Research ArticleOn the Influence of Soret and Dufour Effects on MHD FreeConvective Heat and Mass Transfer Flow over a Vertical Channelwith Constant Suction and Viscous Dissipation

Ime Jimmy Uwanta and Halima Usman

Department of Mathematics Usmanu Danfodiyo University PMB 2346 Sokoto State Nigeria

Correspondence should be addressed to Halima Usman hhalimausmanyahoocom

Received 28 March 2014 Revised 10 June 2014 Accepted 7 July 2014 Published 29 October 2014

Academic Editor Abhik Sanyal

Copyright copy 2014 I J Uwanta and H Usman This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The present paper investigates the combined effects of Soret and Dufour on free convective heat and mass transfer on theunsteady one-dimensional boundary layer flow over a vertical channel in the presence of viscous dissipation and constant suctionThe governing partial differential equations are solved numerically using the implicit Crank-Nicolson method The velocitytemperature and concentration distributions are discussed numerically and presented through graphs Numerical values of theskin-friction coefficient Nusselt number and Sherwood number at the plate are discussed numerically for various values of physicalparameters and are presented through tables It has been observed that the velocity and temperature increase with the increase inthe viscous dissipation parameter and Dufour number while an increase in Soret number causes a reduction in temperature and arise in the velocity and concentration

1 Introduction

The phenomenon of coupled heat and mass transfer byfree convection in a fluid saturated porous medium occursin many engineering and technological and manufacturingindustries such as hydrology geosciences electronic devicescooled by fans geothermal energy utilization petroleumreservoirs and design of steel rolling and nuclear powerplants A comprehensive account of the available informationis provided in the recent books Neild and Bejan [1] andIngham and Pop [2]

In recent years considerable attention has been devotedto study the MHD flows of heat and mass transfer becauseof the applications in geophysics aeronautics and chemicalengineering Palani and Srikanth [3] studied the MHD flowof an electrically conducting fluid over a semi-infinite verticalplate under the influence of the transversely appliedmagneticfieldMakinde [4] investigated theMHDboundary layer flowwith heat and mass transfer over a moving vertical plate inthe presence of magnetic field and convective heat exchangeat the surface Additionally Duwairi [5] analyzed viscous and

joule-heating effects on forced convection flow from radiateisothermal surfaces The effect of viscous dissipation is usu-ally characterized by the Eckert number and has played a veryimportant role in geophysical flow and in nuclear engineeringthat was studied by Alim et al [6] It also plays an importantrole in free convection in various processes on large scales orfor large planetsThe effects of suction on boundary layer flowalso have greater influence over the engineering applicationand have been widely investigated by numerous researchersVarious authors have studied the effects of viscous dissipationand constant suction in different surface geometries Uwanta[7] studied the effects of chemical reaction and radiation onheat and mass transfer past a semi-infinite vertical porousplate with constant mass flux and dissipation Mansour et al[8] described the influence of chemical reaction and viscousdissipation on MHD natural convection flow The effectof chemical reaction and heat and mass transfer along awedge with heat source and concentration in the presence ofsuction or injection has been examined by Kandasamy et al[9] Govardhan et al [10] presented a theoretical studyon the influence of radiation on a steady free convection

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2014 Article ID 639159 11 pageshttpdxdoiorg1011552014639159

2 International Scholarly Research Notices

heat and mass transfer over an isothermal stretching sheetin the presence of a uniform magnetic field with viscousdissipation effect Sattar [11] analyzed the effect of free andforced convection boundary layer flow through a porousmedium with large suction Similarly Mohammed et al [12]investigated the effect of similarity solution for MHD flowthrough vertical porous plate with suction Jai [13] presentedthe study of a viscous dissipation and chemical reactioneffects on flow past a stretching porous surface in a porousmedium In another article a detailed numerical study onthe combined effects of radiation and mass transfer on asteady MHD two-dimensional marangoni convection flowover a flat surface in presence of joule-heating and viscousdissipation under influence of suction and injection is studiedby Ibrahim [14] Khaleque and Samad [15] described theeffects of radiation heat generation and viscous dissipationon MHD free convection flow along a stretching sheet

When heat and mass transfer occur simultaneously ina moving fluid affecting each other causes a cross diffusioneffect the mass transfer caused by temperature gradient iscalled the Soret effect while the heat transfer caused byconcentration effect is called the Dufour effect Soret andDufour effects are important phenomena in areas such ashydrology petrology and geosciences The Soret effect forinstance has been utilized for isotope separation and ina mixture between gases with very light molecular weight(HeH

2) and of medium molecular weight (N

2 air) The

Dufour effect was recently found to be of order of con-siderable magnitude so that it cannot be neglected Eckertand Drake [16] Many researchers studied Soret and Dufoureffects for example Postelnicu [17] analyzed the effect ofSoret and Dufour on heat and mass transfer Chamkha andEl-Kabeir [18] presented a theoretical study of Soret andDufour effects on unsteady coupled heat and mass transferby mixed convection flow over a vertical cone rotating in anambient fluid in the presence of amagnetic field and chemicalreaction Usman and Uwanta [19] have considered the effectof thermal conductivity onMHD heat andmass transfer flowpast an infinite vertical plate with Soret and Dufour effectsSimilarly Uwanta et al [20] have analyzed MHD fluid flowover a vertical plate with Dufour and Soret effectsThe effectsof Soret and Dufour on an unsteady MHD free convectionflow past a vertical porous plate in the presence of suctionor injection have been investigated by Sarada and Shankar[21] A numerical approach has been carried out for the studyof Soret and Dufour effects on mixed convection heat andmass transfer past a vertical heated plate with variable fluidproperties by Nalinakshi et al [22] Subhakar and Gangadhar[23] investigated the combined effects of the free convectiveheat and mass transfer on the unsteady two-dimensionalboundary layer flow over a stretching vertical plate in thepresence of heat generationabsorption and Soret andDufoureffects In another article Srinivasacharya and Reddy [24]examined Soret and Dufour effects on mixed convection ina non-Darcy porous medium saturated with micro polarfluid Additionally Sivaraman et al [25] considered Soretand Dufour effects on MHD free convective heat and masstransfer with thermopheresis and chemical reaction overa porous stretching surface Recently Srinivasacharya and

Upendar [26] analyzed the flow and heat and mass transfercharacteristics of the mixed convection on a vertical plate inamicropolar fluid in the presence of Soret andDufour effectsMost recently a boundary layer analysis has been presentedto study heat and mass transfer in the laminar viscousand incompressible fluid past a continuously moving platesaturated in a non-Darcy porous medium in the presenceof Soret and Dufour effects with temperature dependentviscosity and thermal conductivity by El-Kabeir et al [27]Finally Olanrewaju et al [28] have investigated Dufour andSoret effects on convection heat and mass transfer in anelectrically conducting power law flow over a heated porousplate

In view of the above studies the purpose of currentinvestigation is to examine the influence of Soret and Dufoureffects on MHD free convective heat and mass transfer flowover a vertical channel with constant suction and viscousdissipation

2 Mathematical Formulation

Consider the flow of an unsteady laminar coupled freeconvective heat and mass transfer of an incompressiblefluid past a vertical channel in a porous medium underthe influence of a uniform transverse magnetic field andconstant suction with viscous dissipation in the presenceof Soret and Dufour effects The 119909

1015840-axis is taken on thefinite plate and parallel to the free stream velocity which isvertical and the 1199101015840-axis is taken normal to the plate All fluidproperties are assumed to be constant The magnetic field ofsmall intensity is induced along the 119910 direction The fluidis assumed to be slightly conducting hence the magneticReynolds number is much less than unity and therefore theinduced magnetic field is neglected in comparison with theapplied magnetic field Under the above assumptions thegeneral equations governing the flow can be expressed asfollows

120597V1015840

120597119910

1015840= 0 (1)

120597119906

1015840

120597119905

1015840minus V0

120597119906

1015840

120597119910

1015840= V

120597

2119906

1015840

120597119910

10158402minus

120590119861

2

0

120588

119906

1015840minus

V119896

lowast119906

1015840

+ 119892120573 (119879

1015840minus 119879

1015840

0) + 119892120573

lowast(119862

1015840minus 119862

1015840

0)

(2)

120597119879

1015840

120597119905

1015840minus V0

120597119879

1015840

120597119910

1015840=

1

120588119862119901

120597

120597119910

1015840[119870 (119879)

120597119879

1015840

120597119910

1015840] +

119863119872

120588119862119901

120597

2119862

1015840

120597119910

10158402

+ 119887

lowast119906

10158402+

1

120588119862119901

(

120597119906

1015840

120597119910

1015840)

2

(3)

120597119862

1015840

120597119905

1015840minus V0

120597119862

1015840

120597119910

1015840= 119863

120597

2119862

1015840

120597119910

10158402+ 119863119879

120597

2119879

1015840

120597119910

10158402 (4)

119870 (119879) = 1198960[1 + 120572 (119879

1015840

119908minus 119879)] (5)

International Scholarly Research Notices 3

B0

T998400

120596

C998400

120596

T998400

0

C998400

0

x998400

y998400

g

u998400= 0 u

998400= 0

y998400= Hy

998400= 0

Figure 1 Geometry of the problem

The corresponding initial and boundary conditions are pre-scribed as follows

119905 le 0 119906

1015840= 0 119879

1015840gt 1198790 119862

1015840gt 1198620

forall119910

1015840

119905 gt 0 119906

1015840= 0 119879

1015840= 119879

1015840

119908 119862

1015840= 119862

1015840

119908at 1199101015840 = 0

119906

1015840= 0 119879

1015840= 1198790 119862

1015840= 1198620

at 1199101015840 = 119867

(6)

The geometry of the problem is shown in Figure 1From continuity equation it is clear that the suction

velocity is either a constant or a function of time Henceon integrating (1) the suction velocity normal to the plate isassumed in the form

V1015840 = minusV0 (7)

where V0is a scale of suction velocity which is nonzero

positive constantThe negative sign indicates that the suctionis towards the plate and V

0gt 0 corresponds to steady

suction velocity normal at the surface The fourth and fifthterms on the right hand side of (2) denote the thermal andconcentration buoyancy effects respectively 1199061015840 and V1015840 are thevelocity components in the 119909- and 119910-directions respectively119905 is the time ] is the kinematic viscosity 119892 is the accelerationdue to gravity 120573 is the coefficient of volume expansion 120588is the density 120573lowast is the volumetric coefficient of expansionwith concentration 119870(119879) is the thermal conductivity 119862

119901

is the specific heat capacity at constant pressure 119896lowast is thepermeability of the porous medium 119863

119872is the coefficient

of mass diffusivity 1198960is the thermal conductivity of the

ambient fluid 120572 is a constant depending on the nature of thefluid 119863 is the coefficient of molecular diffusivity 119863

119879is the

coefficient of temperature diffusivity 119887lowast is the dimensionlessjoule-heating parameter 120590 is the electric conductivity and 119861

0

is the magnetic field of constant strength 1198791015840 and 119879

1015840

0are the

temperature of the fluid inside the thermal boundary layerand the fluid temperature in the free stream respectively 1198621015840and 1198621015840

0are the corresponding concentrations

On introducing the following nondimensional quantities

119906 =

119906

1015840

1199060

119905 =

119905

10158401199060

119867

119910 =

119910

1015840

119867

120579 =

119879

1015840minus 119879

1015840

0

119879

1015840

119908minus 119879

1015840

0

119862 =

119862

1015840minus 119862

1015840

0

119862

1015840

119908minus 119862

1015840

0

Pr =1199060120588119862119901

1198960

Sc =1199060

119863

119896 =

119896

lowast1199060

]119867 119872 =

120590119861

2

0119867

1205881199060

Gr =119867119892120573 (119879

1015840

119908minus 119879

1015840

0)

119906

2

0

Gc =119867119892120573

lowast(119862

1015840

119908minus 119862

1015840

0)

119906

2

0

120582 = 120572 (119879

1015840

119908minus 119879

1015840

0) 120574 =

]0119867

1199060

Sr =119863119879(119879

1015840

119908minus 119879

1015840

0)

1199060(119862

1015840

119908minus 119862

1015840

0)

Ec = 119867

1199060120588119862119901(119879

1015840

119908minus 119879

1015840

0)

Du =119863119872(119862

1015840

119908minus 119862

1015840

0)

1199060120588119862119901(119879

1015840

119908minus 119879

1015840

0)

1198871=

119887

lowast119867

1199060(119879

1015840

119908minus 119879

1015840

0)

(8)

Applying (8) the set of (2) (3) (4) (5) and (6) reduces to thefollowing

120597119906

120597119905

minus 120574

120597119906

120597119910

=

120597

2119906

120597119910

2minus119872119906 minus

1

119896

119906 + Gr120579 + Gc119862

120597120579

120597119905

minus 120574

120597120579

120597119910

=

120582

Pr(

120597120579

120597119910

)

2

+

1

Pr(1 + 120582120579)

120597

2120579

120597119910

2

+ Du1205972119862

120597119910

2+ 1198871119906

2+ Ec(120597119906

120597119910

)

2

120597119862

120597119905

minus 120574

120597119862

120597119910

=

1

Sc120597

2119862

120597119910

2+ Sr120597

2120579

120597119910

2

(9)

with the following initial and boundary conditions

119905 le 0 119906 = 0 120579 = 0 119862 = 0 forall119910

119905 gt 0 119906 = 0 120579 = 1 119862 = 1 at 119910 = 0

119906 = 0 120579 = 0 119862 = 0 at 119910 = 1

(10)

where Ec is the Eckert number Pr is the Prandtl numberSc is the Schmidt number Sr is the Soret number Du is theDufour number119872 is the Magnetic field parameter Gr is thethermal Grashof number Gc is the Solutal Grashof number119896 is the porous parameter 119887

1is the joule-heating parameter

120582 is the variable thermal conductivity and 120574 is the variablesuction parameter while 119906 and V are dimensionless velocitycomponents in 119909- and 119910-directions respectively and 119905 is thedimensionless time

4 International Scholarly Research Notices

The skin friction Nusselt number and Sherwood numberare important physical parameters for this type of boundarylayer flow and are given by

119862119891= (

120597119906

120597119910

)

119910=0

Nu = minus(

120597120579

120597119910

)

119910=0

Sh = (

120597119862

120597119910

)

119910=0

(11)

3 Numerical Solution Procedure

The set of coupled nonlinear governing boundary layer equa-tion (9) together with boundary conditions (10) are solvednumerically by using the implicit finite difference methodof Crank-Nicolson typeThe finite difference approximationsequivalent to (9) are as follows

(

119906

119895+1

119894minus 119906

119895

119894

Δ119905

) minus

120574

2Δ119910

(119906

119895

119894+1minus 119906

119895

119894minus1)

=

1

2(Δ119910)

2(119906

119895+1

119894+1minus 2119906

119895+1

119894+ 119906

119895+1

119894minus1+ 119906

119895

119894+1minus 2119906

119895

119894+ 119906

119895

119894minus1)

minus (119872 +

1

119896

) 119906

119895

119894+ Gr(120579)119895

119894+ Gc(119862)119895

119894

(

120579

119895+1

119894minus 120579

119895

119894

Δ119905

) minus

120574

2Δ119910

(120579

119895

119894+1minus 120579

119895

119894minus1)

=

119867

2Pr(Δ119910)2(120579

119895+1

119894+1minus 2120579

119895+1

119894+ 120579

119895+1

119894minus1+ 120579

119895

119894+1minus 2120579

119895

119894+ 120579

119895

119894minus1)

+

Du2(Δ119910)

2(119862

119895+1

119894+1minus 2119862

119895+1

119894+ 119862

119895+1

119894minus1+ 119862

119895

119894+1minus 2119862

119895

119894+ 119862

119895

119894minus1)

+

120582

Pr(Δ119910)2(120579

119895

119894+1minus 120579

119895

119894)

2

+

Ec(Δ119910)

2(119906

119895

119894+1minus 119906

119895

119894)

2

+ 1198871(119906

119895

119894)

2

(

119862

119895+1

119894minus 119862

119895

119894

Δ119905

) minus

120574

2Δ119910

(119862

119895

119894+1minus 119862

119895

119894minus1)

=

1

2Sc(Δ119910)2

times (119862

119895+1

119894+1minus 2119862

119895+1

119894+ 119862

119895+1

119894minus1+ 119862

119895

119894+1minus 2119862

119895

119894+ 119862

119895

119894minus1)

+

Sr2(Δ119910)

2(120579

119895+1

119894+1minus 2120579

119895+1

119894+ 120579

119895+1

119894minus1+ 120579

119895

119894+1minus 2120579

119895

119894+ 120579

119895

119894minus1)

(12)

The initial and boundary conditions take the following forms

119906119894119895= 0 120579

119894119895= 0 119862

119894119895= 0

1199060119895

= 0 1205790119895

= 1 1198620119895

= 1

119906119867119895

= 0 120579119867119895

= 0 119862119867119895

= 0

(13)

where119867 corresponds to 1Equations (12) are simplified as follows

minus 1198801119906

119895+1

119894minus1+ 1198802119906

119895+1

119894minus 1198801119906

119895+1

119894+1

= 1198803119906

119895

119894minus1+ 1198804119906

119895

119894+ 1198805119906

119895

119894+1+ 1198806120579

119895

119894+ 1198807119862

119895

119894

(14)

minus 1198791120579

119895+1

119894minus1+ 1198792120579

119895+1

119894minus 1198791120579

119895+1

119894+1

= 1198793120579

119895

119894minus1+ 1198794120579

119895

119894+ 1198795120579

119895

119894+1+ 1198796(120579

119895

119894+1minus 120579

119895

119894)

2

+ 1198797(119906

119895

119894+1minus 119906

119895

119894)

2

+ 1198798(119906

119895

119894)

2

+ 1198799(119862

119895+1

119894+1minus 2119862

119895+1

119894+ 119862

119895+1

119894minus1+ 119862

119895

119894+1minus 2119862

119895

119894+ 119862

119895

119894minus1)

(15)

minus 1198811119862

119895+1

119894minus1+ 1198812119862

119895+1

119894minus 1198811119862

119895+1

119894+1

= 1198813119862

119895

119894minus1+ 1198814119862

119895

119894+ 1198815119862

119895

119894+1

+ 1198816(120579

119895+1

119894+1minus 2120579

119895+1

119894+ 120579

119895+1

119894minus1+ 120579

119895

119894+1minus 2120579

119895

119894+ 120579

119895

119894minus1)

(16)

The index 119894 corresponds to space 119910 and 119895 corresponds totime 119905 Δ119910 and Δ119905 are the mesh sizes along 119910-direction andtime 119905-direction respectively The finite difference equations(14)ndash(16) at every internal nodal point on a particular 119899-levelconstitute a tridiagonal system of equations which are solvedby usingThomas algorithmDuring the computations in eachtime step the coefficients in (14)ndash(16) are treated as constantsThe values of 120579 119862 and 119906 are known at all grid points at theinitial time 119905 = 0The values of 120579119862 and 119906 at time level (119895+1)using the known values at previous time level 119895 are calculatedas follows

Knowing the values of 120579 119862 and 119906 at a time 119905 = 119895calculate 120579 and 119862 at time 119905 = 119895 + 1 using the finite differenceequations (15) and (16) and solving the tridiagonal systemof equations by using Thomas algorithm as discussed byCarnahan et al [29] Knowing the values of 120579 and 119862 at time119905 = 119895 and 119905 = 119895 + 1 and the values of 119906 at time 119905 = 119895solve (14) using tridiagonal matrix inversion to obtain thevalues of 119906 at time 119905 = 119895 + 1 This process is repeated forvarious 119894 levels Thus the values of 120579 119862 and 119906 are knownat all grid points in the rectangular region at (119895 + 1)th timelevel Computations are carried out until the steady state isreached The Implicit Crank-Nicolson method is a secondordermethod119874(Δ119905)2 in time and has no restrictions on spaceΔ119910 and time stepΔ119905 that is the method is compatible Hencethe finite difference scheme is unconditionally stable andtherefore compatibility and stability ensures the convergenceof the scheme Computations are carried out for differentvalues of physical parameters involved in the problem

International Scholarly Research Notices 5

0 02 04 06 08 10

05

1

15

y

Gr = 50 10 15 20

Velo

city

(U)

Figure 2 Velocity profile for different values of Gr with Gc = 2120574 = 3 Sc = 022 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

05

1

Velo

city

(U)

y

Gc = 50 10 15 20

Figure 3 Velocity profile for different values of Gc with Gr = 2120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

4 Results and Discussion

In this paper numerical values are assigned physically tothe embedded parameters in the system in order to reporton the analysis of the fluid flow structure with respect tovelocity temperature and concentration profiles Numericalresults for velocity temperature and concentration profilesare presented on graphs while the skin-friction coefficientNusselt number and Sherwood number are shown in tabularform The Prandtl number is taken to be (Pr = 071 70)which corresponds to air and water The value of Schmidtnumber is taken as (Sc = 022 066 094 262) representingdiffusing chemical species of most common interest in airfor hydrogen oxygen carbon dioxide and propyl benzenewhile for Soret numbers (250) and (689) correspondingto thermal diffusion ratio of (H

2ndashCO2) and (HendashAr) are

chosen respectively while other parameters in the floware chosen arbitrarily The influence of thermal Grashofnumber Gr and solutal Grashof number Gc on the velocityis presented in Figures 2 and 3 The thermal Grashof numbersignifies the relative effect of the thermal buoyancy forceto the viscous hydrodynamic force in the boundary layerwhile the solutal Grashof number defines the ratio of thespecies buoyancy force to the viscous hydrodynamic forceAs expected the fluid velocity increases due to the enhance-ment of thermal and species buoyancy forces The velocitydistribution increases rapidly near the porous plate and thendecreases smoothly to the free stream value For different

0 02 04 06 08 10

005

01

015

02

y

M = 10 30 50 70

Velo

city

(U)

Figure 4 Velocity profile for different values of 119872 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

k = 01 02 03 04

Velo

city

(U)

Figure 5 Velocity profile for different values of 119896with Gr = 2 Gc =2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119872 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

values of magnetic parameter119872 and porous parameter 119896 thevelocity profiles are plotted in Figures 4 and 5 respectivelyIt can be seen that as 119872 increases the velocity decreasesThis results agrees with the expectations since the magneticfield exerts a restraining force on the fluid which tends toimpede it is motion Figure 5 depicts the effect of the porousparameter 119896 on the velocity profile An increase in 119896 increasesthe resistance of the porous medium which will tend toaccelerate the flow and therefore increase the velocity

For various values of suction parameter 120574 the velocitytemperature and concentration profiles are plotted in Figures6(a)ndash6(c) It is found out that an increase in the suctionparameter causes a fall in the velocity and concentrationprofiles throughout the boundary layer while increase inthe temperature profiles This is due to the fact that suctionparameter stabilizes the boundary layer growth The effect ofSoret number Sr on velocity temperature and concentrationprofiles is illustrated in Figures 7(a)ndash7(c) respectively TheSoret number defines the effect of the temperature gradientsinducing significant mass diffusion effects It can be seenthat the velocity and concentration profiles increase with anincrease in Sr while a rise in Sr causes a fall in the temperatureprofiles within the boundary layer These behaviors areevident from Figures 7(a)ndash7(c) The influence of viscousdissipation parameter that is Eckert number Ec on thevelocity profiles is depicted in Figure 8 The Eckert numberexpresses the relationship between the kinetic energy in theflow and the enthalpy It embodies the conversion of kinetic

6 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

S = 10 50 10 15

Velo

city

(U)

(a)

S = 10 50 10 15

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

(b)

0 02 04 06 08 1

0

05

1

y

S = 10 50 10 15

minus05

conc

entr

atio

n (C

)

(c)

Figure 6 (a) Velocity profile for different values of (120574 = 119878) with Gr = 2 Gc = 2 119896 = 1 Sc = 022 1198871= 001 120582 = 05119872 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of (120574 = 119878) with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1

119872 = 1 119896 = 1 Sr = 02 Du = 02 Ec = 001 Pr = 071 and 119905 = 002 (c) Concentration profile for different values of (120574 = 119878) with Gr = 2Gc = 2 Sc = 022 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 3 Du = 01 Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

Sr = 166 250 368 689

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Sr = 166 250 368 689

(b)

0 02 04 06 08 10

05

1

Con

cent

ratio

n (C

)

y

Sr = 166 250 368 689

(c)

Figure 7 (a) Velocity profile for different values of Sr with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1119872 = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Sr with Gr = 5 Gc = 2 120574 = 01 Sc = 022 1198871= 01 120582 = 1

119896 = 1119872 = 1 Du = 02 Ec = 001 Pr = 071 and 119905 = 002 (c) Concentration profile for different values of Sr with Gr = 2 Gc = 2 Sc = 0221198871= 001 120582 = 05119872 = 1 119896 = 1 120574 = 3 Du = 01 Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 7

0 02 04 06 08 10

05

1

15

y

Ec = 10 20 30 40

Velo

city

(U)

Figure 8 Velocity profile for different values of Ec with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

119872 = 1 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

t = 002 004 006 008Pr = 071

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 071

(b)

0 02 04 06 08 10

05

1

Con

cent

ratio

n (C

)

y

t = 002 004 006 008Pr = 071

(c)

Figure 9 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

0 02 04 06 08 10

005

01

y

t = 002 004 006 008

Velo

city

(U) Pr = 70

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 70

(b)

0 02 04 06 08 10

05

1C

once

ntra

tion

(C)

y

t = 002 004 006 008Pr = 70

(c)

Figure 10 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

energy into internal energy by work done against the viscousfluid stress It is observed that greater viscous dissipative heatcauses an increase in the velocity profiles across the boundarylayer Figures 9(a)ndash10(c) describe the behavior for velocitytemperature and concentration profiles for different values ofnondimensional time 119905 for two working fluids air (Pr = 071)and water (Pr = 70) It is evident from these figures that astime increases the velocity temperature and concentrationincrease The consequence of temperature increase resultsfrom increase in time since convection current becomesstronger and hence velocity and concentration increase withtime

Figures 11(a) and 11(b) graphically show the influence ofDufour number Du on the velocity and temperature profilesrespectively The Dufour number signifies the contributionof the concentration gradients to the thermal energy flux inthe flow It is seen that as Du increases there is monotonic

