P.O. Olanrewaju, Nonlinear Sci. Lett.A, Vol.4, No.1, 21-34, 2013

Copyright 2013 Asian Academic Publisher Ltd. Journal Homepage: www.NonlinearScience.com

Dufour and Soret effects on MHD free convection with thermalradiation and mass transfer past a vertical plate embedded in a porousmedium

1P.O. Olanrewaju, 2A. Adeniyan

1Department of Mathematics, Covenant University, Ota, Ogun State. NigeriaEmail: oladapo_anu@yahoo.ie2Department of Mathematics, University of Lagos, Lagos, NigeriaEmail: aadeniyan@unilag.edu.ng

Abstract

The thermal-diffusion and diffusion-thermo effects on the heat and mass transfer characteristics of freeconvection past a moving vertical plate embedded in a porous medium in the presence of magneticfield, blowing/suction and thermal radiation is investigated. Using appropriate similarity variables andboundary-layer approximations, the fluid equations for continuity, momentum, energy and chemicalspecies concentration balance governing the problem are formulated. These equations are solvednumerically by using the most effective Newton–Raphson shooting method along with sixth-orderRunge–Kutta integration algorithm. . It was found that the fluid temperature increases while both thefluid velocity and chemical species concentration decreases as Soret number decreases and Dufournumber increases. Moreover, both magnetic field and porous medium permeability parameters have aweakening effect on the local skin friction coefficient, local Nusselt number and the local Sherwoodnumber. Comparison with previously published work is performed for a special case and fullagreement is obtained. Our analysis shows that for some kinds of mixtures (for example, H2-air) withthe light and medium molecular weight, the Soret and Dufour’s effects play a significant role andshould be taken into consideration. This type of study has potential to serve as a prototype forindustrial processes involving isotope separation as well as understanding the flow structure, heat andmass transfer characteristics of mixture between gases with very light/medium molecular weight.

Keywords: Magnetohydrodynamic (MHD), Boundary layer flow; Porous medium; Heat and masstransfer; Dufour and Soret effects; Thermal radiation

1. Introduction

The problem of hydromagnetic flow with heat and mass transfer over a moving surface could be verypracticable in many applications in the engineering and industries. For instance, in annealing andthinning of copper wires, the properties of the final product depend to a great extent on the rate ofcooling. By drawing such strips in an electrically conducting fluid subject to a magnetic field, the rateof cooling can be controlled and final products of desired characteristics might be achieved. And also,in several engineering processes, materials manufactured by extrusion processes and heat treatedmaterials travelling between a feed roll and a wind up roll on convey belts possess the characteristicsof a moving continuous surface. Meanwhile, the range of free convective flows that occur in natureand in engineering practice is very large and has been extensively considered by many researchers

ISSN 2076-2275: Nonlinear Science Letters A- Mathematics, Physics and Mechanics22

