Research ArticleMixed Convective Heat Transfer for MHD Viscoelastic FluidFlow over a Porous Wedge with Thermal Radiation

M M Rashidi12 M Ali3 N Freidoonimehr4 B Rostami4 and M Anwar Hossain5

1 Mechanical Engineering Department Engineering Faculty of Bu-Ali Sina University Hamedan Iran2University of Michigan-Shanghai Jiao Tong University Joint Institute Shanghai Jiao Tong University Shanghai China3Mechanical Engineering Department College of Engineering King Saud University PO Box 800 Riyadh 11421 Saudi Arabia4Young Researchers amp Elite Club Hamedan Branch Islamic Azad University PO Box 6518115743 Hamedan Iran5Department of Mathematics University of Dhaka Dhaka 1000 Bangladesh

Correspondence should be addressed to M M Rashidi mm rashidiyahoocom

Received 13 July 2013 Accepted 5 December 2013 Published 11 February 2014

Academic Editor Rama Subba Reddy Gorla

Copyright copy 2014 M M Rashidi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The main concern of the present paper is to study the MHD mixed convective heat transfer for an incompressible laminarand electrically conducting viscoelastic fluid flow past a permeable wedge with thermal radiation via a semianalyticalnumericalmethod called Homotopy Analysis Method (HAM) The boundary-layer governing partial differential equations (PDEs) aretransformed into highly nonlinear coupled ordinary differential equations (ODEs) consisting of the momentum and energyequations using similarity solutionThe currentHAMsolution demonstrates very good agreementwith previously published studiesfor some special casesThe effects of different physical flow parameters such as wedge angle (120573) magnetic field (M) viscoelastic (k

1)

suctioninjection (119891119908) thermal radiation (Nr) and Prandtl number (Pr) on the fluid velocity component (1198911015840(120578)) and temperature

distribution (120579(120578)) are illustrated graphically and discussed in detail

1 Introduction

Flows over the tips of rockets aircrafts and submarines aresome common examples of stagnation flow applications [1]Since the pioneer work of Hiemenz [2] who reduced theNavier-Stokes equations for the forced convection problemto an ordinary form of third order via a similarity trans-formation Eckert [3] studied a similar solution consideringthe momentum and energy equations Ariel [4] presented anumerical algorithm for computing the laminar two-dime-nsional flow of a second grade fluid near a stagnation pointAbel et al [5] carried out a study on heat transfer in a visco-elastic boundary-layer flow over a stretching sheet consider-ing two types of different heating processes PST and PHFPrasad et al [6] analyzed the effect of variable viscosity onMHD viscoelastic fluid flow and heat transfer over a stretch-ing sheet and revealed that with the increase of magnetic fieldparameter the wall temperature profile decreases Datti et al[7] depicted a notable increase in the thickness of the the-rmal boundary-layer and fluid temperature with the increase

in thermal radiation parameter in a viscoelastic fluid flowover a nonisothermal stretching sheet Aliakbar et al [8]realized that any increase in the elasticity number decreasesthe total amount of heat transfer from the sheet to the fluidFurthermore study of viscoelastic fluid flow and heat transferover a stretching sheet with variable viscosity is carried outby Subhas Abel et al [9] Their study showed that the effectof fluid viscosity parameter is to decrease the temperatureprofile through either porous or nonporous medium due tothe decreasing of thermal boundary-layer thickness Anwaret al [10] demonstrated that there is a value for the mixedconvection parameter in heated cylinder which boundary-layer does not separate at all and the value of this parameterincreases with the increase of the viscoelastic parameterSonth et al [11] presented hypergeometric (Kummers) func-tion for a viscoelastic fluid flow over a stretching surfacein presence of heat sourcesink viscous dissipation andsuctionblowing Pal [12] presented a numerical solution in astagnation-point flow over a stretching surface with thermalradiation and illustrated the influence of different parameters

Hindawi Publishing CorporationAdvances in Mechanical EngineeringVolume 2014 Article ID 735939 10 pageshttpdxdoiorg1011552014735939

2 Advances in Mechanical Engineering

on velocity temperature and concentration profiles for bothcases of assisting and opposing flows Hsiao [13] showed thatheat andmass transfer ofMHDviscoelasticmixed convectionflow decreases the heat transfer efficiency Rashidi et al[14] employed HAM to study a non-Newtonian flow over anonisothermal wedge Chamkha et al [15] investigatedMHDforced convection flow adjacent to a nonisothermal wedge inthe presence of a heat source or sink by the implicit finite-difference method In another study of MHD convectionof viscoelastic fluid past a porous wedge by Hsiao [16] heshowed that the elastic effect increases the local heat transfercoefficient and heat transfer of a wedge Erfani et al [17]solved an off-centered stagnation flow towards a rotating discusing the Modified Differential Transform Method Rashidiet al [18] investigated a steady incompressible and laminar-free convective flow of a two-dimensional electrically con-ducting viscoelastic fluid over a moving stretching surfacethrough a porous medium analytically Mahapatra et al [19]studied an inclined stagnation point of viscoelastic fluid flowover a stretching sheet with three different temperature dis-tributions Other researchers have studied different aspects ofstagnation point flows [20 21]

HAM is known as one of the most reliable techniquesto solve nonlinear problems HAM was employed by Liaowho was the first to offer a general analytical method fornonlinear problems [20 21] Considering the effects of Brow-nian motion and thermophoresis Mustafa et al [22] studiedstagnation point flow of a nanofluid towards a stretchingsheet using HAM Rashidi et al [23] perused partial slipthermal-diffusion and diffusion-thermo on MHD flow overa rotating disk with viscous dissipation and Ohmic heatingThe mixed convection of an incompressible Maxwell fluidflow over a vertical stretching surface was studied by Abbaset al [24] via HAM considering both cases of assisting andopposing flows Thermal radiation effect on an exponentialstretching surface was perused by Sajid and Hayat [25]via HAM Rashidi et al [26] demonstrated the parametricanalysis and optimization of entropy generation in unsteadyMHD flow past a stretching rotating disk using artificialneural network (ANN) particle swarm optimization (PSO)algorithm and HAM Dinarvand et al [27] employed HAMto investigate the unsteady laminar (MHD) flow near theforward stagnation point of a rotating and translating sphereAbbasbandy et al [28] employed HAM for nonlinear bound-ary value problems Nowadays HAM has been employed byresearchers for different nonlinear problems Shahmohamadiet al [29] investigated the flow of a viscous incompressiblefluid between two parallel plates due to the normal motionof the plates using HAM In another study Rashidi et al [30]presented the homotopy simulation for nanofluid dynamicsfrom a nonlinearly stretching isothermal permeable sheetwith transpiration

In the present studywe examine the analytical solution fortwo-dimensional MHD mixed convection viscoelastic fluidflow over a porous wedge with thermal radiation Analyticalsolutions for the velocity and the temperature distributionare obtained using a powerful technique namely HAMThe profiles are plotted and discussed for the variations ofdifferent involved parameters

Tinfin

y

u

m

Ω = 120587120573

B

x

ue(x) = axm

Tw(x) = Tinfin + cxm

Figure 1 Configuration of the flow and geometrical coordinates

2 Problem Statement andMathematical Formulation

We assume the steady 2D MHD mixed convective heattransfer in an incompressible and electrically conductingviscoelastic fluid flow past a permeable wedge in the neigh-borhood of a stagnation point flow with a variable magneticfield 119861(119909) = 119861

0119909(119898minus1)2 normally applied in the y-direction

The coordinate system and geometry of the problem areshown in Figure 1 The velocity of the external flow far awayfrom the wedge is 119906

119890(119909) = 119886119909119898 The induced magnetic

field can be neglected in comparison to the applied magneticfieldThe governing equations for the continuitymomentumand energy using the Boussinesq and the boundary-layerapproximations and the above assumptions can be presentedrespectively as follows (for more details see [31 32])

120597119906

120597119909+120597V120597119910= 0 (1)

119906120597119906

120597119909+ V120597119906

120597119910

= 119906119890

119889119906119890

119889119909+ ]1205972119906

1205971199102

+ 1198960(119906

1205973119906

1205971199091205971199102+120597119906

120597119909

1205972119906

1205971199102+120597119906

120597119910

1205972V1205971199102

+ V1205973119906

1205971199103)

minus1205901198612

120588(119906 minus 119906

119890)

(2)

120588119862119901(119906120597119879

120597119909+ V120597119879

120597119910) = 119896

1205972119879

1205971199102minus

1

120588119862119901

120597119902119903

120597119910 (3)

where 119906 and V are the velocity components in the 119909 and 119910directions along and normal to the wedge surface respec-tively ] is the kinematic viscosity 119896

0is the viscoelasticity

parameter 120590 is the electrical conductivity 120588 is the fluid den-sity 119879 is the fluid temperature 119896 is the thermal conductivity119862119901is the specific heat at constant pressure and 119902

119903is the

radiative heat flux term

Advances in Mechanical Engineering 3

By applying the Rosseland approximation for radiationthe radiative heat flux 119902

119903is introduced as

119902119903= minus

4120590lowast

3119896lowast1205971198794

120597119910 (4)

where 120590lowast and 119896lowast are the Stephan-Boltzman constant and themean absorption coefficient respectivelyWe assume that thetemperature difference within the flow is such that the term1198794 can be expressed as a linear function of temperature This

is accomplished by expanding it in a Taylor series about 119879infin

as follows [33]

1198794= 1198794

infin+ 41198793

infin(119879 minus 119879

infin) + 6119879

2

infin(119879 minus 119879

infin)2

+ sdot sdot sdot (5)

By neglecting the second and higher-order terms in theabove equation beyond the first degree in (119879minus119879

infin) we obtain

1198794cong 41198793

infin119879 minus 3119879

4

infin (6)

Applying the above approximation to (4) we have

119902119903= minus

16120590lowast1198793infin

3119896lowast120597119879

120597119910 (7)

The appropriate boundary conditions are introduced as

119906 = 0 V = V119908

119879 = 119879119908(119909) at 119910 = 0

119906 997888rarr 119906119890(119909)

120597119906

120597119910997888rarr 0 119879 997888rarr 119879

infinas 119910 997888rarr infin

(8)

The suctioninjection velocity distribution across thewedge surface is assumed to have a function form of V

119908=

minus 119891119908(]119886)12((119898 + 1)2)119909(119898minus1)2 The wedge surface temper-

ature is equal to 119879119908(119909) = 119879

infin+ 119888119909119898 that 120573 = 2119898119898 + 1

is the wedge angle parameter which corresponds to Ω =

120587120573 for a total angle of the wedge 120573 = 0 and 120573 = 1

correspond to the horizontal wall case and the vertical wallcase respectively and also 119879

infinis the temperature of the

ambient fluidThenondimensional forms of flow velocity andtemperature distribution of (1)ndash(3) are given by introducingthe stream function 120595 and similarity variable 120578

120578 = radic119906119890(119909)

]119909119910 120595 = radic]119909119906

119890(119909)119891 (120578)

120579 (120578) =119879 minus 119879infin

119879119908minus 119879infin

(9)

where120595(119909 119910) satisfies the continuity equation and the streamfunction defined as 119906 = 120597120595120597119910 and V = minus120597120595120597119909 Bysubstituting (7) and (9) into (2)-(3) the momentum and

energy equations are transformed into a nonlinear coupledsystem of similar equations

119898(11989110158402(120578) minus 1) minus

119898 + 1

2119891 (120578) 119891

10158401015840(120578)

minus 119891101584010158401015840(120578) + 119872(119891

1015840(120578) minus 1)

minus 1198961 (3119898 minus 1) 119891

1015840(120578) 119891101584010158401015840(120578)

minus(3119898 minus 1)

2119891101584010158402(120578)

minus(119898 + 1)

2119891 (120578) 119891

(119868119881)(120578) = 0

(1 + 119873119903) 12057910158401015840(120578)

+ Pr(119898 + 12119891 (120578) 120579

1015840(120578) minus 119898119891

1015840(120578) 120579 (120578)) = 0

(10)

where 119872 = 12059011986120119886120588 is the magnetic parameter 119896

1=

1198960119886119909119898minus1] is the viscoelastic parameter (when119898 = 1 (120573 = 1)

the viscoelastic parameter takes the form of 1198961= 1198960119886]

similar to the viscoelastic parameter obtained by Hayatet al [34]) 119873119903 = 16120590

lowast1198793

infin3119896lowast120572 is the thermal radiation

parameter Pr = 120583119862119901119896 is the Prandtl number and primes

denote differentiation with respect to 120578 The correspondingboundary conditions become