8 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

Du = 10 20 30 40

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Du = 10 20 30 40

(b)

Figure 11 (a) Velocity profile for different values of Du with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1 Sr = 1 119872 = 1

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Du with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1119872 = 1

119896 = 1 Sr = 02 120574 = 01 Ec = 001 Pr = 071 and 119905 = 002

increase in the velocity and temperature profiles For differentvalues of joule-heating parameter 119887

1and thermal conduc-

tivity parameter 120582 the temperature profiles are plotted inFigures 12 and 13 respectively It is observed that increas-ing the joule-heating parameter and thermal conductivityparameter produces significant increase in the thermal con-duction of the fluid which is physically true because as thethermal conductivity increases the temperature within thefluid increases Figure 14 describes the behavior of variousvalues of Scmidt number Sc on the concentration profilesThe Schmidt number characterizes the ratio of thicknesses ofviscous to the mass diffusivityThe Scmidt number quantifiesthe relative effectiveness ofmomentumandmass transport bydiffusion in the velocity and concentration boundary layers Itis observed that increase in the values of Sc causes the speciesconcentration and its boundary layer thickness to decreasesignificantly

The effects of various governing parameters on skin-friction coefficient 119862

119891 Nusselt number Nu and Sherwood

number Sh are shown in Tables 1 and 2 In order to highlightthe contributions of each parameter one parameter is variedwhile the rest take default fixed values It is observed fromTable 1 that an increase in any of the parameters 119872 and 120574

causes reduction in the skin-friction while increasing any ofthe parameters Gr Gc 119896 and 120582 resulted in correspondingincrease in the skin-friction coefficient It is also seen thatas 120582 and 120574 increase there is a rise in Nusselt number andSherwood number respectively From Table 2 it is observedthat an increase in Ec 119887

1 and Du leads to a rise in the skin-

friction coefficient Nusselt number and Sherwood numberrespectively while an increase in Pr leads to a fall in the skin-friction coefficient Nusselt number and Sherwood numberrespectively It is also seen that as Sr and Sc increase there isa fall in the skin-friction coefficient and a rise in Nusselt andSherwood numbers respectively

5 Conclusions

The present paper analyzes the influence of Soret and Dufoureffects on MHD free convective heat and mass transferflow over a vertical channel with constant suction and vis-cous dissipation The resulting partial differential equations

0 02 04 06 08 10

05

1

15

Tem

pera

ture

(T)

y

b1 = 10 50 10 15

Figure 12 Temperature profile for different values of 1198871with Gr = 5

Gc = 2 Sc = 022 120574 = 01 120582 = 1119872 = 1 119896 = 1 Sr = 02 Du = 02Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

V = 05 10 20 30

Figure 13 Temperature profile for different values of (120582 = V) withGr = 5 Gc = 2 Sc = 022 119887

1= 01 120574 = 01119872 = 1 119896 = 1 Sr = 02

Du = 02 Ec = 001 Pr = 071 and 119905 = 002

0 02 04 06 08 10

05

1

Con

cent

artio

n (C

)

y

Sc = 022 066 094 262

Figure 14 Concentration profile for different values of Sc with Gr =2 Gc = 2 120574 = 01 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 9

Table 1 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various values of physical parameters

(GrGc119872 119896 120582 and 120574)

Gr Gc 119872 119896 120582 120574 119862119891

Nu Sh20 20 10 05 10 2 02611 33328 0617140 20 10 05 10 2 04720 33451 0622120 40 10 05 10 2 03119 33386 0619520 20 30 05 10 2 02353 33317 0616720 20 10 10 10 2 02757 33334 0617420 20 10 05 20 2 02897 81676 3433420 20 10 05 10 30 02072 45600 13689With fixed values of (Pr = 071 Ec = 001 Sc = 06 Du = 02 1198871 = 10 Sr = 10 and 119905 = 02)

Table 2 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various physical parameters (Pr 119887

1 Ec Sr

Du and Sc)

Pr 1198871

Ec Sr Du Sc 119862119891

Nu Sh071 10 20 10 02 022 04038 20257 1434430 10 20 10 02 022 02898 06168 14296071 20 20 10 02 022 04041 20311 14354071 10 30 10 02 022 04053 20729 14713071 10 20 166 02 022 04019 20523 15730071 10 20 10 10 022 04133 32477 16418071 10 20 10 02 094 03519 20983 17267With fixed values of (Gr = 20 Gc = 20119872 = 10 120574 = 05 120582 = 10 119896 = 05 and 119905 = 02)

are nondimensionalised simplified and solved by implicitfinite difference method of Crank-Nicolson type From thepresent numerical study the following conclusions can bedrawn

(1) Velocity profiles increased due to increase in thermalGrashof number solutal Grashof number porousparameter Eckert number Soret number Dufournumber and dimensionless time while it decreaseddue to increase in magnetic parameter and suctionparameter

(2) An increase in temperature profiles is a function ofan increase in thermal conductivity suction parame-ter Eckert number joule-heating parameter Dufournumber and dimensionless time while it decreaseddue to increase in Soret number

(3) Concentration profiles decreased due to increasesin Schmidt number and suction parameter whileit increased due to increase in Soret number anddimensionless time

(4) There is a rise in the skin-friction coefficient Nusseltnumber and Sherwood number due to increasein Eckert number thermal conductivity parameterjoule-heating parameter and Dufour number whilea fall is observed in skin-friction coefficient withincrease in Soret number and Prandtl number

Nomenclature

119862 Concentration119862119901 Specific heat at constant pressure

119863 Mass diffusivity119863119872 Coefficient of Mass diffusivity

119863119879 Coefficient of temperature diffusivity

Du Dufour number119892 Acceleration due to gravityGr Grashof numberGc Solutal Grashof number119896 Permeability of the porous mediumNu Nusselt numberPr Prandtl numberSc Schmidt numberSr Soret number119879 Temperature119906 V Velocities in the 119909- and 119910-direction

respectively119909 119910 Cartesian coordinates along the plate

and normal to it respectively1198610 Magnetic field of constant strength

Ec Eckert number119872 Magnetic field parameter1198871 Joule-heating parameter

Greek Letters

120573

lowast Coefficient of expansion with concentration120573 Coefficient of thermal expansion120588 Density of fluid

10 International Scholarly Research Notices

120590 Electric conductivity120579 Dimensionless temperature] Kinematic viscosity120574 Suction parameter120582 Variable thermal conductivity

Subscripts

119908 Condition at wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D A Neild and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006

[2] D B Ingham and I PopTransport Phenomena in PorousMediaElsevier Oxford UK 2005

[3] G Palani and U Srikanth ldquoMHD flow past a semi-infinitevertical plate with mass transferrdquo Nonlinear Analysis Modellingand Control vol 14 no 3 pp 345ndash356 2009

[4] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoThe Canadian Journal of Chemical Engineering vol 88 no 6pp 983ndash990 2010

[5] H M Duwairi ldquoViscous and Joule heating effects on forcedconvection flow from radiate isothermal porous surfacesrdquoInternational Journal of Numerical Methods for Heat and FluidFlow vol 15 no 5 pp 429ndash440 2005

[6] M A Alim M D Alam and A Mamun ldquoJoule heating effecton the coupling of conduction with magneto hydrodynamicfree convection flow from a vertical platerdquo Non-Linear AnalysisModeling and Control vol 12 no 3 pp 307ndash316 2007

[7] I J Uwanta ldquoEffects of chemical reaction and radiation onHeatand Mass Transfer past a semi-infinite vertical porous platewith constant mass flux and dissipationrdquo European Journal ofScientific Research vol 87 no 2 pp 190ndash200 2012

[8] M A Mansour N F El-Anssary and A M Aly ldquoEffect ofchemical reaction and viscous dissipation on MHD naturalconvection flows saturated in porous media with suction orinjectionrdquo International Journal of Applied Mathematics andMechanics vol 4 no 2 pp 60ndash70 2008

[9] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[10] K Govardhan N Kishan and B Balaswamy ldquoEffect of viscousdissipation and radiation on MHD gas flow and heat and masstransfer over a stretching surface with a uniform free streamrdquoJournal of Engineering Physics andThermophysics vol 85 no 4pp 909ndash916 2012

[11] M A Sattar ldquoFree and forced convection boundary layer flowthrough a porous medium with large suctionrdquo InternationalJournal of Energy Research vol 17 no 1 pp 1ndash7 1993

[12] F Mohammad O Masatiro S Abdus and A MohamudldquoSimilarity solutions for MHD flow through vertical porous

plate with suctionrdquo Journal of Computational and AppliedMechanics vol 6 no 1 pp 15ndash25 2005

[13] S Jai ldquoViscous dissipation and chemical reaction effects on flowpast a stretching surface in a porous mediumrdquo Advance Theoryof Applied Mechanics vol 5 pp 323ndash331 2012

[14] S M Ibrahim ldquoEffects of mass transfer radiation Joule heatingand viscous dissipation on steady MHDmarangoni convectionflow over a flat surface with suction and injectionrdquo InternationalJournal of EngineeringMathematics vol 2013 Article ID 9038189 pages 2013

[15] T S Khaleque and M A Samad ldquoEffects of radiation heatgeneration and viscous dissipation on MHD free convectionflow along a stretching sheetrdquo Research Journal of AppliedSciences Engineering and Technology vol 2 no 4 pp 368ndash3772010

[16] E R G Eckert and R M Drake Analysis of Heat and MassTransfer MC Graw-Hill New York NY USA 1974

[17] A Postelnicu ldquoHeat and mass transfer by natural convection ata stagnation point in a porous medium considering Soret andDufour effectsrdquoHeat andMass Transfer vol 46 no 8-9 pp 831ndash840 2010

[18] A J Chamkha and S M M El-Kabeir ldquoUnsteady Heat andMass Transfer by MHD mixed convection flow from a rotatingvertical cone with chemical reaction and Soret and Dufoureffectsrdquo Chemical Engineering Communications vol 200 no 9pp 1220ndash1236 2013

[19] H Usman and I J Uwanta ldquoEffect of thermal conductivity onMHDHeat andMass transfer flow past an infinite vertical platewith Soret and Dufour effectsrdquo American Journal of AppliedMathematics vol 1 no 3 pp 28ndash38 2013

[20] I J Uwanta K K Asogwa and U A Ali ldquoMHD fluid flowover a vertical plate withDufour and Soret effectsrdquo InternationalJournal of Computer Applications vol 45 no 2 pp 8ndash16 2008

[21] K Sarada andB Shankar ldquoThe effects of Soret andDufour on anunsteadyMHD free convection flow past a vertical porous platein the presence of suction or injectionrdquo International Journal ofEngineering and Science vol 2 no 7 pp 13ndash25 2013

[22] N Nalinakshi P A Dinesh and D V Chandrashekar ldquoSoretand dufour effects on mixed convection heat and mass transferwith variable fluid propertiesrdquo International Journal of Mathe-matical Archive vol 4 no 11 pp 203ndash215 2013

[23] M J Subhakar and K Gangadhar ldquoSoret and Dufour effectson MHD free convection heat and mass transfer flow overa stretching vertical plate with suction and heat sourcesinkrdquoInternational Journal ofModern Engineering Research vol 2 no5 pp 3458ndash3468 2012

[24] D Srinivasacharya and C R Reddy ldquoSoret and Dufour effectson mixed convection in a non-DARcy porous medium satu-rated with micropolar fluidrdquo Nonlinear Analysis Modelling andControl vol 16 no 1 pp 100ndash115 2011

[25] N Sivaraman P K Sivagnana and R Kandasamy ldquoSoret andDufour effects on MHD free convective heat and mass transferwith thermopheresis and chemical reaction over a porousstretching surface group theory transformationrdquo Journal ofApplied Mechanics and Technical Physics vol 53 no 6 pp 871ndash879 2012

[26] D Srinivasacharya and M Upendar ldquoSoret and Dufour effectson MHD mixed convection heat and mass transfer in amicropolar fluidrdquo Central European Journal Engineering vol 3no 4 pp 679ndash689 2013

[27] S MM El-Kabeir MModathar and A M Rashed ldquoSoret anddufour effects on heat and mass transfer from a continuously

International Scholarly Research Notices 11

moving plate embedded in porous media with temperaturedependent viscosity and thermal conductivityrdquo Journal of Mod-ern Methods in Numerical Mathematics vol 4 no 2 pp 10ndash222013

[28] P O Olanrewaju O J Fenuga J A Gbadeyan and T GOkedayo ldquoDufour and Soret effects on convection heat andmass transfer in an electrical conducting power law flow overa heated porous platerdquo International Journal for ComputationalMethods in Engineering Science andMechanics vol 14 no 1 pp32ndash39 2013

[29] B Carnahan H A Luther and J O Wilkes Applied NumericalMethods John Wiley amp Sons New York NY USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Scholarly Research Notices

heat and mass transfer over an isothermal stretching sheetin the presence of a uniform magnetic field with viscousdissipation effect Sattar [11] analyzed the effect of free andforced convection boundary layer flow through a porousmedium with large suction Similarly Mohammed et al [12]investigated the effect of similarity solution for MHD flowthrough vertical porous plate with suction Jai [13] presentedthe study of a viscous dissipation and chemical reactioneffects on flow past a stretching porous surface in a porousmedium In another article a detailed numerical study onthe combined effects of radiation and mass transfer on asteady MHD two-dimensional marangoni convection flowover a flat surface in presence of joule-heating and viscousdissipation under influence of suction and injection is studiedby Ibrahim [14] Khaleque and Samad [15] described theeffects of radiation heat generation and viscous dissipationon MHD free convection flow along a stretching sheet

When heat and mass transfer occur simultaneously ina moving fluid affecting each other causes a cross diffusioneffect the mass transfer caused by temperature gradient iscalled the Soret effect while the heat transfer caused byconcentration effect is called the Dufour effect Soret andDufour effects are important phenomena in areas such ashydrology petrology and geosciences The Soret effect forinstance has been utilized for isotope separation and ina mixture between gases with very light molecular weight(HeH

2) and of medium molecular weight (N

2 air) The

Dufour effect was recently found to be of order of con-siderable magnitude so that it cannot be neglected Eckertand Drake [16] Many researchers studied Soret and Dufoureffects for example Postelnicu [17] analyzed the effect ofSoret and Dufour on heat and mass transfer Chamkha andEl-Kabeir [18] presented a theoretical study of Soret andDufour effects on unsteady coupled heat and mass transferby mixed convection flow over a vertical cone rotating in anambient fluid in the presence of amagnetic field and chemicalreaction Usman and Uwanta [19] have considered the effectof thermal conductivity onMHD heat andmass transfer flowpast an infinite vertical plate with Soret and Dufour effectsSimilarly Uwanta et al [20] have analyzed MHD fluid flowover a vertical plate with Dufour and Soret effectsThe effectsof Soret and Dufour on an unsteady MHD free convectionflow past a vertical porous plate in the presence of suctionor injection have been investigated by Sarada and Shankar[21] A numerical approach has been carried out for the studyof Soret and Dufour effects on mixed convection heat andmass transfer past a vertical heated plate with variable fluidproperties by Nalinakshi et al [22] Subhakar and Gangadhar[23] investigated the combined effects of the free convectiveheat and mass transfer on the unsteady two-dimensionalboundary layer flow over a stretching vertical plate in thepresence of heat generationabsorption and Soret andDufoureffects In another article Srinivasacharya and Reddy [24]examined Soret and Dufour effects on mixed convection ina non-Darcy porous medium saturated with micro polarfluid Additionally Sivaraman et al [25] considered Soretand Dufour effects on MHD free convective heat and masstransfer with thermopheresis and chemical reaction overa porous stretching surface Recently Srinivasacharya and

Upendar [26] analyzed the flow and heat and mass transfercharacteristics of the mixed convection on a vertical plate inamicropolar fluid in the presence of Soret andDufour effectsMost recently a boundary layer analysis has been presentedto study heat and mass transfer in the laminar viscousand incompressible fluid past a continuously moving platesaturated in a non-Darcy porous medium in the presenceof Soret and Dufour effects with temperature dependentviscosity and thermal conductivity by El-Kabeir et al [27]Finally Olanrewaju et al [28] have investigated Dufour andSoret effects on convection heat and mass transfer in anelectrically conducting power law flow over a heated porousplate

In view of the above studies the purpose of currentinvestigation is to examine the influence of Soret and Dufoureffects on MHD free convective heat and mass transfer flowover a vertical channel with constant suction and viscousdissipation

2 Mathematical Formulation

Consider the flow of an unsteady laminar coupled freeconvective heat and mass transfer of an incompressiblefluid past a vertical channel in a porous medium underthe influence of a uniform transverse magnetic field andconstant suction with viscous dissipation in the presenceof Soret and Dufour effects The 119909

1015840-axis is taken on thefinite plate and parallel to the free stream velocity which isvertical and the 1199101015840-axis is taken normal to the plate All fluidproperties are assumed to be constant The magnetic field ofsmall intensity is induced along the 119910 direction The fluidis assumed to be slightly conducting hence the magneticReynolds number is much less than unity and therefore theinduced magnetic field is neglected in comparison with theapplied magnetic field Under the above assumptions thegeneral equations governing the flow can be expressed asfollows

120597V1015840

120597119910

1015840= 0 (1)

120597119906

1015840

120597119905

1015840minus V0

120597119906

1015840

120597119910

1015840= V

120597

2119906

1015840

120597119910

10158402minus

120590119861

2

0

120588

119906

1015840minus

V119896

lowast119906

1015840

+ 119892120573 (119879

1015840minus 119879

1015840

0) + 119892120573

lowast(119862

1015840minus 119862

1015840

0)

(2)

120597119879

1015840

120597119905

1015840minus V0

120597119879

1015840

120597119910

1015840=

1

120588119862119901

120597

120597119910

1015840[119870 (119879)

120597119879

1015840

120597119910

1015840] +

119863119872

120588119862119901

120597

2119862

1015840

120597119910

10158402

+ 119887

lowast119906

10158402+

1

120588119862119901

(

120597119906

1015840

120597119910

1015840)

2

(3)

120597119862

1015840

120597119905

1015840minus V0

120597119862

1015840

120597119910

1015840= 119863

120597

2119862

1015840

120597119910

10158402+ 119863119879

120597

2119879

1015840

120597119910

10158402 (4)

119870 (119879) = 1198960[1 + 120572 (119879

1015840

119908minus 119879)] (5)

International Scholarly Research Notices 3

B0

T998400

120596

C998400

120596

T998400

0

C998400

0

x998400

y998400

g

u998400= 0 u

998400= 0

y998400= Hy

998400= 0

Figure 1 Geometry of the problem

The corresponding initial and boundary conditions are pre-scribed as follows

119905 le 0 119906

1015840= 0 119879

1015840gt 1198790 119862

1015840gt 1198620

forall119910

1015840

119905 gt 0 119906

1015840= 0 119879

1015840= 119879

1015840

119908 119862

1015840= 119862

1015840

119908at 1199101015840 = 0

119906

1015840= 0 119879

1015840= 1198790 119862

1015840= 1198620

at 1199101015840 = 119867

(6)

The geometry of the problem is shown in Figure 1From continuity equation it is clear that the suction

velocity is either a constant or a function of time Henceon integrating (1) the suction velocity normal to the plate isassumed in the form

V1015840 = minusV0 (7)

where V0is a scale of suction velocity which is nonzero

positive constantThe negative sign indicates that the suctionis towards the plate and V

0gt 0 corresponds to steady

suction velocity normal at the surface The fourth and fifthterms on the right hand side of (2) denote the thermal andconcentration buoyancy effects respectively 1199061015840 and V1015840 are thevelocity components in the 119909- and 119910-directions respectively119905 is the time ] is the kinematic viscosity 119892 is the accelerationdue to gravity 120573 is the coefficient of volume expansion 120588is the density 120573lowast is the volumetric coefficient of expansionwith concentration 119870(119879) is the thermal conductivity 119862

119901

is the specific heat capacity at constant pressure 119896lowast is thepermeability of the porous medium 119863

119872is the coefficient

of mass diffusivity 1198960is the thermal conductivity of the

ambient fluid 120572 is a constant depending on the nature of thefluid 119863 is the coefficient of molecular diffusivity 119863

119879is the

coefficient of temperature diffusivity 119887lowast is the dimensionlessjoule-heating parameter 120590 is the electric conductivity and 119861

0

is the magnetic field of constant strength 1198791015840 and 119879

1015840

0are the

temperature of the fluid inside the thermal boundary layerand the fluid temperature in the free stream respectively 1198621015840and 1198621015840

0are the corresponding concentrations

On introducing the following nondimensional quantities

119906 =

119906

1015840

1199060

119905 =

119905

10158401199060

119867

119910 =

119910

1015840

119867

120579 =

119879

1015840minus 119879

1015840

0

119879

1015840

119908minus 119879

1015840

0

119862 =

119862

1015840minus 119862

1015840

0

119862

1015840

119908minus 119862

1015840

0

Pr =1199060120588119862119901

1198960

Sc =1199060

119863

119896 =

119896

lowast1199060

]119867 119872 =

120590119861

2

0119867

1205881199060

Gr =119867119892120573 (119879

1015840

119908minus 119879

1015840

0)

119906

2

0

Gc =119867119892120573

lowast(119862

1015840

119908minus 119862

1015840

0)

119906

2

0

120582 = 120572 (119879

1015840

119908minus 119879

1015840

0) 120574 =

]0119867

1199060

Sr =119863119879(119879

1015840

119908minus 119879

1015840

0)

1199060(119862

1015840

119908minus 119862

1015840

0)

Ec = 119867

1199060120588119862119901(119879

1015840

119908minus 119879

1015840

0)

Du =119863119872(119862

1015840

119908minus 119862

1015840

0)

1199060120588119862119901(119879

1015840

119908minus 119879

1015840

0)

1198871=

119887

lowast119867

1199060(119879

1015840

119908minus 119879

1015840

0)

(8)

Applying (8) the set of (2) (3) (4) (5) and (6) reduces to thefollowing

120597119906

120597119905

minus 120574

120597119906

120597119910

=

120597

2119906

120597119910

2minus119872119906 minus

1

119896

119906 + Gr120579 + Gc119862

120597120579

120597119905

minus 120574

120597120579

120597119910

=

120582

Pr(

120597120579

120597119910

)

2

+

1

Pr(1 + 120582120579)

120597

2120579

120597119910

2

+ Du1205972119862

120597119910

2+ 1198871119906

2+ Ec(120597119906

120597119910

)

2

120597119862

120597119905

minus 120574

120597119862

120597119910

=

1

Sc120597

2119862

120597119910

2+ Sr120597

2120579

120597119910

2

(9)

with the following initial and boundary conditions

119905 le 0 119906 = 0 120579 = 0 119862 = 0 forall119910

119905 gt 0 119906 = 0 120579 = 1 119862 = 1 at 119910 = 0

119906 = 0 120579 = 0 119862 = 0 at 119910 = 1

(10)

where Ec is the Eckert number Pr is the Prandtl numberSc is the Schmidt number Sr is the Soret number Du is theDufour number119872 is the Magnetic field parameter Gr is thethermal Grashof number Gc is the Solutal Grashof number119896 is the porous parameter 119887

1is the joule-heating parameter

120582 is the variable thermal conductivity and 120574 is the variablesuction parameter while 119906 and V are dimensionless velocitycomponents in 119909- and 119910-directions respectively and 119905 is thedimensionless time

4 International Scholarly Research Notices

The skin friction Nusselt number and Sherwood numberare important physical parameters for this type of boundarylayer flow and are given by

119862119891= (

120597119906

120597119910

)

119910=0

Nu = minus(

120597120579

120597119910

)

119910=0

Sh = (

120597119862

120597119910

)

119910=0

(11)

3 Numerical Solution Procedure

The set of coupled nonlinear governing boundary layer equa-tion (9) together with boundary conditions (10) are solvednumerically by using the implicit finite difference methodof Crank-Nicolson typeThe finite difference approximationsequivalent to (9) are as follows

(

119906

119895+1

119894minus 119906

119895

119894

Δ119905

) minus

120574

2Δ119910

(119906

119895

119894+1minus 119906

119895

119894minus1)

=

1

2(Δ119910)

2(119906

119895+1

119894+1minus 2119906

119895+1

119894+ 119906

119895+1

119894minus1+ 119906

119895

119894+1minus 2119906

119895

119894+ 119906

119895

119894minus1)

minus (119872 +

1

119896

) 119906

119895

119894+ Gr(120579)119895

119894+ Gc(119862)119895

119894

(

120579

119895+1

119894minus 120579

119895

119894

Δ119905

) minus

120574

2Δ119910

(120579

119895

119894+1minus 120579

119895

119894minus1)

=

119867

2Pr(Δ119910)2(120579

119895+1

119894+1minus 2120579

119895+1

119894+ 120579

119895+1

119894minus1+ 120579

119895

119894+1minus 2120579

119895

119894+ 120579

119895

119894minus1)

+

Du2(Δ119910)

2(119862

119895+1

119894+1minus 2119862

119895+1

119894+ 119862

119895+1

119894minus1+ 119862

119895

119894+1minus 2119862

119895

119894+ 119862

119895

119894minus1)

+

120582

Pr(Δ119910)2(120579

119895

119894+1minus 120579

119895

119894)

2

+

Ec(Δ119910)

2(119906

119895

119894+1minus 119906

119895

119894)

2

+ 1198871(119906

119895

119894)

2

(

119862

119895+1

119894minus 119862

119895

119894

Δ119905

) minus

120574

2Δ119910

(119862

119895

119894+1minus 119862

119895

119894minus1)

=

1

2Sc(Δ119910)2

times (119862

119895+1

119894+1minus 2119862

119895+1

119894+ 119862

119895+1

119894minus1+ 119862

119895

119894+1minus 2119862

119895

119894+ 119862

119895

119894minus1)

+

Sr2(Δ119910)

2(120579

119895+1

119894+1minus 2120579

119895+1

119894+ 120579

119895+1

119894minus1+ 120579

119895

119894+1minus 2120579

119895

119894+ 120579

119895

119894minus1)

(12)

