Research Article MHD Flow of a Viscous Fluid over an ......Research Article MHD Flow of a Viscous Fluid over an Exponentially Stretching Sheet in a Porous Medium IftikharAhmad, 1 MuhmmadSajid,

Research ArticleMHD Flow of a Viscous Fluid over an Exponentially StretchingSheet in a Porous Medium

Iftikhar Ahmad1 Muhmmad Sajid2 Wasim Awan1 Muhammad Rafique3 Wajid Aziz4

Manzoor Ahmed1 Aamar Abbasi1 and Moeen Taj1

1 Department of Mathematics University of Azad Jammu and Kashmir Muzaffarabad Azad Kashmir 13100 Pakistan2Theoretical Physics Division PINSTECH PO Nilore Islamabad 44000 Pakistan3Department of Physics University of Azad Jammu and Kashmir Muzaffarabad Azad Kashmir 13100 Pakistan4Department of Computer Sciences and Information Technology University of Azad Kashmir Azad Kashmir 13100 Pakistan

Correspondence should be addressed to Iftikhar Ahmad aaiftikharyahoocom

Received 18 January 2014 Revised 2 April 2014 Accepted 8 April 2014 Published 30 April 2014

Academic Editor Ning Hu

Copyright copy 2014 Iftikhar Ahmad et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Radiation effects onmagnetohydrodynamic (MHD) boundary-layer flow and heat transfer characteristic through a porousmediumdue to an exponentially stretching sheet have been studied Formulation of the problem is based upon the variable thermalconductivity The heat transfer analysis is carried out for both prescribed surface temperature (PST) and prescribed heat flux(PHF) casesThe developed system of nonlinear coupled partial differential equations is transformed to nonlinear coupled ordinarydifferential equations by using similarity transformationsThe series solutions for the transformed of the transformed flow and heattransfer problem were constructed by homotopy analysis method (HAM)The obtained results are analyzed under the influence ofvarious physical parameters

1 Introduction

Since the revolutionary work of Sakiadis [1 2] on boundary-layer flow past a moving surface a great deal of research workhas been carried out for the two-dimensional boundary-layerflows Crane [3] extended the Sakiadis [1 2] flow problemby assuming a stretching boundary Cranersquos problem is oneof the flow problems in boundary layer theory that possessesan exact solution The stretching velocity in Cranersquos problemis linearly proportional to the distance from origin The heatandmass transfer on the flow past a porous stretching surfaceis discussed by P S Gupta and A S Gupta [4] Brady andAcrivos [5] proved the existence and uniqueness of the solu-tion for the stretching flow Three-dimensional flow due toa stretching surface was analyzed by McLeod and Rajagopal[6] In another paper Wang [7] extended Cranersquos problemfor a stretching cylinder Flow problems due to a stretchingsurface have important applications in technology geother-mal energy recovery and manufacturing process such asoil recovery artificial fibers metal extrusion and metalspinning Extensive literature is available regarding the steady

flows in Newtonian and non-Newtonian fluid flows over astretching sheetThe readers are referred to the studies [8ndash14]and references thereinThe transportation of heat in a porousmedium has applications in geothermal systems rough oilmining soil-water contamination and biomechanical prob-lems Vajravelu [15] reported steady flow and heat transfer ofviscous fluids by considering different heating process in aporous medium

The above mentioned problems deal with linear stretch-ing of the surface Magyari and Keller [16] initiated the workby assuming an exponentially stretching surface The heattransfer analysis for flow past an exponentially stretching sur-face was carried out by Elbashbeshy [17] The same problemby considering the constitutive equation of a viscoelastic fluidwas investigated by Khan [18] Due to the applications inelectrical power generator astrophysical flows solar powertechnology space vehicle reentry and so forth the heat trans-fer analysis in the presence of radiation is another importantarea of research Raptis [19] investigated the radiation effectsfor the flow past a semi-infinite flat plate The influence ofthermal radiation in a viscoelastic fluid past a stretching sheet

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 256761 8 pageshttpdxdoiorg1011552014256761

2 Journal of Applied Mathematics

is illustrated by Chen [20] Sajid and Hayat [21] discussed thehomotopy series solution for the influence of radiation onthe flow past an exponentially stretching sheet Chiam [22]investigated the heat transfer analysis with variable thermalconductivity in a stagnation point flow towards a stretchingsheet In another paper the analysis of effects of variablethermal conductivity was discussed by Chiam [23]

The purpose of the present paper is to demonstrate ananalysis for the MHD flow and heat transfer analysis ofa viscous fluid towards an exponentially stretching sheetin the presence of radiation effects and Darcyrsquos resistanceThe analytic series solution is presented by using homotopyanalysis method [24ndash30]

2 Mathematical Formulation of the Problem

Consider a steady incompressible two-dimensional MHDflow of an electrically conducting viscous fluid towards anexponentially stretching sheet in a porous mediumThe fluidoccupies the space 119910 gt 0 and is flowing in the 119909 directionand 119910-axis is normal to the flow A constant magnetic field ofstrength119861

0is applied in the normal direction and the induced

magnetic field is neglected which is a valid assumption whenthe magnetic Reynolds number is small The boundary-layerequations that govern the present flow and heat transferproblem are

120597119906

120597119909+120597V120597119910

= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= minus

1

120588

120597119901

120597119909+ ]

1205972119906

1205971199102minus

]1198961015840119906 minus

1205901198612

0

120588119906 (2)

120597119901

120597119910= 0 (3)

120588119888119901(119906

120597119879

120597119909+ V

120597119879

120597119910) =

120597

120597119910(119896

120597119879

120597119910) minus

120597119902119903

120597119910 (4)

where 119906 and V are the components of velocity in 119909 and 119910

directions respectively 119901 is the pressure 120588 is the density] represents kinematic viscosity 1198961015840 is the permeability ofporous medium 120590 is the electrical conductivity 119879 is the fluidtemperature 119888

119901is the specific heat 119902

119903is radiative heat flux

and 119896 is the variable thermal conductivity defined in thefollowing way

119896 = 119896infin[1 + 120598120579 (120578)] in PST case

119896infin[1 + 120598120601 (120578)] in PHF case

(5)

in which 120598 is the small parameter (120598 gt 0 for gases and120598 lt 0 for solids and liquids) and 120579(120578) and 120601(120578) aredimensionless temperature distributions for PST and PHF

cases respectively The relevant boundary conditions for thepresent problem are

119906 = 1198800119890119909119897 V = 0

119879 = 119879infin+ 1198601198901198861199092119897

(PST)

minus119896120597119879

120597119910= 119861119890(119887+119897)1199092119897

(PHF) at 119910 = 0

(6)

119906 997888rarr 0 119879 997888rarr 119879infin

as 119910 997888rarr infin (7)

where 1198800is a characteristic velocity 119897 is the characteristic

length scale and 119860 119886 119861 and 119887 are the parameters oftemperature profiles depending on the properties of theliquid Equation (7) suggests that 120597119901120597119909 = 0 far away from theplate Since there is no variation in the pressure in 119910 directionas evident from (3) therefore 120597119901120597119909 = 0 is zero throughoutthe flow field and we have from (2)

119906120597119906

120597119909+ V

120597119906

120597119910= ]

