Research Article Radiation Effects on Mass Transfer...

Hindawi Publishing CorporationISRN Computational MathematicsVolume 2013 Article ID 765408 9 pageshttpdxdoiorg1011552013765408

Research ArticleRadiation Effects on Mass Transfer Flow through a HighlyPorous Medium with Heat Generation and Chemical Reaction

S Mohammed Ibrahim

Department of Mathematics Priyadarshini College of Engineering and Technology Nellore Andhra Pradesh 524004 India

Correspondence should be addressed to S Mohammed Ibrahim ibrahimsvugmailcom

Received 13 November 2012 Accepted 9 January 2013

Academic Editors O Kuksenok and W G Weng

Copyright copy 2013 S Mohammed Ibrahim 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 is concerned to analyze the influence of the unsteady free convection flow of a viscous incompressible fluidthrough a porous medium with high porosity bounded by a vertical infinite moving plate in the presence of thermal radiationheat generation and chemical reactionThe fluid is considered to be gray absorbing and emitting but nonscattering medium andRosseland approximation is considered to describe the radiative heat flux in the energy equation The dimensionless governingequations for this investigation are solved analytically using perturbation technique The effects of various governing parameterson the velocity temperature concentration skin-friction coefficient Nusselt number and Sherwood number are shown in figuresand tables and analyzed in detail

1 Introduction

Transport of momentum and energy in fluid-saturatedporous media with low porosities are commonly describedby Darcyrsquos model for conservation of momentum and byan energy equation based on the velocity field found fromthis model by Kaviany [1] In contrast to rocks soil sandand other media that do fall in this category certain porousmaterials such as foam metals and fibrous media usuallyhave high porosity Vajravelu [2] examined the steady flowof heat transfer in a porous medium with high porosityRaptis [3] studied mathematically the case of time varyingtwo-dimensional natural convection heat transfer of anincompressible electrically conducting viscous fluid througha high porousmediumbounded by an infinite vertical porousplate Hong et al [4] Chen and Lin [5] and Jaiswal andSoundalgekar [6] studied the natural convection in a porousmedium with high porosity Hiremath and Patil [7] studiedthe effect of free convection currents on the oscillatory flow ofthe polar fluid through a porous medium which is boundedby the vertical plane surface with constant temperature

Many processes in engineering areas occur in hightemperature and consequently the radiation plays a sig-nificant role Chandrasekhara and Nagaraju [8] examined

the composite heat transfer in a variable porosity mediumbounded by an infinite vertical flat plate in the presence ofradiation Yih [9] studied the radiation effects on naturalconvection over a cylinder embedded in porous mediaMohammadein and El-Amin [10] considered the thermalradiation effects on power law fluids over a horizontal plateembedded in a porous medium Raptis [11] studied the heattransfer in a porous medium with high porosity in the pres-ence of radiation Raptis and Perdikis [12] studied unsteadyflow through a porous medium with high porosity boundedby a vertical infinite plate in the presence of radiation

In the processes such as drying Evaporation at thesurface of water body Energy transfer in a wet coolingtower and the flow in a desert cooler heat and mass transferoccur simultaneously Possible applications of this type offlow can be found in many industries For example in thepower industry among the methods of generating electricpower is one in which electrical energy is extracted directlyfrom a moving conducting fluid The study of heat andmass transfer with chemical reaction is of great practicalimportance to engineers and scientists because of its frequentoccurrence in many branches of science and engineeringChambre and Young [13] presented a first order chemicalreaction in the neighborhood of a horizontal plate Das et al

2 ISRN Computational Mathematics

[14] investigated the effect of the first order homogeneouschemical reaction on the process of an unsteady flow pasta vertical plate with a constant heat and mass transferMuthucumarswamy and Ganesan [15] studied the effect ofchemical reaction and injection on the flow characteristics inan unsteady upward motion of an isothermal plate AnandRao and Shivaiah [16] noticed that the chemical reactioneffects on an unsteady MHD free convective flow past aninfinite vertical porous plate with constant suction and heatsource or generation

The study of heat generation or absorption effects inmoving fluids is important in view of several physical prob-lems such as fluids undergoing exothermic or endothermicchemical reactions Vajravelu and Hadjinicolaou [17] studiedthe heat transfer characteristics in the laminar boundarylayer of a viscous fluid over a stretching sheet with viscousdissipation or frictional heating and internal heat generationMolla et al [18] studied the problem of natural convectionflow along a vertical wavy surface with uniform surfacetemperature in the presence of heat generationabsorptionAlam et al [19] considered the problem of free convectiveheat and mass transfer flow past an inclined semi-infiniteheated surface of a steady viscous incompressible fluid in thepresence of magnetic field and heat generation Chamkha[20] investigated an unsteady convective heat and masstransfer flow past a semi-infinite porous moving plate withheat absorption Hady et al [21] studied the problem of freeconvection flow along a vertical wavy surface embedded ina saturated porous media in the presence of internal heatgeneration or absorption effect Ambethkar [22] investigatedthe numerical solutions of heat and mass transfer effectsof an unsteady MHD free convective flow past an infinitevertical plate with constant suction and heat source or sinkMohammed Ibrahim and Bhaskar Reddy [23] studied theradiation and mass transfer effects on MHD free convectionflow along a stretching surface with viscous dissipation andheat generation

In view of the above studies an unsteady free convectiveheat and mass transfer flow of a viscous incompressibleradiating fluid through a porous medium with high porositybounded by an infinite vertical moving plate is consideredin the presence of heat generation and chemical reaction Itis assumed that the plate is embedded in porous mediumand moves with constant velocity in the flow directionThe equations of continuity linear momentum energy anddiffusion which govern the flow field are solved by using aregular perturbation method The behavior of the velocitytemperature concentration skin friction Nusselt numberand Sherwood number has been discussed for variations inthe governing parameters

2 Mathematical Analysis

An unsteady two-dimensional laminar free convective masstransfer flow of a viscous incompressible fluid through ahighly porous medium past an infinite vertical movingporous plate in the presence of thermal radiation heatgeneration and chemical reaction is considered The fluid

and the porous structure are assumed to be in local thermalequilibrium It is also assumed that there is radiation onlyfrom the fluidThe fluid is a gray emitting and absorbing butnon-scattering medium and the Rosseland approximation isused to describe the radiative heat flux in the energy equationA homogeneous first order chemical reaction between fluidand the species concentration is considered in which the rateof chemical reaction is directly proportional to the speciesconcentration All the fluid properties are assumed to beconstant except that the influence of the density variationwith temperature is considered only in the body force term(Boussinesqrsquos approximation) The 119909

1015840-axis is chosen alongthe plate in the direction opposite to the direction of gravityand the 1199101015840-axis is taken normal to it Since the flow field isof extreme size all the variables are functions of 1199101015840 and 119905

1015840

only Hence under the usual Boussinesqrsquos approximation theequations of mass linear momentum energy and diffusionare

continuity equation

1205971199071015840

1205971199101015840= 0 (1)

momentum equation

1205971199061015840

1205971199051015840+ 1199071015840 1205971199061015840

1205971199101015840= minus

1

120588

1205971199011015840

1205971199091015840+ 120584

12059721199061015840

12059711991010158402

+ 119892120573 (1198791015840minus 1198791015840

infin)

+ 119892120573lowast(1198621015840minus 1198621015840

infin) minus

120584

11987010158401205931199061015840

(2)

energy equation

120590

1205971198791015840

1205971199051015840+ 1205931199071015840 1205971198791015840

1205971199101015840

=

119896

120588119888119901

12059721198791015840

12059711991010158402

minus

120593

120588119888119901

120597119902119903

1205971199101015840+

1198760

120588119888119901

(1198791015840minus 1198791015840

infin)

(3)

diffusion equation

1205971198621015840

1205971199051015840+ 1199071015840 1205971198621015840

1205971199101015840= 119863

12059721198621015840

12059711991010158402

minus 1198701015840

119903(1198621015840minus 1198621015840

infin) (4)

where 1199091015840 1199101015840 and 1199051015840 are the dimensional distances along and

perpendicular to the plate and dimensional time respec-tively 1199061015840 and 119907

1015840 the components of dimensional velocitiesalong 119909

1015840 and 1199101015840 directions respectively 119862

1015840 and 1198791015840 the

dimensional concentration and temperature of the fluidrespectively 120588 the fluid density 120584 the kinematic viscosity 119888

119901

the specific heat at constant pressure120590 the heat capacity ratio119892 the acceleration due to gravity 120573 and 120573

lowast the volumetriccoefficient of thermal and concentration expansion 1198701015840 thepermeability of the porous medium 120593 the porosity 119863 themolecular diffusivity 1198701015840

119903the chemical reaction parameter

ISRN Computational Mathematics 3

and 119896 the fluid thermal conductivity The third and fourthterms on the right hand side of the momentum equation(2) denote the thermal and concentration buoyancy effectsrespectively and the fifth term represents the bulk matrixlinear resistance that is Darcy term Also the second termon the right hand side of the energy equation (3) representsthermal radiation The radiative heat flux term by using theRosseland approximation (Brewster [24]) is given by