119891 (120578) = 119891119908

1198911015840(120578) = 0 120579 (120578) = 1 at 120578 = 0

1198911015840(120578) = 1 119891

10158401015840(120578) = 0 120579 (120578) = 0 as 120578 997888rarr infin

(11)

where 119891119908is the suctioninjection parameter with 119891

119908gt 0

showing a uniform suction through the wedge surface

3 HAM Solution

We select the initial approximations such that the boundaryconditions are satisfied as follows

1198910(120578) = 119890

minus120578+ 120578 + 119891

119908minus 1

1205790(120578) = 119890

minus120578

(12)

The linear operatorsL119891(119891) andL

120579(120579) are introduced as

L119891(119891) =

1205974119891

1205971205784+1205973119891

1205971205783

L120579(120579) =

1205972120579

1205971205782+120597120579

120597120578

(13)

with the following properties

L119891(1198881+ 1198882120578 + 11988831205782+ 1198884119890minus120578) = 0

L120579(1198885+ 1198886119890minus120578) = 0

(14)

4 Advances in Mechanical Engineering

where 119888119894 119894 = 1 minus 6 are the arbitrary constants The nonlinear

operators according to (10) are defined as

N119891[119891 (120578 119902)] = 119898((

120597119891 (120578 119902)

120597120578)

2

minus 1)

minus(119898 + 1)

2119891 (120578 119902)

1205972119891 (120578 119902)

1205971205782minus1205973119891 (120578 119902)

1205971205783

minus 1198961(3119898 minus 1)

120597119891 (120578 119902)

120597120578

1205973119891 (120578 119902)

1205971205783

minus(3119898 minus 1)

2(1205972119891 (120578 119902)

1205971205782)

2

minus(119898 + 1)

2119891 (120578 119902)

1205974119891 (120578 119902)

1205971205784

+119872(120597119891 (120578 119902)

120597120578minus 1)

N120579[119891 (120578 119902) 120579 (120578 119902)]

= (1 + 119873119903)1205972120579 (120578 119902)

1205971205782

+ Pr((119898 + 1)2

119891 (120578 119902)120597120579 (120578 119902)

120597120578

minus 119898120597119891 (120578 119902)

120597120578120579 (120578 119902))

(15)

The auxiliary functions become

119867119891(120578) = 119867

120579(120578) = 119890

minus120578 (16)

The symbolic software Mathematica is employed to solvethe 119894th order deformation equations

L119891[119891119894(120578) minus 120594

119894119891119894minus1(120578)] = ℎH

119891(120578) 119877119891119894(120578)

L120579[120579119894(120578) minus 120594

119894120579119894minus1(120578)] = ℎH

120579(120578) 119877120579119894(120578)

(17)

where ℎ is the auxiliary nonzero parameter and

119877119891119894(120578) = 119898(

119894minus1

sum119895=0

(120597119891119895(120578)

120597120578

120597119891119894minus1minus119895

(120578)

120597120578) minus 1)

minus(119898 + 1)

2

119894minus1

sum119895=0

(119891119895(120578)

1205972119891119894minus1minus119895

(120578)

1205971205782)

minus1205973119891119894minus1(120578)

1205971205783

minus 1198961

119894minus1

sum119895=0

((3119898 minus 1)120597119891119895(120578)

120597120578

1205973119891119894minus1minus119895

(120578)

1205971205783

minus(3119898 minus 1)

2

1205972119891119895(120578)

1205971205782

1205972119891119894minus1minus119895

(120578)

1205971205782

minus(119898 + 1)

2119891119895(120578)

1205974119891119894minus1minus119895

(120578)

1205971205784)

+119872(120597119891119894minus1(120578)

120597120578minus 1)

119877120579119894(120578) = (1 + 119873

119903)1205972120579119894minus1(120578)

1205971205782

+ Pr119894minus1

sum119895=0

((119898 + 1)

2119891119895(120578)

120597120579119894minus1minus119895

(120578)

120597120578

minus 119898120579119895(120578)

120597119891119894minus1minus119895

(120578)

120597120578)

120594119894=

0 119894 le 1

1 119894 gt 1

(18)

are the involved parameters in HAM theory (for more infor-mation about the different steps of HAM see [20 35 36])To control and speed the convergence of the approximationseries by the help of the so-called ℎ-curve it is significant tochoose a proper value of auxiliary parameter The ℎ-curvesof 119891101584010158401015840(0) and 1205791015840(0) obtained by the 18th order of HAMsolution are shown in Figure 2 To obtain the optimal valuesof auxiliary parameters the averaged residual errors aredefined as

Res119891= 119898((

119889119891 (120578)

119889120578)

2

minus 1) minus(119898 + 1)

2119891 (120578)

1198892119891 (120578)

1198891205782

minus1198893119891 (120578)

1198891205783+119872(

119889119891 (120578)

119889120578minus 1)

minus 1198961(3119898 minus 1)

119889119891 (120578)

119889120578

1198893119891 (120578)

1198891205783minus(3119898 minus 1)

2

times (1198892119891 (120578)

1198891205782)

2

minus(119898 + 1)

2119891 (120578)

1198894119891 (120578)

1198891205784

(19)

Res120579= (1 + 119873

119903)1198892120579 (120578)

1198891205782

+ Pr((119898 + 1)2

119891 (120578)119889120579 (120578)

119889120578minus 119898

119889119891 (120578)

119889120578120579 (120578))

(20)

Advances in Mechanical Engineering 5

h

minus12

minus09

minus06

minus03

0

minus15minus12 minus09 minus06 minus03 0

120579998400(0)

f998400998400998400(0)ℏ-c

urve

Figure 2 The ℎ-curves of 119891101584010158401015840(0) and 1205791015840(0) obtained by the 18thorder approximation of HAM solution when119872 = 119891

119908= Pr = 119873119903 =

1 120573 = 13 and 1198961= 3

Resid

ual e

rror

0 2 4 6 8 10

h = minus08

h = minus09

h = minus10

h = minus11

h = minus12 (optimal value)h = minus13

minus00025

minus0002

minus00015

minus0001

minus00005

0

00005

0001

120578

Figure 3 The residual error of (19) when 119891119908= 1 120573 = 13 119896

1= 05

and119872 = 5

In order to survey the accuracy of the present methodthe residual errors for the 18th order of HAM solutions of(19) and (20) are illustrated in Figures 3 and 4 respectivelyIn addition we compare some of our results with the resultsof the previously published studies of [37 38] to highlight

Resid

ual e

rror

0 2 4 6 8 10120578

minus0002

minus00015

minus0001

minus00005

0

00005

h = minus065

h = minus070

h = minus075

h = minus080

h = minus085 (optimal value)h = minus090

Figure 4 The residual error of (20) when 119872 = 119891119908= 119873119903 = 1

120573 = 13 1198961= 05 and Pr = 5

Table 1 Comparison results of 11989110158401015840(0) for different values ofsuctioninjection parameter (119891

119908) when119872 = 119896

1= 0 and 120573 = 1

119891119908

Reference [37] Reference [38] Present resultsminus1 07566 075658 075658018minus05 09692 096923 0969229820 12326 123259 12325936505 15418 154175 1541751721 18893 188931 188931809

the validity of the applied method for some values of fixedparameters 119872 = 119896

1= 0 and 120573 = 1 A very excellent

agreement can be observed between them as seen in Table 1

4 Results and Discussion

The nonlinear ordinary differential equations (10) subjectedto the boundary conditions (11) are solved for some valuesof the wedge angle parameter 120573 magnetic parameter 119872viscoelastic parameter 119896

1 suction parameter 119891

119908 thermal

radiation parameter 119873119903 and Prandtl number Pr via HAMThis section discusses the effects of above flow physicalparameters on the velocity and temperature profiles 1198911015840(120578)and 120579(120578) It should be mentioned that some representativephysical parameters are used to simulate realistic flows

Figures 5 and 6 illustrate the effect of the wedge angleparameter on the velocity profiles and temperature distribu-tions when 119872 = 119891

119908= Pr = 119873119903 = 1 and 119896

1= 05 It

should be noticed that 119898 = 1 (120573 = 1) permits completesimilarity solutions of (10) where 119896

1is constant and not 119891(119909)

6 Advances in Mechanical Engineering

0 1 2 3 4 5 60

02

04

06

08

1

f998400 (120578)

120573 = 0

120573 = 13

120573 = 23

120573 = 1

120578

Figure 5 Effect of wedge angle parameter on the velocity profilewhen119872 = 119891

119908= 1 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

120573 = 0

120573 = 13

120573 = 23

120573 = 1

120578

120579(120578)

Figure 6 Effect of wedge angle parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 119873119903 = 1 and 119896

1= 05

as shown in Figures 5 and 6 However if 119898 = 1 (120573 = 1) solu-tions can be obtained but it will be local in other words localsimilarity is sought as seen in other figures in this sectionAn increase in the wedge angle parameter leads to increase inthe free stream velocity and the Reynolds number and conse-quently the velocity boundary-layer thickness decreases The

0 1 2 3 4 5 60

02

04

06

08

1

M = 0

M = 1

M = 2

M = 3

M = 4

M = 5

120578

f998400 (120578)

Figure 7 Effect of magnetic parameter on the velocity profile when119891119908= 1 120573 = 13 and 119896

1= 05

temperature distribution and the thermal boundary-layerthickness decrease as the wedge angle parameter increasesIndeed increase in the wedge angle parameter causes theincrease in the heat transfer coefficient and the rate of heattransfer

The effect of magnetic parameter on the velocity profilesand temperature distributions is displayed in Figures 7 and 8with 119891

119908= Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05 A drag-

like force named Lorentz force is created by the infliction ofthe vertical magnetic field to the electrically conducting fluidThis force has the tendency to slow down the flow over thewedge Accordingly the velocity and temperature boundary-layer thickness decrease with the increasing of the magneticinteraction parameter

Figures 9 and 10 show the effect of the viscoelastic param-eter on the velocity profile and temperature distribution withthe constant values of other parameters 119872 = 119891

119908= Pr =

119873119903 = 1 and 120573 = 13 As the viscoelastic parameter increasesthe fluid velocity profile decreases and also the temperaturedistribution increasesThis occurs due to the development ofthe tensile stress This behavior is similar to that reported byAnwar et al [10]

The effect of suction parameter on the velocity andtemperature profiles is demonstrated in Figures 11 and 12with 119872 = Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05 In

this study the suction case has been considered in all figuresbased on the boundary-layer assumption which stated thatthe boundary-layer thickness is supposed to be very thinand it will not be allowed to increase as it will violate theboundary-layer assumption displayed by Prandtl in 1904

Advances in Mechanical Engineering 7

0 1 2 3 4 5 60

02

04

06

08

1

M = 0

M = 1

M = 2

M = 3

M = 4

M = 5

120578

120579(120578)

Figure 8 Effect of magnetic parameter on the temperature distri-bution when 119891

119908= Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

k1 = 00

k1 = 05

k1 = 10

k1 = 15

k1 = 20

k1 = 30

120578

f998400 (120578)

Figure 9 Effect of viscoelastic parameter on the velocity profilewhen119872 = 119891

119908= 1 and 120573 = 13

0 1 2 3 4 5 60

02

04

06

08

1

k1 = 00

k1 = 05

k1 = 10

k1 = 15

k1 = 20

k1 = 30

120578

120579(120578)

Figure 10 Effect of viscoelastic parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 119873119903 = 1 and 120573 = 13

0 1 2 3 4 5 60

02

04

06

08

1

120578

f998400 (120578)

fw = 00

fw = 05

fw = 10

fw = 20

fw = 30

Figure 11 Effect of suction parameter on the velocity profile when119872 = 1 120573 = 13 and 119896

1= 05

8 Advances in Mechanical Engineering

0 1 2 3 4 5 60

02

04

06

08

1

120578

fw = 00

fw = 05

fw = 10

fw = 20

fw = 30

120579(120578)

Figure 12 Effect of suction parameter on the temperature distribu-tion when119872 = Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05

Applying the suction at the wedge surface causes the amountof the fluid to draw into the surface and consequently thehydrodynamic boundary-layer becomes thinner In additionthe thermal boundary-layer gets depressed by increasing thesuction parameter