1205972119906

1205971199102minus

]1198961015840119906 minus

1205901198612

0

120588119906 (8)

The radiative heat flux is defined as

119902119903= minus

41205901

3119898

1205971198794

120597119910 (9)

in which 1205901is the Stefan-Boltzmann constant and 119898 is the

coefficient of mean absorption Assuming 1198794 as a linear

combination of temperature then

1198794cong minus3119879

4

infin+ 41198793

infin119879 (10)

Substituting (9) and (10) in (4) one gets

120588119888119901(119906

120597119879

120597119909+ V

120597119879

120597119910) =

120597

120597119910(119896

120597119879

120597119910) +

161198793

infin1205901

3119898

1205972119879

1205971199102 (11)

Introducing the dimensionless variables

120578 = radic1198800

2]1198971198901199092119897

119910 120595 (119909 119910) = radic2]11989711988001198901199092119897

119891 (120578) (12)

120579 (120578) =119879 minus 119879infin

119879119908minus 119879infin

(PST) 120601 (120578) =119879 minus 119879infin

119879119908minus 119879infin

(PHF)

(13)

In terms of above variables flow and heat transfer problemstake the following form

119891101584010158401015840minus 211989110158402+ 11989111989110158401015840minus 2 (119877 +119872)119891

1015840= 0

(1 + Tr+120598120579) 12057910158401015840 + Pr1198911205791015840 minus 119886Pr1198911015840120579 + 12059812057910158402= 0 (PST)

(1 + Tr+120598120601) 12060110158401015840 + Pr1198911206011015840 minus 119887Pr1198911015840120601 + 12059812060110158402= 0 (PHF)

119891 (0) = 0 1198911015840(0) = 1

120579 (0) = 1 1206011015840(0) = minus

1

1 + 120598

1198911015840(infin) = 120579 (infin) = 120601 (infin) = 0

(14)

Journal of Applied Mathematics 3

where the dimensionless parameters are given as

119877 =]12059311989711989610158401198800

119890minus119909119897

Pr =120583119888119901

119896infin

119872 =1198971205901198612

∘

1205881198800

119890minus119909119897

Tr =1612059011198793

infin

3119896infin119870

(15)

3 HAM Solution

To find the series solutions for velocity and temperatureprofiles given in (14) we use homotopy analysis method(HAM) The velocity and temperature distributions can beexpressed in terms of the following base functions

120578119895 exp (minus119899120578) | 119895 ge 0 119899 ge 0 (16)

in the form

119891 (120578) = 1198860

00+

infin

sum

119899=0

infin

sum

119896=0

119886119896

119898119899120578119896 exp (minus119899120578)

120579 (120578) =

infin

sum

119899=0

infin

sum

119896=0

119887119896

119898119899120578119896 exp (minus119899120578)

120601 (120578) =

infin

sum

119899=0

infin

sum

119896=0

119888119896

119898119899120578119896 exp (minus119899120578)

(17)

where 119886119896119898119899

119887119896119898119899

and 119888119896

119898119899are the coefficients The following

initial guesses are selected for solution expressions of 119891(120578)120579(120578) and 120601(120578)

1198910(120578) = 1 minus 119890

minus120578 120579

0(120578) = 119890

minus120578

1206010(120578) =

1

1 + 120598119890minus120578

(18)

and auxiliary linear operators are

L1(119891) = 119891

10158401015840minus 1198911015840 L

2(120579) = 120579

10158401015840minus 120579

L3(120601) = 120601

10158401015840minus 120601

(19)

which satisfy

L1[1198621+ 1198622119890120578+ 1198623119890minus120578] = 0 L

2[1198624119890120578+ 1198625119890minus120578] = 0

L3[1198626119890120578+ 1198627119890minus120578] = 0

(20)

where 119862119894(119894 = 1 2 7) are arbitrary constants The zero-

order deformation problems are

(1 minus 119901)L1[119891 (120578 119901) minus 119891

0(120578)] = 119901ℎ

119891N119891[119891 (120578 119901)]

(1 minus 119901)L2[120579 (120578 119901) minus 120579

0(120578)] = 119901ℎ

120579N120579[119891 (120578 119901)]

(1 minus 119901)L3[120601 (120578 119901) minus 120601

0(120578)] = 119901ℎ

120601N120601[119891 (120578 119901)]

119891 (0 119901) = 0 1198911015840(0 119901) = 1

120579 (0 119901) = 1120597120601 (0 119901)

120597120578= minus

1

1 + 120598

1198911015840(infin 119901) = 120579 (infin 119901) = 120601 (infin 119901) = 0

(21)

The nonlinear operators are given by

N119891[119891 (120578 119901)] =

1205973119891 (120578 119901)

1205971205783minus 2(

120597119891 (120578 119901)

120597120578)

2

+ 119891 (120578 119901)1205972119891 (120578 119901)

1205971205782minus 2 (119877 +119872)

120597119891 (120578 119901)

120597120578

(22)

N120579[119891 (120578 119901) 120579 (120578 119901)]

=

[[[[[

[

1 + Tr+120598120579 (120578 119901)1205972120579 (120578 119901)

1205971205782+ Pr119891 (120578 119901)

120597120579 (120578 119901)

120597120578

minus119886Pr120597119891 (120578 119901)

120597120578120579 (120578 119901) + 120598(

120597120579 (120578 119901)

120597120578)

2

]]]]]

]

(23)

N120601[119891 (120578 119901) 120601 (120578 119901)]

=

[[[[

[

1 + Tr+120598120601 (120578 119901)1205972120601 (120578 119901)

1205971205782+ Pr119891 (120578 119901)

120597120601 (120578 119901)

120597120578

minus119887Pr120597119891 (120578 119901)

120597120578120601 (120578 119901) + 120598(

120597120601 (120578 119901)

120597120578)

2

]]]]

]

(24)

inwhich119901 isin [0 1] is an embedding parameter andℎ119891ℎ120579 and

ℎ120601are nonzero auxiliary parameters For119901 = 0 and119901 = 1 we

respectively have

119891 (120578 0) = 1198910(120578) 119891 (120578 1) = 119891 (120578)

120579 (120578 0) = 1205790(120578) 120579 (120578 1) = 120579 (120578)

120601 (120578 0) = 1206010(120578) 120601 (120578 1) = 120601 (120578)

(25)


As 119901 varies from 0 to 1 119891(120578 119901) 120579(120578 119901) and 120601(120578 119901) vary from1198910(120578) 1205790(120578) and 120601

0(120578) to 119891(120578) 120579(120578) and 120601(120578) respectively

Using Taylorrsquos series one can write

119891 (120578 119901) = 1198910(120578) +

infin

sum

119898=1

119891119898(120578) 119901119898

120579 (120578 119901) = 1205790(120578) +

infin

sum

119898=1

120579119898(120578) 119901119898

120601 (120578 119901) = 1206010(120578) +

infin

sum

119898=1

120601119898(120578) 119901119898

(26)

where

119891119898(120578) =

1

119898

120597119898119891 (120578 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

120579119898(120578) =

1

119898

120597119898120579 (120578 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

120601119898(120578) =

1

119898

120597119898120601 (120578 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

(27)

Note that the zero-order deformation equations contain threeauxiliary parameters ℎ