119902119903=

minus412059011990412059711987910158404

31198701198901205971199101015840 (5)

where 120590119904is the Stefan-Boltzmann constant and 119870

119890is the

mean absorption coefficient It should be noted that by usingthe Rosseland approximation the present analysis is limitedto optically thick fluids If temperature differences withinthe flow are sufficiently small then (6) can be linearised byexpanding 119879

10158404 into the Taylor series about119879infin which after

neglecting higher order terms takes the form

11987910158404

asymp 41198793

infin119879 minus 3119879

4

infin (6)

It is assumed that the permeable plate moves withconstant velocity in the direction of fluid flow It is alsoassumed that the plate temperature and concentration areexponentially varying with time Under these assumptionsthe appropriate boundary conditions for the velocity temper-ature and concentration fields are

1199061015840= 1198801015840

119901 119879

1015840= 1198791015840

119908+ 120576 (119879

1015840

119908minus 1198791015840

infin) 11989011989910158401199051015840

1198621015840= 1198621015840

119908+ 120576 (119879

1015840

119908minus 1198791015840

infin) 11989011989910158401199051015840

at 1199101015840= 0

1199061015840997888rarr 119880

1015840

infin 1198791015840997888rarr 119879

1015840

infin 1198621015840997888rarr 119862

1015840

infinas 1199101015840997888rarr infin

(7)

where 1198801015840

119901is the wall dimensional velocity 1198621015840

119908and 119879

1015840

119908are

the wall dimensional concentration and temperature respec-tively 1198801015840

infin 1198621015840infin and 119879

1015840

infinare the free stream dimensional

velocity concentration and temperature respectively and 1198991015840is the constant

It is clear from (1) that the suction velocity normal to theplate is either a constant or a function of time Hence thesuction velocity normal to the plate is taken as

1199071015840= minus1198810 (8)

where 1198810is a scale of suction velocity which is a nonzero

positive constantThe negative sign indicates that the suctionis towards the plate

Outside the boundary layer (2) gives

1

120588

1198891199011015840

1198891199091015840= minus

120593120584

11987010158401198801015840

infin (9)

In order to write the governing equations and the bound-ary conditions in dimensionless form the following nondi-mensional quantities are introduced

119906 =

1199061015840

1198801015840

infin

119910 =

11988101199101015840

119907

119880119901=

1198801015840

119901

1198801015840

infin

119899 =

1198991015840119907

1198812

0

119905 =

11990510158401198812

0

119907

120582 =

120590

120593

120579 =

1198791015840minus 1198791015840

infin

1198791015840

119908minus 1198791015840

infin

119862 =

1198621015840minus 1198621015840

infin

1198621015840

119908minus 1198621015840

infin

Gr =120584119892120573 (119879

1015840

119908minus 1198791015840

infin)

1198801015840

infin1198812

0

Gc =120584119892120573lowast(1198621015840

119908minus 1198621015840

infin)

1198801015840

infin1198812

0

119870 =

11987010158401198812

0

1205931205842

Pr =120588119888119901120593120592

119896

119877 =

119870119890119896

41205931205901199041198793

infin

119876 =

1198760120584

1205931205881198881199011198812

0

Sc = 120584

119863

119870119903=

1198701015840

119903120584

1198812

0

(10)

In view of (5)ndash(10) (2)ndash(4) reduce to the following non-dimensional form

120597119906

120597119905

minus

120597119906

120597119910

=

1205972119906

1205971199102+ Gr120579 + Gc119862 +

1

119870

(1 minus 119906)

120582

120597120579

120597119905

minus

120597120579

120597119910

=

1

Γ

1205972120579

1205971199102+ 119876120579

120597119862

120597119905

minus

120597119862

120597119910

=

1

Sc1205972120601

1205971199102minus 119870119903119862

(11)

where Γ = (1 minus 4(3119877 + 4))Pr and Gr Gc Pr 119877 Sc119876119870 and119870119903are the thermal Grashof number solutal Grashof number

Prandtl number radiation parameter Schmidt number heatgeneration parameter permeability of the porous mediumand chemical reaction parameter respectively

The corresponding dimensionless boundary conditionsare

119906 = 119880119901

120579 = 1 + 120576119890119899119905

119862 = 1 + 120576119890119899119905 at 119910 = 0

119906 997888rarr 1 120579 997888rarr 0 119862 997888rarr 0 as 119910 997888rarr infin

(12)

3 Solution of the Problem

Equations (11) are coupled nonlinear partial differentialequations and these cannot be solved in closed form How-ever these equations can be reduced to a set of ordinarydifferential equations which can be solved analytically This


can be done by representing the velocity temperature andconcentration of the fluid in the neighborhood of the plate as

119906 (119910 119905) = 1199060(119910) + 120576119890

1198991199051199061(119910) + 0(120576)

2+ sdot sdot sdot

120579 (119910 119905) = 1205790(119910) + 120576119890

1198991199051205791(119910) + 0(120576)

2+ sdot sdot sdot

119862 (119910 119905) = 1198620(119910) + 120576119890

1198991199051198621(119910) + 0(120576)

2+ sdot sdot sdot

(13)

Substituting (13) in (11) equating the harmonic andnonharmonic terms and neglecting the higher order termsof 0(120576)2 we obtain

11990610158401015840

0+ 1199061015840

0minus

1

119870

1199060= minus

1

119870

minus Gr1205790minus Gc119862

0

11990610158401015840

1+ 1199061015840

1minus (119899 +

1

119870

)1199061= minusGr120579

1minus Gc119862

1

12057910158401015840

0+ Γ1205791015840

0+ Γ119876120579

0= 0

12057910158401015840

1+ Γ1205791015840

1minus 119899120582Γ120579

1+ Γ119876120579

1= 0

11986210158401015840

0+ Sc1198621015840

0minus Sc119870

1199031198620= 0

11986210158401015840

1+ Sc1198621015840

1minus Sc (119870

119903+ 119899)119862

1= 0

(14)

where the prime denotes ordinary differentiationwith respectto 119910

The corresponding boundary conditions can be writtenas

1199060= 119880119901 1199061= 0 120579

0= 1

1205791= 1 119862

0= 1 119862

1= 1 at 119910 = 0

1199060997888rarr 1 119906

1997888rarr 0 120579

0997888rarr 0

1205791997888rarr 0 119862

0997888rarr 0 119862

1997888rarr 0 as 119910 997888rarr infin

(15)

Solving (14) subject to boundary conditions (15) weobtain the velocity temperature and concentration distribu-tions in the boundary layer as

119906 (119910 119905) = 1 + 1198604119890minus1198981119910+ 1198605119890minus1198984119910+ 1198606119890minus1198986119910

+ 120576119890119899119905(1198603119890minus1198985119910+ 1198601119890minus1198982119910+ 1198602119890minus1198983119910)

(16)

120579 (119910 119905) = 119890minus1198981119910+ 120576119890119899119905(119890minus1198982119910) (17)

119862 (119910 119905) = 119890minus1198984119910+ 120576119890119899119905(119890minus1198983119910) (18)

where the expressions for the constants are given in theappendix

The skin-friction Nusselt number and Sherwood num-ber are important physical parameters for this type of bound-ary layer flow These parameters can be defined and deter-mined as follows

Knowing the velocity field the skin-friction at the platecan be obtained which in non-dimensional form is given by

119862119891

=

1205911015840

119908

12058811988001198810

= (

120597119906

120597119910

)

119910=0

= (

1205971199060

120597119910

+ 120576119890119899119905 1205971199061

120597119910

)

119910=0

= minus [11986041198981+ 11986051198984+ 11986061198986

+120576119890119899119905(11986031198985+ 11986011198982+ 11986021198983)]

(19)

Knowing the temperature field the rate of heat transfercoefficient can be obtained which in the non-dimensionalform in terms of the Nusselt number is given by

Nu = minus119909

(1205971198791205971199101015840)1199101015840=0

(1198791015840

119908minus 1198791015840

infin)

997904rArr NuReminus1119909

= minus(

120597120579

120597119910

)

119910=0

= minus(

1205971205790

120597119910

+ 120576119890119899119905 1205971205791

120597119910

)

119910=0

= minus [minus1198981+ 120576119890119899119905(minus1198982)]

(20)

Knowing the concentration field the rate of mass transfercoefficient can be obtained which in the non-dimensionalform in terms of the Sherwood number is given by

Sh = minus119909

(1205971198621205971199101015840)1199101015840=0

(1198621015840

119908minus 1198621015840

infin)

997904rArr ShReminus1119909

= minus(

120597119862

120597119910

)

119910=0

= minus[

1205971198620

120597119910

+ 120576119890119899119905 1205971198621

120597119910

]