The effects of the thermal radiation parameter and thePrandtl number on the temperature distribution are shownin Figures 13 and 14 when 119872 = 119891

119908= 119873119903 = 1 120573 = 13

and 1198961= 05 The rate of energy transport to the fluid

increases by increasing the thermal radiation parameterThus the temperature of the fluid increases On the otherhand the increase of radiation parameter leads to overcom-ing the effect of convective heat transfer Based on the Prandtlnumber definition (Pr = ]120572) this parameter is definedas the ratio between the momentum diffusion to thermaldiffusion Thus with the increase of Prandtl number thethermal diffusion decreases and so the thermal boundary-layer becomes thinner as seen in Figure 14 It physicallymeans that the flow with large Prandtl number dissipates theheat faster to the fluid as the temperature gradient gets steeperand hence increasing the heat transfer coefficient between thesurface and the fluid

5 Conclusions

In this paper the semi-analyticalnumerical techniqueknown as HAM has been implemented to solve the trans-formed differential equations describing the MHD mixedconvective heat transfer for an incompressible laminarand electrically conducting viscoelastic fluid flow over a

0 1 2 3 4 5 60

02

04

06

08

1

120578

f998400 (120578)

Nr = 00

Nr = 05

Nr = 10

Nr = 20

Nr = 30

Figure 13 Effect of thermal radiation parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 1 120573 = 13 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

120578

120579(120578)

Pr = 071

Pr = 100

Pr = 200

Pr = 300

Pr = 500

Figure 14 Effect of Prandtl number on the temperature distributionwhen119872 = 119891

119908= 119873119903 = 1 120573 = 13 and 119896

1= 05

porous wedge in the presence of the thermal radiation effectThe present semi-numericalanalytical simulations agreeclosely with the previous studies for some special casesHAM has been shown to be a very strong and efficient

Advances in Mechanical Engineering 9

technique in finding analytical solutions for nonlineardifferential equations HAM is displayed to illustrate exce-llent convergence and accuracy and is currently beingemployed to extend the present study to mixed convectiveheat transfer simulations The effects of different physicalkey parameters such as wedge angle parameter magneticparameter viscoelastic parameter suction parameterthermal radiation parameter and Prandtl number are plottedand discussedThe results show that as the wedge angle incre-ases the heat transfer to the fluid increases for other constantspecified parameter and for Pr = 1 The magnetic field has aweak effect on the thermal boundary thickness however thesuction has a remarkable effect on it Increasing the thermalradiation parameter reduces the heat transfer coefficientbetween the wedge and the fluid however increasing Prandtlnumber increases it

Conflict of Interests

On behalf of all the authors there is no conflict of interests toreport

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[2] K Hiemenz Die Grenzschicht an einem in den gleichformigenFlussigkeitsstrom eingetauchten geraden Kreiszylinder WeberBerlin Germany 1911

[3] E RG EckertDie Berechnung desWarmeubergangs in der lam-inaren Grenzschicht umstromter Korper VDI Berlin Germany1942

[4] P D Ariel ldquoA numerical algorithm for computing the stagna-tion point flow of a second grade fluid withwithout suctionrdquoJournal of Computational and Applied Mathematics vol 59 no1 pp 9ndash24 1995

[5] M S Abel P G Siddheshwar andMMNandeppanavar ldquoHeattransfer in a viscoelastic boundary layer flow over a stretchingsheet with viscous dissipation and non-uniform heat sourcerdquoInternational Journal of Heat andMass Transfer vol 50 no 5-6pp 960ndash966 2007

[6] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[7] P S Datti K V Prasad M S Abel and A Joshi ldquoMHDvisco-elastic fluid flow over a non-isothermal stretching sheetrdquoInternational Journal of Engineering Science vol 42 no 8-9 pp935ndash946 2004

[8] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[9] M Subhas Abel S K Khan and K V Prasad ldquoStudy ofvisco-elastic fluid flow and heat transfer over a stretching sheetwith variable viscosityrdquo International Journal of Non-LinearMechanics vol 37 no 1 pp 81ndash88 2002

[10] I Anwar N Amin and I Pop ldquoMixed convection boundarylayer flow of a viscoelastic fluid over a horizontal circular cyli-nderrdquo International Journal ofNon-LinearMechanics vol 43 no9 pp 814ndash821 2008

[11] R M Sonth S K KhanM S Abel and K V Prasad ldquoHeat andmass transfer in a visco-elastic fluid flow over an acceleratingsurface with heat sourcesink and viscous dissipationrdquoHeat andMass Transfer vol 38 no 3 pp 213ndash220 2002

[12] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[13] K-L Hsiao ldquoHeat and mass mixed convection for MHD visco-elastic fluid past a stretching sheet with ohmic dissipationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 7 pp 1803ndash1812 2010

[14] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012

[15] A J Chamkha M Mujtaba A Quadri and C Issa ldquoThermalradiation effects on MHD forced convection flow adjacent to anon-isothermal wedge in the presence of a heat source or sinkrdquoHeat and Mass Transfer vol 39 no 4 pp 305ndash312 2003

[16] K-L Hsiao ldquoMHDmixed convection for viscoelastic fluid pasta porouswedgerdquo International Journal of Non-LinearMechanicsvol 46 no 1 pp 1ndash8 2011

[17] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010

[18] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuously moving stretching surface by HomotopyAnalysis Method with two auxiliary parametersrdquo Journal ofApplied Mathematics vol 2012 Article ID 780415 19 pages2012

[19] T R Mahapatra S Dholey and A S Gupta ldquoOblique stag-nation-point flow of an incompressible visco-elastic fluid towa-rds a stretching surfacerdquo International Journal of Non-LinearMechanics vol 42 no 3 pp 484ndash499 2007

[20] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC New York NY USA2004

[21] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[22] MMustafa T Hayat I Pop S Asghar and S Obaidat ldquoStagna-tion-point flow of a nanofluid towards a stretching sheetrdquo Inter-national Journal of Heat and Mass Transfer vol 54 no 25-26pp 5588ndash5594 2011

[23] M M Rashidi T Hayat E Erfani S A Mohimanian Pour andA A Hendi ldquoSimultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steadyMHDconvective flow

10 Advances in Mechanical Engineering

due to a rotating diskrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[24] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[25] 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

[26] M M Rashidi M Ali N Freidoonimehr and F Nazari ldquoPara-metric analysis and optimization of entropy generation inunsteady MHD flow over a stretching rotating disk using arti-ficial neural network and particle swarm optimization algori-thmrdquo Energy vol 55 pp 497ndash510 2013

[27] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[28] S Abbasbandy E Magyari and E Shivanian ldquoThe homotopyanalysis method for multiple solutions of nonlinear boundaryvalue problemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 9-10 pp 3530ndash3536 2009

[29] H Shahmohamadi M M Rashidi and S Dinarvand ldquoAna-lytic approximate solutions for unsteady two-dimensional andaxisymmetric squeezing flows between parallel platesrdquo Mathe-matical Problems in Engineering vol 2008 Article ID 93509512 pages 2008

[30] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica 2013

[31] X Su L Zheng X Zhang and J Zhang ldquoMHD mixed conve-ctive heat transfer over a permeable stretching wedge withthermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[32] M Massoudi ldquoLocal non-similarity solutions for the flow ofa non-Newtonian fluid over a wedgerdquo International Journal ofNon-Linear Mechanics vol 36 no 6 pp 961ndash976 2001

[33] D Pal ldquoCombined effects of non-uniform heat sourcesink andthermal radiation on heat transfer over an unsteady stretchingpermeable surfacerdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 4 pp 1890ndash1904 2011

[34] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[35] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[36] M M Rashidi S A Mohimanian pour and S AbbasbandyldquoAnalytic approximate solutions for heat transfer of a microp-olar fluid through a porous mediumwith radiationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 16no 4 pp 1874ndash1889 2011

[37] A Ishak R Nazar and I Pop ldquoFalkner-Skan equation for flowpast a moving wedge with suction or injectionrdquo Journal ofApplied Mathematics and Computing vol 25 no 1-2 pp 67ndash832007

[38] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

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Shock and Vibration

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Mechanical Engineering

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Civil EngineeringAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

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DistributedSensor Networks

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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thinspJournalthinspofthinsp

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Navigation and Observation

International Journal of

Advances in Mechanical Engineering 3

By applying the Rosseland approximation for radiationthe radiative heat flux 119902

119903is introduced as

119902119903= minus

4120590lowast

3119896lowast1205971198794

120597119910 (4)

where 120590lowast and 119896lowast are the Stephan-Boltzman constant and themean absorption coefficient respectivelyWe assume that thetemperature difference within the flow is such that the term1198794 can be expressed as a linear function of temperature This

is accomplished by expanding it in a Taylor series about 119879infin

as follows [33]

1198794= 1198794

infin+ 41198793

infin(119879 minus 119879

infin) + 6119879

2

infin(119879 minus 119879

infin)2

+ sdot sdot sdot (5)

By neglecting the second and higher-order terms in theabove equation beyond the first degree in (119879minus119879

infin) we obtain

1198794cong 41198793

infin119879 minus 3119879

4

infin (6)

Applying the above approximation to (4) we have

119902119903= minus

16120590lowast1198793infin

3119896lowast120597119879

120597119910 (7)

The appropriate boundary conditions are introduced as

119906 = 0 V = V119908

119879 = 119879119908(119909) at 119910 = 0

119906 997888rarr 119906119890(119909)

120597119906

120597119910997888rarr 0 119879 997888rarr 119879

infinas 119910 997888rarr infin

(8)

The suctioninjection velocity distribution across thewedge surface is assumed to have a function form of V

119908=

minus 119891119908(]119886)12((119898 + 1)2)119909(119898minus1)2 The wedge surface temper-

ature is equal to 119879119908(119909) = 119879

infin+ 119888119909119898 that 120573 = 2119898119898 + 1

is the wedge angle parameter which corresponds to Ω =

120587120573 for a total angle of the wedge 120573 = 0 and 120573 = 1

correspond to the horizontal wall case and the vertical wallcase respectively and also 119879

infinis the temperature of the

ambient fluidThenondimensional forms of flow velocity andtemperature distribution of (1)ndash(3) are given by introducingthe stream function 120595 and similarity variable 120578

120578 = radic119906119890(119909)

]119909119910 120595 = radic]119909119906

119890(119909)119891 (120578)

120579 (120578) =119879 minus 119879infin

119879119908minus 119879infin

(9)

where120595(119909 119910) satisfies the continuity equation and the streamfunction defined as 119906 = 120597120595120597119910 and V = minus120597120595120597119909 Bysubstituting (7) and (9) into (2)-(3) the momentum and

energy equations are transformed into a nonlinear coupledsystem of similar equations

119898(11989110158402(120578) minus 1) minus

119898 + 1

2119891 (120578) 119891

10158401015840(120578)

minus 119891101584010158401015840(120578) + 119872(119891

1015840(120578) minus 1)

minus 1198961 (3119898 minus 1) 119891

1015840(120578) 119891101584010158401015840(120578)

minus(3119898 minus 1)

2119891101584010158402(120578)

minus(119898 + 1)

2119891 (120578) 119891

(119868119881)(120578) = 0

(1 + 119873119903) 12057910158401015840(120578)

+ Pr(119898 + 12119891 (120578) 120579

1015840(120578) minus 119898119891

1015840(120578) 120579 (120578)) = 0

(10)

where 119872 = 12059011986120119886120588 is the magnetic parameter 119896

1=

1198960119886119909119898minus1] is the viscoelastic parameter (when119898 = 1 (120573 = 1)

the viscoelastic parameter takes the form of 1198961= 1198960119886]

similar to the viscoelastic parameter obtained by Hayatet al [34]) 119873119903 = 16120590

lowast1198793

infin3119896lowast120572 is the thermal radiation

parameter Pr = 120583119862119901119896 is the Prandtl number and primes

denote differentiation with respect to 120578 The correspondingboundary conditions become

119891 (120578) = 119891119908

1198911015840(120578) = 0 120579 (120578) = 1 at 120578 = 0

1198911015840(120578) = 1 119891

10158401015840(120578) = 0 120579 (120578) = 0 as 120578 997888rarr infin

(11)

where 119891119908is the suctioninjection parameter with 119891

119908gt 0

showing a uniform suction through the wedge surface

3 HAM Solution

We select the initial approximations such that the boundaryconditions are satisfied as follows

1198910(120578) = 119890

minus120578+ 120578 + 119891

119908minus 1

1205790(120578) = 119890

minus120578

(12)