119891 ℎ120579 and ℎ

120601The convergence of series

solutions strongly depends upon these three parametersAssuming ℎ

119891 ℎ120579 and ℎ

120601are chosen in such a way that the

above series are convergent at 119901 = 1 then

119891 (120578) = 1198910(120578) +

infin

sum

119898=1

119891119898(120578)

120579 (120578) = 1205790(120578) +

infin

sum

119898=1

120579119898(120578)

120601 (120578) = 1206010(120578) +

infin

sum

119898=1

120601119898(120578)

(28)

Taking the derivative of (21)ndash(24) 119898-times with respect to119901 then selecting 119901 = 0 and dividing by 119898 the followingexpressions for 119898th order deformation problems are obtain-ed

L1[119891119898(120578) minus 120594

119898119891119898minus1

(120578)] = ℎ119891N119891

119898(120578)

L2[120579119898(120578) minus 120594

119898120579119898minus1

(120578)] = ℎ120579N120579

119898(120578)

L3[120601119898(120578) minus 120594

119898120601119898minus1

(120578)] = ℎ120601N120601

119898(120578)

119891119898 (0) = 119891

1015840

119898(0) = 119891

1015840

119898(infin) = 120579

119898 (0) = 120579119898 (infin)

= 1206011015840

119898(0) = 120601

119898(infin) = 0

(29)

in which

N119891

119898(120578) = 119891

101584010158401015840

119898minus1minus 2 (119877 +119872)119891

1015840

119898minus1

+

119898minus1

sum

119896=0

[119891119898minus1minus119896

11989110158401015840

119896minus 21198911015840

119898minus1minus1198961198911015840

119896]

(30)

N120579

119898(120578) = (1 + Tr) 12057910158401015840

119898minus1+

119898minus1

sum

119896=0

[120598120579119898minus1minus119896

12057910158401015840

119896+ Pr119891

119898minus1minus1198961205791015840

119896

minus 119886Pr1198911015840119898minus1minus119896

120579119896+ 1205981205791015840

119898minus1minus1198961205791015840

119896]

(31)

N120601

119898(120578) = (1 + Tr) 12060110158401015840

119898minus1+

119898minus1

sum

119896=0

[120598120601119898minus1minus119896

12060110158401015840

119896+ Pr119891

119898minus1minus1198961206011015840

119896


120601119896+ 1205981206011015840

119898minus1minus1198961206011015840

119896]

(32)

120594119898=

0 119898 le 1

1 119898 gt 1(33)

The system of linear nonhomogeneous equations (29)ndash(32) issolved using Mathematica in the order119898 = 1 2 3

4 Results and Discussion

The governing nonlinear equations for the MHD flow andheat transfer analysis for a viscous fluid in a porous mediumwith radiation effects and variable thermal conductivity aresolved analytically using homotopy analysis method for bothPST and PHF cases The solutions are given in the form ofanalytic expressions in (28) These analytic expressions con-tain the auxiliary parameters ℎ

119891 ℎ120579 and ℎ

120601 The convergence

and rate of approximation of the obtained series solutionsstrongly depend upon these parameters To examine therange of acceptable values of these parameters we draw the ℎ-curves in Figure 1 for the functions 119891(120578) 120579(120578) and 120601(120578) Thisfigure depicts that the admissible values of ℎ

119891 ℎ120579 and ℎ

120601are

in the interval [minus06 minus03]To see the influence of porosity parameter 119877 Prandtl

number Pr thermal conductivity 120598 temperature parameters119886 and 119887 and thermal radiation Tr on the temperature distri-butions for both PST and PHF cases Figures 2ndash6 have beenplotted Figures (a) are for the PST case and Figures (b) are forPHF case In Figures 2(a) and 2(b) variation of temperaturedistributions with variation in thermal conductivity param-eter 120598 is shown Figure 2(a) illustrates that temperature andthermal boundary-layer thickness in the PST case increasesby increasing thermal conductivity The results in the PHFcase are opposite to that of PST case and are shown in Figure2(b) Figure 3 elucidates that the temperature and thermalboundary-layer thickness increases by an increase in theporosity parameter Since in the governing equationmagneticparameter and porosity parameter appear in the same waytherefore the effects of magnetic parameter are similar tothose of porosity parameter The behavior of temperaturewith changing Prandtl number is quite opposite to that of


minus1 minus08 minus06 minus04 minus02 0

minus15

minus1

minus05

0

05

1Tr = 02 R = 02 M = 03 120598 = 01 Pr = 03 a = b = 03

ℏ

f998400998400(0)120579998400 (0)120601

998400998400(0)

f998400998400(0)

120579998400(0)

120601998400998400(0)

Figure 1 ℎ-curves for the functions 119891(120578) 120579(120578) and 120601(120578) at 20th-order of approximation

0 1 2 3 4 5 6 7

02

04

06

08

1

Tr = 02M = 03 R = 02 Pr = 03 a = 05

120578

120579(120578

)

120598 = 00 025 05 10

(a)

0 1 2 3 4 5 6 7

025

05

075

1

125

15

Tr = 02M = 03 R = 02 Pr = 03 b = 05

120578

120598 = 10 05 025 0

120601(120578

)

(b)

Figure 2 Influence of thermal conductivity parameter 120598 on temperature profiles (a) for PST case and (b) for PHF case

0 1 2 3 4 5 6 7

02

04

06

08

1

120578

120579(120578

)

Tr = 02M = 03 120598 = 03 Pr = 03 a = 05

R = 0 4 8 12

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

14

120578

Tr = 02M = 03 120598 = 03 Pr = 03 b = 05

R = 0 4 8 12

120601(120578

)

(b)

Figure 3 Influence of porosity parameter 119877 on temperature profiles (a) for PST case and (b) for PHF case


0 1 2 3 4 5 6 7

02

04

06

08

1Tr = 02M = 03 120598 = 03 R = 02 a = 05

120578

120579(120578

) Pr = 12 09 06 03

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

14Tr = 02M = 03 120598 = 03 R = 02 b = 05

120578

Pr = 12 09 06 03

120601(120578

)

(b)

Figure 4 Influence of Prandtl number Pr on temperature profiles (a) for PST case and (b) for PHF case

0 1 2 3 4 5 6 7

02

04

06

08

1Tr = 02M = 03 120598 = 03 R = 02 Pr = 05

120578

120579(120578

) a = 8 6 4 2

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

Tr = 02M = 03 120598 = 03 R = 02 Pr = 05

120578

b = 8 6 4 2120601(120578

)

(b)

Figure 5 Influence of parameters 119886 and 119887 on temperature profiles (a) for PST case and (b) for PHF case

0 1 2 3 4 5 6 7

02

04

06

08

1

120578

120579(120578

)

a = 05M = 03 120598 = 03 R = 02 Pr = 05

Tr = 05 10 15 20

(a)

0 1 2 3 4 5 6 7

025

05

075

1

125

15

175

120578

b = 05M = 03 120598 = 03 R = 02 Pr = 05

120601(120578

) Tr = 05 10 15 20

(b)

Figure 6 Influence of thermal radiation parameter Tr on temperature profiles (a) for PST case and (b) for PHF case


porosity parameter and is given in Figure 4 Figure 5 is madeto show the influence of parameter 119886 in the PST case andparameter 119887 in the PHF case on the temperature distributionsIt is evident from Figure 5 that effects of these parameters arethe same as that of Prandtl number The influence of thermalradiation on the temperature profiles for both PST and PHFcases is depicted in Figure 6 The plotted profiles show thatthe temperature and thermal boundary-layer thickness is anincreasing function of the thermal radiation