119910=0


(21)

where Re119909= 1198810119909120584 is the local Reynolds number

4 Results and Discussion

In the preceding section the problem of an unsteady freeconvective flow of a viscous incompressible thermally radi-ating and chemically reacting fluid past a semi-infinite platein the presence of heat generation was formulated and solvedby means of a perturbation method The expressions for thevelocity temperature and concentration were obtained Toillustrate the behavior of these physical quantities numericalvalues of these quantities were computed with respect tothe variations in the governing parameters namely thethermal Grashof number Gr the solutal Grashof numberGc Prandtl number Pr Schmidt number Sc the radiationparameter 119877 the permeability of the porous medium 119870the heat generation parameter 119876 and the chemical reaction


5

4

3

2

1

00 2 4 6 8 10

119906

119910

Gr = 1 2 3 4

Figure 1 Velocity profiles for different values of Gr

parameter119870119903 In the present study the following default

parametric values are adopted Gr = 20 Gc = 20 119870 = 50120582 = 14 Sc = 02 119877 = 50 119870

119903= 20 119876 = 01 Pr = 071

119880119901= 04 119860 = 05 119905 = 10 119899 = 01 and 120576 = 001 All the

graphs and tables therefore correspond to these values unlessspecifically indicated on the appropriate graph

Figure 1 presents the typical velocity profiles in theboundary layer for various values of the thermal Grashofnumber Gr The thermal Grashof number Gr signifies therelative effect of the thermal buoyancy force to the viscoushydrodynamic force in the boundary layer It is observed thatan increase in Gr leads to a rise in the values of velocitydue to enhancement of thermal buoyancy force Here thepositive values of Gr correspond to cooling of the surfaceIt is observed that velocity increases rapidly near the wall ofthe porous plate as Gr increases and then decays to the freestream velocity

For the case of different values of the solutal Grahofnumber Gc the velocity profiles in the boundary layer areshown in Figure 2 The solutal Grashof number Gc definesthe ratio of the species buoyancy force to the viscous hydro-dynamic force As expected as Gc increases the fluid velocityincreases and the peak value is more distinctive maximumvalue in the vicinity of the plate and then decreases properlyto approach the free stream value Figure 3 shows the velocityprofiles for different values of the permeability of the porousmedium 119870 Clearly as 119870 increases the velocity tends toincrease

For different values of the radiation parameter 119877 thevelocity and temperature profiles are plotted in Figures 4(a)and 4(b) The radiation parameter 119877 defines the relativecontribution of conduction heat transfer to thermal radiationtransfer It is obvious that an increase in the radiation param-eter 119877 results in a decrease in the velocity and temperaturewithin the boundary layer as well as decreased thickness ofthe velocity and temperature boundary layers

Figures 5(a) and 5(b) illustrate the velocity and temper-ature profiles for different values of Prandtl number Pr Thenumerical results show that the effect of increasing valuesof Prandtl number results in a decreasing velocity From

Gc = 1 2 3 4

45

4

35

3

25

2

15

1

05

0

119906

0 1 2 3 4 5 6 7 8 9 10119910

Figure 2 Velocity profiles for different values of Gc

119870 = 2 5 7 10

4

3

2

1

0

119906

0 2 4 6 8 10119910

Figure 3 Velocity profiles for different values of 119870

Figure 5(b) as expected the numerical results show thatan increase in the Prandtl number results in a decrease ofthe thermal boundary layer and in general lower averagetemperature with in the boundary layer The reason is thatsmaller values of Pr are equivalent to increase in the thermalconductivity of the fluid and therefore heat is able to diffuseaway from the heated surface more rapidly for higher valuesof Pr Hence in the case of smaller Prandtl numbers thethermal boundary layer is thicker and the rate of heat transferis reduced

Figures 6(a) and 6(b) display the effects of the Schmidtnumber Sc on velocity and concentration respectively TheSchmidt number Sc embodies the ratio of the momentum tothemass diffusivityThe Schmidt number therefore quantifiesthe relative effectiveness ofmomentumandmass transport bydiffusion in the hydrodynamic (velocity) and concentration(species) boundary layers As the Schmidt number increasesthe concentration decreases This causes the concentrationbuoyancy effects to decrease yielding a reduction in the fluidvelocity The reductions in the velocity and concentrationprofiles are accompanied by simultaneous reductions in the


0 2 4 6 8 10119910

119877 = 3 5 7 10

4

3

2

1

0

119906

(a)

0 2 4 6 8 10119910

119877 = 3 5 7 10120579

1

08

06

04

02

0

(b)

Figure 4 (a) Velocity profiles for different values of119877 (b) Temperature profiles for different values of 119877

0 2 4 6 8 10119910

35

3

2

25

1

15

0

05

119906

Pr = 071 08 1 125

(a)

0 2 4 6 8 10119910

120579

1

08

06

04

02

0

Pr = 071 08 1 125

(b)

Figure 5 (a) Velocity profiles for different values of Pr (b) Temperature profiles for different values of Pr

velocity and concentration boundary layers These behaviorsare evident from Figures 6(a) and 6(b)

The influences of chemical reaction parameter 119870119903on the

velocity and concentration across the boundary layer arepresented in Figures 7(a) and 7(b) It is seen that the velocityas well as concentration across the boundary layer decreaseswith an increase in the chemical reaction parameter119870

119903

Figures 8(a) and 8(b) depict the effect of heat generationparameter 119876 on the velocity and temperature It is noticedthat the velocity as well as temperature across the boundarylayer increases with an increase in the heat generationparameter 119876

Tables 1ndash7 show the effects of the thermal Grashofnumber Gr solutal Grashof number Gc radiation parameter119877 Prandtl number Pr Schmidt number Sc chemical reactionparameter 119870

119903 and heat generation parameter 119876 on the

skin friction coefficient 119862119891 Nusselt number Nu and the

Sherwood number Sh FromTables 1 and 2 it is observed thatas Gr or Gc increases the skin-friction coefficient increasesFrom Table 3 it can be seen that as the radiation parameter

Table 1 Effects of Gr on skin-friction 119862119891

Gr 119862119891

10 4596620 6273530 7950340 96272

increases the skin-friction decreases and the Nusselt numberincreases From Table 4 it is found that an increase in Prleads to a decrease in the skin-friction and an increase inthe Nusselt number From Table 5 it is observed that as theSchmidt number increases the skin-friction decreases andthe Sherwood number increases From Table 6 it is seenthat as the chemical reaction parameter 119870

119903increases the

skin-friction decreases and the Sherwood number increasesFrom Table 7 it is observed that as the heat generationparameter 119876increases the skin-friction increases and theNusselt number decreases


3

25

2

15

1

05

0

119906

119910

10 2 3 4 5 6 7 8 9

Sc = 02 04 06 09

(a)

119862

1

08

06

04

02

0

119910

10 2 3 4 5 6

Sc = 02 04 06 09

(b)

Figure 6 (a) Velocity profiles for different values of Sc (b) Concentration profiles for different values of Sc

35

3

25

2

15

1

05

0

119906

0 2 4 6 8 10119910

119870119903 = 2 5 7 10

(a)

1

08

06

04

02

0

119862

0 1 2 3 4 5 6119910

119870119903 = 2 5 7 10

(b)

Figure 7 (a) Velocity profiles for different values of 119870119903 (b) Concentration profiles for different values of119870

119903

Table 2 Effects of Gc on skin-friction 119862119891

Gc 119862119891

10 5164920 6273530 7382140 84907

Table 3 Effects of R on skin-friction 119862119891and Nusselt number

NuReminus1119909

119877 119862119891

NuReminus1119909

30 67751 0357750 62735 0436970 60693 04763100 59186 05089

Table 4 Effects of Pr on skin-friction 119862119891and Nusselt number

NuReminus1119909

Pr 119862119891

NuReminus1119909

071 62735 0436908 58946 0514410 53134 06811125 48515 08853

Table 5 Effects of Sc on skin-friction 119862119891and Sherwood number

Sh Reminus1119909

Sc 119862119891

ShReminus1119909

02 62735 0748704 56259 1129106 53141 1451909 50489 18861


0 2 4 6 8 10119910

119876 = 001 005 01 015

45

4

35

3

25

2

15

1

05

0

119906

(a)

1

08

06

04

02

00 2 4 6 8 10

119910

119876 = 001 005 01 015120579

(b)

Figure 8 (a) Velocity profiles for different values of 119876 (b) Temperature profiles for different values of 119876

Table 6 Effects of 119870119903on skin-friction 119862

119891and Sherwood number

Sh Reminus1119909

119870119903

119862119891

ShReminus1119909

20 62735 0748750 56402 1117370 54421 13018100 52532 15346

Table 7 Effects of Q on skin-friction 119862119891and Nusselt number

NuReminus1119909

119876 119862119891

NuReminus1119909

001 57173 05577005 59040 0512101 62735 04369015 72641 02863

5 Conclusions

The problem of unsteady two-dimensional laminar freeconvective mass transfer flow of a viscous incompressiblefluid through a highly porous medium past an infinitevertical moving porous plate in the thermal radiation heatgeneration and chemical reaction has been studied Thenondimensional governing equations were solved by pertur-bation technique Numerical results are presented to illustratethe details of the flow and heat transfer characteristics andtheir dependence on the material parameters We observethat the velocity increases as the thermal Grashof numberGr solutal Grashof number Gc permeability of the porousmedium 119870 or heat generation parameter 119876 increases whileit decreases as the Prndtl number Pr radiation parameterR Schmidt number Sc or the chemical reaction parameter119870119903 It is observed that the temperature decreases as the