(see, Jaluria, 1980; Gupta and Gupta, 1977; Makinde, 2010, among others). When heat and masstransfer occur simultaneously between the fluxes, the driving potentials are of more intricate natureMakinde and Aziz (2010). An energy flux can be generated not only by temperature gradients but bycomposition gradients. The energy flux caused by a composition is called Dufour or diffusion-thermoeffect. Temperature gradients can also create mass fluxes, and this is the Soret or thermal-diffusioneffect Kafoussias and Williams (1995). Generally, the thermal-diffusion and the diffusion-thermoeffects are of smaller-order magnitude than the effects prescribed by Fourier’s or Fick’s laws and areoften neglected in heat and mass transfer processes (see, Chamkha, 1997; Mohamed and Abo-Dahab,2009). However, there are exceptions, the thermal-diffusion effect, for instance, has been utilized forisotope separation. In mixture between gases with very light molecular weight (Hydrogen-Hellium)and of medium molecular weight (Nitrogen-air) the diffusion-thermo effect was found to be of amagnitude such that it cannot be neglected. Afify (2009) carried out an analysis to study freeconvective heat and mass transfer of an incompressible, electrically conducting fluid over a stretchingsheet in the presence of suction and injection with thermal-diffusion and diffusion-thermo effects.Alam et al. (2006), studied Dufour and Soret effects on steady free convection and mass transfer flowpast a semi-infinite vertical porous plate in a porous medium.Moreover, interaction of buoyancy with thermal radiation has also increased greatly during the lastdecade due to its importance in many practical applications (see, Sparrow and Cess, 1978; Makindeand Olanrewaju, 2010; Abd El-Aziz, 2009). The thermal radiation effect is important under manyisothermal and nonisothermal situations. If the entire system involving the polymer extrusion processis placed in a thermally controlled environment, then thermal radiation could be important. Theknowledge of radiation heat transfer in the system can, perhaps, lead to a desired product with asought characteristic Makinde, (2005) studied the thermal-diffusion and diffusion-thermo effects onthe heat and mass transfer characteristics of free convection past a continuously stretching permeablesurface in the presence of magnetic field, blowing/suction, and radiation. Hayat et al. (2010) examinedthe effects of radiation and magnetic field on the mixed convection stagnation-point flow over avertical stretching sheet in a porous medium. Although there are numerous practical applications ofbuoyancy interaction with magnetic field, thermal radiation, thermal-diffusion and diffusion-thermoon the heat and mass transfer characteristics in industrial processes, few previous published papersdiscussed the combined relation.Motivated by the above referenced work and the numerous possible industrial applications of theproblem (like in isotope separation), it is of paramount interest in this study to investigate thecombined effects of the thermal radiation, magnetic field, Soret and Dufour number on boundary layerfree convection flow past a vertical moving porous surface embedded in a porous medium. By usingsimilarity transformations, the set of governing equations and the boundary condition are reduced tonon-linear ordinary differential equations with appropriate boundary conditions. Furthermore, thesimilarity equations are solved numerically by using shooting technique with sixth-order Runge–Kuttaintegration scheme. Numerical results of the local skin friction coefficient, the local Nusselt number,local Sherwood number as well as the velocity, temperature and the concentration profiles arepresented for different physical parameters.

2. Mathematical Formulation

Consider a two-dimensional steady free convection and mass transfer flow past a continuously movingsemi-infinite vertical porous flat plate in a porous medium. over a vertical porous plate embedded of aviscous incompressible fluid past a continuously moving semi-infinite vertical porous flat plate in aporous medium is considered. The fluid is assumed to be Newtonian, incompressible, viscous andelectrically conducting. Its property variations due to temperature and chemical species concentration

P.O. Olanrewaju, Nonlinear Sci. Lett.A, Vol.4, No.1, 21-34, 2013 23

are limited to fluid density. The density variation and the effects of the buoyancy are taken intoaccount in the momentum equation (Boussinesq approximation). In addition, there is no appliedelectric field and all of the Hall effects and Joule heating are neglected (see Figure 1). The flow isassumed to be in the x-direction which is taken along the plate in the upward direction and the y-axis istaken normal to it.

x

T g U , C, T

u=U0 T

v =v0(x) C

Cw u

y

Figure 1. Flow configuration and coordinate system

Then under the usual Boussinesq’s approximation, the governing equations relevant to the problemare:Continuity equation

0

y

v

x

u(1)

Momentum equation

uB

uK

CCgTTgy

u

y

uv

x

uu e

20*

2

2

)()(

(2)

Energy equation

y

q

ky

C

cc

kD

y

T

y

Tv

x

Tu r

ps

Tm

2

2

2

2

(3)

Concentration equation

,2

2

2

2

y

T

T

kD

y

CD

y

Cv

x

Cu

m

Tmm

(4)

BO

ISSN 2076-2275: Nonlinear Science Letters A- Mathematics, Physics and Mechanics24

where (u, v) are the velocity components in the x and y directions respectively, υ is the kinematicviscosity, g is the acceleration due to gravity, ρ is the density, β is the coefficient of thermal expansion,β* is the coefficient of concentration expansion, T is the fluid temperature within the boundary layer, Cis the diffusion species concentration within the boundary layer. The plate surface is kept at a constanttemperature Tw and constant concentration Cw of the diffusion species, which are higher than thecorresponding values