The initial and boundary conditions take the following forms

119906119894119895= 0 120579

119894119895= 0 119862

119894119895= 0

1199060119895

= 0 1205790119895

= 1 1198620119895

= 1

119906119867119895

= 0 120579119867119895

= 0 119862119867119895

= 0

(13)

where119867 corresponds to 1Equations (12) are simplified as follows

minus 1198801119906

119895+1

119894minus1+ 1198802119906

119895+1

119894minus 1198801119906

119895+1

119894+1

= 1198803119906

119895

119894minus1+ 1198804119906

119895

119894+ 1198805119906

119895

119894+1+ 1198806120579

119895

119894+ 1198807119862

119895

119894

(14)

minus 1198791120579

119895+1

119894minus1+ 1198792120579

119895+1

119894minus 1198791120579

119895+1

119894+1

= 1198793120579

119895

119894minus1+ 1198794120579

119895

119894+ 1198795120579

119895

119894+1+ 1198796(120579

119895

119894+1minus 120579

119895

119894)

2

+ 1198797(119906

119895

119894+1minus 119906

119895

119894)

2

+ 1198798(119906

119895

119894)

2

+ 1198799(119862

119895+1

119894+1minus 2119862

119895+1

119894+ 119862

119895+1

119894minus1+ 119862

119895

119894+1minus 2119862

119895

119894+ 119862

119895

119894minus1)

(15)

minus 1198811119862

119895+1

119894minus1+ 1198812119862

119895+1

119894minus 1198811119862

119895+1

119894+1

= 1198813119862

119895

119894minus1+ 1198814119862

119895

119894+ 1198815119862

119895

119894+1

+ 1198816(120579

119895+1

119894+1minus 2120579

119895+1

119894+ 120579

119895+1

119894minus1+ 120579

119895

119894+1minus 2120579

119895

119894+ 120579

119895

119894minus1)

(16)

The index 119894 corresponds to space 119910 and 119895 corresponds totime 119905 Δ119910 and Δ119905 are the mesh sizes along 119910-direction andtime 119905-direction respectively The finite difference equations(14)ndash(16) at every internal nodal point on a particular 119899-levelconstitute a tridiagonal system of equations which are solvedby usingThomas algorithmDuring the computations in eachtime step the coefficients in (14)ndash(16) are treated as constantsThe values of 120579 119862 and 119906 are known at all grid points at theinitial time 119905 = 0The values of 120579119862 and 119906 at time level (119895+1)using the known values at previous time level 119895 are calculatedas follows

Knowing the values of 120579 119862 and 119906 at a time 119905 = 119895calculate 120579 and 119862 at time 119905 = 119895 + 1 using the finite differenceequations (15) and (16) and solving the tridiagonal systemof equations by using Thomas algorithm as discussed byCarnahan et al [29] Knowing the values of 120579 and 119862 at time119905 = 119895 and 119905 = 119895 + 1 and the values of 119906 at time 119905 = 119895solve (14) using tridiagonal matrix inversion to obtain thevalues of 119906 at time 119905 = 119895 + 1 This process is repeated forvarious 119894 levels Thus the values of 120579 119862 and 119906 are knownat all grid points in the rectangular region at (119895 + 1)th timelevel Computations are carried out until the steady state isreached The Implicit Crank-Nicolson method is a secondordermethod119874(Δ119905)2 in time and has no restrictions on spaceΔ119910 and time stepΔ119905 that is the method is compatible Hencethe finite difference scheme is unconditionally stable andtherefore compatibility and stability ensures the convergenceof the scheme Computations are carried out for differentvalues of physical parameters involved in the problem

International Scholarly Research Notices 5

0 02 04 06 08 10

05

1

15

y

Gr = 50 10 15 20

Velo

city

(U)

Figure 2 Velocity profile for different values of Gr with Gc = 2120574 = 3 Sc = 022 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

05

1

Velo

city

(U)

y

Gc = 50 10 15 20

Figure 3 Velocity profile for different values of Gc with Gr = 2120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

4 Results and Discussion

In this paper numerical values are assigned physically tothe embedded parameters in the system in order to reporton the analysis of the fluid flow structure with respect tovelocity temperature and concentration profiles Numericalresults for velocity temperature and concentration profilesare presented on graphs while the skin-friction coefficientNusselt number and Sherwood number are shown in tabularform The Prandtl number is taken to be (Pr = 071 70)which corresponds to air and water The value of Schmidtnumber is taken as (Sc = 022 066 094 262) representingdiffusing chemical species of most common interest in airfor hydrogen oxygen carbon dioxide and propyl benzenewhile for Soret numbers (250) and (689) correspondingto thermal diffusion ratio of (H

2ndashCO2) and (HendashAr) are

chosen respectively while other parameters in the floware chosen arbitrarily The influence of thermal Grashofnumber Gr and solutal Grashof number Gc on the velocityis presented in Figures 2 and 3 The thermal Grashof numbersignifies the relative effect of the thermal buoyancy forceto the viscous hydrodynamic force in the boundary layerwhile the solutal Grashof number defines the ratio of thespecies buoyancy force to the viscous hydrodynamic forceAs expected the fluid velocity increases due to the enhance-ment of thermal and species buoyancy forces The velocitydistribution increases rapidly near the porous plate and thendecreases smoothly to the free stream value For different

0 02 04 06 08 10

005

01

015

02

y

M = 10 30 50 70

Velo

city

(U)

Figure 4 Velocity profile for different values of 119872 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

k = 01 02 03 04

Velo

city

(U)

Figure 5 Velocity profile for different values of 119896with Gr = 2 Gc =2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119872 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

values of magnetic parameter119872 and porous parameter 119896 thevelocity profiles are plotted in Figures 4 and 5 respectivelyIt can be seen that as 119872 increases the velocity decreasesThis results agrees with the expectations since the magneticfield exerts a restraining force on the fluid which tends toimpede it is motion Figure 5 depicts the effect of the porousparameter 119896 on the velocity profile An increase in 119896 increasesthe resistance of the porous medium which will tend toaccelerate the flow and therefore increase the velocity

For various values of suction parameter 120574 the velocitytemperature and concentration profiles are plotted in Figures6(a)ndash6(c) It is found out that an increase in the suctionparameter causes a fall in the velocity and concentrationprofiles throughout the boundary layer while increase inthe temperature profiles This is due to the fact that suctionparameter stabilizes the boundary layer growth The effect ofSoret number Sr on velocity temperature and concentrationprofiles is illustrated in Figures 7(a)ndash7(c) respectively TheSoret number defines the effect of the temperature gradientsinducing significant mass diffusion effects It can be seenthat the velocity and concentration profiles increase with anincrease in Sr while a rise in Sr causes a fall in the temperatureprofiles within the boundary layer These behaviors areevident from Figures 7(a)ndash7(c) The influence of viscousdissipation parameter that is Eckert number Ec on thevelocity profiles is depicted in Figure 8 The Eckert numberexpresses the relationship between the kinetic energy in theflow and the enthalpy It embodies the conversion of kinetic

6 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

S = 10 50 10 15

Velo

city

(U)

(a)

S = 10 50 10 15

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

(b)

0 02 04 06 08 1

0

05

1

y

S = 10 50 10 15

minus05

conc

entr

atio

n (C

)

(c)

Figure 6 (a) Velocity profile for different values of (120574 = 119878) with Gr = 2 Gc = 2 119896 = 1 Sc = 022 1198871= 001 120582 = 05119872 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of (120574 = 119878) with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1

119872 = 1 119896 = 1 Sr = 02 Du = 02 Ec = 001 Pr = 071 and 119905 = 002 (c) Concentration profile for different values of (120574 = 119878) with Gr = 2Gc = 2 Sc = 022 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 3 Du = 01 Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

Sr = 166 250 368 689

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Sr = 166 250 368 689

(b)

0 02 04 06 08 10

05

1

Con

cent

ratio

n (C

)

y

Sr = 166 250 368 689

(c)

Figure 7 (a) Velocity profile for different values of Sr with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1119872 = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Sr with Gr = 5 Gc = 2 120574 = 01 Sc = 022 1198871= 01 120582 = 1

119896 = 1119872 = 1 Du = 02 Ec = 001 Pr = 071 and 119905 = 002 (c) Concentration profile for different values of Sr with Gr = 2 Gc = 2 Sc = 0221198871= 001 120582 = 05119872 = 1 119896 = 1 120574 = 3 Du = 01 Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 7

0 02 04 06 08 10

05

1

15

y

Ec = 10 20 30 40

Velo

city

(U)

Figure 8 Velocity profile for different values of Ec with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

119872 = 1 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

t = 002 004 006 008Pr = 071

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 071

(b)

0 02 04 06 08 10

05

1

Con

cent

ratio

n (C

)

y

t = 002 004 006 008Pr = 071

(c)

Figure 9 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

0 02 04 06 08 10

005

01

y

t = 002 004 006 008

Velo

city

(U) Pr = 70

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 70

(b)

0 02 04 06 08 10

05

1C

once

ntra

tion

(C)

y

t = 002 004 006 008Pr = 70

(c)

Figure 10 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

energy into internal energy by work done against the viscousfluid stress It is observed that greater viscous dissipative heatcauses an increase in the velocity profiles across the boundarylayer Figures 9(a)ndash10(c) describe the behavior for velocitytemperature and concentration profiles for different values ofnondimensional time 119905 for two working fluids air (Pr = 071)and water (Pr = 70) It is evident from these figures that astime increases the velocity temperature and concentrationincrease The consequence of temperature increase resultsfrom increase in time since convection current becomesstronger and hence velocity and concentration increase withtime

Figures 11(a) and 11(b) graphically show the influence ofDufour number Du on the velocity and temperature profilesrespectively The Dufour number signifies the contributionof the concentration gradients to the thermal energy flux inthe flow It is seen that as Du increases there is monotonic

8 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

Du = 10 20 30 40

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Du = 10 20 30 40

(b)

Figure 11 (a) Velocity profile for different values of Du with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1 Sr = 1 119872 = 1

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Du with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1119872 = 1

119896 = 1 Sr = 02 120574 = 01 Ec = 001 Pr = 071 and 119905 = 002

increase in the velocity and temperature profiles For differentvalues of joule-heating parameter 119887

1and thermal conduc-

tivity parameter 120582 the temperature profiles are plotted inFigures 12 and 13 respectively It is observed that increas-ing the joule-heating parameter and thermal conductivityparameter produces significant increase in the thermal con-duction of the fluid which is physically true because as thethermal conductivity increases the temperature within thefluid increases Figure 14 describes the behavior of variousvalues of Scmidt number Sc on the concentration profilesThe Schmidt number characterizes the ratio of thicknesses ofviscous to the mass diffusivityThe Scmidt number quantifiesthe relative effectiveness ofmomentumandmass transport bydiffusion in the velocity and concentration boundary layers Itis observed that increase in the values of Sc causes the speciesconcentration and its boundary layer thickness to decreasesignificantly

The effects of various governing parameters on skin-friction coefficient 119862

119891 Nusselt number Nu and Sherwood

number Sh are shown in Tables 1 and 2 In order to highlightthe contributions of each parameter one parameter is variedwhile the rest take default fixed values It is observed fromTable 1 that an increase in any of the parameters 119872 and 120574

causes reduction in the skin-friction while increasing any ofthe parameters Gr Gc 119896 and 120582 resulted in correspondingincrease in the skin-friction coefficient It is also seen thatas 120582 and 120574 increase there is a rise in Nusselt number andSherwood number respectively From Table 2 it is observedthat an increase in Ec 119887

1 and Du leads to a rise in the skin-

friction coefficient Nusselt number and Sherwood numberrespectively while an increase in Pr leads to a fall in the skin-friction coefficient Nusselt number and Sherwood numberrespectively It is also seen that as Sr and Sc increase there isa fall in the skin-friction coefficient and a rise in Nusselt andSherwood numbers respectively

5 Conclusions

The present paper analyzes the influence of Soret and Dufoureffects on MHD free convective heat and mass transferflow over a vertical channel with constant suction and vis-cous dissipation The resulting partial differential equations

0 02 04 06 08 10

05

1

15

Tem

pera

ture

(T)

y

b1 = 10 50 10 15

Figure 12 Temperature profile for different values of 1198871with Gr = 5

Gc = 2 Sc = 022 120574 = 01 120582 = 1119872 = 1 119896 = 1 Sr = 02 Du = 02Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

V = 05 10 20 30

Figure 13 Temperature profile for different values of (120582 = V) withGr = 5 Gc = 2 Sc = 022 119887

1= 01 120574 = 01119872 = 1 119896 = 1 Sr = 02

Du = 02 Ec = 001 Pr = 071 and 119905 = 002

0 02 04 06 08 10

05

1

Con

cent

artio

n (C

)

y

Sc = 022 066 094 262

Figure 14 Concentration profile for different values of Sc with Gr =2 Gc = 2 120574 = 01 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 9

Table 1 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various values of physical parameters

(GrGc119872 119896 120582 and 120574)

Gr Gc 119872 119896 120582 120574 119862119891

Nu Sh20 20 10 05 10 2 02611 33328 0617140 20 10 05 10 2 04720 33451 0622120 40 10 05 10 2 03119 33386 0619520 20 30 05 10 2 02353 33317 0616720 20 10 10 10 2 02757 33334 0617420 20 10 05 20 2 02897 81676 3433420 20 10 05 10 30 02072 45600 13689With fixed values of (Pr = 071 Ec = 001 Sc = 06 Du = 02 1198871 = 10 Sr = 10 and 119905 = 02)

Table 2 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various physical parameters (Pr 119887

1 Ec Sr

Du and Sc)

Pr 1198871

Ec Sr Du Sc 119862119891

Nu Sh071 10 20 10 02 022 04038 20257 1434430 10 20 10 02 022 02898 06168 14296071 20 20 10 02 022 04041 20311 14354071 10 30 10 02 022 04053 20729 14713071 10 20 166 02 022 04019 20523 15730071 10 20 10 10 022 04133 32477 16418071 10 20 10 02 094 03519 20983 17267With fixed values of (Gr = 20 Gc = 20119872 = 10 120574 = 05 120582 = 10 119896 = 05 and 119905 = 02)

are nondimensionalised simplified and solved by implicitfinite difference method of Crank-Nicolson type From thepresent numerical study the following conclusions can bedrawn

(1) Velocity profiles increased due to increase in thermalGrashof number solutal Grashof number porousparameter Eckert number Soret number Dufournumber and dimensionless time while it decreaseddue to increase in magnetic parameter and suctionparameter

(2) An increase in temperature profiles is a function ofan increase in thermal conductivity suction parame-ter Eckert number joule-heating parameter Dufournumber and dimensionless time while it decreaseddue to increase in Soret number

(3) Concentration profiles decreased due to increasesin Schmidt number and suction parameter whileit increased due to increase in Soret number anddimensionless time

(4) There is a rise in the skin-friction coefficient Nusseltnumber and Sherwood number due to increasein Eckert number thermal conductivity parameterjoule-heating parameter and Dufour number whilea fall is observed in skin-friction coefficient withincrease in Soret number and Prandtl number

Nomenclature

119862 Concentration119862119901 Specific heat at constant pressure

119863 Mass diffusivity119863119872 Coefficient of Mass diffusivity

119863119879 Coefficient of temperature diffusivity

Du Dufour number119892 Acceleration due to gravityGr Grashof numberGc Solutal Grashof number119896 Permeability of the porous mediumNu Nusselt numberPr Prandtl numberSc Schmidt numberSr Soret number119879 Temperature119906 V Velocities in the 119909- and 119910-direction

respectively119909 119910 Cartesian coordinates along the plate

and normal to it respectively1198610 Magnetic field of constant strength

Ec Eckert number119872 Magnetic field parameter1198871 Joule-heating parameter

Greek Letters

120573

lowast Coefficient of expansion with concentration120573 Coefficient of thermal expansion120588 Density of fluid

10 International Scholarly Research Notices

120590 Electric conductivity120579 Dimensionless temperature] Kinematic viscosity120574 Suction parameter120582 Variable thermal conductivity

Subscripts

119908 Condition at wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D A Neild and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006

[2] D B Ingham and I PopTransport Phenomena in PorousMediaElsevier Oxford UK 2005

[3] G Palani and U Srikanth ldquoMHD flow past a semi-infinitevertical plate with mass transferrdquo Nonlinear Analysis Modellingand Control vol 14 no 3 pp 345ndash356 2009

[4] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoThe Canadian Journal of Chemical Engineering vol 88 no 6pp 983ndash990 2010

[5] H M Duwairi ldquoViscous and Joule heating effects on forcedconvection flow from radiate isothermal porous surfacesrdquoInternational Journal of Numerical Methods for Heat and FluidFlow vol 15 no 5 pp 429ndash440 2005

[6] M A Alim M D Alam and A Mamun ldquoJoule heating effecton the coupling of conduction with magneto hydrodynamicfree convection flow from a vertical platerdquo Non-Linear AnalysisModeling and Control vol 12 no 3 pp 307ndash316 2007

[7] I J Uwanta ldquoEffects of chemical reaction and radiation onHeatand Mass Transfer past a semi-infinite vertical porous platewith constant mass flux and dissipationrdquo European Journal ofScientific Research vol 87 no 2 pp 190ndash200 2012

[8] M A Mansour N F El-Anssary and A M Aly ldquoEffect ofchemical reaction and viscous dissipation on MHD naturalconvection flows saturated in porous media with suction orinjectionrdquo International Journal of Applied Mathematics andMechanics vol 4 no 2 pp 60ndash70 2008

[9] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[10] K Govardhan N Kishan and B Balaswamy ldquoEffect of viscousdissipation and radiation on MHD gas flow and heat and masstransfer over a stretching surface with a uniform free streamrdquoJournal of Engineering Physics andThermophysics vol 85 no 4pp 909ndash916 2012

[11] M A Sattar ldquoFree and forced convection boundary layer flowthrough a porous medium with large suctionrdquo InternationalJournal of Energy Research vol 17 no 1 pp 1ndash7 1993

[12] F Mohammad O Masatiro S Abdus and A MohamudldquoSimilarity solutions for MHD flow through vertical porous

plate with suctionrdquo Journal of Computational and AppliedMechanics vol 6 no 1 pp 15ndash25 2005

[13] S Jai ldquoViscous dissipation and chemical reaction effects on flowpast a stretching surface in a porous mediumrdquo Advance Theoryof Applied Mechanics vol 5 pp 323ndash331 2012

[14] S M Ibrahim ldquoEffects of mass transfer radiation Joule heatingand viscous dissipation on steady MHDmarangoni convectionflow over a flat surface with suction and injectionrdquo InternationalJournal of EngineeringMathematics vol 2013 Article ID 9038189 pages 2013

[15] T S Khaleque and M A Samad ldquoEffects of radiation heatgeneration and viscous dissipation on MHD free convectionflow along a stretching sheetrdquo Research Journal of AppliedSciences Engineering and Technology vol 2 no 4 pp 368ndash3772010

[16] E R G Eckert and R M Drake Analysis of Heat and MassTransfer MC Graw-Hill New York NY USA 1974

[17] A Postelnicu ldquoHeat and mass transfer by natural convection ata stagnation point in a porous medium considering Soret andDufour effectsrdquoHeat andMass Transfer vol 46 no 8-9 pp 831ndash840 2010

[18] A J Chamkha and S M M El-Kabeir ldquoUnsteady Heat andMass Transfer by MHD mixed convection flow from a rotatingvertical cone with chemical reaction and Soret and Dufoureffectsrdquo Chemical Engineering Communications vol 200 no 9pp 1220ndash1236 2013

[19] H Usman and I J Uwanta ldquoEffect of thermal conductivity onMHDHeat andMass transfer flow past an infinite vertical platewith Soret and Dufour effectsrdquo American Journal of AppliedMathematics vol 1 no 3 pp 28ndash38 2013

[20] I J Uwanta K K Asogwa and U A Ali ldquoMHD fluid flowover a vertical plate withDufour and Soret effectsrdquo InternationalJournal of Computer Applications vol 45 no 2 pp 8ndash16 2008

[21] K Sarada andB Shankar ldquoThe effects of Soret andDufour on anunsteadyMHD free convection flow past a vertical porous platein the presence of suction or injectionrdquo International Journal ofEngineering and Science vol 2 no 7 pp 13ndash25 2013

[22] N Nalinakshi P A Dinesh and D V Chandrashekar ldquoSoretand dufour effects on mixed convection heat and mass transferwith variable fluid propertiesrdquo International Journal of Mathe-matical Archive vol 4 no 11 pp 203ndash215 2013

[23] M J Subhakar and K Gangadhar ldquoSoret and Dufour effectson MHD free convection heat and mass transfer flow overa stretching vertical plate with suction and heat sourcesinkrdquoInternational Journal ofModern Engineering Research vol 2 no5 pp 3458ndash3468 2012

[24] D Srinivasacharya and C R Reddy ldquoSoret and Dufour effectson mixed convection in a non-DARcy porous medium satu-rated with micropolar fluidrdquo Nonlinear Analysis Modelling andControl vol 16 no 1 pp 100ndash115 2011

[25] N Sivaraman P K Sivagnana and R Kandasamy ldquoSoret andDufour effects on MHD free convective heat and mass transferwith thermopheresis and chemical reaction over a porousstretching surface group theory transformationrdquo Journal ofApplied Mechanics and Technical Physics vol 53 no 6 pp 871ndash879 2012

[26] D Srinivasacharya and M Upendar ldquoSoret and Dufour effectson MHD mixed convection heat and mass transfer in amicropolar fluidrdquo Central European Journal Engineering vol 3no 4 pp 679ndash689 2013

[27] S MM El-Kabeir MModathar and A M Rashed ldquoSoret anddufour effects on heat and mass transfer from a continuously

International Scholarly Research Notices 11

moving plate embedded in porous media with temperaturedependent viscosity and thermal conductivityrdquo Journal of Mod-ern Methods in Numerical Mathematics vol 4 no 2 pp 10ndash222013

[28] P O Olanrewaju O J Fenuga J A Gbadeyan and T GOkedayo ldquoDufour and Soret effects on convection heat andmass transfer in an electrical conducting power law flow overa heated porous platerdquo International Journal for ComputationalMethods in Engineering Science andMechanics vol 14 no 1 pp32ndash39 2013

[29] B Carnahan H A Luther and J O Wilkes Applied NumericalMethods John Wiley amp Sons New York NY USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 3

B0

T998400

120596

C998400

120596

T998400

0

C998400

0

x998400

y998400

g

u998400= 0 u

998400= 0

y998400= Hy

998400= 0

Figure 1 Geometry of the problem

The corresponding initial and boundary conditions are pre-scribed as follows

119905 le 0 119906

1015840= 0 119879

1015840gt 1198790 119862

1015840gt 1198620

forall119910

1015840

119905 gt 0 119906

1015840= 0 119879

1015840= 119879

1015840

119908 119862

1015840= 119862

1015840

119908at 1199101015840 = 0

119906

1015840= 0 119879

1015840= 1198790 119862

1015840= 1198620

at 1199101015840 = 119867

(6)

The geometry of the problem is shown in Figure 1From continuity equation it is clear that the suction

velocity is either a constant or a function of time Henceon integrating (1) the suction velocity normal to the plate isassumed in the form

V1015840 = minusV0 (7)

where V0is a scale of suction velocity which is nonzero

positive constantThe negative sign indicates that the suctionis towards the plate and V

0gt 0 corresponds to steady

suction velocity normal at the surface The fourth and fifthterms on the right hand side of (2) denote the thermal andconcentration buoyancy effects respectively 1199061015840 and V1015840 are thevelocity components in the 119909- and 119910-directions respectively119905 is the time ] is the kinematic viscosity 119892 is the accelerationdue to gravity 120573 is the coefficient of volume expansion 120588is the density 120573lowast is the volumetric coefficient of expansionwith concentration 119870(119879) is the thermal conductivity 119862

119901

is the specific heat capacity at constant pressure 119896lowast is thepermeability of the porous medium 119863

119872is the coefficient

of mass diffusivity 1198960is the thermal conductivity of the

ambient fluid 120572 is a constant depending on the nature of thefluid 119863 is the coefficient of molecular diffusivity 119863

119879is the

coefficient of temperature diffusivity 119887lowast is the dimensionlessjoule-heating parameter 120590 is the electric conductivity and 119861

0

is the magnetic field of constant strength 1198791015840 and 119879

1015840

0are the

temperature of the fluid inside the thermal boundary layerand the fluid temperature in the free stream respectively 1198621015840and 1198621015840

0are the corresponding concentrations

On introducing the following nondimensional quantities

119906 =

119906

1015840

1199060

119905 =

119905

10158401199060

119867

119910 =

119910

1015840

119867

120579 =

119879

1015840minus 119879

1015840

0

119879

1015840

119908minus 119879

1015840

0

119862 =

119862

1015840minus 119862

1015840

0

119862

1015840

119908minus 119862

1015840

0

Pr =1199060120588119862119901

1198960

Sc =1199060

119863

119896 =

119896

lowast1199060

]119867 119872 =

120590119861

2

0119867

1205881199060

Gr =119867119892120573 (119879

1015840

119908minus 119879

1015840

0)

119906

2

0

Gc =119867119892120573

lowast(119862

1015840

119908minus 119862

1015840

0)

119906

2

0

120582 = 120572 (119879

1015840

119908minus 119879

1015840

0) 120574 =

]0119867

1199060

Sr =119863119879(119879

1015840

119908minus 119879

1015840

0)

1199060(119862

1015840

119908minus 119862

1015840

0)

Ec = 119867

1199060120588119862119901(119879

1015840

119908minus 119879

1015840

0)

Du =119863119872(119862

1015840

119908minus 119862

1015840

0)

1199060120588119862119901(119879

1015840

119908minus 119879

1015840

0)

1198871=

119887

lowast119867

1199060(119879

1015840

119908minus 119879

1015840

0)

(8)

Applying (8) the set of (2) (3) (4) (5) and (6) reduces to thefollowing

120597119906

120597119905

minus 120574

120597119906

120597119910

=

120597

2119906

120597119910

2minus119872119906 minus

1

119896

119906 + Gr120579 + Gc119862

120597120579

120597119905

minus 120574

120597120579

120597119910

=

120582

Pr(

120597120579

120597119910

)