Prandtl number Pr or radiation parameter 119877 increases whileit increases as the heat generation parameter119876 increasesThe

concentration decreases as the Schmidt number Sc or thechemical reaction parameter119870

119903increases

Appendix

One has

1198981=

Γ + radicΓ2minus 4Γ119876

2

1198982=

Γ + radicΓ2+ 4Γ (119899120582 minus 119876)

2

1198983=

Sc + radicSc2 + 4Sc (119899 + 119870119903)

2

1198984=

Sc + radicSc2 + 4119870119903Sc

2

1198985=

1 + radic1 + 4 (119899 + (1119870))

2

1198986=

1 + radic1 + (4119870)

2

1198601=

minusGr1198982

2minus 1198982minus (119899 + (1119870))

1198602=

minusGc1198982

3minus 1198983minus (119899 + (1119870))

1198603= minus (119860

1+ 1198602)

1198604=

minusGr1198982

1minus 1198981minus (1119870)

1198605=

minusGc1198982

4minus 1198984minus (1119870)

1198606= 119880119901minus (1 + 119860

4+ 1198605)

(A1)


References

[1] M Kaviany Principles of Heat Transfer in a Porous MediaSpriger New York NY USA 2nd edition 1999

[2] K Vajravelu ldquoFlow and heat transfer in a saturated over astretching surfacerdquo Zeitschrift fur Angewandte Mathematik undMechanik vol 74 pp 605ndash614 1994

[3] A A Raptis ldquoFlow through a porous medium in the presenceof magnetic fieldrdquo International Journal of Energy Research vol10 no 1 pp 97ndash100 1986

[4] J T Hong C L Tien and M Kaviany ldquoNon-Darcian effectson vertical-plate natural convection in porous media with highporositiesrdquo International Journal of Heat andMass Transfer vol28 no 11 pp 2149ndash2157 1985

[5] C K Chen and C R Lin ldquoNatural convection from anisothermal vertical surface embedded in a thermally stratifiedhigh-porosity mediumrdquo International Journal of EngineeringScience vol 33 no 1 pp 131ndash138 1995

[6] B S Jaiswal and V M Soundalgekar ldquoOscillating plate temper-ature effects on a flow past an infinite vertical porous plate withconstant suction and embedded in a porousmediumrdquoHeat andMass Transfer vol 37 no 2-3 pp 125ndash131 2001

[7] P S Hiremath and P M Patil ldquoFree convection effects onthe oscillatory flow of a couple stress fluid through a porousmediumrdquo Acta Mechanica vol 98 no 1ndash4 pp 143ndash158 1993

[8] BCChandrasekhara andPNagaraju ldquoComposite heat transferin the case of a steady laminar flow of a gray fluid with smalloptical density past a horizontal plate embedded in a saturatedporous mediumrdquo Warme- und Stoffubertragung vol 23 no 6pp 343ndash352 1988

[9] K A Yih ldquoRadiation effect on natural convection over avertical cylinder embedded in porous mediardquo InternationalCommunications in Heat and Mass Transfer vol 26 no 2 pp259ndash267 1999

[10] A A Mohammadein and M F El-Amin ldquoThermal radiationeffects on power-law fluids over a horizontal plate embedded ina porous mediumrdquo International Communications in Heat andMass Transfer vol 27 no 7 pp 1025ndash1035 2000

[11] A Raptis ldquoRadiation and flow through a porous mediumrdquoJournal of Porous Media vol 4 no 3 pp 271ndash273 2001

[12] A Raptis and C Perdikis ldquoUnsteady flow through a highlyporous medium in the presence of radiationrdquo Transport inPorous Media vol 57 no 2 pp 171ndash179 2004

[13] P L Chambre and J D Young ldquoOn the diffusion of a chemicallyreactive species in a laminar boundary layer flowrdquo Physics ofFluids vol 1 no 1 pp 48ndash54 1958

[14] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesen vol 60 no 10 pp 284ndash287 1994

[15] R Muthucumarswamy and P Ganesan ldquoEffect of the chemicalreaction and injection on the flow characteristics in an unsteadyupward motion of an isothermal platerdquo Journal of AppliedMechanics and Technical Physics vol 42 pp 665ndash671 2001

[16] J Anand Rao and S Shivaiah ldquoChemical reaction effectson an unsteady MHD free convective flow past an infinitevertical porous plate with constant suction and heat sourcerdquoInternational Journal of Applied Mathematics and Mechanicsvol 7 no 8 pp 98ndash118 2011

[17] K Vajravelu and A Hadjinicolaou ldquoHeat transfer in a viscousfluid over a stretching sheet with viscous dissipation and

internal heat generationrdquo International Communications inHeatand Mass Transfer vol 20 no 3 pp 417ndash430 1993

[18] M M Molla M A Hossain and L S Yao ldquoNatural convectionflow along a vertical wavy surface with uniform surface temper-ature in presence of heat generationabsorptionrdquo InternationalJournal of Thermal Sciences vol 43 no 2 pp 157ndash163 2004

[19] M S Alam M M Rahman and M A Sattar ldquoMHD Freeconvection heat and mass transfer flow past an inclined surfacewith heat generationrdquo Thammasat International Journal ofScience and Technology vol 11 no 4 pp 1ndash8 2006

[20] A J Chamkha ldquoUnsteady MHD convective heat and masstransfer past a semi-infinite vertical permeable moving platewith heat absorptionrdquo International Journal of EngineeringScience vol 42 no 2 pp 217ndash230 2004

[21] F M Hady R A Mohamed and A Mahdy ldquoMHD free con-vection flow along a vertical wavy surface with heat generationor absorption effectrdquo International Communications inHeat andMass Transfer vol 33 no 10 pp 1253ndash1263 2006

[22] V Ambethkar ldquoNumerical solutions of heat and mass transfereffects of an unsteadyMHD free convective flow past an infinitevertical plate with constant suction and heat source of sinkrdquoInternational Journal of Applied Mathematics and Mechanicsvol 5 no 3 pp 96ndash115 2009

[23] S Mohammed Ibrahim and N Bhaskar Reddy ldquoRadiation andmass transfer effects on MHD free convection flow along astretching surfacewith viscous dissipation and heat generationrdquoInternational Journal of Applied Mathematics and Mechanicsvol 8 no 8 pp 1ndash21 2012

[24] M Q BrewsterThermal Radiative Transfer and Properties JohnWiley amp Sons New York NY USA 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


[14] investigated the effect of the first order homogeneouschemical reaction on the process of an unsteady flow pasta vertical plate with a constant heat and mass transferMuthucumarswamy and Ganesan [15] studied the effect ofchemical reaction and injection on the flow characteristics inan unsteady upward motion of an isothermal plate AnandRao and Shivaiah [16] noticed that the chemical reactioneffects on an unsteady MHD free convective flow past aninfinite vertical porous plate with constant suction and heatsource or generation

The study of heat generation or absorption effects inmoving fluids is important in view of several physical prob-lems such as fluids undergoing exothermic or endothermicchemical reactions Vajravelu and Hadjinicolaou [17] studiedthe heat transfer characteristics in the laminar boundarylayer of a viscous fluid over a stretching sheet with viscousdissipation or frictional heating and internal heat generationMolla et al [18] studied the problem of natural convectionflow along a vertical wavy surface with uniform surfacetemperature in the presence of heat generationabsorptionAlam et al [19] considered the problem of free convectiveheat and mass transfer flow past an inclined semi-infiniteheated surface of a steady viscous incompressible fluid in thepresence of magnetic field and heat generation Chamkha[20] investigated an unsteady convective heat and masstransfer flow past a semi-infinite porous moving plate withheat absorption Hady et al [21] studied the problem of freeconvection flow along a vertical wavy surface embedded ina saturated porous media in the presence of internal heatgeneration or absorption effect Ambethkar [22] investigatedthe numerical solutions of heat and mass transfer effectsof an unsteady MHD free convective flow past an infinitevertical plate with constant suction and heat source or sinkMohammed Ibrahim and Bhaskar Reddy [23] studied theradiation and mass transfer effects on MHD free convectionflow along a stretching surface with viscous dissipation andheat generation

In view of the above studies an unsteady free convectiveheat and mass transfer flow of a viscous incompressibleradiating fluid through a porous medium with high porositybounded by an infinite vertical moving plate is consideredin the presence of heat generation and chemical reaction Itis assumed that the plate is embedded in porous mediumand moves with constant velocity in the flow directionThe equations of continuity linear momentum energy anddiffusion which govern the flow field are solved by using aregular perturbation method The behavior of the velocitytemperature concentration skin friction Nusselt numberand Sherwood number has been discussed for variations inthe governing parameters