The linear operatorsL119891(119891) andL

120579(120579) are introduced as

L119891(119891) =

1205974119891

1205971205784+1205973119891

1205971205783

L120579(120579) =

1205972120579

1205971205782+120597120579

120597120578

(13)

with the following properties

L119891(1198881+ 1198882120578 + 11988831205782+ 1198884119890minus120578) = 0

L120579(1198885+ 1198886119890minus120578) = 0

(14)

4 Advances in Mechanical Engineering

where 119888119894 119894 = 1 minus 6 are the arbitrary constants The nonlinear

operators according to (10) are defined as

N119891[119891 (120578 119902)] = 119898((

120597119891 (120578 119902)

120597120578)

2

minus 1)

minus(119898 + 1)

2119891 (120578 119902)

1205972119891 (120578 119902)

1205971205782minus1205973119891 (120578 119902)

1205971205783

minus 1198961(3119898 minus 1)

120597119891 (120578 119902)

120597120578

1205973119891 (120578 119902)

1205971205783

minus(3119898 minus 1)

2(1205972119891 (120578 119902)

1205971205782)

2

minus(119898 + 1)

2119891 (120578 119902)

1205974119891 (120578 119902)

1205971205784

+119872(120597119891 (120578 119902)

120597120578minus 1)

N120579[119891 (120578 119902) 120579 (120578 119902)]

= (1 + 119873119903)1205972120579 (120578 119902)

1205971205782

+ Pr((119898 + 1)2

119891 (120578 119902)120597120579 (120578 119902)

120597120578

minus 119898120597119891 (120578 119902)

120597120578120579 (120578 119902))

(15)

The auxiliary functions become

119867119891(120578) = 119867

120579(120578) = 119890

minus120578 (16)

The symbolic software Mathematica is employed to solvethe 119894th order deformation equations

L119891[119891119894(120578) minus 120594

119894119891119894minus1(120578)] = ℎH

119891(120578) 119877119891119894(120578)

L120579[120579119894(120578) minus 120594

119894120579119894minus1(120578)] = ℎH

120579(120578) 119877120579119894(120578)

(17)

where ℎ is the auxiliary nonzero parameter and

119877119891119894(120578) = 119898(

119894minus1

sum119895=0

(120597119891119895(120578)

120597120578

120597119891119894minus1minus119895

(120578)

120597120578) minus 1)

minus(119898 + 1)

2

119894minus1

sum119895=0

(119891119895(120578)

1205972119891119894minus1minus119895

(120578)

1205971205782)

minus1205973119891119894minus1(120578)

1205971205783

minus 1198961

119894minus1

sum119895=0

((3119898 minus 1)120597119891119895(120578)

120597120578

1205973119891119894minus1minus119895

(120578)

1205971205783

minus(3119898 minus 1)

2

1205972119891119895(120578)

1205971205782

1205972119891119894minus1minus119895

(120578)

1205971205782

minus(119898 + 1)

2119891119895(120578)

1205974119891119894minus1minus119895

(120578)

1205971205784)

+119872(120597119891119894minus1(120578)

120597120578minus 1)

119877120579119894(120578) = (1 + 119873

119903)1205972120579119894minus1(120578)

1205971205782

+ Pr119894minus1

sum119895=0

((119898 + 1)

2119891119895(120578)

120597120579119894minus1minus119895

(120578)

120597120578

minus 119898120579119895(120578)

120597119891119894minus1minus119895

(120578)

120597120578)

120594119894=

0 119894 le 1

1 119894 gt 1

(18)

are the involved parameters in HAM theory (for more infor-mation about the different steps of HAM see [20 35 36])To control and speed the convergence of the approximationseries by the help of the so-called ℎ-curve it is significant tochoose a proper value of auxiliary parameter The ℎ-curvesof 119891101584010158401015840(0) and 1205791015840(0) obtained by the 18th order of HAMsolution are shown in Figure 2 To obtain the optimal valuesof auxiliary parameters the averaged residual errors aredefined as

Res119891= 119898((

119889119891 (120578)

119889120578)

2

minus 1) minus(119898 + 1)

2119891 (120578)

1198892119891 (120578)

1198891205782

minus1198893119891 (120578)

1198891205783+119872(

119889119891 (120578)

119889120578minus 1)

minus 1198961(3119898 minus 1)

119889119891 (120578)

119889120578

1198893119891 (120578)

1198891205783minus(3119898 minus 1)

2

times (1198892119891 (120578)

1198891205782)

2

minus(119898 + 1)

2119891 (120578)

1198894119891 (120578)

1198891205784

(19)

Res120579= (1 + 119873

119903)1198892120579 (120578)

1198891205782

+ Pr((119898 + 1)2

119891 (120578)119889120579 (120578)

119889120578minus 119898

119889119891 (120578)

119889120578120579 (120578))

(20)

Advances in Mechanical Engineering 5

h

minus12

minus09

minus06

minus03

0

minus15minus12 minus09 minus06 minus03 0

120579998400(0)

f998400998400998400(0)ℏ-c

urve

Figure 2 The ℎ-curves of 119891101584010158401015840(0) and 1205791015840(0) obtained by the 18thorder approximation of HAM solution when119872 = 119891

119908= Pr = 119873119903 =

1 120573 = 13 and 1198961= 3

Resid

ual e

rror

0 2 4 6 8 10

h = minus08

h = minus09

h = minus10

h = minus11

h = minus12 (optimal value)h = minus13

minus00025

minus0002

minus00015

minus0001

minus00005

0

00005

0001

120578

Figure 3 The residual error of (19) when 119891119908= 1 120573 = 13 119896

1= 05

and119872 = 5

In order to survey the accuracy of the present methodthe residual errors for the 18th order of HAM solutions of(19) and (20) are illustrated in Figures 3 and 4 respectivelyIn addition we compare some of our results with the resultsof the previously published studies of [37 38] to highlight

Resid

ual e

rror

0 2 4 6 8 10120578

minus0002

minus00015

minus0001

minus00005

0

00005

h = minus065

h = minus070

h = minus075

h = minus080

h = minus085 (optimal value)h = minus090

Figure 4 The residual error of (20) when 119872 = 119891119908= 119873119903 = 1

120573 = 13 1198961= 05 and Pr = 5

Table 1 Comparison results of 11989110158401015840(0) for different values ofsuctioninjection parameter (119891

119908) when119872 = 119896

1= 0 and 120573 = 1

119891119908

Reference [37] Reference [38] Present resultsminus1 07566 075658 075658018minus05 09692 096923 0969229820 12326 123259 12325936505 15418 154175 1541751721 18893 188931 188931809

the validity of the applied method for some values of fixedparameters 119872 = 119896

1= 0 and 120573 = 1 A very excellent

agreement can be observed between them as seen in Table 1

4 Results and Discussion

The nonlinear ordinary differential equations (10) subjectedto the boundary conditions (11) are solved for some valuesof the wedge angle parameter 120573 magnetic parameter 119872viscoelastic parameter 119896

1 suction parameter 119891

119908 thermal

radiation parameter 119873119903 and Prandtl number Pr via HAMThis section discusses the effects of above flow physicalparameters on the velocity and temperature profiles 1198911015840(120578)and 120579(120578) It should be mentioned that some representativephysical parameters are used to simulate realistic flows

Figures 5 and 6 illustrate the effect of the wedge angleparameter on the velocity profiles and temperature distribu-tions when 119872 = 119891

119908= Pr = 119873119903 = 1 and 119896

1= 05 It

should be noticed that 119898 = 1 (120573 = 1) permits completesimilarity solutions of (10) where 119896

1is constant and not 119891(119909)

6 Advances in Mechanical Engineering

0 1 2 3 4 5 60

02

04

06

08

1

f998400 (120578)

120573 = 0

120573 = 13

120573 = 23

120573 = 1

120578

Figure 5 Effect of wedge angle parameter on the velocity profilewhen119872 = 119891

119908= 1 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

120573 = 0

120573 = 13

120573 = 23

120573 = 1

120578

120579(120578)

Figure 6 Effect of wedge angle parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 119873119903 = 1 and 119896

1= 05

as shown in Figures 5 and 6 However if 119898 = 1 (120573 = 1) solu-tions can be obtained but it will be local in other words localsimilarity is sought as seen in other figures in this sectionAn increase in the wedge angle parameter leads to increase inthe free stream velocity and the Reynolds number and conse-quently the velocity boundary-layer thickness decreases The

0 1 2 3 4 5 60

02

04

06

08

1

M = 0

M = 1

M = 2

M = 3

M = 4

M = 5

120578

f998400 (120578)

Figure 7 Effect of magnetic parameter on the velocity profile when119891119908= 1 120573 = 13 and 119896

1= 05

temperature distribution and the thermal boundary-layerthickness decrease as the wedge angle parameter increasesIndeed increase in the wedge angle parameter causes theincrease in the heat transfer coefficient and the rate of heattransfer

The effect of magnetic parameter on the velocity profilesand temperature distributions is displayed in Figures 7 and 8with 119891

119908= Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05 A drag-

like force named Lorentz force is created by the infliction ofthe vertical magnetic field to the electrically conducting fluidThis force has the tendency to slow down the flow over thewedge Accordingly the velocity and temperature boundary-layer thickness decrease with the increasing of the magneticinteraction parameter

Figures 9 and 10 show the effect of the viscoelastic param-eter on the velocity profile and temperature distribution withthe constant values of other parameters 119872 = 119891

119908= Pr =

119873119903 = 1 and 120573 = 13 As the viscoelastic parameter increasesthe fluid velocity profile decreases and also the temperaturedistribution increasesThis occurs due to the development ofthe tensile stress This behavior is similar to that reported byAnwar et al [10]

The effect of suction parameter on the velocity andtemperature profiles is demonstrated in Figures 11 and 12with 119872 = Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05 In

this study the suction case has been considered in all figuresbased on the boundary-layer assumption which stated thatthe boundary-layer thickness is supposed to be very thinand it will not be allowed to increase as it will violate theboundary-layer assumption displayed by Prandtl in 1904

Advances in Mechanical Engineering 7

0 1 2 3 4 5 60

02

04

06

08

1

M = 0

M = 1

M = 2

M = 3

M = 4

M = 5

120578

120579(120578)

Figure 8 Effect of magnetic parameter on the temperature distri-bution when 119891

119908= Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

k1 = 00

k1 = 05

k1 = 10

k1 = 15

k1 = 20

k1 = 30

120578

f998400 (120578)

Figure 9 Effect of viscoelastic parameter on the velocity profilewhen119872 = 119891

119908= 1 and 120573 = 13

0 1 2 3 4 5 60

02

04

06

08

1

k1 = 00

k1 = 05

k1 = 10

k1 = 15

k1 = 20

k1 = 30

120578

120579(120578)

Figure 10 Effect of viscoelastic parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 119873119903 = 1 and 120573 = 13

0 1 2 3 4 5 60

02

04

06

08

1

120578

f998400 (120578)

fw = 00

fw = 05

fw = 10

fw = 20

fw = 30

Figure 11 Effect of suction parameter on the velocity profile when119872 = 1 120573 = 13 and 119896

1= 05

8 Advances in Mechanical Engineering

0 1 2 3 4 5 60

02

04

06

08

1

120578

fw = 00

fw = 05

fw = 10

fw = 20

fw = 30

120579(120578)

Figure 12 Effect of suction parameter on the temperature distribu-tion when119872 = Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05

Applying the suction at the wedge surface causes the amountof the fluid to draw into the surface and consequently thehydrodynamic boundary-layer becomes thinner In additionthe thermal boundary-layer gets depressed by increasing thesuction parameter

The effects of the thermal radiation parameter and thePrandtl number on the temperature distribution are shownin Figures 13 and 14 when 119872 = 119891

119908= 119873119903 = 1 120573 = 13

and 1198961= 05 The rate of energy transport to the fluid

increases by increasing the thermal radiation parameterThus the temperature of the fluid increases On the otherhand the increase of radiation parameter leads to overcom-ing the effect of convective heat transfer Based on the Prandtlnumber definition (Pr = ]120572) this parameter is definedas the ratio between the momentum diffusion to thermaldiffusion Thus with the increase of Prandtl number thethermal diffusion decreases and so the thermal boundary-layer becomes thinner as seen in Figure 14 It physicallymeans that the flow with large Prandtl number dissipates theheat faster to the fluid as the temperature gradient gets steeperand hence increasing the heat transfer coefficient between thesurface and the fluid