T andC of the ambient fluid. Also, K is the permeability of a porous

medium, α is the thermal diffusivity, Dm is the coefficient of mass diffusivity, cp is the specific heat atconstant pressure, Tm is the mean fluid temperature, kT is the thermal diffusion ratio, and cs is theconcentration susceptibility. The boundary conditions for the model are

,,,0,0

,0,,, 00

yasCCTTvu

yatCCTTxvvUu

ww

ww (5)

where U0 is the plate uniform velocity and v0(x) is the velocity of suction at the plate surface. Theradiative heat flux qr is described by Roseland approximation [10] such that

,3

4 4*

y

T

Kqr

(6)

where * and K are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively.Following Chamkha (1997), we assume that the temperature differences within the flow aresufficiently small so that the T4 can be expressed as a linear function after using Taylor series toexpand T4 about the free stream temperature T and neglecting higher-order terms. This results in thefollowing approximation:

.34 434 TTTT (7)

Using (6) and (7) in (3), we obtain

.3

162

23*

y

T

K

T

y

qr

(8)

From Eq. (1) we introduce stream function ψ such that

yu /and

xv / .

Following Alam et al. [9], it is convenient to use the following similarity transformations:

.2

),(,

,,,2

00

00

ffx

UvfUu

CC

CC

TT

TTfvU

x

Uy

ww

w

(9)

P.O. Olanrewaju, Nonlinear Sci. Lett.A, Vol.4, No.1, 21-34, 2013 25

Upon substituting the expressions in Eq. (9) into Eqs. (1)–(4), we obtain the following transformedequations:

,0 fMfKGmGrfff (10)

,0PrPr3

41

fd DfR (11)

.0 ScSrScf (12)

The transformed boundary conditions are:

,0,0,0

,01,1,1,

asf

atfff w

(13)

where a prime denotes ordinary differentiation with respect to η, and

20

2

U

xTTgGr w

is the local thermal Grashof number,

0

2* 2

U

xCCgGm w

is the local solutal Grashof number,

0

2

UK

xK

is the local permeability parameter,

Pr is the Prandtl number,

mDSc

is the Schmidt number,

CCT

TTkDSr

wm

wTm

is the Soret number,

TTcc

CCkDD

wps

wTmf

is the Dufour number,

k

TRd

3*4 is the Radiation parameter,

0

20

*2

U

xBM

is local magnetic field parameter,

00

2

U

xvf w is the suction velocity parameter.

Of special significance with respect to application of free convection problems are the local skin-friction coefficient, the local Nusselt number and the local Sherwood number, which are givenrespectively by the following expressions;

ISSN 2076-2275: Nonlinear Science Letters A- Mathematics, Physics and Mechanics26

,0Re2

1 21

fC f ,0Re 21

Nu 0Re 2

1

Sh , (14)

where /Re 0 xU is the local Reynold’s number.

3. Numerical procedure

The set of Eqs. (11)-(13) together with the boundary conditions (14) have been solved numerically byapplying Nachtsheim-Swigert shooting iteration technique along side with Runge-Kutta sixth-orderintegration method (see, Makinde, 2010; Na, 1979). The numerical procedure is implemented oncomputer using MAPLE (Heck, 2003) and the grid mesh of = 0.001 is selected to be satisfactory

for a convergence criterion of 10-7 in nearly all cases and the maximum values of = 8.0 which issufficient large for the velocity to approach the relevant stream velocity. From the process ofnumerical computation, the local skin-friction coefficient, the local Nusselt number and the localSherwood number, which are respectively proportional to 0,0 f and )0( are also sortedout and their numerical values are presented in a tabular form. In order to check the method ofnumerical accuracy, Table 1 are the data comparisons with the previous published paper [9] and theresults are found in good agreement.