2

+

1

Pr(1 + 120582120579)

120597

2120579

120597119910

2

+ Du1205972119862

120597119910

2+ 1198871119906

2+ Ec(120597119906

120597119910

)

2

120597119862

120597119905

minus 120574

120597119862

120597119910

=

1

Sc120597

2119862

120597119910

2+ Sr120597

2120579

120597119910

2

(9)

with the following initial and boundary conditions

119905 le 0 119906 = 0 120579 = 0 119862 = 0 forall119910

119905 gt 0 119906 = 0 120579 = 1 119862 = 1 at 119910 = 0

119906 = 0 120579 = 0 119862 = 0 at 119910 = 1

(10)

where Ec is the Eckert number Pr is the Prandtl numberSc is the Schmidt number Sr is the Soret number Du is theDufour number119872 is the Magnetic field parameter Gr is thethermal Grashof number Gc is the Solutal Grashof number119896 is the porous parameter 119887

1is the joule-heating parameter

120582 is the variable thermal conductivity and 120574 is the variablesuction parameter while 119906 and V are dimensionless velocitycomponents in 119909- and 119910-directions respectively and 119905 is thedimensionless time

4 International Scholarly Research Notices

The skin friction Nusselt number and Sherwood numberare important physical parameters for this type of boundarylayer flow and are given by

119862119891= (

120597119906

120597119910

)

119910=0

Nu = minus(

120597120579

120597119910

)

119910=0

Sh = (

120597119862

120597119910

)

119910=0

(11)

3 Numerical Solution Procedure

The set of coupled nonlinear governing boundary layer equa-tion (9) together with boundary conditions (10) are solvednumerically by using the implicit finite difference methodof Crank-Nicolson typeThe finite difference approximationsequivalent to (9) are as follows

(

119906

119895+1

119894minus 119906

119895

119894

Δ119905

) minus

120574

2Δ119910

(119906

119895

119894+1minus 119906

119895

119894minus1)

=

1

2(Δ119910)

2(119906

119895+1

119894+1minus 2119906

119895+1

119894+ 119906

119895+1

119894minus1+ 119906

119895

119894+1minus 2119906

119895

119894+ 119906

119895

119894minus1)

minus (119872 +

1

119896

) 119906

119895

119894+ Gr(120579)119895

119894+ Gc(119862)119895

119894

(

120579

119895+1

119894minus 120579

119895

119894

Δ119905

) minus

120574

2Δ119910

(120579

119895

119894+1minus 120579

119895

119894minus1)

=

119867

2Pr(Δ119910)2(120579

119895+1

119894+1minus 2120579

119895+1

119894+ 120579

119895+1

119894minus1+ 120579

119895

119894+1minus 2120579

119895

119894+ 120579

119895

119894minus1)

+

Du2(Δ119910)

2(119862

119895+1

119894+1minus 2119862

119895+1

119894+ 119862

119895+1

119894minus1+ 119862

119895

119894+1minus 2119862

119895

119894+ 119862

119895

119894minus1)

+

120582

Pr(Δ119910)2(120579

119895

119894+1minus 120579

119895

119894)

2

+

Ec(Δ119910)

2(119906

119895

119894+1minus 119906

119895

119894)

2

+ 1198871(119906

119895

119894)

2

(

119862

119895+1

119894minus 119862

119895

119894

Δ119905

) minus

120574

2Δ119910

(119862

119895

119894+1minus 119862

119895

119894minus1)

=

1

2Sc(Δ119910)2

times (119862

119895+1

119894+1minus 2119862

119895+1

119894+ 119862

119895+1

119894minus1+ 119862

119895

119894+1minus 2119862

119895

119894+ 119862

119895

119894minus1)

+

Sr2(Δ119910)

2(120579

119895+1

119894+1minus 2120579

119895+1

119894+ 120579

119895+1

119894minus1+ 120579

119895

119894+1minus 2120579

119895

119894+ 120579

119895

119894minus1)

(12)

The initial and boundary conditions take the following forms

119906119894119895= 0 120579

119894119895= 0 119862

119894119895= 0

1199060119895

= 0 1205790119895

= 1 1198620119895

= 1

119906119867119895

= 0 120579119867119895

= 0 119862119867119895

= 0

(13)

where119867 corresponds to 1Equations (12) are simplified as follows

minus 1198801119906

119895+1

119894minus1+ 1198802119906

119895+1

119894minus 1198801119906

119895+1

119894+1

= 1198803119906

119895

119894minus1+ 1198804119906

119895

119894+ 1198805119906

119895

119894+1+ 1198806120579

119895

119894+ 1198807119862

119895

119894

(14)

minus 1198791120579

119895+1

119894minus1+ 1198792120579

119895+1

119894minus 1198791120579

119895+1

119894+1

= 1198793120579

119895

119894minus1+ 1198794120579

119895

119894+ 1198795120579

119895

119894+1+ 1198796(120579

119895

119894+1minus 120579

119895

119894)

2

+ 1198797(119906

119895

119894+1minus 119906

119895

119894)

2

+ 1198798(119906

119895

119894)

2

+ 1198799(119862

119895+1

119894+1minus 2119862

119895+1

119894+ 119862

119895+1

119894minus1+ 119862

119895

119894+1minus 2119862

119895

119894+ 119862

119895

119894minus1)

(15)

minus 1198811119862

119895+1

119894minus1+ 1198812119862

119895+1

119894minus 1198811119862

119895+1

119894+1

= 1198813119862

119895

119894minus1+ 1198814119862

119895

119894+ 1198815119862

119895

119894+1

+ 1198816(120579

119895+1

119894+1minus 2120579

119895+1

119894+ 120579

119895+1

119894minus1+ 120579

119895

119894+1minus 2120579

119895

119894+ 120579

119895

119894minus1)

(16)

The index 119894 corresponds to space 119910 and 119895 corresponds totime 119905 Δ119910 and Δ119905 are the mesh sizes along 119910-direction andtime 119905-direction respectively The finite difference equations(14)ndash(16) at every internal nodal point on a particular 119899-levelconstitute a tridiagonal system of equations which are solvedby usingThomas algorithmDuring the computations in eachtime step the coefficients in (14)ndash(16) are treated as constantsThe values of 120579 119862 and 119906 are known at all grid points at theinitial time 119905 = 0The values of 120579119862 and 119906 at time level (119895+1)using the known values at previous time level 119895 are calculatedas follows

Knowing the values of 120579 119862 and 119906 at a time 119905 = 119895calculate 120579 and 119862 at time 119905 = 119895 + 1 using the finite differenceequations (15) and (16) and solving the tridiagonal systemof equations by using Thomas algorithm as discussed byCarnahan et al [29] Knowing the values of 120579 and 119862 at time119905 = 119895 and 119905 = 119895 + 1 and the values of 119906 at time 119905 = 119895solve (14) using tridiagonal matrix inversion to obtain thevalues of 119906 at time 119905 = 119895 + 1 This process is repeated forvarious 119894 levels Thus the values of 120579 119862 and 119906 are knownat all grid points in the rectangular region at (119895 + 1)th timelevel Computations are carried out until the steady state isreached The Implicit Crank-Nicolson method is a secondordermethod119874(Δ119905)2 in time and has no restrictions on spaceΔ119910 and time stepΔ119905 that is the method is compatible Hencethe finite difference scheme is unconditionally stable andtherefore compatibility and stability ensures the convergenceof the scheme Computations are carried out for differentvalues of physical parameters involved in the problem

International Scholarly Research Notices 5

0 02 04 06 08 10

05

1

15

y

Gr = 50 10 15 20

Velo

city

(U)

Figure 2 Velocity profile for different values of Gr with Gc = 2120574 = 3 Sc = 022 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

05

1

Velo

city

(U)

y

Gc = 50 10 15 20

Figure 3 Velocity profile for different values of Gc with Gr = 2120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

4 Results and Discussion

In this paper numerical values are assigned physically tothe embedded parameters in the system in order to reporton the analysis of the fluid flow structure with respect tovelocity temperature and concentration profiles Numericalresults for velocity temperature and concentration profilesare presented on graphs while the skin-friction coefficientNusselt number and Sherwood number are shown in tabularform The Prandtl number is taken to be (Pr = 071 70)which corresponds to air and water The value of Schmidtnumber is taken as (Sc = 022 066 094 262) representingdiffusing chemical species of most common interest in airfor hydrogen oxygen carbon dioxide and propyl benzenewhile for Soret numbers (250) and (689) correspondingto thermal diffusion ratio of (H

2ndashCO2) and (HendashAr) are

chosen respectively while other parameters in the floware chosen arbitrarily The influence of thermal Grashofnumber Gr and solutal Grashof number Gc on the velocityis presented in Figures 2 and 3 The thermal Grashof numbersignifies the relative effect of the thermal buoyancy forceto the viscous hydrodynamic force in the boundary layerwhile the solutal Grashof number defines the ratio of thespecies buoyancy force to the viscous hydrodynamic forceAs expected the fluid velocity increases due to the enhance-ment of thermal and species buoyancy forces The velocitydistribution increases rapidly near the porous plate and thendecreases smoothly to the free stream value For different

0 02 04 06 08 10

005

01

015

02

y

M = 10 30 50 70

Velo

city

(U)

Figure 4 Velocity profile for different values of 119872 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

k = 01 02 03 04

Velo

city

(U)

Figure 5 Velocity profile for different values of 119896with Gr = 2 Gc =2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119872 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

values of magnetic parameter119872 and porous parameter 119896 thevelocity profiles are plotted in Figures 4 and 5 respectivelyIt can be seen that as 119872 increases the velocity decreasesThis results agrees with the expectations since the magneticfield exerts a restraining force on the fluid which tends toimpede it is motion Figure 5 depicts the effect of the porousparameter 119896 on the velocity profile An increase in 119896 increasesthe resistance of the porous medium which will tend toaccelerate the flow and therefore increase the velocity

For various values of suction parameter 120574 the velocitytemperature and concentration profiles are plotted in Figures6(a)ndash6(c) It is found out that an increase in the suctionparameter causes a fall in the velocity and concentrationprofiles throughout the boundary layer while increase inthe temperature profiles This is due to the fact that suctionparameter stabilizes the boundary layer growth The effect ofSoret number Sr on velocity temperature and concentrationprofiles is illustrated in Figures 7(a)ndash7(c) respectively TheSoret number defines the effect of the temperature gradientsinducing significant mass diffusion effects It can be seenthat the velocity and concentration profiles increase with anincrease in Sr while a rise in Sr causes a fall in the temperatureprofiles within the boundary layer These behaviors areevident from Figures 7(a)ndash7(c) The influence of viscousdissipation parameter that is Eckert number Ec on thevelocity profiles is depicted in Figure 8 The Eckert numberexpresses the relationship between the kinetic energy in theflow and the enthalpy It embodies the conversion of kinetic

6 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

S = 10 50 10 15

Velo

city

(U)

(a)

S = 10 50 10 15

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

(b)

0 02 04 06 08 1

0

05

1

y

S = 10 50 10 15

minus05

conc

entr

atio

n (C

)

(c)

Figure 6 (a) Velocity profile for different values of (120574 = 119878) with Gr = 2 Gc = 2 119896 = 1 Sc = 022 1198871= 001 120582 = 05119872 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of (120574 = 119878) with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1

119872 = 1 119896 = 1 Sr = 02 Du = 02 Ec = 001 Pr = 071 and 119905 = 002 (c) Concentration profile for different values of (120574 = 119878) with Gr = 2Gc = 2 Sc = 022 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 3 Du = 01 Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

Sr = 166 250 368 689

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Sr = 166 250 368 689

(b)

0 02 04 06 08 10

05

1

Con

cent

ratio

n (C

)

y

Sr = 166 250 368 689

(c)

Figure 7 (a) Velocity profile for different values of Sr with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1119872 = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Sr with Gr = 5 Gc = 2 120574 = 01 Sc = 022 1198871= 01 120582 = 1

119896 = 1119872 = 1 Du = 02 Ec = 001 Pr = 071 and 119905 = 002 (c) Concentration profile for different values of Sr with Gr = 2 Gc = 2 Sc = 0221198871= 001 120582 = 05119872 = 1 119896 = 1 120574 = 3 Du = 01 Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 7

0 02 04 06 08 10

05

1

15

y

Ec = 10 20 30 40

Velo

city

(U)

Figure 8 Velocity profile for different values of Ec with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

119872 = 1 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

t = 002 004 006 008Pr = 071

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 071

(b)

0 02 04 06 08 10

05

1

Con

cent

ratio

n (C

)

y

t = 002 004 006 008Pr = 071

(c)

Figure 9 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

0 02 04 06 08 10

005

01

y

t = 002 004 006 008

Velo

city

(U) Pr = 70

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 70

(b)

0 02 04 06 08 10

05

1C

once

ntra

tion

(C)

y

t = 002 004 006 008Pr = 70

(c)

Figure 10 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

energy into internal energy by work done against the viscousfluid stress It is observed that greater viscous dissipative heatcauses an increase in the velocity profiles across the boundarylayer Figures 9(a)ndash10(c) describe the behavior for velocitytemperature and concentration profiles for different values ofnondimensional time 119905 for two working fluids air (Pr = 071)and water (Pr = 70) It is evident from these figures that astime increases the velocity temperature and concentrationincrease The consequence of temperature increase resultsfrom increase in time since convection current becomesstronger and hence velocity and concentration increase withtime

Figures 11(a) and 11(b) graphically show the influence ofDufour number Du on the velocity and temperature profilesrespectively The Dufour number signifies the contributionof the concentration gradients to the thermal energy flux inthe flow It is seen that as Du increases there is monotonic

8 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

Du = 10 20 30 40

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Du = 10 20 30 40

(b)

Figure 11 (a) Velocity profile for different values of Du with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1 Sr = 1 119872 = 1

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Du with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1119872 = 1

119896 = 1 Sr = 02 120574 = 01 Ec = 001 Pr = 071 and 119905 = 002

increase in the velocity and temperature profiles For differentvalues of joule-heating parameter 119887

1and thermal conduc-

tivity parameter 120582 the temperature profiles are plotted inFigures 12 and 13 respectively It is observed that increas-ing the joule-heating parameter and thermal conductivityparameter produces significant increase in the thermal con-duction of the fluid which is physically true because as thethermal conductivity increases the temperature within thefluid increases Figure 14 describes the behavior of variousvalues of Scmidt number Sc on the concentration profilesThe Schmidt number characterizes the ratio of thicknesses ofviscous to the mass diffusivityThe Scmidt number quantifiesthe relative effectiveness ofmomentumandmass transport bydiffusion in the velocity and concentration boundary layers Itis observed that increase in the values of Sc causes the speciesconcentration and its boundary layer thickness to decreasesignificantly

The effects of various governing parameters on skin-friction coefficient 119862

119891 Nusselt number Nu and Sherwood

number Sh are shown in Tables 1 and 2 In order to highlightthe contributions of each parameter one parameter is variedwhile the rest take default fixed values It is observed fromTable 1 that an increase in any of the parameters 119872 and 120574

causes reduction in the skin-friction while increasing any ofthe parameters Gr Gc 119896 and 120582 resulted in correspondingincrease in the skin-friction coefficient It is also seen thatas 120582 and 120574 increase there is a rise in Nusselt number andSherwood number respectively From Table 2 it is observedthat an increase in Ec 119887

1 and Du leads to a rise in the skin-

friction coefficient Nusselt number and Sherwood numberrespectively while an increase in Pr leads to a fall in the skin-friction coefficient Nusselt number and Sherwood numberrespectively It is also seen that as Sr and Sc increase there isa fall in the skin-friction coefficient and a rise in Nusselt andSherwood numbers respectively

5 Conclusions

The present paper analyzes the influence of Soret and Dufoureffects on MHD free convective heat and mass transferflow over a vertical channel with constant suction and vis-cous dissipation The resulting partial differential equations

0 02 04 06 08 10

05

1

15

Tem

pera

ture

(T)

y

b1 = 10 50 10 15

Figure 12 Temperature profile for different values of 1198871with Gr = 5

Gc = 2 Sc = 022 120574 = 01 120582 = 1119872 = 1 119896 = 1 Sr = 02 Du = 02Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

V = 05 10 20 30

Figure 13 Temperature profile for different values of (120582 = V) withGr = 5 Gc = 2 Sc = 022 119887

1= 01 120574 = 01119872 = 1 119896 = 1 Sr = 02

Du = 02 Ec = 001 Pr = 071 and 119905 = 002

0 02 04 06 08 10

05

1

Con

cent

artio

n (C

)

y

Sc = 022 066 094 262

Figure 14 Concentration profile for different values of Sc with Gr =2 Gc = 2 120574 = 01 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 9

Table 1 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various values of physical parameters

(GrGc119872 119896 120582 and 120574)

Gr Gc 119872 119896 120582 120574 119862119891

Nu Sh20 20 10 05 10 2 02611 33328 0617140 20 10 05 10 2 04720 33451 0622120 40 10 05 10 2 03119 33386 0619520 20 30 05 10 2 02353 33317 0616720 20 10 10 10 2 02757 33334 0617420 20 10 05 20 2 02897 81676 3433420 20 10 05 10 30 02072 45600 13689With fixed values of (Pr = 071 Ec = 001 Sc = 06 Du = 02 1198871 = 10 Sr = 10 and 119905 = 02)

Table 2 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various physical parameters (Pr 119887

1 Ec Sr

Du and Sc)

Pr 1198871

Ec Sr Du Sc 119862119891

Nu Sh071 10 20 10 02 022 04038 20257 1434430 10 20 10 02 022 02898 06168 14296071 20 20 10 02 022 04041 20311 14354071 10 30 10 02 022 04053 20729 14713071 10 20 166 02 022 04019 20523 15730071 10 20 10 10 022 04133 32477 16418071 10 20 10 02 094 03519 20983 17267With fixed values of (Gr = 20 Gc = 20119872 = 10 120574 = 05 120582 = 10 119896 = 05 and 119905 = 02)

are nondimensionalised simplified and solved by implicitfinite difference method of Crank-Nicolson type From thepresent numerical study the following conclusions can bedrawn

(1) Velocity profiles increased due to increase in thermalGrashof number solutal Grashof number porousparameter Eckert number Soret number Dufournumber and dimensionless time while it decreaseddue to increase in magnetic parameter and suctionparameter

(2) An increase in temperature profiles is a function ofan increase in thermal conductivity suction parame-ter Eckert number joule-heating parameter Dufournumber and dimensionless time while it decreaseddue to increase in Soret number

(3) Concentration profiles decreased due to increasesin Schmidt number and suction parameter whileit increased due to increase in Soret number anddimensionless time

(4) There is a rise in the skin-friction coefficient Nusseltnumber and Sherwood number due to increasein Eckert number thermal conductivity parameterjoule-heating parameter and Dufour number whilea fall is observed in skin-friction coefficient withincrease in Soret number and Prandtl number

Nomenclature

119862 Concentration119862119901 Specific heat at constant pressure

119863 Mass diffusivity119863119872 Coefficient of Mass diffusivity

119863119879 Coefficient of temperature diffusivity

Du Dufour number119892 Acceleration due to gravityGr Grashof numberGc Solutal Grashof number119896 Permeability of the porous mediumNu Nusselt numberPr Prandtl numberSc Schmidt numberSr Soret number119879 Temperature119906 V Velocities in the 119909- and 119910-direction

respectively119909 119910 Cartesian coordinates along the plate

and normal to it respectively1198610 Magnetic field of constant strength

Ec Eckert number119872 Magnetic field parameter1198871 Joule-heating parameter

Greek Letters

120573

lowast Coefficient of expansion with concentration120573 Coefficient of thermal expansion120588 Density of fluid

10 International Scholarly Research Notices

120590 Electric conductivity120579 Dimensionless temperature] Kinematic viscosity120574 Suction parameter120582 Variable thermal conductivity

Subscripts

119908 Condition at wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D A Neild and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006

[2] D B Ingham and I PopTransport Phenomena in PorousMediaElsevier Oxford UK 2005

[3] G Palani and U Srikanth ldquoMHD flow past a semi-infinitevertical plate with mass transferrdquo Nonlinear Analysis Modellingand Control vol 14 no 3 pp 345ndash356 2009

[4] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoThe Canadian Journal of Chemical Engineering vol 88 no 6pp 983ndash990 2010

[5] H M Duwairi ldquoViscous and Joule heating effects on forcedconvection flow from radiate isothermal porous surfacesrdquoInternational Journal of Numerical Methods for Heat and FluidFlow vol 15 no 5 pp 429ndash440 2005

[6] M A Alim M D Alam and A Mamun ldquoJoule heating effecton the coupling of conduction with magneto hydrodynamicfree convection flow from a vertical platerdquo Non-Linear AnalysisModeling and Control vol 12 no 3 pp 307ndash316 2007

[7] I J Uwanta ldquoEffects of chemical reaction and radiation onHeatand Mass Transfer past a semi-infinite vertical porous platewith constant mass flux and dissipationrdquo European Journal ofScientific Research vol 87 no 2 pp 190ndash200 2012

[8] M A Mansour N F El-Anssary and A M Aly ldquoEffect ofchemical reaction and viscous dissipation on MHD naturalconvection flows saturated in porous media with suction orinjectionrdquo International Journal of Applied Mathematics andMechanics vol 4 no 2 pp 60ndash70 2008

[9] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[10] K Govardhan N Kishan and B Balaswamy ldquoEffect of viscousdissipation and radiation on MHD gas flow and heat and masstransfer over a stretching surface with a uniform free streamrdquoJournal of Engineering Physics andThermophysics vol 85 no 4pp 909ndash916 2012

[11] M A Sattar ldquoFree and forced convection boundary layer flowthrough a porous medium with large suctionrdquo InternationalJournal of Energy Research vol 17 no 1 pp 1ndash7 1993

[12] F Mohammad O Masatiro S Abdus and A MohamudldquoSimilarity solutions for MHD flow through vertical porous

plate with suctionrdquo Journal of Computational and AppliedMechanics vol 6 no 1 pp 15ndash25 2005

[13] S Jai ldquoViscous dissipation and chemical reaction effects on flowpast a stretching surface in a porous mediumrdquo Advance Theoryof Applied Mechanics vol 5 pp 323ndash331 2012

[14] S M Ibrahim ldquoEffects of mass transfer radiation Joule heatingand viscous dissipation on steady MHDmarangoni convectionflow over a flat surface with suction and injectionrdquo InternationalJournal of EngineeringMathematics vol 2013 Article ID 9038189 pages 2013

[15] T S Khaleque and M A Samad ldquoEffects of radiation heatgeneration and viscous dissipation on MHD free convectionflow along a stretching sheetrdquo Research Journal of AppliedSciences Engineering and Technology vol 2 no 4 pp 368ndash3772010

[16] E R G Eckert and R M Drake Analysis of Heat and MassTransfer MC Graw-Hill New York NY USA 1974

[17] A Postelnicu ldquoHeat and mass transfer by natural convection ata stagnation point in a porous medium considering Soret andDufour effectsrdquoHeat andMass Transfer vol 46 no 8-9 pp 831ndash840 2010

[18] A J Chamkha and S M M El-Kabeir ldquoUnsteady Heat andMass Transfer by MHD mixed convection flow from a rotatingvertical cone with chemical reaction and Soret and Dufoureffectsrdquo Chemical Engineering Communications vol 200 no 9pp 1220ndash1236 2013

[19] H Usman and I J Uwanta ldquoEffect of thermal conductivity onMHDHeat andMass transfer flow past an infinite vertical platewith Soret and Dufour effectsrdquo American Journal of AppliedMathematics vol 1 no 3 pp 28ndash38 2013

[20] I J Uwanta K K Asogwa and U A Ali ldquoMHD fluid flowover a vertical plate withDufour and Soret effectsrdquo InternationalJournal of Computer Applications vol 45 no 2 pp 8ndash16 2008

[21] K Sarada andB Shankar ldquoThe effects of Soret andDufour on anunsteadyMHD free convection flow past a vertical porous platein the presence of suction or injectionrdquo International Journal ofEngineering and Science vol 2 no 7 pp 13ndash25 2013

[22] N Nalinakshi P A Dinesh and D V Chandrashekar ldquoSoretand dufour effects on mixed convection heat and mass transferwith variable fluid propertiesrdquo International Journal of Mathe-matical Archive vol 4 no 11 pp 203ndash215 2013

[23] M J Subhakar and K Gangadhar ldquoSoret and Dufour effectson MHD free convection heat and mass transfer flow overa stretching vertical plate with suction and heat sourcesinkrdquoInternational Journal ofModern Engineering Research vol 2 no5 pp 3458ndash3468 2012

[24] D Srinivasacharya and C R Reddy ldquoSoret and Dufour effectson mixed convection in a non-DARcy porous medium satu-rated with micropolar fluidrdquo Nonlinear Analysis Modelling andControl vol 16 no 1 pp 100ndash115 2011

[25] N Sivaraman P K Sivagnana and R Kandasamy ldquoSoret andDufour effects on MHD free convective heat and mass transferwith thermopheresis and chemical reaction over a porousstretching surface group theory transformationrdquo Journal ofApplied Mechanics and Technical Physics vol 53 no 6 pp 871ndash879 2012

[26] D Srinivasacharya and M Upendar ldquoSoret and Dufour effectson MHD mixed convection heat and mass transfer in amicropolar fluidrdquo Central European Journal Engineering vol 3no 4 pp 679ndash689 2013

[27] S MM El-Kabeir MModathar and A M Rashed ldquoSoret anddufour effects on heat and mass transfer from a continuously

International Scholarly Research Notices 11

moving plate embedded in porous media with temperaturedependent viscosity and thermal conductivityrdquo Journal of Mod-ern Methods in Numerical Mathematics vol 4 no 2 pp 10ndash222013

[28] P O Olanrewaju O J Fenuga J A Gbadeyan and T GOkedayo ldquoDufour and Soret effects on convection heat andmass transfer in an electrical conducting power law flow overa heated porous platerdquo International Journal for ComputationalMethods in Engineering Science andMechanics vol 14 no 1 pp32ndash39 2013

[29] B Carnahan H A Luther and J O Wilkes Applied NumericalMethods John Wiley amp Sons New York NY USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Scholarly Research Notices