5 Conclusions

In this paper the semi-analyticalnumerical techniqueknown as HAM has been implemented to solve the trans-formed differential equations describing the MHD mixedconvective heat transfer for an incompressible laminarand electrically conducting viscoelastic fluid flow over a

0 1 2 3 4 5 60

02

04

06

08

1

120578

f998400 (120578)

Nr = 00

Nr = 05

Nr = 10

Nr = 20

Nr = 30

Figure 13 Effect of thermal radiation parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 1 120573 = 13 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

120578

120579(120578)

Pr = 071

Pr = 100

Pr = 200

Pr = 300

Pr = 500

Figure 14 Effect of Prandtl number on the temperature distributionwhen119872 = 119891

119908= 119873119903 = 1 120573 = 13 and 119896

1= 05

porous wedge in the presence of the thermal radiation effectThe present semi-numericalanalytical simulations agreeclosely with the previous studies for some special casesHAM has been shown to be a very strong and efficient

Advances in Mechanical Engineering 9

technique in finding analytical solutions for nonlineardifferential equations HAM is displayed to illustrate exce-llent convergence and accuracy and is currently beingemployed to extend the present study to mixed convectiveheat transfer simulations The effects of different physicalkey parameters such as wedge angle parameter magneticparameter viscoelastic parameter suction parameterthermal radiation parameter and Prandtl number are plottedand discussedThe results show that as the wedge angle incre-ases the heat transfer to the fluid increases for other constantspecified parameter and for Pr = 1 The magnetic field has aweak effect on the thermal boundary thickness however thesuction has a remarkable effect on it Increasing the thermalradiation parameter reduces the heat transfer coefficientbetween the wedge and the fluid however increasing Prandtlnumber increases it

Conflict of Interests

On behalf of all the authors there is no conflict of interests toreport

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[2] K Hiemenz Die Grenzschicht an einem in den gleichformigenFlussigkeitsstrom eingetauchten geraden Kreiszylinder WeberBerlin Germany 1911

[3] E RG EckertDie Berechnung desWarmeubergangs in der lam-inaren Grenzschicht umstromter Korper VDI Berlin Germany1942

[4] P D Ariel ldquoA numerical algorithm for computing the stagna-tion point flow of a second grade fluid withwithout suctionrdquoJournal of Computational and Applied Mathematics vol 59 no1 pp 9ndash24 1995

[5] M S Abel P G Siddheshwar andMMNandeppanavar ldquoHeattransfer in a viscoelastic boundary layer flow over a stretchingsheet with viscous dissipation and non-uniform heat sourcerdquoInternational Journal of Heat andMass Transfer vol 50 no 5-6pp 960ndash966 2007

[6] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[7] P S Datti K V Prasad M S Abel and A Joshi ldquoMHDvisco-elastic fluid flow over a non-isothermal stretching sheetrdquoInternational Journal of Engineering Science vol 42 no 8-9 pp935ndash946 2004

[8] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[9] M Subhas Abel S K Khan and K V Prasad ldquoStudy ofvisco-elastic fluid flow and heat transfer over a stretching sheetwith variable viscosityrdquo International Journal of Non-LinearMechanics vol 37 no 1 pp 81ndash88 2002

[10] I Anwar N Amin and I Pop ldquoMixed convection boundarylayer flow of a viscoelastic fluid over a horizontal circular cyli-nderrdquo International Journal ofNon-LinearMechanics vol 43 no9 pp 814ndash821 2008

[11] R M Sonth S K KhanM S Abel and K V Prasad ldquoHeat andmass transfer in a visco-elastic fluid flow over an acceleratingsurface with heat sourcesink and viscous dissipationrdquoHeat andMass Transfer vol 38 no 3 pp 213ndash220 2002

[12] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[13] K-L Hsiao ldquoHeat and mass mixed convection for MHD visco-elastic fluid past a stretching sheet with ohmic dissipationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 7 pp 1803ndash1812 2010

[14] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012

[15] A J Chamkha M Mujtaba A Quadri and C Issa ldquoThermalradiation effects on MHD forced convection flow adjacent to anon-isothermal wedge in the presence of a heat source or sinkrdquoHeat and Mass Transfer vol 39 no 4 pp 305ndash312 2003

[16] K-L Hsiao ldquoMHDmixed convection for viscoelastic fluid pasta porouswedgerdquo International Journal of Non-LinearMechanicsvol 46 no 1 pp 1ndash8 2011

[17] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010

[18] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuously moving stretching surface by HomotopyAnalysis Method with two auxiliary parametersrdquo Journal ofApplied Mathematics vol 2012 Article ID 780415 19 pages2012

[19] T R Mahapatra S Dholey and A S Gupta ldquoOblique stag-nation-point flow of an incompressible visco-elastic fluid towa-rds a stretching surfacerdquo International Journal of Non-LinearMechanics vol 42 no 3 pp 484ndash499 2007

[20] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC New York NY USA2004

[21] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[22] MMustafa T Hayat I Pop S Asghar and S Obaidat ldquoStagna-tion-point flow of a nanofluid towards a stretching sheetrdquo Inter-national Journal of Heat and Mass Transfer vol 54 no 25-26pp 5588ndash5594 2011

[23] M M Rashidi T Hayat E Erfani S A Mohimanian Pour andA A Hendi ldquoSimultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steadyMHDconvective flow

10 Advances in Mechanical Engineering

due to a rotating diskrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[24] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[25] 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

[26] M M Rashidi M Ali N Freidoonimehr and F Nazari ldquoPara-metric analysis and optimization of entropy generation inunsteady MHD flow over a stretching rotating disk using arti-ficial neural network and particle swarm optimization algori-thmrdquo Energy vol 55 pp 497ndash510 2013

[27] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[28] S Abbasbandy E Magyari and E Shivanian ldquoThe homotopyanalysis method for multiple solutions of nonlinear boundaryvalue problemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 9-10 pp 3530ndash3536 2009

[29] H Shahmohamadi M M Rashidi and S Dinarvand ldquoAna-lytic approximate solutions for unsteady two-dimensional andaxisymmetric squeezing flows between parallel platesrdquo Mathe-matical Problems in Engineering vol 2008 Article ID 93509512 pages 2008

[30] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica 2013

[31] X Su L Zheng X Zhang and J Zhang ldquoMHD mixed conve-ctive heat transfer over a permeable stretching wedge withthermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[32] M Massoudi ldquoLocal non-similarity solutions for the flow ofa non-Newtonian fluid over a wedgerdquo International Journal ofNon-Linear Mechanics vol 36 no 6 pp 961ndash976 2001

[33] D Pal ldquoCombined effects of non-uniform heat sourcesink andthermal radiation on heat transfer over an unsteady stretchingpermeable surfacerdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 4 pp 1890ndash1904 2011

[34] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[35] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[36] M M Rashidi S A Mohimanian pour and S AbbasbandyldquoAnalytic approximate solutions for heat transfer of a microp-olar fluid through a porous mediumwith radiationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 16no 4 pp 1874ndash1889 2011

[37] A Ishak R Nazar and I Pop ldquoFalkner-Skan equation for flowpast a moving wedge with suction or injectionrdquo Journal ofApplied Mathematics and Computing vol 25 no 1-2 pp 67ndash832007

[38] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mechanical Engineering

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Civil EngineeringAdvances in

Advances inAcoustics ampVibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

thinspJournalthinspofthinsp

Sensors

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Active and Passive Electronic Components

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

RoboticsJournal of

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Chemical EngineeringInternational Journal of

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Control Scienceand Engineering

Journal of

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International Journal of

Antennas andPropagation

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

4 Advances in Mechanical Engineering

where 119888119894 119894 = 1 minus 6 are the arbitrary constants The nonlinear

operators according to (10) are defined as

N119891[119891 (120578 119902)] = 119898((

120597119891 (120578 119902)

120597120578)

2

minus 1)

minus(119898 + 1)

2119891 (120578 119902)

1205972119891 (120578 119902)

1205971205782minus1205973119891 (120578 119902)

1205971205783

minus 1198961(3119898 minus 1)

120597119891 (120578 119902)

120597120578

1205973119891 (120578 119902)

1205971205783

minus(3119898 minus 1)

2(1205972119891 (120578 119902)

1205971205782)

2

minus(119898 + 1)

2119891 (120578 119902)

1205974119891 (120578 119902)

1205971205784

+119872(120597119891 (120578 119902)

120597120578minus 1)

N120579[119891 (120578 119902) 120579 (120578 119902)]

= (1 + 119873119903)1205972120579 (120578 119902)

1205971205782

+ Pr((119898 + 1)2

119891 (120578 119902)120597120579 (120578 119902)

120597120578

minus 119898120597119891 (120578 119902)

120597120578120579 (120578 119902))

(15)

The auxiliary functions become

119867119891(120578) = 119867

120579(120578) = 119890

minus120578 (16)

The symbolic software Mathematica is employed to solvethe 119894th order deformation equations

L119891[119891119894(120578) minus 120594

119894119891119894minus1(120578)] = ℎH

119891(120578) 119877119891119894(120578)

L120579[120579119894(120578) minus 120594

119894120579119894minus1(120578)] = ℎH

120579(120578) 119877120579119894(120578)

(17)

where ℎ is the auxiliary nonzero parameter and

119877119891119894(120578) = 119898(

119894minus1

sum119895=0

(120597119891119895(120578)

120597120578

120597119891119894minus1minus119895

(120578)

120597120578) minus 1)

minus(119898 + 1)

2

119894minus1

sum119895=0

(119891119895(120578)

1205972119891119894minus1minus119895

(120578)

1205971205782)

minus1205973119891119894minus1(120578)

1205971205783

minus 1198961

119894minus1

sum119895=0

((3119898 minus 1)120597119891119895(120578)

120597120578

1205973119891119894minus1minus119895

(120578)

1205971205783

minus(3119898 minus 1)

2

1205972119891119895(120578)

1205971205782

1205972119891119894minus1minus119895

(120578)

1205971205782

minus(119898 + 1)

2119891119895(120578)

1205974119891119894minus1minus119895

(120578)

1205971205784)

+119872(120597119891119894minus1(120578)

120597120578minus 1)

119877120579119894(120578) = (1 + 119873

119903)1205972120579119894minus1(120578)

1205971205782

+ Pr119894minus1

sum119895=0

((119898 + 1)

2119891119895(120578)

120597120579119894minus1minus119895

(120578)

120597120578

minus 119898120579119895(120578)

120597119891119894minus1minus119895

(120578)

120597120578)

120594119894=

0 119894 le 1

1 119894 gt 1

(18)

are the involved parameters in HAM theory (for more infor-mation about the different steps of HAM see [20 35 36])To control and speed the convergence of the approximationseries by the help of the so-called ℎ-curve it is significant tochoose a proper value of auxiliary parameter The ℎ-curvesof 119891101584010158401015840(0) and 1205791015840(0) obtained by the 18th order of HAMsolution are shown in Figure 2 To obtain the optimal valuesof auxiliary parameters the averaged residual errors aredefined as

Res119891= 119898((

119889119891 (120578)

119889120578)

2

minus 1) minus(119898 + 1)

2119891 (120578)

1198892119891 (120578)

1198891205782

minus1198893119891 (120578)

1198891205783+119872(

119889119891 (120578)

119889120578minus 1)

minus 1198961(3119898 minus 1)

119889119891 (120578)

119889120578

1198893119891 (120578)

1198891205783minus(3119898 minus 1)

2

times (1198892119891 (120578)

1198891205782)

2

minus(119898 + 1)

2119891 (120578)

1198894119891 (120578)

1198891205784

(19)

Res120579= (1 + 119873

119903)1198892120579 (120578)

1198891205782

+ Pr((119898 + 1)2

119891 (120578)119889120579 (120578)

119889120578minus 119898

119889119891 (120578)

119889120578120579 (120578))

(20)

Advances in Mechanical Engineering 5

h

minus12

minus09

minus06

minus03

0

minus15minus12 minus09 minus06 minus03 0

120579998400(0)

f998400998400998400(0)ℏ-c

urve

Figure 2 The ℎ-curves of 119891101584010158401015840(0) and 1205791015840(0) obtained by the 18thorder approximation of HAM solution when119872 = 119891