2 Mathematical Analysis

An unsteady two-dimensional laminar free convective masstransfer flow of a viscous incompressible fluid through ahighly porous medium past an infinite vertical movingporous plate in the presence of thermal radiation heatgeneration and chemical reaction is considered The fluid

and the porous structure are assumed to be in local thermalequilibrium It is also assumed that there is radiation onlyfrom the fluidThe fluid is a gray emitting and absorbing butnon-scattering medium and the Rosseland approximation isused to describe the radiative heat flux in the energy equationA homogeneous first order chemical reaction between fluidand the species concentration is considered in which the rateof chemical reaction is directly proportional to the speciesconcentration All the fluid properties are assumed to beconstant except that the influence of the density variationwith temperature is considered only in the body force term(Boussinesqrsquos approximation) The 119909

1015840-axis is chosen alongthe plate in the direction opposite to the direction of gravityand the 1199101015840-axis is taken normal to it Since the flow field isof extreme size all the variables are functions of 1199101015840 and 119905

1015840

only Hence under the usual Boussinesqrsquos approximation theequations of mass linear momentum energy and diffusionare

continuity equation

1205971199071015840

1205971199101015840= 0 (1)

momentum equation

1205971199061015840

1205971199051015840+ 1199071015840 1205971199061015840

1205971199101015840= minus

1

120588

1205971199011015840

1205971199091015840+ 120584

12059721199061015840

12059711991010158402

+ 119892120573 (1198791015840minus 1198791015840

infin)

+ 119892120573lowast(1198621015840minus 1198621015840

infin) minus

120584

11987010158401205931199061015840

(2)

energy equation

120590

1205971198791015840

1205971199051015840+ 1205931199071015840 1205971198791015840

1205971199101015840

=

119896

120588119888119901

12059721198791015840

12059711991010158402

minus

120593

120588119888119901

120597119902119903

1205971199101015840+

1198760

120588119888119901

(1198791015840minus 1198791015840

infin)

(3)

diffusion equation

1205971198621015840

1205971199051015840+ 1199071015840 1205971198621015840

1205971199101015840= 119863

12059721198621015840

12059711991010158402

minus 1198701015840

119903(1198621015840minus 1198621015840

infin) (4)

where 1199091015840 1199101015840 and 1199051015840 are the dimensional distances along and

perpendicular to the plate and dimensional time respec-tively 1199061015840 and 119907

1015840 the components of dimensional velocitiesalong 119909

1015840 and 1199101015840 directions respectively 119862

1015840 and 1198791015840 the

dimensional concentration and temperature of the fluidrespectively 120588 the fluid density 120584 the kinematic viscosity 119888

119901

the specific heat at constant pressure120590 the heat capacity ratio119892 the acceleration due to gravity 120573 and 120573

lowast the volumetriccoefficient of thermal and concentration expansion 1198701015840 thepermeability of the porous medium 120593 the porosity 119863 themolecular diffusivity 1198701015840

119903the chemical reaction parameter



119902119903=

minus412059011990412059711987910158404

31198701198901205971199101015840 (5)


119890is the




11987910158404

asymp 41198793


4

infin (6)


1199061015840= 1198801015840

119901 119879

1015840= 1198791015840

119908+ 120576 (119879

1015840

119908minus 1198791015840

infin) 11989011989910158401199051015840

1198621015840= 1198621015840

119908+ 120576 (119879

1015840

119908minus 1198791015840

infin) 11989011989910158401199051015840

at 1199101015840= 0

1199061015840997888rarr 119880

1015840

infin 1198791015840997888rarr 119879

1015840

infin 1198621015840997888rarr 119862

1015840


(7)

where 1198801015840


119908and 119879

1015840

119908are


infin 1198621015840infin and 119879

1015840




1199071015840= minus1198810 (8)




1

120588

1198891199011015840

1198891199091015840= minus

120593120584

11987010158401198801015840

infin (9)


119906 =

1199061015840

1198801015840

infin

119910 =

11988101199101015840

119907

119880119901=

1198801015840

119901

1198801015840

infin

119899 =

1198991015840119907

1198812

0

119905 =

11990510158401198812

0

119907

120582 =

120590

120593

120579 =

1198791015840minus 1198791015840

infin

1198791015840

119908minus 1198791015840

infin

119862 =

1198621015840minus 1198621015840

infin

1198621015840

119908minus 1198621015840

infin

Gr =120584119892120573 (119879

1015840

119908minus 1198791015840

infin)

1198801015840

infin1198812

0

Gc =120584119892120573lowast(1198621015840

119908minus 1198621015840

infin)

1198801015840

infin1198812

0

119870 =

11987010158401198812

0

1205931205842

Pr =120588119888119901120593120592

119896

119877 =

119870119890119896

41205931205901199041198793

infin

119876 =

1198760120584

1205931205881198881199011198812

0

Sc = 120584

119863

119870119903=

1198701015840

119903120584

1198812

0

(10)


120597119906

120597119905

minus

120597119906

120597119910

=

1205972119906

1205971199102+ Gr120579 + Gc119862 +

1

119870

(1 minus 119906)

120582

120597120579

120597119905

minus

120597120579

120597119910

=

1

Γ

1205972120579

1205971199102+ 119876120579

120597119862

120597119905

minus

120597119862

120597119910

=

1

Sc1205972120601

1205971199102minus 119870119903119862

(11)




119906 = 119880119901

120579 = 1 + 120576119890119899119905

119862 = 1 + 120576119890119899119905 at 119910 = 0


(12)





119906 (119910 119905) = 1199060(119910) + 120576119890

1198991199051199061(119910) + 0(120576)

2+ sdot sdot sdot

120579 (119910 119905) = 1205790(119910) + 120576119890

1198991199051205791(119910) + 0(120576)

2+ sdot sdot sdot

119862 (119910 119905) = 1198620(119910) + 120576119890

1198991199051198621(119910) + 0(120576)

2+ sdot sdot sdot

(13)


11990610158401015840

0+ 1199061015840

0minus

1

119870

1199060= minus

1

119870


0

11990610158401015840

1+ 1199061015840

1minus (119899 +

1

119870

)1199061= minusGr120579

1minus Gc119862

1

12057910158401015840

0+ Γ1205791015840

0+ Γ119876120579

0= 0

12057910158401015840

1+ Γ1205791015840

1minus 119899120582Γ120579

1+ Γ119876120579

1= 0

11986210158401015840

0+ Sc1198621015840

0minus Sc119870

1199031198620= 0

11986210158401015840

1+ Sc1198621015840

1minus Sc (119870

119903+ 119899)119862

1= 0

(14)



1199060= 119880119901 1199061= 0 120579

0= 1

1205791= 1 119862

0= 1 119862

1= 1 at 119910 = 0

1199060997888rarr 1 119906

1997888rarr 0 120579

0997888rarr 0

1205791997888rarr 0 119862

0997888rarr 0 119862


(15)




(16)

120579 (119910 119905) = 119890minus1198981119910+ 120576119890119899119905(119890minus1198982119910) (17)

119862 (119910 119905) = 119890minus1198984119910+ 120576119890119899119905(119890minus1198983119910) (18)




119862119891

=

1205911015840

119908

12058811988001198810

= (

120597119906

120597119910

)

119910=0

= (

1205971199060

120597119910

+ 120576119890119899119905 1205971199061

120597119910

)

119910=0

= minus [11986041198981+ 11986051198984+ 11986061198986

+120576119890119899119905(11986031198985+ 11986011198982+ 11986021198983)]

(19)


Nu = minus119909

(1205971198791205971199101015840)1199101015840=0

(1198791015840

119908minus 1198791015840

infin)


= minus(

120597120579

120597119910

)

119910=0

= minus(

1205971205790

120597119910

+ 120576119890119899119905 1205971205791

120597119910

)

119910=0


(20)


Sh = minus119909

(1205971198621205971199101015840)1199101015840=0

(1198621015840

119908minus 1198621015840

infin)


= minus(

120597119862

120597119910

)

119910=0

= minus[

1205971198620

120597119910

+ 120576119890119899119905 1205971198621

120597119910

]

119910=0


(21)





5

4

3

2

1

00 2 4 6 8 10

119906

119910

Gr = 1 2 3 4




119903= 20 119876 = 01 Pr = 071

119880119901= 04 119860 = 05 119905 = 10 119899 = 01 and 120576 = 001 All the






Gc = 1 2 3 4

45

4

35

3

25

2

15

1

05

0

119906

0 1 2 3 4 5 6 7 8 9 10119910


119870 = 2 5 7 10

4

3

2

1

0

119906

0 2 4 6 8 10119910





0 2 4 6 8 10119910

119877 = 3 5 7 10

4

3

2

1

0

119906

(a)

0 2 4 6 8 10119910

119877 = 3 5 7 10120579

1

08

06

04

02

0

(b)


0 2 4 6 8 10119910

35

3

2

25

1

15

0

05

119906

Pr = 071 08 1 125

(a)

0 2 4 6 8 10119910

120579

1

08

06

04

02

0

Pr = 071 08 1 125

(b)





119903







Gr 119862119891

10 4596620 6273530 7950340 96272


119903increases the



3

25

2

15

1

05

0

119906

119910

10 2 3 4 5 6 7 8 9

Sc = 02 04 06 09

(a)