The skin friction Nusselt number and Sherwood numberare important physical parameters for this type of boundarylayer flow and are given by

119862119891= (

120597119906

120597119910

)

119910=0

Nu = minus(

120597120579

120597119910

)

119910=0

Sh = (

120597119862

120597119910

)

119910=0

(11)

3 Numerical Solution Procedure

The set of coupled nonlinear governing boundary layer equa-tion (9) together with boundary conditions (10) are solvednumerically by using the implicit finite difference methodof Crank-Nicolson typeThe finite difference approximationsequivalent to (9) are as follows

(

119906

119895+1

119894minus 119906

119895

119894

Δ119905

) minus

120574

2Δ119910

(119906

119895

119894+1minus 119906

119895

119894minus1)

=

1

2(Δ119910)

2(119906

119895+1

119894+1minus 2119906

119895+1

119894+ 119906

119895+1

119894minus1+ 119906

119895

119894+1minus 2119906

119895

119894+ 119906

119895

119894minus1)

minus (119872 +

1

119896

) 119906

119895

119894+ Gr(120579)119895

119894+ Gc(119862)119895

119894

(

120579

119895+1

119894minus 120579

119895

119894

Δ119905

) minus

120574

2Δ119910

(120579

119895

119894+1minus 120579

119895

119894minus1)

=

119867

2Pr(Δ119910)2(120579

119895+1

119894+1minus 2120579

119895+1

119894+ 120579

119895+1

119894minus1+ 120579

119895

119894+1minus 2120579

119895

119894+ 120579

119895

119894minus1)

+

Du2(Δ119910)

2(119862

119895+1

119894+1minus 2119862

119895+1

119894+ 119862

119895+1

119894minus1+ 119862

119895

119894+1minus 2119862

119895

119894+ 119862

119895

119894minus1)

+

120582

Pr(Δ119910)2(120579

119895

119894+1minus 120579

119895

119894)

2

+

Ec(Δ119910)

2(119906

119895

119894+1minus 119906

119895

119894)

2

+ 1198871(119906

119895

119894)

2

(

119862

119895+1

119894minus 119862

119895

119894

Δ119905

) minus

120574

2Δ119910

(119862

119895

119894+1minus 119862

119895

119894minus1)

=

1

2Sc(Δ119910)2

times (119862

119895+1

119894+1minus 2119862

119895+1

119894+ 119862

119895+1

119894minus1+ 119862

119895

119894+1minus 2119862

119895

119894+ 119862

119895

119894minus1)

+

Sr2(Δ119910)

2(120579

119895+1

119894+1minus 2120579

119895+1

119894+ 120579

119895+1

119894minus1+ 120579

119895

119894+1minus 2120579

119895

119894+ 120579

119895

119894minus1)

(12)

The initial and boundary conditions take the following forms

119906119894119895= 0 120579

119894119895= 0 119862

119894119895= 0

1199060119895

= 0 1205790119895

= 1 1198620119895

= 1

119906119867119895

= 0 120579119867119895

= 0 119862119867119895

= 0

(13)

where119867 corresponds to 1Equations (12) are simplified as follows

minus 1198801119906

119895+1

119894minus1+ 1198802119906

119895+1

119894minus 1198801119906

119895+1

119894+1

= 1198803119906

119895

119894minus1+ 1198804119906

119895

119894+ 1198805119906

119895

119894+1+ 1198806120579

119895

119894+ 1198807119862

119895

119894

(14)

minus 1198791120579

119895+1

119894minus1+ 1198792120579

119895+1

119894minus 1198791120579

119895+1

119894+1

= 1198793120579

119895

119894minus1+ 1198794120579

119895

119894+ 1198795120579

119895

119894+1+ 1198796(120579

119895

119894+1minus 120579

119895

119894)

2

+ 1198797(119906

119895

119894+1minus 119906

119895

119894)

2

+ 1198798(119906

119895

119894)

2

+ 1198799(119862

119895+1

119894+1minus 2119862

119895+1

119894+ 119862

119895+1

119894minus1+ 119862

119895

119894+1minus 2119862

119895

119894+ 119862

119895

119894minus1)

(15)

minus 1198811119862

119895+1

119894minus1+ 1198812119862

119895+1

119894minus 1198811119862

119895+1

119894+1

= 1198813119862

119895

119894minus1+ 1198814119862

119895

119894+ 1198815119862

119895

119894+1

+ 1198816(120579

119895+1

119894+1minus 2120579

119895+1

119894+ 120579

119895+1

119894minus1+ 120579

119895

119894+1minus 2120579

119895

119894+ 120579

119895

119894minus1)

(16)

The index 119894 corresponds to space 119910 and 119895 corresponds totime 119905 Δ119910 and Δ119905 are the mesh sizes along 119910-direction andtime 119905-direction respectively The finite difference equations(14)ndash(16) at every internal nodal point on a particular 119899-levelconstitute a tridiagonal system of equations which are solvedby usingThomas algorithmDuring the computations in eachtime step the coefficients in (14)ndash(16) are treated as constantsThe values of 120579 119862 and 119906 are known at all grid points at theinitial time 119905 = 0The values of 120579119862 and 119906 at time level (119895+1)using the known values at previous time level 119895 are calculatedas follows

Knowing the values of 120579 119862 and 119906 at a time 119905 = 119895calculate 120579 and 119862 at time 119905 = 119895 + 1 using the finite differenceequations (15) and (16) and solving the tridiagonal systemof equations by using Thomas algorithm as discussed byCarnahan et al [29] Knowing the values of 120579 and 119862 at time119905 = 119895 and 119905 = 119895 + 1 and the values of 119906 at time 119905 = 119895solve (14) using tridiagonal matrix inversion to obtain thevalues of 119906 at time 119905 = 119895 + 1 This process is repeated forvarious 119894 levels Thus the values of 120579 119862 and 119906 are knownat all grid points in the rectangular region at (119895 + 1)th timelevel Computations are carried out until the steady state isreached The Implicit Crank-Nicolson method is a secondordermethod119874(Δ119905)2 in time and has no restrictions on spaceΔ119910 and time stepΔ119905 that is the method is compatible Hencethe finite difference scheme is unconditionally stable andtherefore compatibility and stability ensures the convergenceof the scheme Computations are carried out for differentvalues of physical parameters involved in the problem

International Scholarly Research Notices 5

0 02 04 06 08 10

05

1

15

y

Gr = 50 10 15 20

Velo

city

(U)

Figure 2 Velocity profile for different values of Gr with Gc = 2120574 = 3 Sc = 022 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

05

1

Velo

city

(U)

y

Gc = 50 10 15 20

Figure 3 Velocity profile for different values of Gc with Gr = 2120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

4 Results and Discussion

In this paper numerical values are assigned physically tothe embedded parameters in the system in order to reporton the analysis of the fluid flow structure with respect tovelocity temperature and concentration profiles Numericalresults for velocity temperature and concentration profilesare presented on graphs while the skin-friction coefficientNusselt number and Sherwood number are shown in tabularform The Prandtl number is taken to be (Pr = 071 70)which corresponds to air and water The value of Schmidtnumber is taken as (Sc = 022 066 094 262) representingdiffusing chemical species of most common interest in airfor hydrogen oxygen carbon dioxide and propyl benzenewhile for Soret numbers (250) and (689) correspondingto thermal diffusion ratio of (H

2ndashCO2) and (HendashAr) are

chosen respectively while other parameters in the floware chosen arbitrarily The influence of thermal Grashofnumber Gr and solutal Grashof number Gc on the velocityis presented in Figures 2 and 3 The thermal Grashof numbersignifies the relative effect of the thermal buoyancy forceto the viscous hydrodynamic force in the boundary layerwhile the solutal Grashof number defines the ratio of thespecies buoyancy force to the viscous hydrodynamic forceAs expected the fluid velocity increases due to the enhance-ment of thermal and species buoyancy forces The velocitydistribution increases rapidly near the porous plate and thendecreases smoothly to the free stream value For different

0 02 04 06 08 10

005

01

015

02

y

M = 10 30 50 70

Velo

city

(U)

Figure 4 Velocity profile for different values of 119872 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

k = 01 02 03 04

Velo

city

(U)

Figure 5 Velocity profile for different values of 119896with Gr = 2 Gc =2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119872 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

values of magnetic parameter119872 and porous parameter 119896 thevelocity profiles are plotted in Figures 4 and 5 respectivelyIt can be seen that as 119872 increases the velocity decreasesThis results agrees with the expectations since the magneticfield exerts a restraining force on the fluid which tends toimpede it is motion Figure 5 depicts the effect of the porousparameter 119896 on the velocity profile An increase in 119896 increasesthe resistance of the porous medium which will tend toaccelerate the flow and therefore increase the velocity

For various values of suction parameter 120574 the velocitytemperature and concentration profiles are plotted in Figures6(a)ndash6(c) It is found out that an increase in the suctionparameter causes a fall in the velocity and concentrationprofiles throughout the boundary layer while increase inthe temperature profiles This is due to the fact that suctionparameter stabilizes the boundary layer growth The effect ofSoret number Sr on velocity temperature and concentrationprofiles is illustrated in Figures 7(a)ndash7(c) respectively TheSoret number defines the effect of the temperature gradientsinducing significant mass diffusion effects It can be seenthat the velocity and concentration profiles increase with anincrease in Sr while a rise in Sr causes a fall in the temperatureprofiles within the boundary layer These behaviors areevident from Figures 7(a)ndash7(c) The influence of viscousdissipation parameter that is Eckert number Ec on thevelocity profiles is depicted in Figure 8 The Eckert numberexpresses the relationship between the kinetic energy in theflow and the enthalpy It embodies the conversion of kinetic

6 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

S = 10 50 10 15

Velo

city

(U)

(a)

S = 10 50 10 15

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

(b)

0 02 04 06 08 1

0

05

1

y

S = 10 50 10 15

minus05

conc

entr

atio

n (C

)

(c)

Figure 6 (a) Velocity profile for different values of (120574 = 119878) with Gr = 2 Gc = 2 119896 = 1 Sc = 022 1198871= 001 120582 = 05119872 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of (120574 = 119878) with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1

119872 = 1 119896 = 1 Sr = 02 Du = 02 Ec = 001 Pr = 071 and 119905 = 002 (c) Concentration profile for different values of (120574 = 119878) with Gr = 2Gc = 2 Sc = 022 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 3 Du = 01 Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

Sr = 166 250 368 689

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Sr = 166 250 368 689

(b)

0 02 04 06 08 10

05

1

Con

cent

ratio

n (C

)

y

Sr = 166 250 368 689

(c)

Figure 7 (a) Velocity profile for different values of Sr with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1119872 = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Sr with Gr = 5 Gc = 2 120574 = 01 Sc = 022 1198871= 01 120582 = 1

119896 = 1119872 = 1 Du = 02 Ec = 001 Pr = 071 and 119905 = 002 (c) Concentration profile for different values of Sr with Gr = 2 Gc = 2 Sc = 0221198871= 001 120582 = 05119872 = 1 119896 = 1 120574 = 3 Du = 01 Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 7

0 02 04 06 08 10

05

1

15

y

Ec = 10 20 30 40

Velo

city

(U)

Figure 8 Velocity profile for different values of Ec with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

119872 = 1 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

t = 002 004 006 008Pr = 071

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 071

(b)

0 02 04 06 08 10

05

1

Con

cent

ratio

n (C

)

y

t = 002 004 006 008Pr = 071

(c)

Figure 9 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

0 02 04 06 08 10

005

01

y

t = 002 004 006 008

Velo

city

(U) Pr = 70

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 70

(b)

0 02 04 06 08 10

05

1C

once

ntra

tion

(C)

y

t = 002 004 006 008Pr = 70

(c)

Figure 10 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

energy into internal energy by work done against the viscousfluid stress It is observed that greater viscous dissipative heatcauses an increase in the velocity profiles across the boundarylayer Figures 9(a)ndash10(c) describe the behavior for velocitytemperature and concentration profiles for different values ofnondimensional time 119905 for two working fluids air (Pr = 071)and water (Pr = 70) It is evident from these figures that astime increases the velocity temperature and concentrationincrease The consequence of temperature increase resultsfrom increase in time since convection current becomesstronger and hence velocity and concentration increase withtime

Figures 11(a) and 11(b) graphically show the influence ofDufour number Du on the velocity and temperature profilesrespectively The Dufour number signifies the contributionof the concentration gradients to the thermal energy flux inthe flow It is seen that as Du increases there is monotonic

8 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

Du = 10 20 30 40

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Du = 10 20 30 40

(b)

Figure 11 (a) Velocity profile for different values of Du with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1 Sr = 1 119872 = 1

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Du with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1119872 = 1

119896 = 1 Sr = 02 120574 = 01 Ec = 001 Pr = 071 and 119905 = 002

increase in the velocity and temperature profiles For differentvalues of joule-heating parameter 119887

1and thermal conduc-

tivity parameter 120582 the temperature profiles are plotted inFigures 12 and 13 respectively It is observed that increas-ing the joule-heating parameter and thermal conductivityparameter produces significant increase in the thermal con-duction of the fluid which is physically true because as thethermal conductivity increases the temperature within thefluid increases Figure 14 describes the behavior of variousvalues of Scmidt number Sc on the concentration profilesThe Schmidt number characterizes the ratio of thicknesses ofviscous to the mass diffusivityThe Scmidt number quantifiesthe relative effectiveness ofmomentumandmass transport bydiffusion in the velocity and concentration boundary layers Itis observed that increase in the values of Sc causes the speciesconcentration and its boundary layer thickness to decreasesignificantly

The effects of various governing parameters on skin-friction coefficient 119862

119891 Nusselt number Nu and Sherwood

number Sh are shown in Tables 1 and 2 In order to highlightthe contributions of each parameter one parameter is variedwhile the rest take default fixed values It is observed fromTable 1 that an increase in any of the parameters 119872 and 120574

causes reduction in the skin-friction while increasing any ofthe parameters Gr Gc 119896 and 120582 resulted in correspondingincrease in the skin-friction coefficient It is also seen thatas 120582 and 120574 increase there is a rise in Nusselt number andSherwood number respectively From Table 2 it is observedthat an increase in Ec 119887

1 and Du leads to a rise in the skin-

friction coefficient Nusselt number and Sherwood numberrespectively while an increase in Pr leads to a fall in the skin-friction coefficient Nusselt number and Sherwood numberrespectively It is also seen that as Sr and Sc increase there isa fall in the skin-friction coefficient and a rise in Nusselt andSherwood numbers respectively

5 Conclusions

The present paper analyzes the influence of Soret and Dufoureffects on MHD free convective heat and mass transferflow over a vertical channel with constant suction and vis-cous dissipation The resulting partial differential equations

0 02 04 06 08 10

05

1

15

Tem

pera

ture

(T)

y

b1 = 10 50 10 15

Figure 12 Temperature profile for different values of 1198871with Gr = 5

Gc = 2 Sc = 022 120574 = 01 120582 = 1119872 = 1 119896 = 1 Sr = 02 Du = 02Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

V = 05 10 20 30

Figure 13 Temperature profile for different values of (120582 = V) withGr = 5 Gc = 2 Sc = 022 119887

1= 01 120574 = 01119872 = 1 119896 = 1 Sr = 02

Du = 02 Ec = 001 Pr = 071 and 119905 = 002

0 02 04 06 08 10

05

1

Con

cent

artio

n (C

)

y

Sc = 022 066 094 262

Figure 14 Concentration profile for different values of Sc with Gr =2 Gc = 2 120574 = 01 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 9

Table 1 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various values of physical parameters

(GrGc119872 119896 120582 and 120574)

Gr Gc 119872 119896 120582 120574 119862119891

Nu Sh20 20 10 05 10 2 02611 33328 0617140 20 10 05 10 2 04720 33451 0622120 40 10 05 10 2 03119 33386 0619520 20 30 05 10 2 02353 33317 0616720 20 10 10 10 2 02757 33334 0617420 20 10 05 20 2 02897 81676 3433420 20 10 05 10 30 02072 45600 13689With fixed values of (Pr = 071 Ec = 001 Sc = 06 Du = 02 1198871 = 10 Sr = 10 and 119905 = 02)

Table 2 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various physical parameters (Pr 119887

1 Ec Sr

Du and Sc)

Pr 1198871

Ec Sr Du Sc 119862119891

Nu Sh071 10 20 10 02 022 04038 20257 1434430 10 20 10 02 022 02898 06168 14296071 20 20 10 02 022 04041 20311 14354071 10 30 10 02 022 04053 20729 14713071 10 20 166 02 022 04019 20523 15730071 10 20 10 10 022 04133 32477 16418071 10 20 10 02 094 03519 20983 17267With fixed values of (Gr = 20 Gc = 20119872 = 10 120574 = 05 120582 = 10 119896 = 05 and 119905 = 02)

are nondimensionalised simplified and solved by implicitfinite difference method of Crank-Nicolson type From thepresent numerical study the following conclusions can bedrawn

(1) Velocity profiles increased due to increase in thermalGrashof number solutal Grashof number porousparameter Eckert number Soret number Dufournumber and dimensionless time while it decreaseddue to increase in magnetic parameter and suctionparameter

(2) An increase in temperature profiles is a function ofan increase in thermal conductivity suction parame-ter Eckert number joule-heating parameter Dufournumber and dimensionless time while it decreaseddue to increase in Soret number

(3) Concentration profiles decreased due to increasesin Schmidt number and suction parameter whileit increased due to increase in Soret number anddimensionless time

(4) There is a rise in the skin-friction coefficient Nusseltnumber and Sherwood number due to increasein Eckert number thermal conductivity parameterjoule-heating parameter and Dufour number whilea fall is observed in skin-friction coefficient withincrease in Soret number and Prandtl number

Nomenclature

119862 Concentration119862119901 Specific heat at constant pressure

119863 Mass diffusivity119863119872 Coefficient of Mass diffusivity

119863119879 Coefficient of temperature diffusivity

Du Dufour number119892 Acceleration due to gravityGr Grashof numberGc Solutal Grashof number119896 Permeability of the porous mediumNu Nusselt numberPr Prandtl numberSc Schmidt numberSr Soret number119879 Temperature119906 V Velocities in the 119909- and 119910-direction

respectively119909 119910 Cartesian coordinates along the plate

and normal to it respectively1198610 Magnetic field of constant strength

Ec Eckert number119872 Magnetic field parameter1198871 Joule-heating parameter

Greek Letters

120573

lowast Coefficient of expansion with concentration120573 Coefficient of thermal expansion120588 Density of fluid

10 International Scholarly Research Notices

120590 Electric conductivity120579 Dimensionless temperature] Kinematic viscosity120574 Suction parameter120582 Variable thermal conductivity

Subscripts

119908 Condition at wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D A Neild and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006

[2] D B Ingham and I PopTransport Phenomena in PorousMediaElsevier Oxford UK 2005

[3] G Palani and U Srikanth ldquoMHD flow past a semi-infinitevertical plate with mass transferrdquo Nonlinear Analysis Modellingand Control vol 14 no 3 pp 345ndash356 2009

[4] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoThe Canadian Journal of Chemical Engineering vol 88 no 6pp 983ndash990 2010

[5] H M Duwairi ldquoViscous and Joule heating effects on forcedconvection flow from radiate isothermal porous surfacesrdquoInternational Journal of Numerical Methods for Heat and FluidFlow vol 15 no 5 pp 429ndash440 2005

[6] M A Alim M D Alam and A Mamun ldquoJoule heating effecton the coupling of conduction with magneto hydrodynamicfree convection flow from a vertical platerdquo Non-Linear AnalysisModeling and Control vol 12 no 3 pp 307ndash316 2007

[7] I J Uwanta ldquoEffects of chemical reaction and radiation onHeatand Mass Transfer past a semi-infinite vertical porous platewith constant mass flux and dissipationrdquo European Journal ofScientific Research vol 87 no 2 pp 190ndash200 2012

[8] M A Mansour N F El-Anssary and A M Aly ldquoEffect ofchemical reaction and viscous dissipation on MHD naturalconvection flows saturated in porous media with suction orinjectionrdquo International Journal of Applied Mathematics andMechanics vol 4 no 2 pp 60ndash70 2008

[9] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[10] K Govardhan N Kishan and B Balaswamy ldquoEffect of viscousdissipation and radiation on MHD gas flow and heat and masstransfer over a stretching surface with a uniform free streamrdquoJournal of Engineering Physics andThermophysics vol 85 no 4pp 909ndash916 2012

[11] M A Sattar ldquoFree and forced convection boundary layer flowthrough a porous medium with large suctionrdquo InternationalJournal of Energy Research vol 17 no 1 pp 1ndash7 1993

[12] F Mohammad O Masatiro S Abdus and A MohamudldquoSimilarity solutions for MHD flow through vertical porous

plate with suctionrdquo Journal of Computational and AppliedMechanics vol 6 no 1 pp 15ndash25 2005

[13] S Jai ldquoViscous dissipation and chemical reaction effects on flowpast a stretching surface in a porous mediumrdquo Advance Theoryof Applied Mechanics vol 5 pp 323ndash331 2012

[14] S M Ibrahim ldquoEffects of mass transfer radiation Joule heatingand viscous dissipation on steady MHDmarangoni convectionflow over a flat surface with suction and injectionrdquo InternationalJournal of EngineeringMathematics vol 2013 Article ID 9038189 pages 2013

[15] T S Khaleque and M A Samad ldquoEffects of radiation heatgeneration and viscous dissipation on MHD free convectionflow along a stretching sheetrdquo Research Journal of AppliedSciences Engineering and Technology vol 2 no 4 pp 368ndash3772010

[16] E R G Eckert and R M Drake Analysis of Heat and MassTransfer MC Graw-Hill New York NY USA 1974

[17] A Postelnicu ldquoHeat and mass transfer by natural convection ata stagnation point in a porous medium considering Soret andDufour effectsrdquoHeat andMass Transfer vol 46 no 8-9 pp 831ndash840 2010

[18] A J Chamkha and S M M El-Kabeir ldquoUnsteady Heat andMass Transfer by MHD mixed convection flow from a rotatingvertical cone with chemical reaction and Soret and Dufoureffectsrdquo Chemical Engineering Communications vol 200 no 9pp 1220ndash1236 2013

[19] H Usman and I J Uwanta ldquoEffect of thermal conductivity onMHDHeat andMass transfer flow past an infinite vertical platewith Soret and Dufour effectsrdquo American Journal of AppliedMathematics vol 1 no 3 pp 28ndash38 2013

[20] I J Uwanta K K Asogwa and U A Ali ldquoMHD fluid flowover a vertical plate withDufour and Soret effectsrdquo InternationalJournal of Computer Applications vol 45 no 2 pp 8ndash16 2008

[21] K Sarada andB Shankar ldquoThe effects of Soret andDufour on anunsteadyMHD free convection flow past a vertical porous platein the presence of suction or injectionrdquo International Journal ofEngineering and Science vol 2 no 7 pp 13ndash25 2013

[22] N Nalinakshi P A Dinesh and D V Chandrashekar ldquoSoretand dufour effects on mixed convection heat and mass transferwith variable fluid propertiesrdquo International Journal of Mathe-matical Archive vol 4 no 11 pp 203ndash215 2013

[23] M J Subhakar and K Gangadhar ldquoSoret and Dufour effectson MHD free convection heat and mass transfer flow overa stretching vertical plate with suction and heat sourcesinkrdquoInternational Journal ofModern Engineering Research vol 2 no5 pp 3458ndash3468 2012

[24] D Srinivasacharya and C R Reddy ldquoSoret and Dufour effectson mixed convection in a non-DARcy porous medium satu-rated with micropolar fluidrdquo Nonlinear Analysis Modelling andControl vol 16 no 1 pp 100ndash115 2011

[25] N Sivaraman P K Sivagnana and R Kandasamy ldquoSoret andDufour effects on MHD free convective heat and mass transferwith thermopheresis and chemical reaction over a porousstretching surface group theory transformationrdquo Journal ofApplied Mechanics and Technical Physics vol 53 no 6 pp 871ndash879 2012

[26] D Srinivasacharya and M Upendar ldquoSoret and Dufour effectson MHD mixed convection heat and mass transfer in amicropolar fluidrdquo Central European Journal Engineering vol 3no 4 pp 679ndash689 2013

[27] S MM El-Kabeir MModathar and A M Rashed ldquoSoret anddufour effects on heat and mass transfer from a continuously

International Scholarly Research Notices 11

moving plate embedded in porous media with temperaturedependent viscosity and thermal conductivityrdquo Journal of Mod-ern Methods in Numerical Mathematics vol 4 no 2 pp 10ndash222013

[28] P O Olanrewaju O J Fenuga J A Gbadeyan and T GOkedayo ldquoDufour and Soret effects on convection heat andmass transfer in an electrical conducting power law flow overa heated porous platerdquo International Journal for ComputationalMethods in Engineering Science andMechanics vol 14 no 1 pp32ndash39 2013

[29] B Carnahan H A Luther and J O Wilkes Applied NumericalMethods John Wiley amp Sons New York NY USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 5

0 02 04 06 08 10

05

1

15

y

Gr = 50 10 15 20

Velo

city

(U)

Figure 2 Velocity profile for different values of Gr with Gc = 2120574 = 3 Sc = 022 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

05

1

Velo

city

(U)

y

Gc = 50 10 15 20

Figure 3 Velocity profile for different values of Gc with Gr = 2120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

4 Results and Discussion

In this paper numerical values are assigned physically tothe embedded parameters in the system in order to reporton the analysis of the fluid flow structure with respect tovelocity temperature and concentration profiles Numericalresults for velocity temperature and concentration profilesare presented on graphs while the skin-friction coefficientNusselt number and Sherwood number are shown in tabularform The Prandtl number is taken to be (Pr = 071 70)which corresponds to air and water The value of Schmidtnumber is taken as (Sc = 022 066 094 262) representingdiffusing chemical species of most common interest in airfor hydrogen oxygen carbon dioxide and propyl benzenewhile for Soret numbers (250) and (689) correspondingto thermal diffusion ratio of (H