Table 1: Computations showing comparison of Skin-friction coefficient, Nusselt number andSherwood number with Alam et al. [9] for Gr=10, Gm=4, fw=0.5, K=0.3, Pr=0.71, Sc=0.22, Ra=0,M=0.

Present Present Present Alam etal. [9]

Alam etal. [9]

Alam etal. [9]

Df Sr )0(f )0( )0( )0(f )0( )0( 0.030 2.0 6.2293 1.1564 0.1532 6.2285 1.1565 0.15310.037 1.6 6.1500 1.1500 0.2284 6.1491 1.1501 0.22830.050 1.2 6.0729 1.1426 0.3033 6.0720 1.1428 0.30330.075 0.8 6.0015 1.1332 0.3781 6.0006 1.1333 0.37810.150 0.4 5.9563 1.1155 0.4540 5.9553 1.1157 0.4540

4. Results and discussions

The effects of various embedded thermophysical parameters on the velocity, temperature andconcentration profiles as well as the local skin-friction coefficient, the local Nusselt number and thelocal Sherwood number are tabulated and displayed with the graphical illustrations in table (2) andfigures (2)-(13). The value of Prandtl number Pr is taken to be 0.71, which corresponds, physically toair and the Schmidt number (Sc) is chosen for hydrogen (Sc = 0.22) at approximate mean temperatureof 25oC and one atmospheric pressure. Dufour number Df and Soret number Sr are chosen in such away that their product is constant provided that the mean temperature (Tm) is kept constant as well.Moreover, the numerical values of other parameters like thermal Grashof number Gr, solutal Grashofnumber Gm, suction parameter fw and permeability parameter K, radiation parameter Ra and magneticfield parameter M are chosen arbitrarily. It is noteworthy that positive values of the buoyancyparameters Gr > 0, Gm > 0 corresponds to the cooling problem and also indicates that the chemical

P.O. Olanrewaju, Nonlinear Sci. Lett.A, Vol.4, No.1, 21-34, 2013 27

species concentration in the free stream region is less than the concentration at the boundary surface.Table 2 shows the data of local skin friction, heat and mass transfer rate. From the table, we coulddetermine that as the Soret number decreases and the Dufour number increases, the local skin frictionat the plate surface diminished, the heat transfer rate weakened while mass transfer rate is enhanced.The local skin friction coefficient and the local Sherwood number at the plate surface increase byincreasing of the radiation parameter but the local Nusselt number decreases. Both magnetic field andporous medium permeability parameters have a weakening effect on the local skin friction coefficient,local Nusselt number and the local Sherwood number. Moreover, it is observed that the suctionparameter (fw > 0) tends to decrease the local heat and mass transfer rate, while the opposite trend isobserved for the injection parameter (fw < 0). This is because as the injection is applied at the surface,the momentum transport is reduced near the surface thereby causing a reduction in the local Nusseltnumber and that blowing tends to broaden the concentration distributions, accordingly reducing thewall concentration gradient, and hence the local Sherwood number results.

Table 2: Computations showing Skin-friction coefficient, Nusselt number and Sherwood number forGr=10, Gm=4, Pr=0.71, Sc=0.22.

Df Sr M Ra K fw )0(f )0( )0( 0.030 2.0 0.1 0.1 0.3 0.5 6.23999 1.07146 0.188130.037 1.6 0.1 0.1 0.3 0.5 6.16584 1.06555 0.256660.050 1.2 0.1 0.1 0.3 0.5 6.09386 1.05876 0.324970.075 0.8 0.1 0.1 0.3 0.5 6.02731 1.05003 0.393220.150 0.4 0.1 0.1 0.3 0.5 5.98513 1.03389 0.462480.150 0.4 0.5 0.1 0.3 0.5 5.52688 1.01415 0.450440.150 0.4 1.0 0.1 0.3 0.5 5.00711 0.99095 0.436300.150 0.4 0.1 0.5 0.3 0.5 6.45742 0.83289 0.497870.150 0.4 0.1 1.0 0.3 0.5 6.88763 0.69198 0.528970.150 0.4 0.1 0.1 0.5 0.5 5.75101 1.02389 0.456380.150 0.4 0.1 0.1 1.0 0.5 5.20837 1.00004 0.441840.150 0.4 0.1 0.1 0.3 0.7 5.93166 1.10808 0.476960.150 0.4 0.1 0.1 0.3 1.0 5.81149 1.22421 0.499400.150 0.4 0.1 0.1 0.3 -0.5 5.94164 0.70335 0.394510.150 0.4 0.1 0.1 0.3 -1.0 5.74348 0.56407 0.36238