5 Conclusions

Magnetohydrodynamic boundary-layer flow and heat trans-fer characteristic due to exponentially stretching sheet in thepresence of thermal radiation Darcy resistance and variablethermal conductivity have been studied in the paper Thesimilarity solutions are developed using homotopy analysismethod Following are the findings of current study

(1) The variable thermal conductivity has an impact inenhancing the temperatures profile in PST case andreduction in temperature in PHF case

(2) The effect of porosity parameter 119877 increases thetemperature in both PST and PHF cases

(3) The effect of increasing the values of Prandtl numberis to decrease the thermal boundary-layer thicknessin both of the cases

Conflict of Interests

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

Acknowledgment

Muhammad Sajid acknowledges the support from AS-ICTP

References

[1] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfacesrdquo AIChE Journal vol 7 no 1 pp 26ndash28 1961

[2] B C Sakiadis ldquoBoundary layer behaviour on continuous solidsurfaces II the boundary layer on a continuous flat surfacerdquoAIChE Journal vol 17 pp 221ndash225 1961

[3] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift fur Ange-wandteMathematik und Physik vol 21 no 4 pp 645ndash647 1970

[4] P S Gupta and A S Gupta ldquoHeat and mass transfer on astretching sheet with suction and blowingrdquoThe Canadian Jour-nal of Chemical Engineering vol 55 no 6 pp 744ndash746 1977

[5] J F Brady andA Acrivos ldquoSteady flow in a channel or tube withan accelerating surface velocity An exact solution to theNavier-Stokes equations with reverse flowrdquo Journal of Fluid Mechanicsvol 112 pp 127ndash150 1981

[6] J B McLeod and K R Rajagopal ldquoOn the uniqueness of flowof a Navier-Stokes fluid due to a stretching boundaryrdquo Archivefor Rational Mechanics and Analysis vol 98 no 4 pp 385ndash3931987

[7] C YWang ldquoThe three-dimensional flow due to a stretching flatsurfacerdquo Physics of Fluids vol 27 no 8 pp 1915ndash1917 1984

[8] C Y Wang ldquoFluid flow due to a stretching cylinderrdquo Physics ofFluids vol 31 pp 466ndash468 1988

[9] R Cortell ldquoFlow and heat transfer of a fluid through aporous medium over a stretching surface with internal heatgenerationabsorption and suctionblowingrdquo Fluid DynamicsResearch vol 37 no 4 pp 231ndash245 2005

[10] R Cortell ldquoA note on magnetohydrodynamic flow of a power-law fluid over a stretching sheetrdquo Applied Mathematics andComputation vol 168 no 1 pp 557ndash566 2005

[11] S-J Liao ldquoOn the analytic solution of magnetohydrodynamicflows of Non-Newtonian fluids over a stretching sheetrdquo Journalof Fluid Mechanics no 488 pp 189ndash212 2003

[12] P D Ariel T Hayat and S Asghar ldquoThe flow of an elastico-vis-cous fluid past a stretching sheet with partial sliprdquo Acta Mech-anica vol 187 no 1ndash4 pp 29ndash35 2006

[13] P D Ariel ldquoAxisymmetric flow of a second grade fluid past astretching sheetrdquo International Journal of Engineering Sciencevol 39 no 5 pp 529ndash553 2001

[14] T Hayat and M Sajid ldquoAnalytic solution for axisymmetric flowand heat transfer of a second grade fluid past a stretching sheetrdquoInternational Journal of Heat and Mass Transfer vol 50 no 1-2pp 75ndash84 2007

[15] K Vajravelu ldquoFlow and heat transfer in a saturated porousmed-iumrdquoZeitschrift fur angewandteMathematik undMechanik vol74 pp 605ndash614 1994

[16] E Magyari and B Keller ldquoHeat andmass transfer in the bound-ary layers on an exponentially stretching continuous surfacerdquoJournal of Physics D Applied Physics vol 32 no 5 pp 577ndash5851999

[17] E M A Elbashbeshy ldquoHeat transfer over an exponentiallystretching continuous surface with suctionrdquo Archives of Mech-anics vol 53 no 6 pp 643ndash651 2001

[18] S KKhan ldquoBoundary layer vicoelastic flowover an exponentialstretching sheetrdquo International Journal of AppliedMechanics andEngineering vol 11 pp 321ndash335 2006

[19] A Raptis C Perdikis andH S Takhar ldquoEffect of thermal radia-tion onMHDflowrdquoAppliedMathematics and Computation vol153 no 3 pp 645ndash649 2004

[20] C-H Chen ldquoOn the analytic solution of MHD flow and heattransfer for two types of viscoelastic fluid over a stretchingsheet with energy dissipation internal heat source and thermalradiationrdquo International Journal of Heat and Mass Transfer vol53 no 19-20 pp 4264ndash4273 2010

[21] M Sajid and T Hayat ldquoInfluence of thermal radiation on theboundary layer flow due to an exponentially stretching sheetrdquoInternational Communications in Heat and Mass Transfer vol35 no 3 pp 347ndash356 2008

[22] T C Chiam ldquoHeat transfer with variable conductivity in a stag-nation-point flow towards a stretching sheetrdquo InternationalCommunications in Heat and Mass Transfer vol 23 no 2 pp239ndash248 1996

[23] T C Chiam ldquoHeat transfer in a fluid with variable thermal con-ductivity over a linearly stretching sheetrdquo Acta Mechanica vol129 no 1-2 pp 63ndash72 1998

[24] S J LiaoThe proposed homotopy anslysis technique for the solu-tion of non-linear problems [PhD thesis] Shanghai Jiao TongUniversity 1992

[25] S J Liao Beyond Perturbation Introduction to Homotopy Ana-lysis Method Chapman amp HallCRC Boca Raton Fla USA2003


[26] S Liao ldquoA new branch of solutions of boundary-layer flows overan impermeable stretched platerdquo International Journal of Heatand Mass Transfer vol 48 no 12 pp 2529ndash2539 2005

[27] S J Liao ldquoAn analytic solution of unsteady boundary-layerflows caused by an impulsively stretching platerdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 11 no3 pp 326ndash339 2006

[28] M Sajid T Hayat and S Asghar ldquoOn the analytic solutionof the steady flow of a fourth grade fluidrdquo Physics Letters AGeneral Atomic and Solid State Physics vol 355 no 1 pp 18ndash262006

[29] Z Abbas M Sajid and T Hayat ldquoMHD boundary-layer flow ofan upper-convected Maxwell fluid in a porous channelrdquo Theo-retical and Computational Fluid Dynamics vol 20 no 4 pp229ndash238 2006

[30] S Abbasbandy ldquoThe application of homotopy analysis methodto nonlinear equations arising in heat transferrdquo Physics LettersA vol 360 no 1 pp 109ndash113 2006

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


is illustrated by Chen [20] Sajid and Hayat [21] discussed thehomotopy series solution for the influence of radiation onthe flow past an exponentially stretching sheet Chiam [22]investigated the heat transfer analysis with variable thermalconductivity in a stagnation point flow towards a stretchingsheet In another paper the analysis of effects of variablethermal conductivity was discussed by Chiam [23]