2ndashCO2) and (HendashAr) are

chosen respectively while other parameters in the floware chosen arbitrarily The influence of thermal Grashofnumber Gr and solutal Grashof number Gc on the velocityis presented in Figures 2 and 3 The thermal Grashof numbersignifies the relative effect of the thermal buoyancy forceto the viscous hydrodynamic force in the boundary layerwhile the solutal Grashof number defines the ratio of thespecies buoyancy force to the viscous hydrodynamic forceAs expected the fluid velocity increases due to the enhance-ment of thermal and species buoyancy forces The velocitydistribution increases rapidly near the porous plate and thendecreases smoothly to the free stream value For different

0 02 04 06 08 10

005

01

015

02

y

M = 10 30 50 70

Velo

city

(U)

Figure 4 Velocity profile for different values of 119872 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

k = 01 02 03 04

Velo

city

(U)

Figure 5 Velocity profile for different values of 119896with Gr = 2 Gc =2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119872 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02

values of magnetic parameter119872 and porous parameter 119896 thevelocity profiles are plotted in Figures 4 and 5 respectivelyIt can be seen that as 119872 increases the velocity decreasesThis results agrees with the expectations since the magneticfield exerts a restraining force on the fluid which tends toimpede it is motion Figure 5 depicts the effect of the porousparameter 119896 on the velocity profile An increase in 119896 increasesthe resistance of the porous medium which will tend toaccelerate the flow and therefore increase the velocity

For various values of suction parameter 120574 the velocitytemperature and concentration profiles are plotted in Figures6(a)ndash6(c) It is found out that an increase in the suctionparameter causes a fall in the velocity and concentrationprofiles throughout the boundary layer while increase inthe temperature profiles This is due to the fact that suctionparameter stabilizes the boundary layer growth The effect ofSoret number Sr on velocity temperature and concentrationprofiles is illustrated in Figures 7(a)ndash7(c) respectively TheSoret number defines the effect of the temperature gradientsinducing significant mass diffusion effects It can be seenthat the velocity and concentration profiles increase with anincrease in Sr while a rise in Sr causes a fall in the temperatureprofiles within the boundary layer These behaviors areevident from Figures 7(a)ndash7(c) The influence of viscousdissipation parameter that is Eckert number Ec on thevelocity profiles is depicted in Figure 8 The Eckert numberexpresses the relationship between the kinetic energy in theflow and the enthalpy It embodies the conversion of kinetic

6 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

S = 10 50 10 15

Velo

city

(U)

(a)

S = 10 50 10 15

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

(b)

0 02 04 06 08 1

0

05

1

y

S = 10 50 10 15

minus05

conc

entr

atio

n (C

)

(c)

Figure 6 (a) Velocity profile for different values of (120574 = 119878) with Gr = 2 Gc = 2 119896 = 1 Sc = 022 1198871= 001 120582 = 05119872 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of (120574 = 119878) with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1

119872 = 1 119896 = 1 Sr = 02 Du = 02 Ec = 001 Pr = 071 and 119905 = 002 (c) Concentration profile for different values of (120574 = 119878) with Gr = 2Gc = 2 Sc = 022 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 3 Du = 01 Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

Sr = 166 250 368 689

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Sr = 166 250 368 689

(b)

0 02 04 06 08 10

05

1

Con

cent

ratio

n (C

)

y

Sr = 166 250 368 689

(c)

Figure 7 (a) Velocity profile for different values of Sr with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1119872 = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Sr with Gr = 5 Gc = 2 120574 = 01 Sc = 022 1198871= 01 120582 = 1

119896 = 1119872 = 1 Du = 02 Ec = 001 Pr = 071 and 119905 = 002 (c) Concentration profile for different values of Sr with Gr = 2 Gc = 2 Sc = 0221198871= 001 120582 = 05119872 = 1 119896 = 1 120574 = 3 Du = 01 Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 7

0 02 04 06 08 10

05

1

15

y

Ec = 10 20 30 40

Velo

city

(U)

Figure 8 Velocity profile for different values of Ec with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

119872 = 1 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

t = 002 004 006 008Pr = 071

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 071

(b)

0 02 04 06 08 10

05

1

Con

cent

ratio

n (C

)

y

t = 002 004 006 008Pr = 071

(c)

Figure 9 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

0 02 04 06 08 10

005

01

y

t = 002 004 006 008

Velo

city

(U) Pr = 70

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 70

(b)

0 02 04 06 08 10

05

1C

once

ntra

tion

(C)

y

t = 002 004 006 008Pr = 70

(c)

Figure 10 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

energy into internal energy by work done against the viscousfluid stress It is observed that greater viscous dissipative heatcauses an increase in the velocity profiles across the boundarylayer Figures 9(a)ndash10(c) describe the behavior for velocitytemperature and concentration profiles for different values ofnondimensional time 119905 for two working fluids air (Pr = 071)and water (Pr = 70) It is evident from these figures that astime increases the velocity temperature and concentrationincrease The consequence of temperature increase resultsfrom increase in time since convection current becomesstronger and hence velocity and concentration increase withtime

Figures 11(a) and 11(b) graphically show the influence ofDufour number Du on the velocity and temperature profilesrespectively The Dufour number signifies the contributionof the concentration gradients to the thermal energy flux inthe flow It is seen that as Du increases there is monotonic

8 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

Du = 10 20 30 40

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Du = 10 20 30 40

(b)

Figure 11 (a) Velocity profile for different values of Du with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1 Sr = 1 119872 = 1

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Du with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1119872 = 1

119896 = 1 Sr = 02 120574 = 01 Ec = 001 Pr = 071 and 119905 = 002

increase in the velocity and temperature profiles For differentvalues of joule-heating parameter 119887

1and thermal conduc-

tivity parameter 120582 the temperature profiles are plotted inFigures 12 and 13 respectively It is observed that increas-ing the joule-heating parameter and thermal conductivityparameter produces significant increase in the thermal con-duction of the fluid which is physically true because as thethermal conductivity increases the temperature within thefluid increases Figure 14 describes the behavior of variousvalues of Scmidt number Sc on the concentration profilesThe Schmidt number characterizes the ratio of thicknesses ofviscous to the mass diffusivityThe Scmidt number quantifiesthe relative effectiveness ofmomentumandmass transport bydiffusion in the velocity and concentration boundary layers Itis observed that increase in the values of Sc causes the speciesconcentration and its boundary layer thickness to decreasesignificantly

The effects of various governing parameters on skin-friction coefficient 119862

119891 Nusselt number Nu and Sherwood

number Sh are shown in Tables 1 and 2 In order to highlightthe contributions of each parameter one parameter is variedwhile the rest take default fixed values It is observed fromTable 1 that an increase in any of the parameters 119872 and 120574

causes reduction in the skin-friction while increasing any ofthe parameters Gr Gc 119896 and 120582 resulted in correspondingincrease in the skin-friction coefficient It is also seen thatas 120582 and 120574 increase there is a rise in Nusselt number andSherwood number respectively From Table 2 it is observedthat an increase in Ec 119887

1 and Du leads to a rise in the skin-

friction coefficient Nusselt number and Sherwood numberrespectively while an increase in Pr leads to a fall in the skin-friction coefficient Nusselt number and Sherwood numberrespectively It is also seen that as Sr and Sc increase there isa fall in the skin-friction coefficient and a rise in Nusselt andSherwood numbers respectively

5 Conclusions

The present paper analyzes the influence of Soret and Dufoureffects on MHD free convective heat and mass transferflow over a vertical channel with constant suction and vis-cous dissipation The resulting partial differential equations

0 02 04 06 08 10

05

1

15

Tem

pera

ture

(T)

y

b1 = 10 50 10 15

Figure 12 Temperature profile for different values of 1198871with Gr = 5

Gc = 2 Sc = 022 120574 = 01 120582 = 1119872 = 1 119896 = 1 Sr = 02 Du = 02Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

V = 05 10 20 30

Figure 13 Temperature profile for different values of (120582 = V) withGr = 5 Gc = 2 Sc = 022 119887

1= 01 120574 = 01119872 = 1 119896 = 1 Sr = 02

Du = 02 Ec = 001 Pr = 071 and 119905 = 002

0 02 04 06 08 10

05

1

Con

cent

artio

n (C

)

y

Sc = 022 066 094 262

Figure 14 Concentration profile for different values of Sc with Gr =2 Gc = 2 120574 = 01 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 9

Table 1 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various values of physical parameters

(GrGc119872 119896 120582 and 120574)

Gr Gc 119872 119896 120582 120574 119862119891

Nu Sh20 20 10 05 10 2 02611 33328 0617140 20 10 05 10 2 04720 33451 0622120 40 10 05 10 2 03119 33386 0619520 20 30 05 10 2 02353 33317 0616720 20 10 10 10 2 02757 33334 0617420 20 10 05 20 2 02897 81676 3433420 20 10 05 10 30 02072 45600 13689With fixed values of (Pr = 071 Ec = 001 Sc = 06 Du = 02 1198871 = 10 Sr = 10 and 119905 = 02)

Table 2 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various physical parameters (Pr 119887

1 Ec Sr

Du and Sc)

Pr 1198871

Ec Sr Du Sc 119862119891

Nu Sh071 10 20 10 02 022 04038 20257 1434430 10 20 10 02 022 02898 06168 14296071 20 20 10 02 022 04041 20311 14354071 10 30 10 02 022 04053 20729 14713071 10 20 166 02 022 04019 20523 15730071 10 20 10 10 022 04133 32477 16418071 10 20 10 02 094 03519 20983 17267With fixed values of (Gr = 20 Gc = 20119872 = 10 120574 = 05 120582 = 10 119896 = 05 and 119905 = 02)

are nondimensionalised simplified and solved by implicitfinite difference method of Crank-Nicolson type From thepresent numerical study the following conclusions can bedrawn

(1) Velocity profiles increased due to increase in thermalGrashof number solutal Grashof number porousparameter Eckert number Soret number Dufournumber and dimensionless time while it decreaseddue to increase in magnetic parameter and suctionparameter

(2) An increase in temperature profiles is a function ofan increase in thermal conductivity suction parame-ter Eckert number joule-heating parameter Dufournumber and dimensionless time while it decreaseddue to increase in Soret number

(3) Concentration profiles decreased due to increasesin Schmidt number and suction parameter whileit increased due to increase in Soret number anddimensionless time

(4) There is a rise in the skin-friction coefficient Nusseltnumber and Sherwood number due to increasein Eckert number thermal conductivity parameterjoule-heating parameter and Dufour number whilea fall is observed in skin-friction coefficient withincrease in Soret number and Prandtl number

Nomenclature

119862 Concentration119862119901 Specific heat at constant pressure

119863 Mass diffusivity119863119872 Coefficient of Mass diffusivity

119863119879 Coefficient of temperature diffusivity

Du Dufour number119892 Acceleration due to gravityGr Grashof numberGc Solutal Grashof number119896 Permeability of the porous mediumNu Nusselt numberPr Prandtl numberSc Schmidt numberSr Soret number119879 Temperature119906 V Velocities in the 119909- and 119910-direction

respectively119909 119910 Cartesian coordinates along the plate

and normal to it respectively1198610 Magnetic field of constant strength

Ec Eckert number119872 Magnetic field parameter1198871 Joule-heating parameter

Greek Letters

120573

lowast Coefficient of expansion with concentration120573 Coefficient of thermal expansion120588 Density of fluid

10 International Scholarly Research Notices

120590 Electric conductivity120579 Dimensionless temperature] Kinematic viscosity120574 Suction parameter120582 Variable thermal conductivity

Subscripts

119908 Condition at wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D A Neild and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006

[2] D B Ingham and I PopTransport Phenomena in PorousMediaElsevier Oxford UK 2005

[3] G Palani and U Srikanth ldquoMHD flow past a semi-infinitevertical plate with mass transferrdquo Nonlinear Analysis Modellingand Control vol 14 no 3 pp 345ndash356 2009

[4] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoThe Canadian Journal of Chemical Engineering vol 88 no 6pp 983ndash990 2010

[5] H M Duwairi ldquoViscous and Joule heating effects on forcedconvection flow from radiate isothermal porous surfacesrdquoInternational Journal of Numerical Methods for Heat and FluidFlow vol 15 no 5 pp 429ndash440 2005

[6] M A Alim M D Alam and A Mamun ldquoJoule heating effecton the coupling of conduction with magneto hydrodynamicfree convection flow from a vertical platerdquo Non-Linear AnalysisModeling and Control vol 12 no 3 pp 307ndash316 2007

[7] I J Uwanta ldquoEffects of chemical reaction and radiation onHeatand Mass Transfer past a semi-infinite vertical porous platewith constant mass flux and dissipationrdquo European Journal ofScientific Research vol 87 no 2 pp 190ndash200 2012

[8] M A Mansour N F El-Anssary and A M Aly ldquoEffect ofchemical reaction and viscous dissipation on MHD naturalconvection flows saturated in porous media with suction orinjectionrdquo International Journal of Applied Mathematics andMechanics vol 4 no 2 pp 60ndash70 2008

[9] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[10] K Govardhan N Kishan and B Balaswamy ldquoEffect of viscousdissipation and radiation on MHD gas flow and heat and masstransfer over a stretching surface with a uniform free streamrdquoJournal of Engineering Physics andThermophysics vol 85 no 4pp 909ndash916 2012

[11] M A Sattar ldquoFree and forced convection boundary layer flowthrough a porous medium with large suctionrdquo InternationalJournal of Energy Research vol 17 no 1 pp 1ndash7 1993

[12] F Mohammad O Masatiro S Abdus and A MohamudldquoSimilarity solutions for MHD flow through vertical porous

plate with suctionrdquo Journal of Computational and AppliedMechanics vol 6 no 1 pp 15ndash25 2005

[13] S Jai ldquoViscous dissipation and chemical reaction effects on flowpast a stretching surface in a porous mediumrdquo Advance Theoryof Applied Mechanics vol 5 pp 323ndash331 2012

[14] S M Ibrahim ldquoEffects of mass transfer radiation Joule heatingand viscous dissipation on steady MHDmarangoni convectionflow over a flat surface with suction and injectionrdquo InternationalJournal of EngineeringMathematics vol 2013 Article ID 9038189 pages 2013

[15] T S Khaleque and M A Samad ldquoEffects of radiation heatgeneration and viscous dissipation on MHD free convectionflow along a stretching sheetrdquo Research Journal of AppliedSciences Engineering and Technology vol 2 no 4 pp 368ndash3772010

[16] E R G Eckert and R M Drake Analysis of Heat and MassTransfer MC Graw-Hill New York NY USA 1974

[17] A Postelnicu ldquoHeat and mass transfer by natural convection ata stagnation point in a porous medium considering Soret andDufour effectsrdquoHeat andMass Transfer vol 46 no 8-9 pp 831ndash840 2010

[18] A J Chamkha and S M M El-Kabeir ldquoUnsteady Heat andMass Transfer by MHD mixed convection flow from a rotatingvertical cone with chemical reaction and Soret and Dufoureffectsrdquo Chemical Engineering Communications vol 200 no 9pp 1220ndash1236 2013

[19] H Usman and I J Uwanta ldquoEffect of thermal conductivity onMHDHeat andMass transfer flow past an infinite vertical platewith Soret and Dufour effectsrdquo American Journal of AppliedMathematics vol 1 no 3 pp 28ndash38 2013

[20] I J Uwanta K K Asogwa and U A Ali ldquoMHD fluid flowover a vertical plate withDufour and Soret effectsrdquo InternationalJournal of Computer Applications vol 45 no 2 pp 8ndash16 2008

[21] K Sarada andB Shankar ldquoThe effects of Soret andDufour on anunsteadyMHD free convection flow past a vertical porous platein the presence of suction or injectionrdquo International Journal ofEngineering and Science vol 2 no 7 pp 13ndash25 2013

[22] N Nalinakshi P A Dinesh and D V Chandrashekar ldquoSoretand dufour effects on mixed convection heat and mass transferwith variable fluid propertiesrdquo International Journal of Mathe-matical Archive vol 4 no 11 pp 203ndash215 2013

[23] M J Subhakar and K Gangadhar ldquoSoret and Dufour effectson MHD free convection heat and mass transfer flow overa stretching vertical plate with suction and heat sourcesinkrdquoInternational Journal ofModern Engineering Research vol 2 no5 pp 3458ndash3468 2012

[24] D Srinivasacharya and C R Reddy ldquoSoret and Dufour effectson mixed convection in a non-DARcy porous medium satu-rated with micropolar fluidrdquo Nonlinear Analysis Modelling andControl vol 16 no 1 pp 100ndash115 2011

[25] N Sivaraman P K Sivagnana and R Kandasamy ldquoSoret andDufour effects on MHD free convective heat and mass transferwith thermopheresis and chemical reaction over a porousstretching surface group theory transformationrdquo Journal ofApplied Mechanics and Technical Physics vol 53 no 6 pp 871ndash879 2012

[26] D Srinivasacharya and M Upendar ldquoSoret and Dufour effectson MHD mixed convection heat and mass transfer in amicropolar fluidrdquo Central European Journal Engineering vol 3no 4 pp 679ndash689 2013

[27] S MM El-Kabeir MModathar and A M Rashed ldquoSoret anddufour effects on heat and mass transfer from a continuously

International Scholarly Research Notices 11

moving plate embedded in porous media with temperaturedependent viscosity and thermal conductivityrdquo Journal of Mod-ern Methods in Numerical Mathematics vol 4 no 2 pp 10ndash222013

[28] P O Olanrewaju O J Fenuga J A Gbadeyan and T GOkedayo ldquoDufour and Soret effects on convection heat andmass transfer in an electrical conducting power law flow overa heated porous platerdquo International Journal for ComputationalMethods in Engineering Science andMechanics vol 14 no 1 pp32ndash39 2013

[29] B Carnahan H A Luther and J O Wilkes Applied NumericalMethods John Wiley amp Sons New York NY USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

S = 10 50 10 15

Velo

city

(U)

(a)

S = 10 50 10 15

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

(b)

0 02 04 06 08 1

0

05

1

y

S = 10 50 10 15

minus05

conc

entr

atio

n (C

)

(c)

Figure 6 (a) Velocity profile for different values of (120574 = 119878) with Gr = 2 Gc = 2 119896 = 1 Sc = 022 1198871= 001 120582 = 05119872 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of (120574 = 119878) with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1

119872 = 1 119896 = 1 Sr = 02 Du = 02 Ec = 001 Pr = 071 and 119905 = 002 (c) Concentration profile for different values of (120574 = 119878) with Gr = 2Gc = 2 Sc = 022 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 3 Du = 01 Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

Sr = 166 250 368 689

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Sr = 166 250 368 689

(b)

0 02 04 06 08 10

05

1

Con

cent

ratio

n (C

)

y

Sr = 166 250 368 689

(c)

Figure 7 (a) Velocity profile for different values of Sr with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1119872 = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Sr with Gr = 5 Gc = 2 120574 = 01 Sc = 022 1198871= 01 120582 = 1

119896 = 1119872 = 1 Du = 02 Ec = 001 Pr = 071 and 119905 = 002 (c) Concentration profile for different values of Sr with Gr = 2 Gc = 2 Sc = 0221198871= 001 120582 = 05119872 = 1 119896 = 1 120574 = 3 Du = 01 Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 7

0 02 04 06 08 10

05

1

15

y

Ec = 10 20 30 40

Velo

city

(U)

Figure 8 Velocity profile for different values of Ec with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

119872 = 1 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

t = 002 004 006 008Pr = 071

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 071

(b)

0 02 04 06 08 10

05

1

Con

cent

ratio

n (C

)

y

t = 002 004 006 008Pr = 071

(c)

Figure 9 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

0 02 04 06 08 10

005

01

y

t = 002 004 006 008

Velo

city

(U) Pr = 70

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 70

(b)

0 02 04 06 08 10

05

1C

once

ntra

tion

(C)

y

t = 002 004 006 008Pr = 70

(c)

Figure 10 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

energy into internal energy by work done against the viscousfluid stress It is observed that greater viscous dissipative heatcauses an increase in the velocity profiles across the boundarylayer Figures 9(a)ndash10(c) describe the behavior for velocitytemperature and concentration profiles for different values ofnondimensional time 119905 for two working fluids air (Pr = 071)and water (Pr = 70) It is evident from these figures that astime increases the velocity temperature and concentrationincrease The consequence of temperature increase resultsfrom increase in time since convection current becomesstronger and hence velocity and concentration increase withtime

Figures 11(a) and 11(b) graphically show the influence ofDufour number Du on the velocity and temperature profilesrespectively The Dufour number signifies the contributionof the concentration gradients to the thermal energy flux inthe flow It is seen that as Du increases there is monotonic

8 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

Du = 10 20 30 40

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Du = 10 20 30 40

(b)

Figure 11 (a) Velocity profile for different values of Du with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1 Sr = 1 119872 = 1

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Du with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1119872 = 1

119896 = 1 Sr = 02 120574 = 01 Ec = 001 Pr = 071 and 119905 = 002

increase in the velocity and temperature profiles For differentvalues of joule-heating parameter 119887

1and thermal conduc-

tivity parameter 120582 the temperature profiles are plotted inFigures 12 and 13 respectively It is observed that increas-ing the joule-heating parameter and thermal conductivityparameter produces significant increase in the thermal con-duction of the fluid which is physically true because as thethermal conductivity increases the temperature within thefluid increases Figure 14 describes the behavior of variousvalues of Scmidt number Sc on the concentration profilesThe Schmidt number characterizes the ratio of thicknesses ofviscous to the mass diffusivityThe Scmidt number quantifiesthe relative effectiveness ofmomentumandmass transport bydiffusion in the velocity and concentration boundary layers Itis observed that increase in the values of Sc causes the speciesconcentration and its boundary layer thickness to decreasesignificantly

The effects of various governing parameters on skin-friction coefficient 119862

119891 Nusselt number Nu and Sherwood

number Sh are shown in Tables 1 and 2 In order to highlightthe contributions of each parameter one parameter is variedwhile the rest take default fixed values It is observed fromTable 1 that an increase in any of the parameters 119872 and 120574

causes reduction in the skin-friction while increasing any ofthe parameters Gr Gc 119896 and 120582 resulted in correspondingincrease in the skin-friction coefficient It is also seen thatas 120582 and 120574 increase there is a rise in Nusselt number andSherwood number respectively From Table 2 it is observedthat an increase in Ec 119887

1 and Du leads to a rise in the skin-

friction coefficient Nusselt number and Sherwood numberrespectively while an increase in Pr leads to a fall in the skin-friction coefficient Nusselt number and Sherwood numberrespectively It is also seen that as Sr and Sc increase there isa fall in the skin-friction coefficient and a rise in Nusselt andSherwood numbers respectively

5 Conclusions

The present paper analyzes the influence of Soret and Dufoureffects on MHD free convective heat and mass transferflow over a vertical channel with constant suction and vis-cous dissipation The resulting partial differential equations

0 02 04 06 08 10

05

1

15

Tem

pera

ture

(T)

y

b1 = 10 50 10 15

Figure 12 Temperature profile for different values of 1198871with Gr = 5

Gc = 2 Sc = 022 120574 = 01 120582 = 1119872 = 1 119896 = 1 Sr = 02 Du = 02Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

V = 05 10 20 30

Figure 13 Temperature profile for different values of (120582 = V) withGr = 5 Gc = 2 Sc = 022 119887

1= 01 120574 = 01119872 = 1 119896 = 1 Sr = 02

Du = 02 Ec = 001 Pr = 071 and 119905 = 002

0 02 04 06 08 10

05

1

Con

cent

artio

n (C

)

y

Sc = 022 066 094 262

Figure 14 Concentration profile for different values of Sc with Gr =2 Gc = 2 120574 = 01 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 9

Table 1 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various values of physical parameters

(GrGc119872 119896 120582 and 120574)

Gr Gc 119872 119896 120582 120574 119862119891

Nu Sh20 20 10 05 10 2 02611 33328 0617140 20 10 05 10 2 04720 33451 0622120 40 10 05 10 2 03119 33386 0619520 20 30 05 10 2 02353 33317 0616720 20 10 10 10 2 02757 33334 0617420 20 10 05 20 2 02897 81676 3433420 20 10 05 10 30 02072 45600 13689With fixed values of (Pr = 071 Ec = 001 Sc = 06 Du = 02 1198871 = 10 Sr = 10 and 119905 = 02)

Table 2 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various physical parameters (Pr 119887

1 Ec Sr

Du and Sc)

Pr 1198871

Ec Sr Du Sc 119862119891

Nu Sh071 10 20 10 02 022 04038 20257 1434430 10 20 10 02 022 02898 06168 14296071 20 20 10 02 022 04041 20311 14354071 10 30 10 02 022 04053 20729 14713071 10 20 166 02 022 04019 20523 15730071 10 20 10 10 022 04133 32477 16418071 10 20 10 02 094 03519 20983 17267With fixed values of (Gr = 20 Gc = 20119872 = 10 120574 = 05 120582 = 10 119896 = 05 and 119905 = 02)

are nondimensionalised simplified and solved by implicitfinite difference method of Crank-Nicolson type From thepresent numerical study the following conclusions can bedrawn

(1) Velocity profiles increased due to increase in thermalGrashof number solutal Grashof number porousparameter Eckert number Soret number Dufournumber and dimensionless time while it decreaseddue to increase in magnetic parameter and suctionparameter

(2) An increase in temperature profiles is a function ofan increase in thermal conductivity suction parame-ter Eckert number joule-heating parameter Dufournumber and dimensionless time while it decreaseddue to increase in Soret number

(3) Concentration profiles decreased due to increasesin Schmidt number and suction parameter whileit increased due to increase in Soret number anddimensionless time

(4) There is a rise in the skin-friction coefficient Nusseltnumber and Sherwood number due to increasein Eckert number thermal conductivity parameterjoule-heating parameter and Dufour number whilea fall is observed in skin-friction coefficient withincrease in Soret number and Prandtl number

Nomenclature

119862 Concentration119862119901 Specific heat at constant pressure

119863 Mass diffusivity119863119872 Coefficient of Mass diffusivity

119863119879 Coefficient of temperature diffusivity

Du Dufour number119892 Acceleration due to gravityGr Grashof numberGc Solutal Grashof number119896 Permeability of the porous mediumNu Nusselt numberPr Prandtl numberSc Schmidt numberSr Soret number119879 Temperature119906 V Velocities in the 119909- and 119910-direction

respectively119909 119910 Cartesian coordinates along the plate

and normal to it respectively1198610 Magnetic field of constant strength

Ec Eckert number119872 Magnetic field parameter1198871 Joule-heating parameter

Greek Letters

120573

lowast Coefficient of expansion with concentration120573 Coefficient of thermal expansion120588 Density of fluid