A. Velocity profiles

Figures 2–5 represent the effects of various thermophysical parameters on the velocity profiles.Generally, it is observed that the fluid velocity increasing from the moving plate surface to its peakvalues within the boundary layer and then decreases gradually to its zero free stream value satisfyingthe boundary condition. In figure 2, the effect of radiation parameter is illustrated. The peak value ofthe the fluid velocity within the boundary layer increases as the radiative heat generated in the flowsystem increases. This can be attributed to the fact that as Ra increases the flux of energy transport tothe fluid increases causing the fluid to flow faster. The effects of magnetic field parameter, thermalGrashof number and permeability parameter on the velocity field are shown in figure 3. It is seen fromthis figure that the peak value of the velocity decreases with a combined increase in the magnetic fieldintensity and permeability parameter while the increase of thermal Grashof number increases the peak

ISSN 2076-2275: Nonlinear Science Letters A- Mathematics, Physics and Mechanics28

value of velocity within the boundary layer. As expected, the effects of a transverse magnetic field onan electrically conducting fluid give rise to a resistive-type force called the Lorentz force. This forcehas the tendency to slow down the motion of the fluid and to increase its temperature andconcentration boundary layer [3, 4, 6-8]. In figure 4, the effects of solutal Grashof number andsuction/injection parameter on the velocity field are shown. It is interesting to note that the peak valueof velocity increases with injection as well as when the concentration difference between the mean andfree stream values increases whereas the velocity peak value within the boundary layer decreases withthe increase of suction parameter. The influence of Soret number Sr and Dufour number Df on thevelocity field is shown in Fig.5. It is observed that the peak values of the fluid velocity decreases asSoret number decreases and Dufour number increases.

Figure 2: Velocity profiles for Pr=0.71, Sc=0.22, Gr=10, K =0.3, Gm=4, fw=0.5, Sr=2, Df = 0.03, M=0.1

P.O. Olanrewaju, Nonlinear Sci. Lett.A, Vol.4, No.1, 21-34, 2013 29

Figure 3: Velocity profiles for Pr=0.71, Sc=0.22, Gm=4, fw=0.5, Sr=2, Df = 0.03, Rd = 0.1

Figure 4: Velocity profiles for Pr=0.71, Sc=0.22, Gr=10, K=0.3, Sr = 2, Df = 0.03, M = 0.1, Rd = 0.1

ISSN 2076-2275: Nonlinear Science Letters A- Mathematics, Physics and Mechanics30

Figure 5: Velocity profiles for Pr=0.71, Sc=0.22, Gr=10, K=0.3, Gm = 4, fw = 0.5, M = 0.1, Rd = 0.1

B. Temperature profiles:

Figures 6–9 demonstrate the effect of various parameters on the fluid temperature profiles. Themaximum fluid temperature is attained at the moving plate surface, within the boundary layer, thefluid temperature decreases gradually to its free stream zero value satisfying the boundary condition asexpected. Moreover figure 6 shows that the dimensionless temperature increases with increasingradiation parameter Ra. The physical facts could be explained by the effect of radiation is to increasethe flux of energy transport to the fluid and accordingly increase the fluid temperature. The effects ofmagnetic field intensity, thermal Grashof number and permeability parameter on the temperature fieldare shown in figure 7. From this figure we observe that the temperature increases with a combinedincrease in the magnetic field intensity and permeability parameter while an increase in thermalGrashof number due to buoyancy effect causes a decrease in the thermal boundary layer thickness.Figure 8 shows the effects of solutal Grashof number and suction/injection parameter on thetemperature profiles. We observed that the temperature decreases with the increase of both the suctionparameter and solutal Grashof number while an increase in the fluid injection at the plate surfacecauses an increase in the thermal boundary layer thickness. The influence of Soret number Sr andDufour number Df on the temperature field is shown in figure 9. It is observed that the fluidtemperature increases as Soret number decreases and Dufour number increases.