119862

1

08

06

04

02

0

119910

10 2 3 4 5 6

Sc = 02 04 06 09

(b)


35

3

25

2

15

1

05

0

119906

0 2 4 6 8 10119910

119870119903 = 2 5 7 10

(a)

1

08

06

04

02

0

119862

0 1 2 3 4 5 6119910

119870119903 = 2 5 7 10

(b)


119903


Gc 119862119891

10 5164920 6273530 7382140 84907


NuReminus1119909

119877 119862119891

NuReminus1119909

30 67751 0357750 62735 0436970 60693 04763100 59186 05089


NuReminus1119909

Pr 119862119891

NuReminus1119909

071 62735 0436908 58946 0514410 53134 06811125 48515 08853


Sh Reminus1119909

Sc 119862119891

ShReminus1119909

02 62735 0748704 56259 1129106 53141 1451909 50489 18861


0 2 4 6 8 10119910

119876 = 001 005 01 015

45

4

35

3

25

2

15

1

05

0

119906

(a)

1

08

06

04

02

00 2 4 6 8 10

119910

119876 = 001 005 01 015120579

(b)




Sh Reminus1119909

119870119903

119862119891

ShReminus1119909

20 62735 0748750 56402 1117370 54421 13018100 52532 15346


NuReminus1119909

119876 119862119891

NuReminus1119909

001 57173 05577005 59040 0512101 62735 04369015 72641 02863

5 Conclusions




119903increases

Appendix

One has

1198981=


2

1198982=

Γ + radicΓ2+ 4Γ (119899120582 minus 119876)

2

1198983=

Sc + radicSc2 + 4Sc (119899 + 119870119903)

2

1198984=

Sc + radicSc2 + 4119870119903Sc

2

1198985=

1 + radic1 + 4 (119899 + (1119870))

2

1198986=

1 + radic1 + (4119870)

2

1198601=

minusGr1198982

2minus 1198982minus (119899 + (1119870))

1198602=

minusGc1198982

3minus 1198983minus (119899 + (1119870))

1198603= minus (119860

1+ 1198602)

1198604=

minusGr1198982

1minus 1198981minus (1119870)

1198605=

minusGc1198982

4minus 1198984minus (1119870)

1198606= 119880119901minus (1 + 119860

4+ 1198605)

(A1)


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119902119903=

minus412059011990412059711987910158404

31198701198901205971199101015840 (5)


119890is the




11987910158404

asymp 41198793


4

infin (6)


1199061015840= 1198801015840

119901 119879

1015840= 1198791015840

119908+ 120576 (119879

1015840

119908minus 1198791015840

infin) 11989011989910158401199051015840

1198621015840= 1198621015840

119908+ 120576 (119879

1015840

119908minus 1198791015840

infin) 11989011989910158401199051015840

at 1199101015840= 0

1199061015840997888rarr 119880

1015840

infin 1198791015840997888rarr 119879

1015840

infin 1198621015840997888rarr 119862

1015840


(7)

where 1198801015840


119908and 119879

1015840

119908are


infin 1198621015840infin and 119879

1015840




1199071015840= minus1198810 (8)




1

120588

1198891199011015840

1198891199091015840= minus

120593120584

11987010158401198801015840

infin (9)


119906 =

1199061015840

1198801015840

infin

119910 =

11988101199101015840

119907

119880119901=

1198801015840

119901

1198801015840

infin

119899 =

1198991015840119907

1198812

0

119905 =

11990510158401198812

0

119907

120582 =

120590

120593

120579 =

1198791015840minus 1198791015840

infin

1198791015840

119908minus 1198791015840

infin

119862 =

1198621015840minus 1198621015840

infin

1198621015840

119908minus 1198621015840

infin

Gr =120584119892120573 (119879

1015840

119908minus 1198791015840

infin)

1198801015840

infin1198812

0

Gc =120584119892120573lowast(1198621015840

119908minus 1198621015840

infin)

1198801015840

infin1198812

0

119870 =

11987010158401198812

0

1205931205842

Pr =120588119888119901120593120592

119896

119877 =

119870119890119896

41205931205901199041198793

infin

119876 =

1198760120584

1205931205881198881199011198812

0

Sc = 120584

119863

119870119903=

1198701015840

119903120584

1198812

0

(10)


120597119906

120597119905

minus

120597119906

120597119910

=

1205972119906

1205971199102+ Gr120579 + Gc119862 +

1

119870

(1 minus 119906)

120582

120597120579

120597119905

minus

120597120579

120597119910

=

1

Γ

1205972120579

1205971199102+ 119876120579

120597119862

120597119905

minus

120597119862

120597119910

=

1

Sc1205972120601

1205971199102minus 119870119903119862

(11)




119906 = 119880119901

120579 = 1 + 120576119890119899119905

119862 = 1 + 120576119890119899119905 at 119910 = 0


(12)





119906 (119910 119905) = 1199060(119910) + 120576119890

1198991199051199061(119910) + 0(120576)

2+ sdot sdot sdot

120579 (119910 119905) = 1205790(119910) + 120576119890

1198991199051205791(119910) + 0(120576)

2+ sdot sdot sdot

119862 (119910 119905) = 1198620(119910) + 120576119890

1198991199051198621(119910) + 0(120576)

2+ sdot sdot sdot

(13)


11990610158401015840

0+ 1199061015840

0minus

1

119870

1199060= minus

1

119870


0

11990610158401015840

1+ 1199061015840

1minus (119899 +

1

119870

)1199061= minusGr120579

1minus Gc119862

1

12057910158401015840

0+ Γ1205791015840

0+ Γ119876120579

0= 0

12057910158401015840

1+ Γ1205791015840

1minus 119899120582Γ120579

1+ Γ119876120579

1= 0

11986210158401015840

0+ Sc1198621015840

0minus Sc119870

1199031198620= 0

11986210158401015840

1+ Sc1198621015840

1minus Sc (119870

119903+ 119899)119862

1= 0

(14)



1199060= 119880119901 1199061= 0 120579

0= 1

1205791= 1 119862

0= 1 119862

1= 1 at 119910 = 0

1199060997888rarr 1 119906

1997888rarr 0 120579

0997888rarr 0

1205791997888rarr 0 119862

0997888rarr 0 119862


(15)




(16)

120579 (119910 119905) = 119890minus1198981119910+ 120576119890119899119905(119890minus1198982119910) (17)

119862 (119910 119905) = 119890minus1198984119910+ 120576119890119899119905(119890minus1198983119910) (18)




119862119891

=

1205911015840

119908

12058811988001198810

= (

120597119906

120597119910

)

119910=0

= (

1205971199060

120597119910

+ 120576119890119899119905 1205971199061

120597119910

)

119910=0

= minus [11986041198981+ 11986051198984+ 11986061198986

+120576119890119899119905(11986031198985+ 11986011198982+ 11986021198983)]

(19)


Nu = minus119909

(1205971198791205971199101015840)1199101015840=0

(1198791015840

119908minus 1198791015840

infin)


= minus(

120597120579

120597119910

)

119910=0

= minus(

1205971205790

120597119910

+ 120576119890119899119905 1205971205791

120597119910

)

119910=0


(20)


Sh = minus119909

(1205971198621205971199101015840)1199101015840=0

(1198621015840

119908minus 1198621015840

infin)


= minus(

120597119862

120597119910

)

119910=0

= minus[

1205971198620

120597119910

+ 120576119890119899119905 1205971198621

120597119910

]

119910=0


(21)





5

4

3

2

1

00 2 4 6 8 10

119906

119910

Gr = 1 2 3 4




119903= 20 119876 = 01 Pr = 071

119880119901= 04 119860 = 05 119905 = 10 119899 = 01 and 120576 = 001 All the






Gc = 1 2 3 4

45

4

35

3

25

2

15

1

05

0

119906

0 1 2 3 4 5 6 7 8 9 10119910


119870 = 2 5 7 10

4

3

2

1

0

119906

0 2 4 6 8 10119910





0 2 4 6 8 10119910

119877 = 3 5 7 10

4

3

2

1

0

119906

(a)

0 2 4 6 8 10119910

119877 = 3 5 7 10120579

1

08

06

04

02

0

(b)


0 2 4 6 8 10119910

35

3

2

25

1

15

0

05

119906

Pr = 071 08 1 125

(a)

0 2 4 6 8 10119910

120579

1

08

06

04

02

0

Pr = 071 08 1 125

(b)





119903







Gr 119862119891

10 4596620 6273530 7950340 96272


119903increases the



3

25

2

15

1

05

0

119906

119910

10 2 3 4 5 6 7 8 9

Sc = 02 04 06 09

(a)

119862

1

08

06

04

02

0

119910

10 2 3 4 5 6

Sc = 02 04 06 09

(b)


35

3

25

2

15

1

05

0

119906

0 2 4 6 8 10119910

119870119903 = 2 5 7 10

(a)

1

08

06

04

02

0

119862

0 1 2 3 4 5 6119910

119870119903 = 2 5 7 10

(b)