The purpose of the present paper is to demonstrate ananalysis for the MHD flow and heat transfer analysis ofa viscous fluid towards an exponentially stretching sheetin the presence of radiation effects and Darcyrsquos resistanceThe analytic series solution is presented by using homotopyanalysis method [24ndash30]

2 Mathematical Formulation of the Problem

Consider a steady incompressible two-dimensional MHDflow of an electrically conducting viscous fluid towards anexponentially stretching sheet in a porous mediumThe fluidoccupies the space 119910 gt 0 and is flowing in the 119909 directionand 119910-axis is normal to the flow A constant magnetic field ofstrength119861

0is applied in the normal direction and the induced

magnetic field is neglected which is a valid assumption whenthe magnetic Reynolds number is small The boundary-layerequations that govern the present flow and heat transferproblem are

120597119906

120597119909+120597V120597119910

= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= minus

1

120588

120597119901

120597119909+ ]

1205972119906

1205971199102minus

]1198961015840119906 minus

1205901198612

0

120588119906 (2)

120597119901

120597119910= 0 (3)

120588119888119901(119906

120597119879

120597119909+ V

120597119879

120597119910) =

120597

120597119910(119896

120597119879

120597119910) minus

120597119902119903

120597119910 (4)

where 119906 and V are the components of velocity in 119909 and 119910

directions respectively 119901 is the pressure 120588 is the density] represents kinematic viscosity 1198961015840 is the permeability ofporous medium 120590 is the electrical conductivity 119879 is the fluidtemperature 119888

119901is the specific heat 119902

119903is radiative heat flux

and 119896 is the variable thermal conductivity defined in thefollowing way

119896 = 119896infin[1 + 120598120579 (120578)] in PST case

119896infin[1 + 120598120601 (120578)] in PHF case

(5)

in which 120598 is the small parameter (120598 gt 0 for gases and120598 lt 0 for solids and liquids) and 120579(120578) and 120601(120578) aredimensionless temperature distributions for PST and PHF

cases respectively The relevant boundary conditions for thepresent problem are

119906 = 1198800119890119909119897 V = 0

119879 = 119879infin+ 1198601198901198861199092119897

(PST)

minus119896120597119879

120597119910= 119861119890(119887+119897)1199092119897

(PHF) at 119910 = 0

(6)

119906 997888rarr 0 119879 997888rarr 119879infin

as 119910 997888rarr infin (7)

where 1198800is a characteristic velocity 119897 is the characteristic

length scale and 119860 119886 119861 and 119887 are the parameters oftemperature profiles depending on the properties of theliquid Equation (7) suggests that 120597119901120597119909 = 0 far away from theplate Since there is no variation in the pressure in 119910 directionas evident from (3) therefore 120597119901120597119909 = 0 is zero throughoutthe flow field and we have from (2)

119906120597119906

120597119909+ V

120597119906

120597119910= ]

1205972119906

1205971199102minus

]1198961015840119906 minus

1205901198612

0

120588119906 (8)

The radiative heat flux is defined as

119902119903= minus

41205901

3119898

1205971198794

120597119910 (9)

in which 1205901is the Stefan-Boltzmann constant and 119898 is the

coefficient of mean absorption Assuming 1198794 as a linear

combination of temperature then

1198794cong minus3119879

4

infin+ 41198793

infin119879 (10)

Substituting (9) and (10) in (4) one gets

120588119888119901(119906

120597119879

120597119909+ V

120597119879

120597119910) =

120597

120597119910(119896

120597119879

120597119910) +

161198793

infin1205901

3119898

1205972119879

1205971199102 (11)

Introducing the dimensionless variables

120578 = radic1198800

2]1198971198901199092119897

119910 120595 (119909 119910) = radic2]11989711988001198901199092119897

119891 (120578) (12)

120579 (120578) =119879 minus 119879infin

119879119908minus 119879infin

(PST) 120601 (120578) =119879 minus 119879infin

119879119908minus 119879infin

(PHF)

(13)

In terms of above variables flow and heat transfer problemstake the following form

119891101584010158401015840minus 211989110158402+ 11989111989110158401015840minus 2 (119877 +119872)119891

1015840= 0

(1 + Tr+120598120579) 12057910158401015840 + Pr1198911205791015840 minus 119886Pr1198911015840120579 + 12059812057910158402= 0 (PST)

(1 + Tr+120598120601) 12060110158401015840 + Pr1198911206011015840 minus 119887Pr1198911015840120601 + 12059812060110158402= 0 (PHF)

119891 (0) = 0 1198911015840(0) = 1

120579 (0) = 1 1206011015840(0) = minus

1

1 + 120598

1198911015840(infin) = 120579 (infin) = 120601 (infin) = 0

(14)



119877 =]12059311989711989610158401198800

119890minus119909119897

Pr =120583119888119901

119896infin

119872 =1198971205901198612

∘

1205881198800

119890minus119909119897

Tr =1612059011198793

infin

3119896infin119870

(15)

3 HAM Solution


120578119895 exp (minus119899120578) | 119895 ge 0 119899 ge 0 (16)

in the form

119891 (120578) = 1198860

00+

infin

sum

119899=0

infin

sum

119896=0

119886119896

119898119899120578119896 exp (minus119899120578)

120579 (120578) =

infin

sum

119899=0

infin

sum

119896=0

119887119896

119898119899120578119896 exp (minus119899120578)

120601 (120578) =

infin

sum

119899=0

infin

sum

119896=0

119888119896

119898119899120578119896 exp (minus119899120578)

(17)

where 119886119896119898119899

119887119896119898119899

and 119888119896



1198910(120578) = 1 minus 119890

minus120578 120579

0(120578) = 119890

minus120578

1206010(120578) =

1

1 + 120598119890minus120578

(18)


L1(119891) = 119891

10158401015840minus 1198911015840 L

2(120579) = 120579

10158401015840minus 120579

L3(120601) = 120601

10158401015840minus 120601

(19)

which satisfy

L1[1198621+ 1198622119890120578+ 1198623119890minus120578] = 0 L

2[1198624119890120578+ 1198625119890minus120578] = 0

L3[1198626119890120578+ 1198627119890minus120578] = 0

(20)



(1 minus 119901)L1[119891 (120578 119901) minus 119891

0(120578)] = 119901ℎ

119891N119891[119891 (120578 119901)]

(1 minus 119901)L2[120579 (120578 119901) minus 120579

0(120578)] = 119901ℎ

120579N120579[119891 (120578 119901)]

(1 minus 119901)L3[120601 (120578 119901) minus 120601

0(120578)] = 119901ℎ

120601N120601[119891 (120578 119901)]

119891 (0 119901) = 0 1198911015840(0 119901) = 1

120579 (0 119901) = 1120597120601 (0 119901)

120597120578= minus

1

1 + 120598


(21)


N119891[119891 (120578 119901)] =

1205973119891 (120578 119901)

1205971205783minus 2(

120597119891 (120578 119901)

120597120578)

2

+ 119891 (120578 119901)1205972119891 (120578 119901)

1205971205782minus 2 (119877 +119872)

120597119891 (120578 119901)