P.O. Olanrewaju, Nonlinear Sci. Lett.A, Vol.4, No.1, 21-34, 2013 31

Figure 6: Temperature profiles for Pr=0.71, Sc=0.22, Gr=10, K =0.3, Gm=4, fw=0.5, Sr=2, Df = 0.03,M= 0.1

Figure 7: Temperature profiles for Pr=0.71, Sc=0.22, Gm=4, fw=0.5, Sr=2, Df = 0.03, Rd = 0.1

ISSN 2076-2275: Nonlinear Science Letters A- Mathematics, Physics and Mechanics32

Figure 8: Temperature profiles for Pr=0.71, Sc=0.22, Gr=10, K=0.3, Sr = 2, Df = 0.03, M = 0.1, Rd =0.1

Figure 9: Temperature profiles for Pr=0.71, Sc=0.22, Gr=10, K=0.3, Gm = 4, fw = 0.5, M = 0.1, Rd =0.1

C. Concentration profiles

The concentration profiles are displayed in figures 10–13.It is seen that the chemical speciesconcentration is highest at the moving plate surface and decreases to its zero free stream values. Anincrease in the value radiation parameter Ra causes a decrease in concentration boundary layerthickness as illustrated in figure 10. The effects of magnetic field parameter, thermal Grashof numberand permeability parameter on the concentration field are shown in figure 11. It is observed that theconcentration increases with a combined increase in magnetic field intensity and permeabilityparameter but decreases with the increases of thermal Grashof number due to buoyancy effects. Theeffects of solutal Grashof number and suction/injection parameter on the concentration profiles are

P.O. Olanrewaju, Nonlinear Sci. Lett.A, Vol.4, No.1, 21-34, 2013 33

shown in figure 12. We observed that the concentration decreases with the increase of both the suctionparameter and solutal Grashof number but increases with an increase in fluid injection at the platesurface. From figure 13, we observed that the chemical species concentration within the boundarylayer decreases as Soret number decreases and Dufour number increases.

Figure 10: Concentration profiles for Pr=0.71, Sc=0.22, Gr=10, K =0.3, Gm=4, fw=0.5, Sr=2, Df =0.03, M= 0.1

Figure 11: Concentration profiles for Pr=0.71, Sc=0.22, Gm=4, fw=0.5, Sr=2, Df = 0.03, Rd = 0.1

Conclusions

In this paper, we have studied numerically the Dufour and Soret effects on MHD free convective heatand mass transfer with radiation past a vertical porous plate embedded in a porous medium for ahydrogen-air mixture as the non-chemical reacting fluid pair. Numerical solutions for the governingequations for momentum, energy and concentration are given. Tabulated values and graphicalrepresentations are presented for the velocity, temperature and concentration functions for variousvalues of Soret number with Dufour number, permeability parameter, magnetic field parameter, thebuoyancy forces, Schmidt number, and thermal radiation parameter. From the above analysis we

ISSN 2076-2275: Nonlinear Science Letters A- Mathematics, Physics and Mechanics34

conclude that for some kinds of mixtures (for example, H2-air) with the light and medium molecularweight, the Soret and Dufour’s effects play a significant role and should be taken into consideration aswell.

Figure 12: Concentration profiles for Pr=0.71, Sc=0.22, Gr=10, K=0.3, Sr = 2, Df = 0.03, M = 0.1, Rd

= 0.1

Figure 13: Concentration profiles for Pr=0.71, Sc=0.22, Gr=10, K=0.3, Gm = 4, fw = 0.5, M = 0.1, Rd

= 0.1

Acknowledgement : Olanrewaju wants to thank the financial support of Covenant University, Ota,Nigeria.

