values of Gm

IJSET@2013

M=0.5,1.0,1.5,2.0 0 0

1

2

3 4 5 y Figure (22): Concentration profiles for different values of M

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International Journal of Scientific Engineering and Technology Volume No.2, Issue No.2, pg : 56-72

(ISSN : 2277-1581) 1 Feb. 2013

1 Sc=0.65, 0.8

S 0=0.5,

Ec=0.001,

Gr=5.0

F=2.0,Q=1.0,Kr=0.5,K=0.5,M=2.0,Gm=10.0

2.2

C

0.6 Pr=0.025,0.050,.075,0.10

Sc=0.65,S 0=1.0,Ec=0.001,Pr=0.71,Gr=5.0

2

Gm=1.0,F=3.0,Q=1.0,Kr=0.5,K=1.0,M=2.0

0.4 1.8

0.2

ď ´

1.6 1.4

0 0

1

2

3

4

5 1.2

y Figure (23): Concentration profiles for different values of Pr

1

Pr=0.5,1.0,1.5,2.0

1 Sc=0.65, 0.8

S 0=2.0, Ec=0.001,

0.8 1

1.5 2 2.5 3 3.5 4 4.5 5 Gr Figure (27): Effect of Prandtl number on skin friction versus Gr

Pr=0.025

F=1.0,Q=1.0,Kr=0.5,K=0.5,M=2.0,Gm=2.0 1

C

0.6

0.4

0.9

Q=1.0,2.0,3.0,4.0

0.8

Nu

0.2

0 0

0.7 M=0.5,0.6,0.7,0.8

2

4

6

8

10

y Figure (24): Concentration profiles for different values of Q

0.6 0.5

Ec=0.001,Pr=0.75,F=1.0,Q=4.0,Gr=4.0,M=2.0 Gm=10.0, K=0.5, Sc=0.65, Kr=0.5, S 0=2.0

1 Sc=0.65, S 0=2.0, Ec=0.001, 0.8

0.4 1

Pr=0.025

F=1.0,Q=1.0,Kr=0.5,K=0.5,M=2.0,Gm=2.0

1.5 2 2.5 3 3.5 4 4.5 Gr Figure (28): Effect of Magnetic field on Nusselt number

5

C

0.6 S0=1.0,2.0,3.0,4.0 0.4

0.2

0 0

5 10 15 y Figure (25): Concentration profiles for different values of S 0

1

S0=2.0,Ec=0.001,Pr=0.025,Gm=2.0 0.8

F=1.0,Q=1.0,Kr=0.5,K=0.5,M=2.0

C

0.6

Sc=0.22,0.30,0.60,0.68 0.4

0.2

0 0

5

10

15

y Figure (26): Concemtration profiles for different values of Sc

IJSET@2013

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International Journal of Scientific Engineering and Technology Volume No.2, Issue No.2, pg : 56-72

IJSET@2013

(ISSN : 2277-1581) 1 Feb. 2013

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