10 International Scholarly Research Notices

120590 Electric conductivity120579 Dimensionless temperature] Kinematic viscosity120574 Suction parameter120582 Variable thermal conductivity

Subscripts

119908 Condition at wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D A Neild and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006

[2] D B Ingham and I PopTransport Phenomena in PorousMediaElsevier Oxford UK 2005

[3] G Palani and U Srikanth ldquoMHD flow past a semi-infinitevertical plate with mass transferrdquo Nonlinear Analysis Modellingand Control vol 14 no 3 pp 345ndash356 2009

[4] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoThe Canadian Journal of Chemical Engineering vol 88 no 6pp 983ndash990 2010

[5] H M Duwairi ldquoViscous and Joule heating effects on forcedconvection flow from radiate isothermal porous surfacesrdquoInternational Journal of Numerical Methods for Heat and FluidFlow vol 15 no 5 pp 429ndash440 2005

[6] M A Alim M D Alam and A Mamun ldquoJoule heating effecton the coupling of conduction with magneto hydrodynamicfree convection flow from a vertical platerdquo Non-Linear AnalysisModeling and Control vol 12 no 3 pp 307ndash316 2007

[7] I J Uwanta ldquoEffects of chemical reaction and radiation onHeatand Mass Transfer past a semi-infinite vertical porous platewith constant mass flux and dissipationrdquo European Journal ofScientific Research vol 87 no 2 pp 190ndash200 2012

[8] M A Mansour N F El-Anssary and A M Aly ldquoEffect ofchemical reaction and viscous dissipation on MHD naturalconvection flows saturated in porous media with suction orinjectionrdquo International Journal of Applied Mathematics andMechanics vol 4 no 2 pp 60ndash70 2008

[9] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[10] K Govardhan N Kishan and B Balaswamy ldquoEffect of viscousdissipation and radiation on MHD gas flow and heat and masstransfer over a stretching surface with a uniform free streamrdquoJournal of Engineering Physics andThermophysics vol 85 no 4pp 909ndash916 2012

[11] M A Sattar ldquoFree and forced convection boundary layer flowthrough a porous medium with large suctionrdquo InternationalJournal of Energy Research vol 17 no 1 pp 1ndash7 1993

[12] F Mohammad O Masatiro S Abdus and A MohamudldquoSimilarity solutions for MHD flow through vertical porous

plate with suctionrdquo Journal of Computational and AppliedMechanics vol 6 no 1 pp 15ndash25 2005

[13] S Jai ldquoViscous dissipation and chemical reaction effects on flowpast a stretching surface in a porous mediumrdquo Advance Theoryof Applied Mechanics vol 5 pp 323ndash331 2012

[14] S M Ibrahim ldquoEffects of mass transfer radiation Joule heatingand viscous dissipation on steady MHDmarangoni convectionflow over a flat surface with suction and injectionrdquo InternationalJournal of EngineeringMathematics vol 2013 Article ID 9038189 pages 2013

[15] T S Khaleque and M A Samad ldquoEffects of radiation heatgeneration and viscous dissipation on MHD free convectionflow along a stretching sheetrdquo Research Journal of AppliedSciences Engineering and Technology vol 2 no 4 pp 368ndash3772010

[16] E R G Eckert and R M Drake Analysis of Heat and MassTransfer MC Graw-Hill New York NY USA 1974

[17] A Postelnicu ldquoHeat and mass transfer by natural convection ata stagnation point in a porous medium considering Soret andDufour effectsrdquoHeat andMass Transfer vol 46 no 8-9 pp 831ndash840 2010

[18] A J Chamkha and S M M El-Kabeir ldquoUnsteady Heat andMass Transfer by MHD mixed convection flow from a rotatingvertical cone with chemical reaction and Soret and Dufoureffectsrdquo Chemical Engineering Communications vol 200 no 9pp 1220ndash1236 2013

[19] H Usman and I J Uwanta ldquoEffect of thermal conductivity onMHDHeat andMass transfer flow past an infinite vertical platewith Soret and Dufour effectsrdquo American Journal of AppliedMathematics vol 1 no 3 pp 28ndash38 2013

[20] I J Uwanta K K Asogwa and U A Ali ldquoMHD fluid flowover a vertical plate withDufour and Soret effectsrdquo InternationalJournal of Computer Applications vol 45 no 2 pp 8ndash16 2008

[21] K Sarada andB Shankar ldquoThe effects of Soret andDufour on anunsteadyMHD free convection flow past a vertical porous platein the presence of suction or injectionrdquo International Journal ofEngineering and Science vol 2 no 7 pp 13ndash25 2013

[22] N Nalinakshi P A Dinesh and D V Chandrashekar ldquoSoretand dufour effects on mixed convection heat and mass transferwith variable fluid propertiesrdquo International Journal of Mathe-matical Archive vol 4 no 11 pp 203ndash215 2013

[23] M J Subhakar and K Gangadhar ldquoSoret and Dufour effectson MHD free convection heat and mass transfer flow overa stretching vertical plate with suction and heat sourcesinkrdquoInternational Journal ofModern Engineering Research vol 2 no5 pp 3458ndash3468 2012

[24] D Srinivasacharya and C R Reddy ldquoSoret and Dufour effectson mixed convection in a non-DARcy porous medium satu-rated with micropolar fluidrdquo Nonlinear Analysis Modelling andControl vol 16 no 1 pp 100ndash115 2011

[25] N Sivaraman P K Sivagnana and R Kandasamy ldquoSoret andDufour effects on MHD free convective heat and mass transferwith thermopheresis and chemical reaction over a porousstretching surface group theory transformationrdquo Journal ofApplied Mechanics and Technical Physics vol 53 no 6 pp 871ndash879 2012

[26] D Srinivasacharya and M Upendar ldquoSoret and Dufour effectson MHD mixed convection heat and mass transfer in amicropolar fluidrdquo Central European Journal Engineering vol 3no 4 pp 679ndash689 2013

[27] S MM El-Kabeir MModathar and A M Rashed ldquoSoret anddufour effects on heat and mass transfer from a continuously

International Scholarly Research Notices 11

moving plate embedded in porous media with temperaturedependent viscosity and thermal conductivityrdquo Journal of Mod-ern Methods in Numerical Mathematics vol 4 no 2 pp 10ndash222013

[28] P O Olanrewaju O J Fenuga J A Gbadeyan and T GOkedayo ldquoDufour and Soret effects on convection heat andmass transfer in an electrical conducting power law flow overa heated porous platerdquo International Journal for ComputationalMethods in Engineering Science andMechanics vol 14 no 1 pp32ndash39 2013

[29] B Carnahan H A Luther and J O Wilkes Applied NumericalMethods John Wiley amp Sons New York NY USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 7

0 02 04 06 08 10

05

1

15

y

Ec = 10 20 30 40

Velo

city

(U)

Figure 8 Velocity profile for different values of Ec with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

119872 = 1 Pr = 071 and 119905 = 02

0 02 04 06 08 10

01

02

y

t = 002 004 006 008Pr = 071

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 071

(b)

0 02 04 06 08 10

05

1

Con

cent

ratio

n (C

)

y

t = 002 004 006 008Pr = 071

(c)

Figure 9 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

0 02 04 06 08 10

005

01

y

t = 002 004 006 008

Velo

city

(U) Pr = 70

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

t = 002 004 006 008Pr = 70

(b)

0 02 04 06 08 10

05

1C

once

ntra

tion

(C)

y

t = 002 004 006 008Pr = 70

(c)

Figure 10 (a) Velocity profile for different values of 119905 with Gr = 2Gc = 2 120574 = 3 Sc = 022 119887

1= 001 120582 = 05 119896 = 1 Sr = 1 Du = 01

Ec = 001 and 119872 = 1 (b) Temperature profile for different valuesof 119905 with Gr = 5 Gc = 2 Sc = 022 119887

1= 01 120582 = 1119872 = 1 119896 = 1

Sr = 02 Du = 02 Ec = 001 and 120574 = 01 (c) Concentration profilefor different values of 119905 with Gr = 2 Gc = 2 Sc = 022 119887

1= 001

120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01 Ec = 001 and 120574 = 01

energy into internal energy by work done against the viscousfluid stress It is observed that greater viscous dissipative heatcauses an increase in the velocity profiles across the boundarylayer Figures 9(a)ndash10(c) describe the behavior for velocitytemperature and concentration profiles for different values ofnondimensional time 119905 for two working fluids air (Pr = 071)and water (Pr = 70) It is evident from these figures that astime increases the velocity temperature and concentrationincrease The consequence of temperature increase resultsfrom increase in time since convection current becomesstronger and hence velocity and concentration increase withtime

Figures 11(a) and 11(b) graphically show the influence ofDufour number Du on the velocity and temperature profilesrespectively The Dufour number signifies the contributionof the concentration gradients to the thermal energy flux inthe flow It is seen that as Du increases there is monotonic

8 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

Du = 10 20 30 40

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Du = 10 20 30 40

(b)

Figure 11 (a) Velocity profile for different values of Du with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1 Sr = 1 119872 = 1

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Du with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1119872 = 1

119896 = 1 Sr = 02 120574 = 01 Ec = 001 Pr = 071 and 119905 = 002

increase in the velocity and temperature profiles For differentvalues of joule-heating parameter 119887

1and thermal conduc-

tivity parameter 120582 the temperature profiles are plotted inFigures 12 and 13 respectively It is observed that increas-ing the joule-heating parameter and thermal conductivityparameter produces significant increase in the thermal con-duction of the fluid which is physically true because as thethermal conductivity increases the temperature within thefluid increases Figure 14 describes the behavior of variousvalues of Scmidt number Sc on the concentration profilesThe Schmidt number characterizes the ratio of thicknesses ofviscous to the mass diffusivityThe Scmidt number quantifiesthe relative effectiveness ofmomentumandmass transport bydiffusion in the velocity and concentration boundary layers Itis observed that increase in the values of Sc causes the speciesconcentration and its boundary layer thickness to decreasesignificantly

The effects of various governing parameters on skin-friction coefficient 119862

119891 Nusselt number Nu and Sherwood

number Sh are shown in Tables 1 and 2 In order to highlightthe contributions of each parameter one parameter is variedwhile the rest take default fixed values It is observed fromTable 1 that an increase in any of the parameters 119872 and 120574

causes reduction in the skin-friction while increasing any ofthe parameters Gr Gc 119896 and 120582 resulted in correspondingincrease in the skin-friction coefficient It is also seen thatas 120582 and 120574 increase there is a rise in Nusselt number andSherwood number respectively From Table 2 it is observedthat an increase in Ec 119887

1 and Du leads to a rise in the skin-

friction coefficient Nusselt number and Sherwood numberrespectively while an increase in Pr leads to a fall in the skin-friction coefficient Nusselt number and Sherwood numberrespectively It is also seen that as Sr and Sc increase there isa fall in the skin-friction coefficient and a rise in Nusselt andSherwood numbers respectively

5 Conclusions

The present paper analyzes the influence of Soret and Dufoureffects on MHD free convective heat and mass transferflow over a vertical channel with constant suction and vis-cous dissipation The resulting partial differential equations

0 02 04 06 08 10

05

1

15

Tem

pera

ture

(T)

y

b1 = 10 50 10 15

Figure 12 Temperature profile for different values of 1198871with Gr = 5

Gc = 2 Sc = 022 120574 = 01 120582 = 1119872 = 1 119896 = 1 Sr = 02 Du = 02Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

V = 05 10 20 30

Figure 13 Temperature profile for different values of (120582 = V) withGr = 5 Gc = 2 Sc = 022 119887

1= 01 120574 = 01119872 = 1 119896 = 1 Sr = 02

Du = 02 Ec = 001 Pr = 071 and 119905 = 002

0 02 04 06 08 10

05

1

Con

cent

artio

n (C

)

y

Sc = 022 066 094 262

Figure 14 Concentration profile for different values of Sc with Gr =2 Gc = 2 120574 = 01 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 9

Table 1 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various values of physical parameters

(GrGc119872 119896 120582 and 120574)

Gr Gc 119872 119896 120582 120574 119862119891

Nu Sh20 20 10 05 10 2 02611 33328 0617140 20 10 05 10 2 04720 33451 0622120 40 10 05 10 2 03119 33386 0619520 20 30 05 10 2 02353 33317 0616720 20 10 10 10 2 02757 33334 0617420 20 10 05 20 2 02897 81676 3433420 20 10 05 10 30 02072 45600 13689With fixed values of (Pr = 071 Ec = 001 Sc = 06 Du = 02 1198871 = 10 Sr = 10 and 119905 = 02)

Table 2 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various physical parameters (Pr 119887

1 Ec Sr

Du and Sc)

Pr 1198871

Ec Sr Du Sc 119862119891

Nu Sh071 10 20 10 02 022 04038 20257 1434430 10 20 10 02 022 02898 06168 14296071 20 20 10 02 022 04041 20311 14354071 10 30 10 02 022 04053 20729 14713071 10 20 166 02 022 04019 20523 15730071 10 20 10 10 022 04133 32477 16418071 10 20 10 02 094 03519 20983 17267With fixed values of (Gr = 20 Gc = 20119872 = 10 120574 = 05 120582 = 10 119896 = 05 and 119905 = 02)

are nondimensionalised simplified and solved by implicitfinite difference method of Crank-Nicolson type From thepresent numerical study the following conclusions can bedrawn

(1) Velocity profiles increased due to increase in thermalGrashof number solutal Grashof number porousparameter Eckert number Soret number Dufournumber and dimensionless time while it decreaseddue to increase in magnetic parameter and suctionparameter

(2) An increase in temperature profiles is a function ofan increase in thermal conductivity suction parame-ter Eckert number joule-heating parameter Dufournumber and dimensionless time while it decreaseddue to increase in Soret number

(3) Concentration profiles decreased due to increasesin Schmidt number and suction parameter whileit increased due to increase in Soret number anddimensionless time

(4) There is a rise in the skin-friction coefficient Nusseltnumber and Sherwood number due to increasein Eckert number thermal conductivity parameterjoule-heating parameter and Dufour number whilea fall is observed in skin-friction coefficient withincrease in Soret number and Prandtl number

Nomenclature

119862 Concentration119862119901 Specific heat at constant pressure

119863 Mass diffusivity119863119872 Coefficient of Mass diffusivity

119863119879 Coefficient of temperature diffusivity

Du Dufour number119892 Acceleration due to gravityGr Grashof numberGc Solutal Grashof number119896 Permeability of the porous mediumNu Nusselt numberPr Prandtl numberSc Schmidt numberSr Soret number119879 Temperature119906 V Velocities in the 119909- and 119910-direction

respectively119909 119910 Cartesian coordinates along the plate

and normal to it respectively1198610 Magnetic field of constant strength

Ec Eckert number119872 Magnetic field parameter1198871 Joule-heating parameter

Greek Letters

120573

lowast Coefficient of expansion with concentration120573 Coefficient of thermal expansion120588 Density of fluid

10 International Scholarly Research Notices

120590 Electric conductivity120579 Dimensionless temperature] Kinematic viscosity120574 Suction parameter120582 Variable thermal conductivity

Subscripts

119908 Condition at wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D A Neild and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006

[2] D B Ingham and I PopTransport Phenomena in PorousMediaElsevier Oxford UK 2005

[3] G Palani and U Srikanth ldquoMHD flow past a semi-infinitevertical plate with mass transferrdquo Nonlinear Analysis Modellingand Control vol 14 no 3 pp 345ndash356 2009

[4] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoThe Canadian Journal of Chemical Engineering vol 88 no 6pp 983ndash990 2010

[5] H M Duwairi ldquoViscous and Joule heating effects on forcedconvection flow from radiate isothermal porous surfacesrdquoInternational Journal of Numerical Methods for Heat and FluidFlow vol 15 no 5 pp 429ndash440 2005

[6] M A Alim M D Alam and A Mamun ldquoJoule heating effecton the coupling of conduction with magneto hydrodynamicfree convection flow from a vertical platerdquo Non-Linear AnalysisModeling and Control vol 12 no 3 pp 307ndash316 2007

[7] I J Uwanta ldquoEffects of chemical reaction and radiation onHeatand Mass Transfer past a semi-infinite vertical porous platewith constant mass flux and dissipationrdquo European Journal ofScientific Research vol 87 no 2 pp 190ndash200 2012

[8] M A Mansour N F El-Anssary and A M Aly ldquoEffect ofchemical reaction and viscous dissipation on MHD naturalconvection flows saturated in porous media with suction orinjectionrdquo International Journal of Applied Mathematics andMechanics vol 4 no 2 pp 60ndash70 2008

[9] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[10] K Govardhan N Kishan and B Balaswamy ldquoEffect of viscousdissipation and radiation on MHD gas flow and heat and masstransfer over a stretching surface with a uniform free streamrdquoJournal of Engineering Physics andThermophysics vol 85 no 4pp 909ndash916 2012

[11] M A Sattar ldquoFree and forced convection boundary layer flowthrough a porous medium with large suctionrdquo InternationalJournal of Energy Research vol 17 no 1 pp 1ndash7 1993

[12] F Mohammad O Masatiro S Abdus and A MohamudldquoSimilarity solutions for MHD flow through vertical porous

plate with suctionrdquo Journal of Computational and AppliedMechanics vol 6 no 1 pp 15ndash25 2005

[13] S Jai ldquoViscous dissipation and chemical reaction effects on flowpast a stretching surface in a porous mediumrdquo Advance Theoryof Applied Mechanics vol 5 pp 323ndash331 2012

[14] S M Ibrahim ldquoEffects of mass transfer radiation Joule heatingand viscous dissipation on steady MHDmarangoni convectionflow over a flat surface with suction and injectionrdquo InternationalJournal of EngineeringMathematics vol 2013 Article ID 9038189 pages 2013

[15] T S Khaleque and M A Samad ldquoEffects of radiation heatgeneration and viscous dissipation on MHD free convectionflow along a stretching sheetrdquo Research Journal of AppliedSciences Engineering and Technology vol 2 no 4 pp 368ndash3772010

[16] E R G Eckert and R M Drake Analysis of Heat and MassTransfer MC Graw-Hill New York NY USA 1974

[17] A Postelnicu ldquoHeat and mass transfer by natural convection ata stagnation point in a porous medium considering Soret andDufour effectsrdquoHeat andMass Transfer vol 46 no 8-9 pp 831ndash840 2010

[18] A J Chamkha and S M M El-Kabeir ldquoUnsteady Heat andMass Transfer by MHD mixed convection flow from a rotatingvertical cone with chemical reaction and Soret and Dufoureffectsrdquo Chemical Engineering Communications vol 200 no 9pp 1220ndash1236 2013

[19] H Usman and I J Uwanta ldquoEffect of thermal conductivity onMHDHeat andMass transfer flow past an infinite vertical platewith Soret and Dufour effectsrdquo American Journal of AppliedMathematics vol 1 no 3 pp 28ndash38 2013

[20] I J Uwanta K K Asogwa and U A Ali ldquoMHD fluid flowover a vertical plate withDufour and Soret effectsrdquo InternationalJournal of Computer Applications vol 45 no 2 pp 8ndash16 2008

[21] K Sarada andB Shankar ldquoThe effects of Soret andDufour on anunsteadyMHD free convection flow past a vertical porous platein the presence of suction or injectionrdquo International Journal ofEngineering and Science vol 2 no 7 pp 13ndash25 2013

[22] N Nalinakshi P A Dinesh and D V Chandrashekar ldquoSoretand dufour effects on mixed convection heat and mass transferwith variable fluid propertiesrdquo International Journal of Mathe-matical Archive vol 4 no 11 pp 203ndash215 2013

[23] M J Subhakar and K Gangadhar ldquoSoret and Dufour effectson MHD free convection heat and mass transfer flow overa stretching vertical plate with suction and heat sourcesinkrdquoInternational Journal ofModern Engineering Research vol 2 no5 pp 3458ndash3468 2012

[24] D Srinivasacharya and C R Reddy ldquoSoret and Dufour effectson mixed convection in a non-DARcy porous medium satu-rated with micropolar fluidrdquo Nonlinear Analysis Modelling andControl vol 16 no 1 pp 100ndash115 2011

[25] N Sivaraman P K Sivagnana and R Kandasamy ldquoSoret andDufour effects on MHD free convective heat and mass transferwith thermopheresis and chemical reaction over a porousstretching surface group theory transformationrdquo Journal ofApplied Mechanics and Technical Physics vol 53 no 6 pp 871ndash879 2012

[26] D Srinivasacharya and M Upendar ldquoSoret and Dufour effectson MHD mixed convection heat and mass transfer in amicropolar fluidrdquo Central European Journal Engineering vol 3no 4 pp 679ndash689 2013

[27] S MM El-Kabeir MModathar and A M Rashed ldquoSoret anddufour effects on heat and mass transfer from a continuously

International Scholarly Research Notices 11

moving plate embedded in porous media with temperaturedependent viscosity and thermal conductivityrdquo Journal of Mod-ern Methods in Numerical Mathematics vol 4 no 2 pp 10ndash222013

[28] P O Olanrewaju O J Fenuga J A Gbadeyan and T GOkedayo ldquoDufour and Soret effects on convection heat andmass transfer in an electrical conducting power law flow overa heated porous platerdquo International Journal for ComputationalMethods in Engineering Science andMechanics vol 14 no 1 pp32ndash39 2013

[29] B Carnahan H A Luther and J O Wilkes Applied NumericalMethods John Wiley amp Sons New York NY USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Scholarly Research Notices

0 02 04 06 08 10

01

02

y

Du = 10 20 30 40

Velo

city

(U)

(a)

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

Du = 10 20 30 40

(b)

Figure 11 (a) Velocity profile for different values of Du with Gr = 2 Gc = 2 120574 = 3 Sc = 022 1198871= 001 120582 = 05 119896 = 1 Sr = 1 119872 = 1

Ec = 001 Pr = 071 and 119905 = 02 (b) Temperature profile for different values of Du with Gr = 5 Gc = 2 Sc = 022 1198871= 01 120582 = 1119872 = 1

119896 = 1 Sr = 02 120574 = 01 Ec = 001 Pr = 071 and 119905 = 002

increase in the velocity and temperature profiles For differentvalues of joule-heating parameter 119887

1and thermal conduc-

tivity parameter 120582 the temperature profiles are plotted inFigures 12 and 13 respectively It is observed that increas-ing the joule-heating parameter and thermal conductivityparameter produces significant increase in the thermal con-duction of the fluid which is physically true because as thethermal conductivity increases the temperature within thefluid increases Figure 14 describes the behavior of variousvalues of Scmidt number Sc on the concentration profilesThe Schmidt number characterizes the ratio of thicknesses ofviscous to the mass diffusivityThe Scmidt number quantifiesthe relative effectiveness ofmomentumandmass transport bydiffusion in the velocity and concentration boundary layers Itis observed that increase in the values of Sc causes the speciesconcentration and its boundary layer thickness to decreasesignificantly

The effects of various governing parameters on skin-friction coefficient 119862

119891 Nusselt number Nu and Sherwood

number Sh are shown in Tables 1 and 2 In order to highlightthe contributions of each parameter one parameter is variedwhile the rest take default fixed values It is observed fromTable 1 that an increase in any of the parameters 119872 and 120574

causes reduction in the skin-friction while increasing any ofthe parameters Gr Gc 119896 and 120582 resulted in correspondingincrease in the skin-friction coefficient It is also seen thatas 120582 and 120574 increase there is a rise in Nusselt number andSherwood number respectively From Table 2 it is observedthat an increase in Ec 119887

1 and Du leads to a rise in the skin-

friction coefficient Nusselt number and Sherwood numberrespectively while an increase in Pr leads to a fall in the skin-friction coefficient Nusselt number and Sherwood numberrespectively It is also seen that as Sr and Sc increase there isa fall in the skin-friction coefficient and a rise in Nusselt andSherwood numbers respectively

5 Conclusions

The present paper analyzes the influence of Soret and Dufoureffects on MHD free convective heat and mass transferflow over a vertical channel with constant suction and vis-cous dissipation The resulting partial differential equations

0 02 04 06 08 10

05

1

15

Tem

pera

ture

(T)

y

b1 = 10 50 10 15

Figure 12 Temperature profile for different values of 1198871with Gr = 5

Gc = 2 Sc = 022 120574 = 01 120582 = 1119872 = 1 119896 = 1 Sr = 02 Du = 02Ec = 001 Pr = 071 and 119905 = 02

0 02 04 06 08 10

05

1

Tem

pera

ture

(T)

y

V = 05 10 20 30

Figure 13 Temperature profile for different values of (120582 = V) withGr = 5 Gc = 2 Sc = 022 119887

1= 01 120574 = 01119872 = 1 119896 = 1 Sr = 02

Du = 02 Ec = 001 Pr = 071 and 119905 = 002

0 02 04 06 08 10

05

1

Con

cent

artio

n (C

)

y

Sc = 022 066 094 262

Figure 14 Concentration profile for different values of Sc with Gr =2 Gc = 2 120574 = 01 119887

1= 001 120582 = 05119872 = 1 119896 = 1 Sr = 1 Du = 01

Ec = 001 Pr = 071 and 119905 = 002

International Scholarly Research Notices 9

Table 1 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various values of physical parameters

(GrGc119872 119896 120582 and 120574)

Gr Gc 119872 119896 120582 120574 119862119891

Nu Sh20 20 10 05 10 2 02611 33328 0617140 20 10 05 10 2 04720 33451 0622120 40 10 05 10 2 03119 33386 0619520 20 30 05 10 2 02353 33317 0616720 20 10 10 10 2 02757 33334 0617420 20 10 05 20 2 02897 81676 3433420 20 10 05 10 30 02072 45600 13689With fixed values of (Pr = 071 Ec = 001 Sc = 06 Du = 02 1198871 = 10 Sr = 10 and 119905 = 02)