119908= Pr = 119873119903 =

1 120573 = 13 and 1198961= 3

Resid

ual e

rror

0 2 4 6 8 10

h = minus08

h = minus09

h = minus10

h = minus11

h = minus12 (optimal value)h = minus13

minus00025

minus0002

minus00015

minus0001

minus00005

0

00005

0001

120578

Figure 3 The residual error of (19) when 119891119908= 1 120573 = 13 119896

1= 05

and119872 = 5

In order to survey the accuracy of the present methodthe residual errors for the 18th order of HAM solutions of(19) and (20) are illustrated in Figures 3 and 4 respectivelyIn addition we compare some of our results with the resultsof the previously published studies of [37 38] to highlight

Resid

ual e

rror

0 2 4 6 8 10120578

minus0002

minus00015

minus0001

minus00005

0

00005

h = minus065

h = minus070

h = minus075

h = minus080

h = minus085 (optimal value)h = minus090

Figure 4 The residual error of (20) when 119872 = 119891119908= 119873119903 = 1

120573 = 13 1198961= 05 and Pr = 5

Table 1 Comparison results of 11989110158401015840(0) for different values ofsuctioninjection parameter (119891

119908) when119872 = 119896

1= 0 and 120573 = 1

119891119908

Reference [37] Reference [38] Present resultsminus1 07566 075658 075658018minus05 09692 096923 0969229820 12326 123259 12325936505 15418 154175 1541751721 18893 188931 188931809

the validity of the applied method for some values of fixedparameters 119872 = 119896

1= 0 and 120573 = 1 A very excellent

agreement can be observed between them as seen in Table 1

4 Results and Discussion

The nonlinear ordinary differential equations (10) subjectedto the boundary conditions (11) are solved for some valuesof the wedge angle parameter 120573 magnetic parameter 119872viscoelastic parameter 119896

1 suction parameter 119891

119908 thermal

radiation parameter 119873119903 and Prandtl number Pr via HAMThis section discusses the effects of above flow physicalparameters on the velocity and temperature profiles 1198911015840(120578)and 120579(120578) It should be mentioned that some representativephysical parameters are used to simulate realistic flows

Figures 5 and 6 illustrate the effect of the wedge angleparameter on the velocity profiles and temperature distribu-tions when 119872 = 119891

119908= Pr = 119873119903 = 1 and 119896

1= 05 It

should be noticed that 119898 = 1 (120573 = 1) permits completesimilarity solutions of (10) where 119896

1is constant and not 119891(119909)

6 Advances in Mechanical Engineering

0 1 2 3 4 5 60

02

04

06

08

1

f998400 (120578)

120573 = 0

120573 = 13

120573 = 23

120573 = 1

120578

Figure 5 Effect of wedge angle parameter on the velocity profilewhen119872 = 119891

119908= 1 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

120573 = 0

120573 = 13

120573 = 23

120573 = 1

120578

120579(120578)

Figure 6 Effect of wedge angle parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 119873119903 = 1 and 119896

1= 05

as shown in Figures 5 and 6 However if 119898 = 1 (120573 = 1) solu-tions can be obtained but it will be local in other words localsimilarity is sought as seen in other figures in this sectionAn increase in the wedge angle parameter leads to increase inthe free stream velocity and the Reynolds number and conse-quently the velocity boundary-layer thickness decreases The

0 1 2 3 4 5 60

02

04

06

08

1

M = 0

M = 1

M = 2

M = 3

M = 4

M = 5

120578

f998400 (120578)

Figure 7 Effect of magnetic parameter on the velocity profile when119891119908= 1 120573 = 13 and 119896

1= 05

temperature distribution and the thermal boundary-layerthickness decrease as the wedge angle parameter increasesIndeed increase in the wedge angle parameter causes theincrease in the heat transfer coefficient and the rate of heattransfer

The effect of magnetic parameter on the velocity profilesand temperature distributions is displayed in Figures 7 and 8with 119891

119908= Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05 A drag-

like force named Lorentz force is created by the infliction ofthe vertical magnetic field to the electrically conducting fluidThis force has the tendency to slow down the flow over thewedge Accordingly the velocity and temperature boundary-layer thickness decrease with the increasing of the magneticinteraction parameter

Figures 9 and 10 show the effect of the viscoelastic param-eter on the velocity profile and temperature distribution withthe constant values of other parameters 119872 = 119891

119908= Pr =

119873119903 = 1 and 120573 = 13 As the viscoelastic parameter increasesthe fluid velocity profile decreases and also the temperaturedistribution increasesThis occurs due to the development ofthe tensile stress This behavior is similar to that reported byAnwar et al [10]

The effect of suction parameter on the velocity andtemperature profiles is demonstrated in Figures 11 and 12with 119872 = Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05 In

this study the suction case has been considered in all figuresbased on the boundary-layer assumption which stated thatthe boundary-layer thickness is supposed to be very thinand it will not be allowed to increase as it will violate theboundary-layer assumption displayed by Prandtl in 1904

Advances in Mechanical Engineering 7

0 1 2 3 4 5 60

02

04

06

08

1

M = 0

M = 1

M = 2

M = 3

M = 4

M = 5

120578

120579(120578)

Figure 8 Effect of magnetic parameter on the temperature distri-bution when 119891

119908= Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

k1 = 00

k1 = 05

k1 = 10

k1 = 15

k1 = 20

k1 = 30

120578

f998400 (120578)

Figure 9 Effect of viscoelastic parameter on the velocity profilewhen119872 = 119891

119908= 1 and 120573 = 13

0 1 2 3 4 5 60

02

04

06

08

1

k1 = 00

k1 = 05

k1 = 10

k1 = 15

k1 = 20

k1 = 30

120578

120579(120578)

Figure 10 Effect of viscoelastic parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 119873119903 = 1 and 120573 = 13

0 1 2 3 4 5 60

02

04

06

08

1

120578

f998400 (120578)

fw = 00

fw = 05

fw = 10

fw = 20

fw = 30

Figure 11 Effect of suction parameter on the velocity profile when119872 = 1 120573 = 13 and 119896

1= 05

8 Advances in Mechanical Engineering

0 1 2 3 4 5 60

02

04

06

08

1

120578

fw = 00

fw = 05

fw = 10

fw = 20

fw = 30

120579(120578)

Figure 12 Effect of suction parameter on the temperature distribu-tion when119872 = Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05

Applying the suction at the wedge surface causes the amountof the fluid to draw into the surface and consequently thehydrodynamic boundary-layer becomes thinner In additionthe thermal boundary-layer gets depressed by increasing thesuction parameter

The effects of the thermal radiation parameter and thePrandtl number on the temperature distribution are shownin Figures 13 and 14 when 119872 = 119891

119908= 119873119903 = 1 120573 = 13

and 1198961= 05 The rate of energy transport to the fluid

increases by increasing the thermal radiation parameterThus the temperature of the fluid increases On the otherhand the increase of radiation parameter leads to overcom-ing the effect of convective heat transfer Based on the Prandtlnumber definition (Pr = ]120572) this parameter is definedas the ratio between the momentum diffusion to thermaldiffusion Thus with the increase of Prandtl number thethermal diffusion decreases and so the thermal boundary-layer becomes thinner as seen in Figure 14 It physicallymeans that the flow with large Prandtl number dissipates theheat faster to the fluid as the temperature gradient gets steeperand hence increasing the heat transfer coefficient between thesurface and the fluid

5 Conclusions

In this paper the semi-analyticalnumerical techniqueknown as HAM has been implemented to solve the trans-formed differential equations describing the MHD mixedconvective heat transfer for an incompressible laminarand electrically conducting viscoelastic fluid flow over a

0 1 2 3 4 5 60

02

04

06

08

1

120578

f998400 (120578)

Nr = 00

Nr = 05

Nr = 10

Nr = 20

Nr = 30

Figure 13 Effect of thermal radiation parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 1 120573 = 13 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

120578

120579(120578)

Pr = 071

Pr = 100

Pr = 200

Pr = 300

Pr = 500

Figure 14 Effect of Prandtl number on the temperature distributionwhen119872 = 119891

119908= 119873119903 = 1 120573 = 13 and 119896

1= 05

porous wedge in the presence of the thermal radiation effectThe present semi-numericalanalytical simulations agreeclosely with the previous studies for some special casesHAM has been shown to be a very strong and efficient

Advances in Mechanical Engineering 9

technique in finding analytical solutions for nonlineardifferential equations HAM is displayed to illustrate exce-llent convergence and accuracy and is currently beingemployed to extend the present study to mixed convectiveheat transfer simulations The effects of different physicalkey parameters such as wedge angle parameter magneticparameter viscoelastic parameter suction parameterthermal radiation parameter and Prandtl number are plottedand discussedThe results show that as the wedge angle incre-ases the heat transfer to the fluid increases for other constantspecified parameter and for Pr = 1 The magnetic field has aweak effect on the thermal boundary thickness however thesuction has a remarkable effect on it Increasing the thermalradiation parameter reduces the heat transfer coefficientbetween the wedge and the fluid however increasing Prandtlnumber increases it

Conflict of Interests

On behalf of all the authors there is no conflict of interests toreport

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[2] K Hiemenz Die Grenzschicht an einem in den gleichformigenFlussigkeitsstrom eingetauchten geraden Kreiszylinder WeberBerlin Germany 1911

[3] E RG EckertDie Berechnung desWarmeubergangs in der lam-inaren Grenzschicht umstromter Korper VDI Berlin Germany1942

[4] P D Ariel ldquoA numerical algorithm for computing the stagna-tion point flow of a second grade fluid withwithout suctionrdquoJournal of Computational and Applied Mathematics vol 59 no1 pp 9ndash24 1995

[5] M S Abel P G Siddheshwar andMMNandeppanavar ldquoHeattransfer in a viscoelastic boundary layer flow over a stretchingsheet with viscous dissipation and non-uniform heat sourcerdquoInternational Journal of Heat andMass Transfer vol 50 no 5-6pp 960ndash966 2007

[6] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[7] P S Datti K V Prasad M S Abel and A Joshi ldquoMHDvisco-elastic fluid flow over a non-isothermal stretching sheetrdquoInternational Journal of Engineering Science vol 42 no 8-9 pp935ndash946 2004

[8] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[9] M Subhas Abel S K Khan and K V Prasad ldquoStudy ofvisco-elastic fluid flow and heat transfer over a stretching sheetwith variable viscosityrdquo International Journal of Non-LinearMechanics vol 37 no 1 pp 81ndash88 2002

[10] I Anwar N Amin and I Pop ldquoMixed convection boundarylayer flow of a viscoelastic fluid over a horizontal circular cyli-nderrdquo International Journal ofNon-LinearMechanics vol 43 no9 pp 814ndash821 2008

[11] R M Sonth S K KhanM S Abel and K V Prasad ldquoHeat andmass transfer in a visco-elastic fluid flow over an acceleratingsurface with heat sourcesink and viscous dissipationrdquoHeat andMass Transfer vol 38 no 3 pp 213ndash220 2002

[12] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[13] K-L Hsiao ldquoHeat and mass mixed convection for MHD visco-elastic fluid past a stretching sheet with ohmic dissipationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 7 pp 1803ndash1812 2010

[14] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012

[15] A J Chamkha M Mujtaba A Quadri and C Issa ldquoThermalradiation effects on MHD forced convection flow adjacent to anon-isothermal wedge in the presence of a heat source or sinkrdquoHeat and Mass Transfer vol 39 no 4 pp 305ndash312 2003

[16] K-L Hsiao ldquoMHDmixed convection for viscoelastic fluid pasta porouswedgerdquo International Journal of Non-LinearMechanicsvol 46 no 1 pp 1ndash8 2011

[17] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010

[18] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuously moving stretching surface by HomotopyAnalysis Method with two auxiliary parametersrdquo Journal ofApplied Mathematics vol 2012 Article ID 780415 19 pages2012

[19] T R Mahapatra S Dholey and A S Gupta ldquoOblique stag-nation-point flow of an incompressible visco-elastic fluid towa-rds a stretching surfacerdquo International Journal of Non-LinearMechanics vol 42 no 3 pp 484ndash499 2007

[20] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC New York NY USA2004

[21] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[22] MMustafa T Hayat I Pop S Asghar and S Obaidat ldquoStagna-tion-point flow of a nanofluid towards a stretching sheetrdquo Inter-national Journal of Heat and Mass Transfer vol 54 no 25-26pp 5588ndash5594 2011