119903


Gc 119862119891

10 5164920 6273530 7382140 84907


NuReminus1119909

119877 119862119891

NuReminus1119909

30 67751 0357750 62735 0436970 60693 04763100 59186 05089


NuReminus1119909

Pr 119862119891

NuReminus1119909

071 62735 0436908 58946 0514410 53134 06811125 48515 08853


Sh Reminus1119909

Sc 119862119891

ShReminus1119909

02 62735 0748704 56259 1129106 53141 1451909 50489 18861


0 2 4 6 8 10119910

119876 = 001 005 01 015

45

4

35

3

25

2

15

1

05

0

119906

(a)

1

08

06

04

02

00 2 4 6 8 10

119910

119876 = 001 005 01 015120579

(b)




Sh Reminus1119909

119870119903

119862119891

ShReminus1119909

20 62735 0748750 56402 1117370 54421 13018100 52532 15346


NuReminus1119909

119876 119862119891

NuReminus1119909

001 57173 05577005 59040 0512101 62735 04369015 72641 02863

5 Conclusions




119903increases

Appendix

One has

1198981=


2

1198982=

Γ + radicΓ2+ 4Γ (119899120582 minus 119876)

2

1198983=

Sc + radicSc2 + 4Sc (119899 + 119870119903)

2

1198984=

Sc + radicSc2 + 4119870119903Sc

2

1198985=

1 + radic1 + 4 (119899 + (1119870))

2

1198986=

1 + radic1 + (4119870)

2

1198601=

minusGr1198982

2minus 1198982minus (119899 + (1119870))

1198602=

minusGc1198982

3minus 1198983minus (119899 + (1119870))

1198603= minus (119860

1+ 1198602)

1198604=

minusGr1198982

1minus 1198981minus (1119870)

1198605=

minusGc1198982

4minus 1198984minus (1119870)

1198606= 119880119901minus (1 + 119860

4+ 1198605)

(A1)


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906 (119910 119905) = 1199060(119910) + 120576119890

1198991199051199061(119910) + 0(120576)

2+ sdot sdot sdot

120579 (119910 119905) = 1205790(119910) + 120576119890

1198991199051205791(119910) + 0(120576)

2+ sdot sdot sdot

119862 (119910 119905) = 1198620(119910) + 120576119890

1198991199051198621(119910) + 0(120576)

2+ sdot sdot sdot

(13)


11990610158401015840

0+ 1199061015840

0minus

1

119870

1199060= minus

1

119870


0

11990610158401015840

1+ 1199061015840

1minus (119899 +

1

119870

)1199061= minusGr120579

1minus Gc119862

1

12057910158401015840

0+ Γ1205791015840

0+ Γ119876120579

0= 0

12057910158401015840

1+ Γ1205791015840

1minus 119899120582Γ120579

1+ Γ119876120579

1= 0

11986210158401015840

0+ Sc1198621015840

0minus Sc119870

1199031198620= 0

11986210158401015840

1+ Sc1198621015840

1minus Sc (119870

119903+ 119899)119862

1= 0

(14)



1199060= 119880119901 1199061= 0 120579

0= 1

1205791= 1 119862

0= 1 119862

1= 1 at 119910 = 0

1199060997888rarr 1 119906

1997888rarr 0 120579

0997888rarr 0

1205791997888rarr 0 119862

0997888rarr 0 119862


(15)




(16)

120579 (119910 119905) = 119890minus1198981119910+ 120576119890119899119905(119890minus1198982119910) (17)

119862 (119910 119905) = 119890minus1198984119910+ 120576119890119899119905(119890minus1198983119910) (18)




119862119891

=

1205911015840

119908

12058811988001198810

= (

120597119906

120597119910

)

119910=0

= (

1205971199060

120597119910

+ 120576119890119899119905 1205971199061

120597119910

)

119910=0

= minus [11986041198981+ 11986051198984+ 11986061198986

+120576119890119899119905(11986031198985+ 11986011198982+ 11986021198983)]

(19)


Nu = minus119909

(1205971198791205971199101015840)1199101015840=0

(1198791015840

119908minus 1198791015840

infin)


= minus(

120597120579

120597119910

)

119910=0

= minus(

1205971205790

120597119910

+ 120576119890119899119905 1205971205791

120597119910

)

119910=0


(20)


Sh = minus119909

(1205971198621205971199101015840)1199101015840=0

(1198621015840

119908minus 1198621015840

infin)


= minus(

120597119862

120597119910

)

119910=0

= minus[

1205971198620

120597119910

+ 120576119890119899119905 1205971198621

120597119910

]

119910=0


(21)





5

4

3

2

1

00 2 4 6 8 10

119906

119910

Gr = 1 2 3 4




119903= 20 119876 = 01 Pr = 071

119880119901= 04 119860 = 05 119905 = 10 119899 = 01 and 120576 = 001 All the






Gc = 1 2 3 4

45

4

35

3

25

2

15

1

05

0

119906

0 1 2 3 4 5 6 7 8 9 10119910


119870 = 2 5 7 10

4

3

2

1

0

119906

0 2 4 6 8 10119910





0 2 4 6 8 10119910

119877 = 3 5 7 10

4

3

2

1

0

119906

(a)

0 2 4 6 8 10119910

119877 = 3 5 7 10120579

1

08

06

04

02

0

(b)


0 2 4 6 8 10119910

35

3

2

25

1

15

0

05

119906

Pr = 071 08 1 125

(a)

0 2 4 6 8 10119910

120579

1

08

06

04

02

0

Pr = 071 08 1 125

(b)





119903







Gr 119862119891

10 4596620 6273530 7950340 96272


119903increases the



3

25

2

15

1

05

0

119906

119910

10 2 3 4 5 6 7 8 9

Sc = 02 04 06 09

(a)

119862

1

08

06

04

02

0

119910

10 2 3 4 5 6

Sc = 02 04 06 09

(b)


35

3

25

2

15

1

05

0

119906

0 2 4 6 8 10119910

119870119903 = 2 5 7 10

(a)

1

08

06

04

02

0

119862

0 1 2 3 4 5 6119910

119870119903 = 2 5 7 10

(b)


119903


Gc 119862119891

10 5164920 6273530 7382140 84907


NuReminus1119909

119877 119862119891

NuReminus1119909

30 67751 0357750 62735 0436970 60693 04763100 59186 05089


NuReminus1119909

Pr 119862119891

NuReminus1119909

071 62735 0436908 58946 0514410 53134 06811125 48515 08853


Sh Reminus1119909

Sc 119862119891

ShReminus1119909

02 62735 0748704 56259 1129106 53141 1451909 50489 18861


0 2 4 6 8 10119910

119876 = 001 005 01 015

45

4

35

3

25

2

15

1

05

0

119906

(a)

1

08

06

04

02

00 2 4 6 8 10

119910

119876 = 001 005 01 015120579

(b)




Sh Reminus1119909

119870119903

119862119891

ShReminus1119909

20 62735 0748750 56402 1117370 54421 13018100 52532 15346


NuReminus1119909

119876 119862119891

NuReminus1119909

001 57173 05577005 59040 0512101 62735 04369015 72641 02863

5 Conclusions




119903increases

Appendix

One has

1198981=


2

1198982=

Γ + radicΓ2+ 4Γ (119899120582 minus 119876)

2

1198983=

Sc + radicSc2 + 4Sc (119899 + 119870119903)

2

1198984=

Sc + radicSc2 + 4119870119903Sc

2

1198985=

1 + radic1 + 4 (119899 + (1119870))

2

1198986=

1 + radic1 + (4119870)

2

1198601=

minusGr1198982

2minus 1198982minus (119899 + (1119870))

1198602=

minusGc1198982

3minus 1198983minus (119899 + (1119870))

1198603= minus (119860

1+ 1198602)

1198604=

minusGr1198982

1minus 1198981minus (1119870)

1198605=

minusGc1198982

4minus 1198984minus (1119870)

1198606= 119880119901minus (1 + 119860

4+ 1198605)

(A1)


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5

4

3

2

1

00 2 4 6 8 10

119906

119910

Gr = 1 2 3 4




119903= 20 119876 = 01 Pr = 071

119880119901= 04 119860 = 05 119905 = 10 119899 = 01 and 120576 = 001 All the






Gc = 1 2 3 4

45

4

35

3

25

2

15

1

05

0

119906

0 1 2 3 4 5 6 7 8 9 10119910


119870 = 2 5 7 10

4

3

2

1

0

119906

0 2 4 6 8 10119910





0 2 4 6 8 10119910

119877 = 3 5 7 10

4

3

2

1

0

119906

(a)

0 2 4 6 8 10119910

119877 = 3 5 7 10120579

1

08

06

04

02

0

(b)


0 2 4 6 8 10119910

35

3

2

25

1

15

0

05

119906

Pr = 071 08 1 125

(a)

0 2 4 6 8 10119910

120579

1

08

06

04

02

0

Pr = 071 08 1 125

(b)





119903







Gr 119862119891

10 4596620 6273530 7950340 96272


119903increases the



3

25

2

15

1

05

0

119906

119910

10 2 3 4 5 6 7 8 9

Sc = 02 04 06 09

(a)