Table 2 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various physical parameters (Pr 119887

1 Ec Sr

Du and Sc)

Pr 1198871

Ec Sr Du Sc 119862119891

Nu Sh071 10 20 10 02 022 04038 20257 1434430 10 20 10 02 022 02898 06168 14296071 20 20 10 02 022 04041 20311 14354071 10 30 10 02 022 04053 20729 14713071 10 20 166 02 022 04019 20523 15730071 10 20 10 10 022 04133 32477 16418071 10 20 10 02 094 03519 20983 17267With fixed values of (Gr = 20 Gc = 20119872 = 10 120574 = 05 120582 = 10 119896 = 05 and 119905 = 02)

are nondimensionalised simplified and solved by implicitfinite difference method of Crank-Nicolson type From thepresent numerical study the following conclusions can bedrawn

(1) Velocity profiles increased due to increase in thermalGrashof number solutal Grashof number porousparameter Eckert number Soret number Dufournumber and dimensionless time while it decreaseddue to increase in magnetic parameter and suctionparameter

(2) An increase in temperature profiles is a function ofan increase in thermal conductivity suction parame-ter Eckert number joule-heating parameter Dufournumber and dimensionless time while it decreaseddue to increase in Soret number

(3) Concentration profiles decreased due to increasesin Schmidt number and suction parameter whileit increased due to increase in Soret number anddimensionless time

(4) There is a rise in the skin-friction coefficient Nusseltnumber and Sherwood number due to increasein Eckert number thermal conductivity parameterjoule-heating parameter and Dufour number whilea fall is observed in skin-friction coefficient withincrease in Soret number and Prandtl number

Nomenclature

119862 Concentration119862119901 Specific heat at constant pressure

119863 Mass diffusivity119863119872 Coefficient of Mass diffusivity

119863119879 Coefficient of temperature diffusivity

Du Dufour number119892 Acceleration due to gravityGr Grashof numberGc Solutal Grashof number119896 Permeability of the porous mediumNu Nusselt numberPr Prandtl numberSc Schmidt numberSr Soret number119879 Temperature119906 V Velocities in the 119909- and 119910-direction

respectively119909 119910 Cartesian coordinates along the plate

and normal to it respectively1198610 Magnetic field of constant strength

Ec Eckert number119872 Magnetic field parameter1198871 Joule-heating parameter

Greek Letters

120573

lowast Coefficient of expansion with concentration120573 Coefficient of thermal expansion120588 Density of fluid

10 International Scholarly Research Notices

120590 Electric conductivity120579 Dimensionless temperature] Kinematic viscosity120574 Suction parameter120582 Variable thermal conductivity

Subscripts

119908 Condition at wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D A Neild and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006

[2] D B Ingham and I PopTransport Phenomena in PorousMediaElsevier Oxford UK 2005

[3] G Palani and U Srikanth ldquoMHD flow past a semi-infinitevertical plate with mass transferrdquo Nonlinear Analysis Modellingand Control vol 14 no 3 pp 345ndash356 2009

[4] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoThe Canadian Journal of Chemical Engineering vol 88 no 6pp 983ndash990 2010

[5] H M Duwairi ldquoViscous and Joule heating effects on forcedconvection flow from radiate isothermal porous surfacesrdquoInternational Journal of Numerical Methods for Heat and FluidFlow vol 15 no 5 pp 429ndash440 2005

[6] M A Alim M D Alam and A Mamun ldquoJoule heating effecton the coupling of conduction with magneto hydrodynamicfree convection flow from a vertical platerdquo Non-Linear AnalysisModeling and Control vol 12 no 3 pp 307ndash316 2007

[7] I J Uwanta ldquoEffects of chemical reaction and radiation onHeatand Mass Transfer past a semi-infinite vertical porous platewith constant mass flux and dissipationrdquo European Journal ofScientific Research vol 87 no 2 pp 190ndash200 2012

[8] M A Mansour N F El-Anssary and A M Aly ldquoEffect ofchemical reaction and viscous dissipation on MHD naturalconvection flows saturated in porous media with suction orinjectionrdquo International Journal of Applied Mathematics andMechanics vol 4 no 2 pp 60ndash70 2008

[9] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[10] K Govardhan N Kishan and B Balaswamy ldquoEffect of viscousdissipation and radiation on MHD gas flow and heat and masstransfer over a stretching surface with a uniform free streamrdquoJournal of Engineering Physics andThermophysics vol 85 no 4pp 909ndash916 2012

[11] M A Sattar ldquoFree and forced convection boundary layer flowthrough a porous medium with large suctionrdquo InternationalJournal of Energy Research vol 17 no 1 pp 1ndash7 1993

[12] F Mohammad O Masatiro S Abdus and A MohamudldquoSimilarity solutions for MHD flow through vertical porous

plate with suctionrdquo Journal of Computational and AppliedMechanics vol 6 no 1 pp 15ndash25 2005

[13] S Jai ldquoViscous dissipation and chemical reaction effects on flowpast a stretching surface in a porous mediumrdquo Advance Theoryof Applied Mechanics vol 5 pp 323ndash331 2012

[14] S M Ibrahim ldquoEffects of mass transfer radiation Joule heatingand viscous dissipation on steady MHDmarangoni convectionflow over a flat surface with suction and injectionrdquo InternationalJournal of EngineeringMathematics vol 2013 Article ID 9038189 pages 2013

[15] T S Khaleque and M A Samad ldquoEffects of radiation heatgeneration and viscous dissipation on MHD free convectionflow along a stretching sheetrdquo Research Journal of AppliedSciences Engineering and Technology vol 2 no 4 pp 368ndash3772010

[16] E R G Eckert and R M Drake Analysis of Heat and MassTransfer MC Graw-Hill New York NY USA 1974

[17] A Postelnicu ldquoHeat and mass transfer by natural convection ata stagnation point in a porous medium considering Soret andDufour effectsrdquoHeat andMass Transfer vol 46 no 8-9 pp 831ndash840 2010

[18] A J Chamkha and S M M El-Kabeir ldquoUnsteady Heat andMass Transfer by MHD mixed convection flow from a rotatingvertical cone with chemical reaction and Soret and Dufoureffectsrdquo Chemical Engineering Communications vol 200 no 9pp 1220ndash1236 2013

[19] H Usman and I J Uwanta ldquoEffect of thermal conductivity onMHDHeat andMass transfer flow past an infinite vertical platewith Soret and Dufour effectsrdquo American Journal of AppliedMathematics vol 1 no 3 pp 28ndash38 2013

[20] I J Uwanta K K Asogwa and U A Ali ldquoMHD fluid flowover a vertical plate withDufour and Soret effectsrdquo InternationalJournal of Computer Applications vol 45 no 2 pp 8ndash16 2008

[21] K Sarada andB Shankar ldquoThe effects of Soret andDufour on anunsteadyMHD free convection flow past a vertical porous platein the presence of suction or injectionrdquo International Journal ofEngineering and Science vol 2 no 7 pp 13ndash25 2013

[22] N Nalinakshi P A Dinesh and D V Chandrashekar ldquoSoretand dufour effects on mixed convection heat and mass transferwith variable fluid propertiesrdquo International Journal of Mathe-matical Archive vol 4 no 11 pp 203ndash215 2013

[23] M J Subhakar and K Gangadhar ldquoSoret and Dufour effectson MHD free convection heat and mass transfer flow overa stretching vertical plate with suction and heat sourcesinkrdquoInternational Journal ofModern Engineering Research vol 2 no5 pp 3458ndash3468 2012

[24] D Srinivasacharya and C R Reddy ldquoSoret and Dufour effectson mixed convection in a non-DARcy porous medium satu-rated with micropolar fluidrdquo Nonlinear Analysis Modelling andControl vol 16 no 1 pp 100ndash115 2011

[25] N Sivaraman P K Sivagnana and R Kandasamy ldquoSoret andDufour effects on MHD free convective heat and mass transferwith thermopheresis and chemical reaction over a porousstretching surface group theory transformationrdquo Journal ofApplied Mechanics and Technical Physics vol 53 no 6 pp 871ndash879 2012

[26] D Srinivasacharya and M Upendar ldquoSoret and Dufour effectson MHD mixed convection heat and mass transfer in amicropolar fluidrdquo Central European Journal Engineering vol 3no 4 pp 679ndash689 2013

[27] S MM El-Kabeir MModathar and A M Rashed ldquoSoret anddufour effects on heat and mass transfer from a continuously

International Scholarly Research Notices 11

moving plate embedded in porous media with temperaturedependent viscosity and thermal conductivityrdquo Journal of Mod-ern Methods in Numerical Mathematics vol 4 no 2 pp 10ndash222013

[28] P O Olanrewaju O J Fenuga J A Gbadeyan and T GOkedayo ldquoDufour and Soret effects on convection heat andmass transfer in an electrical conducting power law flow overa heated porous platerdquo International Journal for ComputationalMethods in Engineering Science andMechanics vol 14 no 1 pp32ndash39 2013

[29] B Carnahan H A Luther and J O Wilkes Applied NumericalMethods John Wiley amp Sons New York NY USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 9

Table 1 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various values of physical parameters

(GrGc119872 119896 120582 and 120574)

Gr Gc 119872 119896 120582 120574 119862119891

Nu Sh20 20 10 05 10 2 02611 33328 0617140 20 10 05 10 2 04720 33451 0622120 40 10 05 10 2 03119 33386 0619520 20 30 05 10 2 02353 33317 0616720 20 10 10 10 2 02757 33334 0617420 20 10 05 20 2 02897 81676 3433420 20 10 05 10 30 02072 45600 13689With fixed values of (Pr = 071 Ec = 001 Sc = 06 Du = 02 1198871 = 10 Sr = 10 and 119905 = 02)

Table 2 Values of skin-friction coefficient 119862119891 Nusselt number Nu and Sherwood number Sh for various physical parameters (Pr 119887

1 Ec Sr

Du and Sc)

Pr 1198871

Ec Sr Du Sc 119862119891

Nu Sh071 10 20 10 02 022 04038 20257 1434430 10 20 10 02 022 02898 06168 14296071 20 20 10 02 022 04041 20311 14354071 10 30 10 02 022 04053 20729 14713071 10 20 166 02 022 04019 20523 15730071 10 20 10 10 022 04133 32477 16418071 10 20 10 02 094 03519 20983 17267With fixed values of (Gr = 20 Gc = 20119872 = 10 120574 = 05 120582 = 10 119896 = 05 and 119905 = 02)

are nondimensionalised simplified and solved by implicitfinite difference method of Crank-Nicolson type From thepresent numerical study the following conclusions can bedrawn

(1) Velocity profiles increased due to increase in thermalGrashof number solutal Grashof number porousparameter Eckert number Soret number Dufournumber and dimensionless time while it decreaseddue to increase in magnetic parameter and suctionparameter

(2) An increase in temperature profiles is a function ofan increase in thermal conductivity suction parame-ter Eckert number joule-heating parameter Dufournumber and dimensionless time while it decreaseddue to increase in Soret number

(3) Concentration profiles decreased due to increasesin Schmidt number and suction parameter whileit increased due to increase in Soret number anddimensionless time

(4) There is a rise in the skin-friction coefficient Nusseltnumber and Sherwood number due to increasein Eckert number thermal conductivity parameterjoule-heating parameter and Dufour number whilea fall is observed in skin-friction coefficient withincrease in Soret number and Prandtl number

Nomenclature

119862 Concentration119862119901 Specific heat at constant pressure

119863 Mass diffusivity119863119872 Coefficient of Mass diffusivity

119863119879 Coefficient of temperature diffusivity

Du Dufour number119892 Acceleration due to gravityGr Grashof numberGc Solutal Grashof number119896 Permeability of the porous mediumNu Nusselt numberPr Prandtl numberSc Schmidt numberSr Soret number119879 Temperature119906 V Velocities in the 119909- and 119910-direction

respectively119909 119910 Cartesian coordinates along the plate

and normal to it respectively1198610 Magnetic field of constant strength

Ec Eckert number119872 Magnetic field parameter1198871 Joule-heating parameter

Greek Letters

120573

lowast Coefficient of expansion with concentration120573 Coefficient of thermal expansion120588 Density of fluid

10 International Scholarly Research Notices

120590 Electric conductivity120579 Dimensionless temperature] Kinematic viscosity120574 Suction parameter120582 Variable thermal conductivity

Subscripts

119908 Condition at wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D A Neild and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006

[2] D B Ingham and I PopTransport Phenomena in PorousMediaElsevier Oxford UK 2005

[3] G Palani and U Srikanth ldquoMHD flow past a semi-infinitevertical plate with mass transferrdquo Nonlinear Analysis Modellingand Control vol 14 no 3 pp 345ndash356 2009

[4] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoThe Canadian Journal of Chemical Engineering vol 88 no 6pp 983ndash990 2010

[5] H M Duwairi ldquoViscous and Joule heating effects on forcedconvection flow from radiate isothermal porous surfacesrdquoInternational Journal of Numerical Methods for Heat and FluidFlow vol 15 no 5 pp 429ndash440 2005

[6] M A Alim M D Alam and A Mamun ldquoJoule heating effecton the coupling of conduction with magneto hydrodynamicfree convection flow from a vertical platerdquo Non-Linear AnalysisModeling and Control vol 12 no 3 pp 307ndash316 2007

[7] I J Uwanta ldquoEffects of chemical reaction and radiation onHeatand Mass Transfer past a semi-infinite vertical porous platewith constant mass flux and dissipationrdquo European Journal ofScientific Research vol 87 no 2 pp 190ndash200 2012

[8] M A Mansour N F El-Anssary and A M Aly ldquoEffect ofchemical reaction and viscous dissipation on MHD naturalconvection flows saturated in porous media with suction orinjectionrdquo International Journal of Applied Mathematics andMechanics vol 4 no 2 pp 60ndash70 2008

[9] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[10] K Govardhan N Kishan and B Balaswamy ldquoEffect of viscousdissipation and radiation on MHD gas flow and heat and masstransfer over a stretching surface with a uniform free streamrdquoJournal of Engineering Physics andThermophysics vol 85 no 4pp 909ndash916 2012

[11] M A Sattar ldquoFree and forced convection boundary layer flowthrough a porous medium with large suctionrdquo InternationalJournal of Energy Research vol 17 no 1 pp 1ndash7 1993

[12] F Mohammad O Masatiro S Abdus and A MohamudldquoSimilarity solutions for MHD flow through vertical porous

plate with suctionrdquo Journal of Computational and AppliedMechanics vol 6 no 1 pp 15ndash25 2005

[13] S Jai ldquoViscous dissipation and chemical reaction effects on flowpast a stretching surface in a porous mediumrdquo Advance Theoryof Applied Mechanics vol 5 pp 323ndash331 2012

[14] S M Ibrahim ldquoEffects of mass transfer radiation Joule heatingand viscous dissipation on steady MHDmarangoni convectionflow over a flat surface with suction and injectionrdquo InternationalJournal of EngineeringMathematics vol 2013 Article ID 9038189 pages 2013

[15] T S Khaleque and M A Samad ldquoEffects of radiation heatgeneration and viscous dissipation on MHD free convectionflow along a stretching sheetrdquo Research Journal of AppliedSciences Engineering and Technology vol 2 no 4 pp 368ndash3772010

[16] E R G Eckert and R M Drake Analysis of Heat and MassTransfer MC Graw-Hill New York NY USA 1974

[17] A Postelnicu ldquoHeat and mass transfer by natural convection ata stagnation point in a porous medium considering Soret andDufour effectsrdquoHeat andMass Transfer vol 46 no 8-9 pp 831ndash840 2010

[18] A J Chamkha and S M M El-Kabeir ldquoUnsteady Heat andMass Transfer by MHD mixed convection flow from a rotatingvertical cone with chemical reaction and Soret and Dufoureffectsrdquo Chemical Engineering Communications vol 200 no 9pp 1220ndash1236 2013

[19] H Usman and I J Uwanta ldquoEffect of thermal conductivity onMHDHeat andMass transfer flow past an infinite vertical platewith Soret and Dufour effectsrdquo American Journal of AppliedMathematics vol 1 no 3 pp 28ndash38 2013

[20] I J Uwanta K K Asogwa and U A Ali ldquoMHD fluid flowover a vertical plate withDufour and Soret effectsrdquo InternationalJournal of Computer Applications vol 45 no 2 pp 8ndash16 2008

[21] K Sarada andB Shankar ldquoThe effects of Soret andDufour on anunsteadyMHD free convection flow past a vertical porous platein the presence of suction or injectionrdquo International Journal ofEngineering and Science vol 2 no 7 pp 13ndash25 2013

[22] N Nalinakshi P A Dinesh and D V Chandrashekar ldquoSoretand dufour effects on mixed convection heat and mass transferwith variable fluid propertiesrdquo International Journal of Mathe-matical Archive vol 4 no 11 pp 203ndash215 2013

[23] M J Subhakar and K Gangadhar ldquoSoret and Dufour effectson MHD free convection heat and mass transfer flow overa stretching vertical plate with suction and heat sourcesinkrdquoInternational Journal ofModern Engineering Research vol 2 no5 pp 3458ndash3468 2012

[24] D Srinivasacharya and C R Reddy ldquoSoret and Dufour effectson mixed convection in a non-DARcy porous medium satu-rated with micropolar fluidrdquo Nonlinear Analysis Modelling andControl vol 16 no 1 pp 100ndash115 2011

[25] N Sivaraman P K Sivagnana and R Kandasamy ldquoSoret andDufour effects on MHD free convective heat and mass transferwith thermopheresis and chemical reaction over a porousstretching surface group theory transformationrdquo Journal ofApplied Mechanics and Technical Physics vol 53 no 6 pp 871ndash879 2012

[26] D Srinivasacharya and M Upendar ldquoSoret and Dufour effectson MHD mixed convection heat and mass transfer in amicropolar fluidrdquo Central European Journal Engineering vol 3no 4 pp 679ndash689 2013

[27] S MM El-Kabeir MModathar and A M Rashed ldquoSoret anddufour effects on heat and mass transfer from a continuously

International Scholarly Research Notices 11

moving plate embedded in porous media with temperaturedependent viscosity and thermal conductivityrdquo Journal of Mod-ern Methods in Numerical Mathematics vol 4 no 2 pp 10ndash222013

[28] P O Olanrewaju O J Fenuga J A Gbadeyan and T GOkedayo ldquoDufour and Soret effects on convection heat andmass transfer in an electrical conducting power law flow overa heated porous platerdquo International Journal for ComputationalMethods in Engineering Science andMechanics vol 14 no 1 pp32ndash39 2013

[29] B Carnahan H A Luther and J O Wilkes Applied NumericalMethods John Wiley amp Sons New York NY USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Scholarly Research Notices

120590 Electric conductivity120579 Dimensionless temperature] Kinematic viscosity120574 Suction parameter120582 Variable thermal conductivity

Subscripts

119908 Condition at wall

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D A Neild and A BejanConvection in PorousMedia SpringerNew York NY USA 3rd edition 2006

[2] D B Ingham and I PopTransport Phenomena in PorousMediaElsevier Oxford UK 2005

[3] G Palani and U Srikanth ldquoMHD flow past a semi-infinitevertical plate with mass transferrdquo Nonlinear Analysis Modellingand Control vol 14 no 3 pp 345ndash356 2009

[4] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoThe Canadian Journal of Chemical Engineering vol 88 no 6pp 983ndash990 2010

[5] H M Duwairi ldquoViscous and Joule heating effects on forcedconvection flow from radiate isothermal porous surfacesrdquoInternational Journal of Numerical Methods for Heat and FluidFlow vol 15 no 5 pp 429ndash440 2005

[6] M A Alim M D Alam and A Mamun ldquoJoule heating effecton the coupling of conduction with magneto hydrodynamicfree convection flow from a vertical platerdquo Non-Linear AnalysisModeling and Control vol 12 no 3 pp 307ndash316 2007

[7] I J Uwanta ldquoEffects of chemical reaction and radiation onHeatand Mass Transfer past a semi-infinite vertical porous platewith constant mass flux and dissipationrdquo European Journal ofScientific Research vol 87 no 2 pp 190ndash200 2012

[8] M A Mansour N F El-Anssary and A M Aly ldquoEffect ofchemical reaction and viscous dissipation on MHD naturalconvection flows saturated in porous media with suction orinjectionrdquo International Journal of Applied Mathematics andMechanics vol 4 no 2 pp 60ndash70 2008

[9] R Kandasamy K Periasamy and K K S Prabhu ldquoEffects ofchemical reaction heat and mass transfer along a wedge withheat source and concentration in the presence of suction orinjectionrdquo International Journal of Heat and Mass Transfer vol48 no 7 pp 1388ndash1394 2005

[10] K Govardhan N Kishan and B Balaswamy ldquoEffect of viscousdissipation and radiation on MHD gas flow and heat and masstransfer over a stretching surface with a uniform free streamrdquoJournal of Engineering Physics andThermophysics vol 85 no 4pp 909ndash916 2012

[11] M A Sattar ldquoFree and forced convection boundary layer flowthrough a porous medium with large suctionrdquo InternationalJournal of Energy Research vol 17 no 1 pp 1ndash7 1993

[12] F Mohammad O Masatiro S Abdus and A MohamudldquoSimilarity solutions for MHD flow through vertical porous

plate with suctionrdquo Journal of Computational and AppliedMechanics vol 6 no 1 pp 15ndash25 2005

[13] S Jai ldquoViscous dissipation and chemical reaction effects on flowpast a stretching surface in a porous mediumrdquo Advance Theoryof Applied Mechanics vol 5 pp 323ndash331 2012

[14] S M Ibrahim ldquoEffects of mass transfer radiation Joule heatingand viscous dissipation on steady MHDmarangoni convectionflow over a flat surface with suction and injectionrdquo InternationalJournal of EngineeringMathematics vol 2013 Article ID 9038189 pages 2013

[15] T S Khaleque and M A Samad ldquoEffects of radiation heatgeneration and viscous dissipation on MHD free convectionflow along a stretching sheetrdquo Research Journal of AppliedSciences Engineering and Technology vol 2 no 4 pp 368ndash3772010

[16] E R G Eckert and R M Drake Analysis of Heat and MassTransfer MC Graw-Hill New York NY USA 1974

[17] A Postelnicu ldquoHeat and mass transfer by natural convection ata stagnation point in a porous medium considering Soret andDufour effectsrdquoHeat andMass Transfer vol 46 no 8-9 pp 831ndash840 2010

[18] A J Chamkha and S M M El-Kabeir ldquoUnsteady Heat andMass Transfer by MHD mixed convection flow from a rotatingvertical cone with chemical reaction and Soret and Dufoureffectsrdquo Chemical Engineering Communications vol 200 no 9pp 1220ndash1236 2013

[19] H Usman and I J Uwanta ldquoEffect of thermal conductivity onMHDHeat andMass transfer flow past an infinite vertical platewith Soret and Dufour effectsrdquo American Journal of AppliedMathematics vol 1 no 3 pp 28ndash38 2013

[20] I J Uwanta K K Asogwa and U A Ali ldquoMHD fluid flowover a vertical plate withDufour and Soret effectsrdquo InternationalJournal of Computer Applications vol 45 no 2 pp 8ndash16 2008

[21] K Sarada andB Shankar ldquoThe effects of Soret andDufour on anunsteadyMHD free convection flow past a vertical porous platein the presence of suction or injectionrdquo International Journal ofEngineering and Science vol 2 no 7 pp 13ndash25 2013

[22] N Nalinakshi P A Dinesh and D V Chandrashekar ldquoSoretand dufour effects on mixed convection heat and mass transferwith variable fluid propertiesrdquo International Journal of Mathe-matical Archive vol 4 no 11 pp 203ndash215 2013

[23] M J Subhakar and K Gangadhar ldquoSoret and Dufour effectson MHD free convection heat and mass transfer flow overa stretching vertical plate with suction and heat sourcesinkrdquoInternational Journal ofModern Engineering Research vol 2 no5 pp 3458ndash3468 2012

[24] D Srinivasacharya and C R Reddy ldquoSoret and Dufour effectson mixed convection in a non-DARcy porous medium satu-rated with micropolar fluidrdquo Nonlinear Analysis Modelling andControl vol 16 no 1 pp 100ndash115 2011

[25] N Sivaraman P K Sivagnana and R Kandasamy ldquoSoret andDufour effects on MHD free convective heat and mass transferwith thermopheresis and chemical reaction over a porousstretching surface group theory transformationrdquo Journal ofApplied Mechanics and Technical Physics vol 53 no 6 pp 871ndash879 2012

[26] D Srinivasacharya and M Upendar ldquoSoret and Dufour effectson MHD mixed convection heat and mass transfer in amicropolar fluidrdquo Central European Journal Engineering vol 3no 4 pp 679ndash689 2013

[27] S MM El-Kabeir MModathar and A M Rashed ldquoSoret anddufour effects on heat and mass transfer from a continuously

International Scholarly Research Notices 11

moving plate embedded in porous media with temperaturedependent viscosity and thermal conductivityrdquo Journal of Mod-ern Methods in Numerical Mathematics vol 4 no 2 pp 10ndash222013

[28] P O Olanrewaju O J Fenuga J A Gbadeyan and T GOkedayo ldquoDufour and Soret effects on convection heat andmass transfer in an electrical conducting power law flow overa heated porous platerdquo International Journal for ComputationalMethods in Engineering Science andMechanics vol 14 no 1 pp32ndash39 2013

[29] B Carnahan H A Luther and J O Wilkes Applied NumericalMethods John Wiley amp Sons New York NY USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Scholarly Research Notices 11

moving plate embedded in porous media with temperaturedependent viscosity and thermal conductivityrdquo Journal of Mod-ern Methods in Numerical Mathematics vol 4 no 2 pp 10ndash222013

[28] P O Olanrewaju O J Fenuga J A Gbadeyan and T GOkedayo ldquoDufour and Soret effects on convection heat andmass transfer in an electrical conducting power law flow overa heated porous platerdquo International Journal for ComputationalMethods in Engineering Science andMechanics vol 14 no 1 pp32ndash39 2013

[29] B Carnahan H A Luther and J O Wilkes Applied NumericalMethods John Wiley amp Sons New York NY USA 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of