120597120578

(22)

N120579[119891 (120578 119901) 120579 (120578 119901)]

=

[[[[[

[

1 + Tr+120598120579 (120578 119901)1205972120579 (120578 119901)

1205971205782+ Pr119891 (120578 119901)

120597120579 (120578 119901)

120597120578

minus119886Pr120597119891 (120578 119901)

120597120578120579 (120578 119901) + 120598(

120597120579 (120578 119901)

120597120578)

2

]]]]]

]

(23)

N120601[119891 (120578 119901) 120601 (120578 119901)]

=

[[[[

[

1 + Tr+120598120601 (120578 119901)1205972120601 (120578 119901)

1205971205782+ Pr119891 (120578 119901)

120597120601 (120578 119901)

120597120578

minus119887Pr120597119891 (120578 119901)

120597120578120601 (120578 119901) + 120598(

120597120601 (120578 119901)

120597120578)

2

]]]]

]

(24)



respectively have

119891 (120578 0) = 1198910(120578) 119891 (120578 1) = 119891 (120578)

120579 (120578 0) = 1205790(120578) 120579 (120578 1) = 120579 (120578)

120601 (120578 0) = 1206010(120578) 120601 (120578 1) = 120601 (120578)

(25)





119891 (120578 119901) = 1198910(120578) +

infin

sum

119898=1

119891119898(120578) 119901119898

120579 (120578 119901) = 1205790(120578) +

infin

sum

119898=1

120579119898(120578) 119901119898

120601 (120578 119901) = 1206010(120578) +

infin

sum

119898=1

120601119898(120578) 119901119898

(26)

where

119891119898(120578) =

1

119898

120597119898119891 (120578 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

120579119898(120578) =

1

119898

120597119898120579 (120578 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

120601119898(120578) =

1

119898

120597119898120601 (120578 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

(27)


119891 ℎ120579 and ℎ



119891 ℎ120579 and ℎ



119891 (120578) = 1198910(120578) +

infin

sum

119898=1

119891119898(120578)

120579 (120578) = 1205790(120578) +

infin

sum

119898=1

120579119898(120578)

120601 (120578) = 1206010(120578) +

infin

sum

119898=1

120601119898(120578)

(28)


L1[119891119898(120578) minus 120594

119898119891119898minus1

(120578)] = ℎ119891N119891

119898(120578)

L2[120579119898(120578) minus 120594

119898120579119898minus1

(120578)] = ℎ120579N120579

119898(120578)

L3[120601119898(120578) minus 120594

119898120601119898minus1

(120578)] = ℎ120601N120601

119898(120578)

119891119898 (0) = 119891

1015840

119898(0) = 119891

1015840

119898(infin) = 120579

119898 (0) = 120579119898 (infin)

= 1206011015840

119898(0) = 120601

119898(infin) = 0

(29)

in which

N119891

119898(120578) = 119891

101584010158401015840

119898minus1minus 2 (119877 +119872)119891

1015840

119898minus1

+

119898minus1

sum

119896=0

[119891119898minus1minus119896

11989110158401015840

119896minus 21198911015840

119898minus1minus1198961198911015840

119896]

(30)

N120579

119898(120578) = (1 + Tr) 12057910158401015840

119898minus1+

119898minus1

sum

119896=0

[120598120579119898minus1minus119896

12057910158401015840

119896+ Pr119891

119898minus1minus1198961205791015840

119896


120579119896+ 1205981205791015840

119898minus1minus1198961205791015840

119896]

(31)

N120601

119898(120578) = (1 + Tr) 12060110158401015840

119898minus1+

119898minus1

sum

119896=0

[120598120601119898minus1minus119896

12060110158401015840

119896+ Pr119891

119898minus1minus1198961206011015840

119896


120601119896+ 1205981206011015840

119898minus1minus1198961206011015840

119896]

(32)

120594119898=

0 119898 le 1

1 119898 gt 1(33)




119891 ℎ120579 and ℎ



119891 ℎ120579 and ℎ

120601are





minus15

minus1

minus05

0

05

1Tr = 02 R = 02 M = 03 120598 = 01 Pr = 03 a = b = 03

ℏ

f998400998400(0)120579998400 (0)120601

998400998400(0)

f998400998400(0)

120579998400(0)

120601998400998400(0)


0 1 2 3 4 5 6 7

02

04

06

08

1

Tr = 02M = 03 R = 02 Pr = 03 a = 05

120578

120579(120578

)

120598 = 00 025 05 10

(a)

0 1 2 3 4 5 6 7

025

05

075

1

125

15

Tr = 02M = 03 R = 02 Pr = 03 b = 05

120578

120598 = 10 05 025 0

120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1

120578

120579(120578

)

Tr = 02M = 03 120598 = 03 Pr = 03 a = 05

R = 0 4 8 12

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

14

120578

Tr = 02M = 03 120598 = 03 Pr = 03 b = 05

R = 0 4 8 12

120601(120578

)

(b)



0 1 2 3 4 5 6 7

02

04

06

08

1Tr = 02M = 03 120598 = 03 R = 02 a = 05

120578

120579(120578

) Pr = 12 09 06 03

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

14Tr = 02M = 03 120598 = 03 R = 02 b = 05

120578

Pr = 12 09 06 03

120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1Tr = 02M = 03 120598 = 03 R = 02 Pr = 05

120578

120579(120578

) a = 8 6 4 2

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

Tr = 02M = 03 120598 = 03 R = 02 Pr = 05

120578

b = 8 6 4 2120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1

120578

120579(120578

)

a = 05M = 03 120598 = 03 R = 02 Pr = 05

Tr = 05 10 15 20

(a)

0 1 2 3 4 5 6 7

025

05

075

1

125

15

175

120578

b = 05M = 03 120598 = 03 R = 02 Pr = 05

120601(120578

) Tr = 05 10 15 20

(b)




5 Conclusions







Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877 =]12059311989711989610158401198800

119890minus119909119897

Pr =120583119888119901

119896infin

119872 =1198971205901198612

∘

1205881198800

119890minus119909119897

Tr =1612059011198793

infin

3119896infin119870

(15)

3 HAM Solution


120578119895 exp (minus119899120578) | 119895 ge 0 119899 ge 0 (16)

in the form

119891 (120578) = 1198860

00+

infin

sum

119899=0

infin

sum

119896=0

119886119896

119898119899120578119896 exp (minus119899120578)

120579 (120578) =

infin

sum

119899=0

infin

sum

119896=0

119887119896

119898119899120578119896 exp (minus119899120578)

120601 (120578) =

infin

sum

119899=0

infin

sum

119896=0

119888119896

119898119899120578119896 exp (minus119899120578)

(17)

where 119886119896119898119899

119887119896119898119899

and 119888119896



1198910(120578) = 1 minus 119890

minus120578 120579

0(120578) = 119890

minus120578

1206010(120578) =

1

1 + 120598119890minus120578

(18)


L1(119891) = 119891

10158401015840minus 1198911015840 L

2(120579) = 120579

10158401015840minus 120579

L3(120601) = 120601

10158401015840minus 120601

(19)

which satisfy

L1[1198621+ 1198622119890120578+ 1198623119890minus120578] = 0 L

2[1198624119890120578+ 1198625119890minus120578] = 0

L3[1198626119890120578+ 1198627119890minus120578] = 0

(20)



