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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)
4 Journal of Applied Mathematics
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
minus 119887Pr1198911015840119898minus1minus119896
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
Journal of Applied Mathematics 5
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
6 Journal of Applied Mathematics
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
Journal of Applied Mathematics 7
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
8 Journal of Applied Mathematics
[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
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Stochastic AnalysisInternational Journal of
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)
4 Journal of Applied Mathematics
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
minus 119887Pr1198911015840119898minus1minus119896
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
Journal of Applied Mathematics 5
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
6 Journal of Applied Mathematics
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
Journal of Applied Mathematics 7
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
8 Journal of Applied Mathematics
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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)
4 Journal of Applied Mathematics
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
minus 119887Pr1198911015840119898minus1minus119896
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
Journal of Applied Mathematics 5
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
6 Journal of Applied Mathematics
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
Journal of Applied Mathematics 7
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
8 Journal of Applied Mathematics
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
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
minus 119887Pr1198911015840119898minus1minus119896
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
Journal of Applied Mathematics 5
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
6 Journal of Applied Mathematics
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
Journal of Applied Mathematics 7
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
8 Journal of Applied Mathematics
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
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
6 Journal of Applied Mathematics
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
Journal of Applied Mathematics 7
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
8 Journal of Applied Mathematics
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
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
Journal of Applied Mathematics 7
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
8 Journal of Applied Mathematics
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
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
8 Journal of Applied Mathematics
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of