119862

1

08

06

04

02

0

119910

10 2 3 4 5 6

Sc = 02 04 06 09

(b)


35

3

25

2

15

1

05

0

119906

0 2 4 6 8 10119910

119870119903 = 2 5 7 10

(a)

1

08

06

04

02

0

119862

0 1 2 3 4 5 6119910

119870119903 = 2 5 7 10

(b)


119903


Gc 119862119891

10 5164920 6273530 7382140 84907


NuReminus1119909

119877 119862119891

NuReminus1119909

30 67751 0357750 62735 0436970 60693 04763100 59186 05089


NuReminus1119909

Pr 119862119891

NuReminus1119909

071 62735 0436908 58946 0514410 53134 06811125 48515 08853


Sh Reminus1119909

Sc 119862119891

ShReminus1119909

02 62735 0748704 56259 1129106 53141 1451909 50489 18861


0 2 4 6 8 10119910

119876 = 001 005 01 015

45

4

35

3

25

2

15

1

05

0

119906

(a)

1

08

06

04

02

00 2 4 6 8 10

119910

119876 = 001 005 01 015120579

(b)




Sh Reminus1119909

119870119903

119862119891

ShReminus1119909

20 62735 0748750 56402 1117370 54421 13018100 52532 15346


NuReminus1119909

119876 119862119891

NuReminus1119909

001 57173 05577005 59040 0512101 62735 04369015 72641 02863

5 Conclusions




119903increases

Appendix

One has

1198981=


2

1198982=

Γ + radicΓ2+ 4Γ (119899120582 minus 119876)

2

1198983=

Sc + radicSc2 + 4Sc (119899 + 119870119903)

2

1198984=

Sc + radicSc2 + 4119870119903Sc

2

1198985=

1 + radic1 + 4 (119899 + (1119870))

2

1198986=

1 + radic1 + (4119870)

2

1198601=

minusGr1198982

2minus 1198982minus (119899 + (1119870))

1198602=

minusGc1198982

3minus 1198983minus (119899 + (1119870))

1198603= minus (119860

1+ 1198602)

1198604=

minusGr1198982

1minus 1198981minus (1119870)

1198605=

minusGc1198982

4minus 1198984minus (1119870)

1198606= 119880119901minus (1 + 119860

4+ 1198605)

(A1)


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10119910

119877 = 3 5 7 10

4

3

2

1

0

119906

(a)

0 2 4 6 8 10119910

119877 = 3 5 7 10120579

1

08

06

04

02

0

(b)


0 2 4 6 8 10119910

35

3

2

25

1

15

0

05

119906

Pr = 071 08 1 125

(a)

0 2 4 6 8 10119910

120579

1

08

06

04

02

0

Pr = 071 08 1 125

(b)





119903







Gr 119862119891

10 4596620 6273530 7950340 96272


119903increases the



3

25

2

15

1

05

0

119906

119910

10 2 3 4 5 6 7 8 9

Sc = 02 04 06 09

(a)

119862

1

08

06

04

02

0

119910

10 2 3 4 5 6

Sc = 02 04 06 09

(b)


35

3

25

2

15

1

05

0

119906

0 2 4 6 8 10119910

119870119903 = 2 5 7 10

(a)

1

08

06

04

02

0

119862

0 1 2 3 4 5 6119910

119870119903 = 2 5 7 10

(b)


119903


Gc 119862119891

10 5164920 6273530 7382140 84907


NuReminus1119909

119877 119862119891

NuReminus1119909

30 67751 0357750 62735 0436970 60693 04763100 59186 05089


NuReminus1119909

Pr 119862119891

NuReminus1119909

071 62735 0436908 58946 0514410 53134 06811125 48515 08853


Sh Reminus1119909

Sc 119862119891

ShReminus1119909

02 62735 0748704 56259 1129106 53141 1451909 50489 18861


0 2 4 6 8 10119910

119876 = 001 005 01 015

45

4

35

3

25

2

15

1

05

0

119906

(a)

1

08

06

04

02

00 2 4 6 8 10

119910

119876 = 001 005 01 015120579

(b)




Sh Reminus1119909

119870119903

119862119891

ShReminus1119909

20 62735 0748750 56402 1117370 54421 13018100 52532 15346


NuReminus1119909

119876 119862119891

NuReminus1119909

001 57173 05577005 59040 0512101 62735 04369015 72641 02863

5 Conclusions




119903increases

Appendix

One has

1198981=


2

1198982=

Γ + radicΓ2+ 4Γ (119899120582 minus 119876)

2

1198983=

Sc + radicSc2 + 4Sc (119899 + 119870119903)

2

1198984=

Sc + radicSc2 + 4119870119903Sc

2

1198985=

1 + radic1 + 4 (119899 + (1119870))

2

1198986=

1 + radic1 + (4119870)

2

1198601=

minusGr1198982

2minus 1198982minus (119899 + (1119870))

1198602=

minusGc1198982

3minus 1198983minus (119899 + (1119870))

1198603= minus (119860

1+ 1198602)

1198604=

minusGr1198982

1minus 1198981minus (1119870)

1198605=

minusGc1198982

4minus 1198984minus (1119870)

1198606= 119880119901minus (1 + 119860

4+ 1198605)

(A1)


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










3

25

2

15

1

05

0

119906

119910

10 2 3 4 5 6 7 8 9

Sc = 02 04 06 09

(a)

119862

1

08

06

04

02

0

119910

10 2 3 4 5 6

Sc = 02 04 06 09

(b)


35

3

25

2

15

1

05

0

119906

0 2 4 6 8 10119910

119870119903 = 2 5 7 10

(a)

1

08

06

04

02

0

119862

0 1 2 3 4 5 6119910

119870119903 = 2 5 7 10

(b)


119903


Gc 119862119891

10 5164920 6273530 7382140 84907


NuReminus1119909

119877 119862119891

NuReminus1119909

30 67751 0357750 62735 0436970 60693 04763100 59186 05089


NuReminus1119909

Pr 119862119891

NuReminus1119909

071 62735 0436908 58946 0514410 53134 06811125 48515 08853


Sh Reminus1119909

Sc 119862119891

ShReminus1119909

02 62735 0748704 56259 1129106 53141 1451909 50489 18861


0 2 4 6 8 10119910

119876 = 001 005 01 015

45

4

35

3

25

2

15

1

05

0

119906

(a)

1

08

06

04

02

00 2 4 6 8 10

119910

119876 = 001 005 01 015120579

(b)




Sh Reminus1119909

119870119903

119862119891

ShReminus1119909

20 62735 0748750 56402 1117370 54421 13018100 52532 15346


NuReminus1119909

119876 119862119891

NuReminus1119909

001 57173 05577005 59040 0512101 62735 04369015 72641 02863

5 Conclusions




119903increases

Appendix

One has

1198981=


2

1198982=

Γ + radicΓ2+ 4Γ (119899120582 minus 119876)

2

1198983=

Sc + radicSc2 + 4Sc (119899 + 119870119903)

2

1198984=

Sc + radicSc2 + 4119870119903Sc

2

1198985=

1 + radic1 + 4 (119899 + (1119870))

2

1198986=

1 + radic1 + (4119870)

2

1198601=

minusGr1198982

2minus 1198982minus (119899 + (1119870))

1198602=

minusGc1198982

3minus 1198983minus (119899 + (1119870))

1198603= minus (119860

1+ 1198602)

1198604=

minusGr1198982

1minus 1198981minus (1119870)

1198605=

minusGc1198982

4minus 1198984minus (1119870)

1198606= 119880119901minus (1 + 119860

4+ 1198605)

(A1)


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10119910

119876 = 001 005 01 015

45

4

35

3

25

2

15

1

05

0

119906

(a)

1

08

06

04

02

00 2 4 6 8 10

119910

119876 = 001 005 01 015120579

(b)




Sh Reminus1119909

119870119903

119862119891

ShReminus1119909

20 62735 0748750 56402 1117370 54421 13018100 52532 15346


NuReminus1119909

119876 119862119891

NuReminus1119909

001 57173 05577005 59040 0512101 62735 04369015 72641 02863

5 Conclusions




119903increases

Appendix

One has

1198981=


2

1198982=

Γ + radicΓ2+ 4Γ (119899120582 minus 119876)

2

1198983=

Sc + radicSc2 + 4Sc (119899 + 119870119903)

2

1198984=

Sc + radicSc2 + 4119870119903Sc

2

1198985=

1 + radic1 + 4 (119899 + (1119870))

2

1198986=

1 + radic1 + (4119870)

2

1198601=

minusGr1198982

2minus 1198982minus (119899 + (1119870))

1198602=

minusGc1198982

3minus 1198983minus (119899 + (1119870))

1198603= minus (119860

1+ 1198602)

1198604=

minusGr1198982

1minus 1198981minus (1119870)

1198605=

minusGc1198982

4minus 1198984minus (1119870)

1198606= 119880119901minus (1 + 119860

4+ 1198605)

(A1)


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Radiation Effects on Mass Transfer...

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