References

P.O. Olanrewaju, Nonlinear Sci. Lett.A, Vol.4, No.1, 21-34, 2013 35

[1] M. Abd El-Aziz, Thermal-diffusion and diffusion-thermo effects on combined heat and masstransfer by hydromagnetic three-dimensional free convection over a permeable stretchingsurface with radiation, Physics Letters, Section A, vol. 372, no. 3, pp. 263-272, 2009.

[2] A. A. Afify, Similarity solution in MHD: effects of thermal diffusion and diffusion thermo onfree convective heat and mass transfer over a stretching surface considering suction or injection,Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2202-2214, 2009.

[3] M. S. Alam, M. Ferdows, M. Ota, M. A. Maleque, Dufour and Soret effects on steady freeconvection and mass transfer flow past a semi-infinite vertical porous plate in a porousmedium, Int. J. of Applied Mechanics and Engineering, vol. 11, no. 3, pp. 535-545, 2006.

[4] A. J. Chamkha, Hydromagnetic natural convection from an isothermal inclined surface adjacentto a thermally stratified porous medium, International Journal of Engineering Science, vol. 37,no. 10-11, pp. 975-986, 1997.

[5] P.S. Gupta, A.S. Gupta, Heat and mass transfer on a stretching sheet with suction or blowing,The Canadian Journal of Chemical Engineering, vol. 55, pp. 744-746, 1977.

[6] T. Hayat, Z. Abbas, I. Pop, S. Asghar, Effects of radiation and magnetic field on the mixedconvection stagnation-point flow over a vertical stretching sheet in a porous medium,International Journal of Heat and Mass Transfer, Vol.53, pp.466-474, 2010.

[7] A. Heck, Introduction to Maple, 3rd Edition, Springer-Verlag, 2003.[8] Y. Jaluria, Natural convection heat and mass transfer, Pergamon Press, Oxford, UK, 1980.[9] N.G. Kafoussias, E. W. Williams, Thermal-diffusion and diffusion-thermo effects on mixed

free forced convective and mass transfer boundary layer flow with temperature dependentviscosity, International Journal of Engineering Science, vol. 33, no. pp. 1369-1384, 1995.

[10] O. D. Makinde: Similarity solution of hydromagnetic heat and mass transfer over a verticalplate with a convective surface boundary condition, International Journal of Physical SciencesVol. 5(6), pp.700-710, 2010.

[11] O. D. Makinde, A. Aziz: MHD mixed convection from a vertical plate embedded in a porousmedium with a convective boundary condition, International Journal of Thermal Sciences, Vol.49, pp.1813-1820, 2010.

[12] O. D. Makinde, P. O. Olanrewaju: Buoyancy effects on thermal boundary layer over a verticalplate with a convective surface boundary condition, Transaction of ASME –Journal of FluidEngineering, Vol. 132, pp.044502(1-4), 2010.

[13] O. D. Makinde, Free convection flow with thermal radiation and mass transfer pasta movingvertical porous plate, International Communications in Heat and Mass Transfer 32 (2005) pp.1411–1419, 2005.

[14] O. D. Makinde: On MHD heat and mass transfer over a moving vertical plate with a convectivesurface boundary condition, Canadian Journal of Chemical Engineering, Vol. 9999, pp. 1-8,2010.

[15] R.A. Mohamed, S.M. Abo-Dahab, Influence of chemical reaction and thermal radiation on theheat and mass transfer in MHD micropolar flow over a vertical moving porous plate in a porousmedium with heat generation, International Journal of Thermal Sciences Vol.48, pp.1800-1813,2009.

[16] T.Y. Na, Computational Methods in Engineering Boundary Value Problems, Academic Press,New York, 1979.

[17] E. M. Sparrow, R.D. Cess, Radiation Heat Transfer, Augmented edition, HemispherePublishing Corp, Washington, DC, 1978.