(1 minus 119901)L1[119891 (120578 119901) minus 119891

0(120578)] = 119901ℎ

119891N119891[119891 (120578 119901)]

(1 minus 119901)L2[120579 (120578 119901) minus 120579

0(120578)] = 119901ℎ

120579N120579[119891 (120578 119901)]

(1 minus 119901)L3[120601 (120578 119901) minus 120601

0(120578)] = 119901ℎ

120601N120601[119891 (120578 119901)]

119891 (0 119901) = 0 1198911015840(0 119901) = 1

120579 (0 119901) = 1120597120601 (0 119901)

120597120578= minus

1

1 + 120598


(21)


N119891[119891 (120578 119901)] =

1205973119891 (120578 119901)

1205971205783minus 2(

120597119891 (120578 119901)

120597120578)

2

+ 119891 (120578 119901)1205972119891 (120578 119901)

1205971205782minus 2 (119877 +119872)

120597119891 (120578 119901)

120597120578

(22)

N120579[119891 (120578 119901) 120579 (120578 119901)]

=

[[[[[

[

1 + Tr+120598120579 (120578 119901)1205972120579 (120578 119901)

1205971205782+ Pr119891 (120578 119901)

120597120579 (120578 119901)

120597120578

minus119886Pr120597119891 (120578 119901)

120597120578120579 (120578 119901) + 120598(

120597120579 (120578 119901)

120597120578)

2

]]]]]

]

(23)

N120601[119891 (120578 119901) 120601 (120578 119901)]

=

[[[[

[

1 + Tr+120598120601 (120578 119901)1205972120601 (120578 119901)

1205971205782+ Pr119891 (120578 119901)

120597120601 (120578 119901)

120597120578

minus119887Pr120597119891 (120578 119901)

120597120578120601 (120578 119901) + 120598(

120597120601 (120578 119901)

120597120578)

2

]]]]

]

(24)



respectively have

119891 (120578 0) = 1198910(120578) 119891 (120578 1) = 119891 (120578)

120579 (120578 0) = 1205790(120578) 120579 (120578 1) = 120579 (120578)

120601 (120578 0) = 1206010(120578) 120601 (120578 1) = 120601 (120578)

(25)





119891 (120578 119901) = 1198910(120578) +

infin

sum

119898=1

119891119898(120578) 119901119898

120579 (120578 119901) = 1205790(120578) +

infin

sum

119898=1

120579119898(120578) 119901119898

120601 (120578 119901) = 1206010(120578) +

infin

sum

119898=1

120601119898(120578) 119901119898

(26)

where

119891119898(120578) =

1

119898

120597119898119891 (120578 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

120579119898(120578) =

1

119898

120597119898120579 (120578 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

120601119898(120578) =

1

119898

120597119898120601 (120578 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

(27)


119891 ℎ120579 and ℎ



119891 ℎ120579 and ℎ



119891 (120578) = 1198910(120578) +

infin

sum

119898=1

119891119898(120578)

120579 (120578) = 1205790(120578) +

infin

sum

119898=1

120579119898(120578)

120601 (120578) = 1206010(120578) +

infin

sum

119898=1

120601119898(120578)

(28)


L1[119891119898(120578) minus 120594

119898119891119898minus1

(120578)] = ℎ119891N119891

119898(120578)

L2[120579119898(120578) minus 120594

119898120579119898minus1

(120578)] = ℎ120579N120579

119898(120578)

L3[120601119898(120578) minus 120594

119898120601119898minus1

(120578)] = ℎ120601N120601

119898(120578)

119891119898 (0) = 119891

1015840

119898(0) = 119891

1015840

119898(infin) = 120579

119898 (0) = 120579119898 (infin)

= 1206011015840

119898(0) = 120601

119898(infin) = 0

(29)

in which

N119891

119898(120578) = 119891

101584010158401015840

119898minus1minus 2 (119877 +119872)119891

1015840

119898minus1

+

119898minus1

sum

119896=0

[119891119898minus1minus119896

11989110158401015840

119896minus 21198911015840

119898minus1minus1198961198911015840

119896]

(30)

N120579

119898(120578) = (1 + Tr) 12057910158401015840

119898minus1+

119898minus1

sum

119896=0

[120598120579119898minus1minus119896

12057910158401015840

119896+ Pr119891

119898minus1minus1198961205791015840

119896


120579119896+ 1205981205791015840

119898minus1minus1198961205791015840

119896]

(31)

N120601

119898(120578) = (1 + Tr) 12060110158401015840

119898minus1+

119898minus1

sum

119896=0

[120598120601119898minus1minus119896

12060110158401015840

119896+ Pr119891

119898minus1minus1198961206011015840

119896


120601119896+ 1205981206011015840

119898minus1minus1198961206011015840

119896]

(32)

120594119898=

0 119898 le 1

1 119898 gt 1(33)




119891 ℎ120579 and ℎ



119891 ℎ120579 and ℎ

120601are





minus15

minus1

minus05

0

05

1Tr = 02 R = 02 M = 03 120598 = 01 Pr = 03 a = b = 03

ℏ

f998400998400(0)120579998400 (0)120601

998400998400(0)

f998400998400(0)

120579998400(0)

120601998400998400(0)


0 1 2 3 4 5 6 7

02

04

06

08

1

Tr = 02M = 03 R = 02 Pr = 03 a = 05

120578

120579(120578

)

120598 = 00 025 05 10

(a)

0 1 2 3 4 5 6 7

025

05

075

1

125

15

Tr = 02M = 03 R = 02 Pr = 03 b = 05

120578

120598 = 10 05 025 0

120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1

120578

120579(120578

)

Tr = 02M = 03 120598 = 03 Pr = 03 a = 05

R = 0 4 8 12

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

14

120578

Tr = 02M = 03 120598 = 03 Pr = 03 b = 05

R = 0 4 8 12

120601(120578

)

(b)



0 1 2 3 4 5 6 7

02

04

06

08

1Tr = 02M = 03 120598 = 03 R = 02 a = 05

120578

120579(120578

) Pr = 12 09 06 03

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

14Tr = 02M = 03 120598 = 03 R = 02 b = 05

120578

Pr = 12 09 06 03

120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1Tr = 02M = 03 120598 = 03 R = 02 Pr = 05

120578

120579(120578

) a = 8 6 4 2

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

Tr = 02M = 03 120598 = 03 R = 02 Pr = 05

120578

b = 8 6 4 2120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1

120578

120579(120578

)

a = 05M = 03 120598 = 03 R = 02 Pr = 05

Tr = 05 10 15 20

(a)

0 1 2 3 4 5 6 7

025

05

075

1

125

15

175

120578

b = 05M = 03 120598 = 03 R = 02 Pr = 05

120601(120578

) Tr = 05 10 15 20

(b)




5 Conclusions







Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119891 (120578 119901) = 1198910(120578) +

infin

sum

119898=1

119891119898(120578) 119901119898

120579 (120578 119901) = 1205790(120578) +

infin

sum

119898=1

120579119898(120578) 119901119898

120601 (120578 119901) = 1206010(120578) +

infin

sum

119898=1

120601119898(120578) 119901119898

(26)

where

119891119898(120578) =

1

119898

120597119898119891 (120578 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

120579119898(120578) =

1

119898

120597119898120579 (120578 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

120601119898(120578) =

1

119898

120597119898120601 (120578 119901)