[23] M M Rashidi T Hayat E Erfani S A Mohimanian Pour andA A Hendi ldquoSimultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steadyMHDconvective flow

10 Advances in Mechanical Engineering

due to a rotating diskrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[24] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[25] 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

[26] M M Rashidi M Ali N Freidoonimehr and F Nazari ldquoPara-metric analysis and optimization of entropy generation inunsteady MHD flow over a stretching rotating disk using arti-ficial neural network and particle swarm optimization algori-thmrdquo Energy vol 55 pp 497ndash510 2013

[27] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[28] S Abbasbandy E Magyari and E Shivanian ldquoThe homotopyanalysis method for multiple solutions of nonlinear boundaryvalue problemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 9-10 pp 3530ndash3536 2009

[29] H Shahmohamadi M M Rashidi and S Dinarvand ldquoAna-lytic approximate solutions for unsteady two-dimensional andaxisymmetric squeezing flows between parallel platesrdquo Mathe-matical Problems in Engineering vol 2008 Article ID 93509512 pages 2008

[30] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica 2013

[31] X Su L Zheng X Zhang and J Zhang ldquoMHD mixed conve-ctive heat transfer over a permeable stretching wedge withthermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[32] M Massoudi ldquoLocal non-similarity solutions for the flow ofa non-Newtonian fluid over a wedgerdquo International Journal ofNon-Linear Mechanics vol 36 no 6 pp 961ndash976 2001

[33] D Pal ldquoCombined effects of non-uniform heat sourcesink andthermal radiation on heat transfer over an unsteady stretchingpermeable surfacerdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 4 pp 1890ndash1904 2011

[34] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[35] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[36] M M Rashidi S A Mohimanian pour and S AbbasbandyldquoAnalytic approximate solutions for heat transfer of a microp-olar fluid through a porous mediumwith radiationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 16no 4 pp 1874ndash1889 2011

[37] A Ishak R Nazar and I Pop ldquoFalkner-Skan equation for flowpast a moving wedge with suction or injectionrdquo Journal ofApplied Mathematics and Computing vol 25 no 1-2 pp 67ndash832007

[38] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mechanical Engineering

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Advances inAcoustics ampVibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

thinspJournalthinspofthinsp

Sensors

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of

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Control Scienceand Engineering

Journal of

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International Journal of

Antennas andPropagation

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Advances in Mechanical Engineering 5

h

minus12

minus09

minus06

minus03

0

minus15minus12 minus09 minus06 minus03 0

120579998400(0)

f998400998400998400(0)ℏ-c

urve

Figure 2 The ℎ-curves of 119891101584010158401015840(0) and 1205791015840(0) obtained by the 18thorder approximation of HAM solution when119872 = 119891

119908= Pr = 119873119903 =

1 120573 = 13 and 1198961= 3

Resid

ual e

rror

0 2 4 6 8 10

h = minus08

h = minus09

h = minus10

h = minus11

h = minus12 (optimal value)h = minus13

minus00025

minus0002

minus00015

minus0001

minus00005

0

00005

0001

120578

Figure 3 The residual error of (19) when 119891119908= 1 120573 = 13 119896

1= 05

and119872 = 5

In order to survey the accuracy of the present methodthe residual errors for the 18th order of HAM solutions of(19) and (20) are illustrated in Figures 3 and 4 respectivelyIn addition we compare some of our results with the resultsof the previously published studies of [37 38] to highlight

Resid

ual e

rror

0 2 4 6 8 10120578

minus0002

minus00015

minus0001

minus00005

0

00005

h = minus065

h = minus070

h = minus075

h = minus080

h = minus085 (optimal value)h = minus090

Figure 4 The residual error of (20) when 119872 = 119891119908= 119873119903 = 1

120573 = 13 1198961= 05 and Pr = 5

Table 1 Comparison results of 11989110158401015840(0) for different values ofsuctioninjection parameter (119891

119908) when119872 = 119896

1= 0 and 120573 = 1

119891119908

Reference [37] Reference [38] Present resultsminus1 07566 075658 075658018minus05 09692 096923 0969229820 12326 123259 12325936505 15418 154175 1541751721 18893 188931 188931809

the validity of the applied method for some values of fixedparameters 119872 = 119896

1= 0 and 120573 = 1 A very excellent

agreement can be observed between them as seen in Table 1

4 Results and Discussion

The nonlinear ordinary differential equations (10) subjectedto the boundary conditions (11) are solved for some valuesof the wedge angle parameter 120573 magnetic parameter 119872viscoelastic parameter 119896

1 suction parameter 119891

119908 thermal

radiation parameter 119873119903 and Prandtl number Pr via HAMThis section discusses the effects of above flow physicalparameters on the velocity and temperature profiles 1198911015840(120578)and 120579(120578) It should be mentioned that some representativephysical parameters are used to simulate realistic flows

Figures 5 and 6 illustrate the effect of the wedge angleparameter on the velocity profiles and temperature distribu-tions when 119872 = 119891

119908= Pr = 119873119903 = 1 and 119896

1= 05 It

should be noticed that 119898 = 1 (120573 = 1) permits completesimilarity solutions of (10) where 119896

1is constant and not 119891(119909)

6 Advances in Mechanical Engineering

0 1 2 3 4 5 60

02

04

06

08

1

f998400 (120578)

120573 = 0

120573 = 13

120573 = 23

120573 = 1

120578

Figure 5 Effect of wedge angle parameter on the velocity profilewhen119872 = 119891

119908= 1 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

120573 = 0

120573 = 13

120573 = 23

120573 = 1

120578

120579(120578)

Figure 6 Effect of wedge angle parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 119873119903 = 1 and 119896

1= 05

as shown in Figures 5 and 6 However if 119898 = 1 (120573 = 1) solu-tions can be obtained but it will be local in other words localsimilarity is sought as seen in other figures in this sectionAn increase in the wedge angle parameter leads to increase inthe free stream velocity and the Reynolds number and conse-quently the velocity boundary-layer thickness decreases The

0 1 2 3 4 5 60

02

04

06

08

1

M = 0

M = 1

M = 2

M = 3

M = 4

M = 5

120578

f998400 (120578)

Figure 7 Effect of magnetic parameter on the velocity profile when119891119908= 1 120573 = 13 and 119896

1= 05

temperature distribution and the thermal boundary-layerthickness decrease as the wedge angle parameter increasesIndeed increase in the wedge angle parameter causes theincrease in the heat transfer coefficient and the rate of heattransfer

The effect of magnetic parameter on the velocity profilesand temperature distributions is displayed in Figures 7 and 8with 119891

119908= Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05 A drag-

like force named Lorentz force is created by the infliction ofthe vertical magnetic field to the electrically conducting fluidThis force has the tendency to slow down the flow over thewedge Accordingly the velocity and temperature boundary-layer thickness decrease with the increasing of the magneticinteraction parameter

Figures 9 and 10 show the effect of the viscoelastic param-eter on the velocity profile and temperature distribution withthe constant values of other parameters 119872 = 119891

119908= Pr =

119873119903 = 1 and 120573 = 13 As the viscoelastic parameter increasesthe fluid velocity profile decreases and also the temperaturedistribution increasesThis occurs due to the development ofthe tensile stress This behavior is similar to that reported byAnwar et al [10]

The effect of suction parameter on the velocity andtemperature profiles is demonstrated in Figures 11 and 12with 119872 = Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05 In

this study the suction case has been considered in all figuresbased on the boundary-layer assumption which stated thatthe boundary-layer thickness is supposed to be very thinand it will not be allowed to increase as it will violate theboundary-layer assumption displayed by Prandtl in 1904

Advances in Mechanical Engineering 7

0 1 2 3 4 5 60

02

04

06

08

1

M = 0

M = 1

M = 2

M = 3

M = 4

M = 5

120578

120579(120578)

Figure 8 Effect of magnetic parameter on the temperature distri-bution when 119891

119908= Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

k1 = 00

k1 = 05

k1 = 10

k1 = 15

k1 = 20

k1 = 30

120578

f998400 (120578)

Figure 9 Effect of viscoelastic parameter on the velocity profilewhen119872 = 119891

119908= 1 and 120573 = 13

0 1 2 3 4 5 60

02

04

06

08

1

k1 = 00

k1 = 05

k1 = 10

k1 = 15

k1 = 20

k1 = 30

120578

120579(120578)

Figure 10 Effect of viscoelastic parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 119873119903 = 1 and 120573 = 13

0 1 2 3 4 5 60

02

04

06

08

1

120578

f998400 (120578)

fw = 00

fw = 05

fw = 10

fw = 20

fw = 30

Figure 11 Effect of suction parameter on the velocity profile when119872 = 1 120573 = 13 and 119896

1= 05

8 Advances in Mechanical Engineering

0 1 2 3 4 5 60

02

04

06

08

1

120578

fw = 00

fw = 05

fw = 10

fw = 20

fw = 30

120579(120578)

Figure 12 Effect of suction parameter on the temperature distribu-tion when119872 = Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05

Applying the suction at the wedge surface causes the amountof the fluid to draw into the surface and consequently thehydrodynamic boundary-layer becomes thinner In additionthe thermal boundary-layer gets depressed by increasing thesuction parameter

The effects of the thermal radiation parameter and thePrandtl number on the temperature distribution are shownin Figures 13 and 14 when 119872 = 119891

119908= 119873119903 = 1 120573 = 13

and 1198961= 05 The rate of energy transport to the fluid

increases by increasing the thermal radiation parameterThus the temperature of the fluid increases On the otherhand the increase of radiation parameter leads to overcom-ing the effect of convective heat transfer Based on the Prandtlnumber definition (Pr = ]120572) this parameter is definedas the ratio between the momentum diffusion to thermaldiffusion Thus with the increase of Prandtl number thethermal diffusion decreases and so the thermal boundary-layer becomes thinner as seen in Figure 14 It physicallymeans that the flow with large Prandtl number dissipates theheat faster to the fluid as the temperature gradient gets steeperand hence increasing the heat transfer coefficient between thesurface and the fluid

5 Conclusions

In this paper the semi-analyticalnumerical techniqueknown as HAM has been implemented to solve the trans-formed differential equations describing the MHD mixedconvective heat transfer for an incompressible laminarand electrically conducting viscoelastic fluid flow over a

0 1 2 3 4 5 60

02

04

06

08

1

120578

f998400 (120578)

Nr = 00

Nr = 05

Nr = 10

Nr = 20

Nr = 30

Figure 13 Effect of thermal radiation parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 1 120573 = 13 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

120578

120579(120578)

Pr = 071

Pr = 100

Pr = 200

Pr = 300

Pr = 500

Figure 14 Effect of Prandtl number on the temperature distributionwhen119872 = 119891

119908= 119873119903 = 1 120573 = 13 and 119896

1= 05

porous wedge in the presence of the thermal radiation effectThe present semi-numericalanalytical simulations agreeclosely with the previous studies for some special casesHAM has been shown to be a very strong and efficient

Advances in Mechanical Engineering 9

technique in finding analytical solutions for nonlineardifferential equations HAM is displayed to illustrate exce-llent convergence and accuracy and is currently beingemployed to extend the present study to mixed convectiveheat transfer simulations The effects of different physicalkey parameters such as wedge angle parameter magneticparameter viscoelastic parameter suction parameterthermal radiation parameter and Prandtl number are plottedand discussedThe results show that as the wedge angle incre-ases the heat transfer to the fluid increases for other constantspecified parameter and for Pr = 1 The magnetic field has aweak effect on the thermal boundary thickness however thesuction has a remarkable effect on it Increasing the thermalradiation parameter reduces the heat transfer coefficientbetween the wedge and the fluid however increasing Prandtlnumber increases it

Conflict of Interests

On behalf of all the authors there is no conflict of interests toreport

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[2] K Hiemenz Die Grenzschicht an einem in den gleichformigenFlussigkeitsstrom eingetauchten geraden Kreiszylinder WeberBerlin Germany 1911

[3] E RG EckertDie Berechnung desWarmeubergangs in der lam-inaren Grenzschicht umstromter Korper VDI Berlin Germany1942

[4] P D Ariel ldquoA numerical algorithm for computing the stagna-tion point flow of a second grade fluid withwithout suctionrdquoJournal of Computational and Applied Mathematics vol 59 no1 pp 9ndash24 1995

[5] M S Abel P G Siddheshwar andMMNandeppanavar ldquoHeattransfer in a viscoelastic boundary layer flow over a stretchingsheet with viscous dissipation and non-uniform heat sourcerdquoInternational Journal of Heat andMass Transfer vol 50 no 5-6pp 960ndash966 2007