120597119901119898

100381610038161003816100381610038161003816100381610038161003816119901=0

(27)


119891 ℎ120579 and ℎ



119891 ℎ120579 and ℎ



119891 (120578) = 1198910(120578) +

infin

sum

119898=1

119891119898(120578)

120579 (120578) = 1205790(120578) +

infin

sum

119898=1

120579119898(120578)

120601 (120578) = 1206010(120578) +

infin

sum

119898=1

120601119898(120578)

(28)


L1[119891119898(120578) minus 120594

119898119891119898minus1

(120578)] = ℎ119891N119891

119898(120578)

L2[120579119898(120578) minus 120594

119898120579119898minus1

(120578)] = ℎ120579N120579

119898(120578)

L3[120601119898(120578) minus 120594

119898120601119898minus1

(120578)] = ℎ120601N120601

119898(120578)

119891119898 (0) = 119891

1015840

119898(0) = 119891

1015840

119898(infin) = 120579

119898 (0) = 120579119898 (infin)

= 1206011015840

119898(0) = 120601

119898(infin) = 0

(29)

in which

N119891

119898(120578) = 119891

101584010158401015840

119898minus1minus 2 (119877 +119872)119891

1015840

119898minus1

+

119898minus1

sum

119896=0

[119891119898minus1minus119896

11989110158401015840

119896minus 21198911015840

119898minus1minus1198961198911015840

119896]

(30)

N120579

119898(120578) = (1 + Tr) 12057910158401015840

119898minus1+

119898minus1

sum

119896=0

[120598120579119898minus1minus119896

12057910158401015840

119896+ Pr119891

119898minus1minus1198961205791015840

119896


120579119896+ 1205981205791015840

119898minus1minus1198961205791015840

119896]

(31)

N120601

119898(120578) = (1 + Tr) 12060110158401015840

119898minus1+

119898minus1

sum

119896=0

[120598120601119898minus1minus119896

12060110158401015840

119896+ Pr119891

119898minus1minus1198961206011015840

119896


120601119896+ 1205981206011015840

119898minus1minus1198961206011015840

119896]

(32)

120594119898=

0 119898 le 1

1 119898 gt 1(33)




119891 ℎ120579 and ℎ



119891 ℎ120579 and ℎ

120601are





minus15

minus1

minus05

0

05

1Tr = 02 R = 02 M = 03 120598 = 01 Pr = 03 a = b = 03

ℏ

f998400998400(0)120579998400 (0)120601

998400998400(0)

f998400998400(0)

120579998400(0)

120601998400998400(0)


0 1 2 3 4 5 6 7

02

04

06

08

1

Tr = 02M = 03 R = 02 Pr = 03 a = 05

120578

120579(120578

)

120598 = 00 025 05 10

(a)

0 1 2 3 4 5 6 7

025

05

075

1

125

15

Tr = 02M = 03 R = 02 Pr = 03 b = 05

120578

120598 = 10 05 025 0

120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1

120578

120579(120578

)

Tr = 02M = 03 120598 = 03 Pr = 03 a = 05

R = 0 4 8 12

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

14

120578

Tr = 02M = 03 120598 = 03 Pr = 03 b = 05

R = 0 4 8 12

120601(120578

)

(b)



0 1 2 3 4 5 6 7

02

04

06

08

1Tr = 02M = 03 120598 = 03 R = 02 a = 05

120578

120579(120578

) Pr = 12 09 06 03

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

14Tr = 02M = 03 120598 = 03 R = 02 b = 05

120578

Pr = 12 09 06 03

120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1Tr = 02M = 03 120598 = 03 R = 02 Pr = 05

120578

120579(120578

) a = 8 6 4 2

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

Tr = 02M = 03 120598 = 03 R = 02 Pr = 05

120578

b = 8 6 4 2120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1

120578

120579(120578

)

a = 05M = 03 120598 = 03 R = 02 Pr = 05

Tr = 05 10 15 20

(a)

0 1 2 3 4 5 6 7

025

05

075

1

125

15

175

120578

b = 05M = 03 120598 = 03 R = 02 Pr = 05

120601(120578

) Tr = 05 10 15 20

(b)




5 Conclusions







Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus15

minus1

minus05

0

05

1Tr = 02 R = 02 M = 03 120598 = 01 Pr = 03 a = b = 03

ℏ

f998400998400(0)120579998400 (0)120601

998400998400(0)

f998400998400(0)

120579998400(0)

120601998400998400(0)


0 1 2 3 4 5 6 7

02

04

06

08

1

Tr = 02M = 03 R = 02 Pr = 03 a = 05

120578

120579(120578

)

120598 = 00 025 05 10

(a)

0 1 2 3 4 5 6 7

025

05

075

1

125

15

Tr = 02M = 03 R = 02 Pr = 03 b = 05

120578

120598 = 10 05 025 0

120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1

120578

120579(120578

)

Tr = 02M = 03 120598 = 03 Pr = 03 a = 05

R = 0 4 8 12

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

14

120578

Tr = 02M = 03 120598 = 03 Pr = 03 b = 05

R = 0 4 8 12

120601(120578

)

(b)



0 1 2 3 4 5 6 7

02

04

06

08

1Tr = 02M = 03 120598 = 03 R = 02 a = 05

120578

120579(120578

) Pr = 12 09 06 03

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

14Tr = 02M = 03 120598 = 03 R = 02 b = 05

120578

Pr = 12 09 06 03

120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1Tr = 02M = 03 120598 = 03 R = 02 Pr = 05

120578

120579(120578

) a = 8 6 4 2

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

Tr = 02M = 03 120598 = 03 R = 02 Pr = 05

120578

b = 8 6 4 2120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1

120578

120579(120578

)

a = 05M = 03 120598 = 03 R = 02 Pr = 05

Tr = 05 10 15 20

(a)

0 1 2 3 4 5 6 7

025

05

075

1

125

15

175

120578

b = 05M = 03 120598 = 03 R = 02 Pr = 05

120601(120578

) Tr = 05 10 15 20

(b)




5 Conclusions







Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7

02

04

06

08

1Tr = 02M = 03 120598 = 03 R = 02 a = 05

120578

120579(120578

) Pr = 12 09 06 03

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

14Tr = 02M = 03 120598 = 03 R = 02 b = 05

120578

Pr = 12 09 06 03

120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1Tr = 02M = 03 120598 = 03 R = 02 Pr = 05

120578

120579(120578

) a = 8 6 4 2

(a)

0 1 2 3 4 5 6 7

02

04

06

08

1

12

Tr = 02M = 03 120598 = 03 R = 02 Pr = 05

120578

b = 8 6 4 2120601(120578

)

(b)


0 1 2 3 4 5 6 7

02

04

06

08

1

120578

120579(120578

)

a = 05M = 03 120598 = 03 R = 02 Pr = 05

Tr = 05 10 15 20

(a)

0 1 2 3 4 5 6 7

025

05

075

1

125

15

175

120578

b = 05M = 03 120598 = 03 R = 02 Pr = 05

120601(120578

) Tr = 05 10 15 20

(b)




5 Conclusions







Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











5 Conclusions







Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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