[6] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[7] P S Datti K V Prasad M S Abel and A Joshi ldquoMHDvisco-elastic fluid flow over a non-isothermal stretching sheetrdquoInternational Journal of Engineering Science vol 42 no 8-9 pp935ndash946 2004

[8] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[9] M Subhas Abel S K Khan and K V Prasad ldquoStudy ofvisco-elastic fluid flow and heat transfer over a stretching sheetwith variable viscosityrdquo International Journal of Non-LinearMechanics vol 37 no 1 pp 81ndash88 2002

[10] I Anwar N Amin and I Pop ldquoMixed convection boundarylayer flow of a viscoelastic fluid over a horizontal circular cyli-nderrdquo International Journal ofNon-LinearMechanics vol 43 no9 pp 814ndash821 2008

[11] R M Sonth S K KhanM S Abel and K V Prasad ldquoHeat andmass transfer in a visco-elastic fluid flow over an acceleratingsurface with heat sourcesink and viscous dissipationrdquoHeat andMass Transfer vol 38 no 3 pp 213ndash220 2002

[12] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[13] K-L Hsiao ldquoHeat and mass mixed convection for MHD visco-elastic fluid past a stretching sheet with ohmic dissipationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 7 pp 1803ndash1812 2010

[14] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012

[15] A J Chamkha M Mujtaba A Quadri and C Issa ldquoThermalradiation effects on MHD forced convection flow adjacent to anon-isothermal wedge in the presence of a heat source or sinkrdquoHeat and Mass Transfer vol 39 no 4 pp 305ndash312 2003

[16] K-L Hsiao ldquoMHDmixed convection for viscoelastic fluid pasta porouswedgerdquo International Journal of Non-LinearMechanicsvol 46 no 1 pp 1ndash8 2011

[17] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010

[18] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuously moving stretching surface by HomotopyAnalysis Method with two auxiliary parametersrdquo Journal ofApplied Mathematics vol 2012 Article ID 780415 19 pages2012

[19] T R Mahapatra S Dholey and A S Gupta ldquoOblique stag-nation-point flow of an incompressible visco-elastic fluid towa-rds a stretching surfacerdquo International Journal of Non-LinearMechanics vol 42 no 3 pp 484ndash499 2007

[20] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC New York NY USA2004

[21] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[22] MMustafa T Hayat I Pop S Asghar and S Obaidat ldquoStagna-tion-point flow of a nanofluid towards a stretching sheetrdquo Inter-national Journal of Heat and Mass Transfer vol 54 no 25-26pp 5588ndash5594 2011

[23] M M Rashidi T Hayat E Erfani S A Mohimanian Pour andA A Hendi ldquoSimultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steadyMHDconvective flow

10 Advances in Mechanical Engineering

due to a rotating diskrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[24] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[25] 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

[26] M M Rashidi M Ali N Freidoonimehr and F Nazari ldquoPara-metric analysis and optimization of entropy generation inunsteady MHD flow over a stretching rotating disk using arti-ficial neural network and particle swarm optimization algori-thmrdquo Energy vol 55 pp 497ndash510 2013

[27] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[28] S Abbasbandy E Magyari and E Shivanian ldquoThe homotopyanalysis method for multiple solutions of nonlinear boundaryvalue problemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 9-10 pp 3530ndash3536 2009

[29] H Shahmohamadi M M Rashidi and S Dinarvand ldquoAna-lytic approximate solutions for unsteady two-dimensional andaxisymmetric squeezing flows between parallel platesrdquo Mathe-matical Problems in Engineering vol 2008 Article ID 93509512 pages 2008

[30] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica 2013

[31] X Su L Zheng X Zhang and J Zhang ldquoMHD mixed conve-ctive heat transfer over a permeable stretching wedge withthermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[32] M Massoudi ldquoLocal non-similarity solutions for the flow ofa non-Newtonian fluid over a wedgerdquo International Journal ofNon-Linear Mechanics vol 36 no 6 pp 961ndash976 2001

[33] D Pal ldquoCombined effects of non-uniform heat sourcesink andthermal radiation on heat transfer over an unsteady stretchingpermeable surfacerdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 4 pp 1890ndash1904 2011

[34] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[35] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[36] M M Rashidi S A Mohimanian pour and S AbbasbandyldquoAnalytic approximate solutions for heat transfer of a microp-olar fluid through a porous mediumwith radiationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 16no 4 pp 1874ndash1889 2011

[37] A Ishak R Nazar and I Pop ldquoFalkner-Skan equation for flowpast a moving wedge with suction or injectionrdquo Journal ofApplied Mathematics and Computing vol 25 no 1-2 pp 67ndash832007

[38] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mechanical Engineering

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Advances inAcoustics ampVibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

thinspJournalthinspofthinsp

Sensors

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Antennas andPropagation

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

6 Advances in Mechanical Engineering

0 1 2 3 4 5 60

02

04

06

08

1

f998400 (120578)

120573 = 0

120573 = 13

120573 = 23

120573 = 1

120578

Figure 5 Effect of wedge angle parameter on the velocity profilewhen119872 = 119891

119908= 1 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

120573 = 0

120573 = 13

120573 = 23

120573 = 1

120578

120579(120578)

Figure 6 Effect of wedge angle parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 119873119903 = 1 and 119896

1= 05

as shown in Figures 5 and 6 However if 119898 = 1 (120573 = 1) solu-tions can be obtained but it will be local in other words localsimilarity is sought as seen in other figures in this sectionAn increase in the wedge angle parameter leads to increase inthe free stream velocity and the Reynolds number and conse-quently the velocity boundary-layer thickness decreases The

0 1 2 3 4 5 60

02

04

06

08

1

M = 0

M = 1

M = 2

M = 3

M = 4

M = 5

120578

f998400 (120578)

Figure 7 Effect of magnetic parameter on the velocity profile when119891119908= 1 120573 = 13 and 119896

1= 05

temperature distribution and the thermal boundary-layerthickness decrease as the wedge angle parameter increasesIndeed increase in the wedge angle parameter causes theincrease in the heat transfer coefficient and the rate of heattransfer

The effect of magnetic parameter on the velocity profilesand temperature distributions is displayed in Figures 7 and 8with 119891

119908= Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05 A drag-

like force named Lorentz force is created by the infliction ofthe vertical magnetic field to the electrically conducting fluidThis force has the tendency to slow down the flow over thewedge Accordingly the velocity and temperature boundary-layer thickness decrease with the increasing of the magneticinteraction parameter

Figures 9 and 10 show the effect of the viscoelastic param-eter on the velocity profile and temperature distribution withthe constant values of other parameters 119872 = 119891

119908= Pr =

119873119903 = 1 and 120573 = 13 As the viscoelastic parameter increasesthe fluid velocity profile decreases and also the temperaturedistribution increasesThis occurs due to the development ofthe tensile stress This behavior is similar to that reported byAnwar et al [10]

The effect of suction parameter on the velocity andtemperature profiles is demonstrated in Figures 11 and 12with 119872 = Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05 In

this study the suction case has been considered in all figuresbased on the boundary-layer assumption which stated thatthe boundary-layer thickness is supposed to be very thinand it will not be allowed to increase as it will violate theboundary-layer assumption displayed by Prandtl in 1904

Advances in Mechanical Engineering 7

0 1 2 3 4 5 60

02

04

06

08

1

M = 0

M = 1

M = 2

M = 3

M = 4

M = 5

120578

120579(120578)

Figure 8 Effect of magnetic parameter on the temperature distri-bution when 119891

119908= Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

k1 = 00

k1 = 05

k1 = 10

k1 = 15

k1 = 20

k1 = 30

120578

f998400 (120578)

Figure 9 Effect of viscoelastic parameter on the velocity profilewhen119872 = 119891

119908= 1 and 120573 = 13

0 1 2 3 4 5 60

02

04

06

08

1

k1 = 00

k1 = 05

k1 = 10

k1 = 15

k1 = 20

k1 = 30

120578

120579(120578)

Figure 10 Effect of viscoelastic parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 119873119903 = 1 and 120573 = 13

0 1 2 3 4 5 60

02

04

06

08

1

120578

f998400 (120578)

fw = 00

fw = 05

fw = 10

fw = 20

fw = 30

Figure 11 Effect of suction parameter on the velocity profile when119872 = 1 120573 = 13 and 119896

1= 05

8 Advances in Mechanical Engineering

0 1 2 3 4 5 60

02

04

06

08

1

120578

fw = 00

fw = 05

fw = 10

fw = 20

fw = 30

120579(120578)

Figure 12 Effect of suction parameter on the temperature distribu-tion when119872 = Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05

Applying the suction at the wedge surface causes the amountof the fluid to draw into the surface and consequently thehydrodynamic boundary-layer becomes thinner In additionthe thermal boundary-layer gets depressed by increasing thesuction parameter

The effects of the thermal radiation parameter and thePrandtl number on the temperature distribution are shownin Figures 13 and 14 when 119872 = 119891

119908= 119873119903 = 1 120573 = 13

and 1198961= 05 The rate of energy transport to the fluid

increases by increasing the thermal radiation parameterThus the temperature of the fluid increases On the otherhand the increase of radiation parameter leads to overcom-ing the effect of convective heat transfer Based on the Prandtlnumber definition (Pr = ]120572) this parameter is definedas the ratio between the momentum diffusion to thermaldiffusion Thus with the increase of Prandtl number thethermal diffusion decreases and so the thermal boundary-layer becomes thinner as seen in Figure 14 It physicallymeans that the flow with large Prandtl number dissipates theheat faster to the fluid as the temperature gradient gets steeperand hence increasing the heat transfer coefficient between thesurface and the fluid

5 Conclusions

In this paper the semi-analyticalnumerical techniqueknown as HAM has been implemented to solve the trans-formed differential equations describing the MHD mixedconvective heat transfer for an incompressible laminarand electrically conducting viscoelastic fluid flow over a

0 1 2 3 4 5 60

02

04

06

08

1

120578

f998400 (120578)

Nr = 00

Nr = 05

Nr = 10

Nr = 20

Nr = 30

Figure 13 Effect of thermal radiation parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 1 120573 = 13 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

120578

120579(120578)

Pr = 071

Pr = 100

Pr = 200

Pr = 300

Pr = 500

Figure 14 Effect of Prandtl number on the temperature distributionwhen119872 = 119891

119908= 119873119903 = 1 120573 = 13 and 119896

1= 05

porous wedge in the presence of the thermal radiation effectThe present semi-numericalanalytical simulations agreeclosely with the previous studies for some special casesHAM has been shown to be a very strong and efficient

Advances in Mechanical Engineering 9

technique in finding analytical solutions for nonlineardifferential equations HAM is displayed to illustrate exce-llent convergence and accuracy and is currently beingemployed to extend the present study to mixed convectiveheat transfer simulations The effects of different physicalkey parameters such as wedge angle parameter magneticparameter viscoelastic parameter suction parameterthermal radiation parameter and Prandtl number are plottedand discussedThe results show that as the wedge angle incre-ases the heat transfer to the fluid increases for other constantspecified parameter and for Pr = 1 The magnetic field has aweak effect on the thermal boundary thickness however thesuction has a remarkable effect on it Increasing the thermalradiation parameter reduces the heat transfer coefficientbetween the wedge and the fluid however increasing Prandtlnumber increases it

Conflict of Interests

On behalf of all the authors there is no conflict of interests toreport

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[2] K Hiemenz Die Grenzschicht an einem in den gleichformigenFlussigkeitsstrom eingetauchten geraden Kreiszylinder WeberBerlin Germany 1911

[3] E RG EckertDie Berechnung desWarmeubergangs in der lam-inaren Grenzschicht umstromter Korper VDI Berlin Germany1942

[4] P D Ariel ldquoA numerical algorithm for computing the stagna-tion point flow of a second grade fluid withwithout suctionrdquoJournal of Computational and Applied Mathematics vol 59 no1 pp 9ndash24 1995

[5] M S Abel P G Siddheshwar andMMNandeppanavar ldquoHeattransfer in a viscoelastic boundary layer flow over a stretchingsheet with viscous dissipation and non-uniform heat sourcerdquoInternational Journal of Heat andMass Transfer vol 50 no 5-6pp 960ndash966 2007

[6] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[7] P S Datti K V Prasad M S Abel and A Joshi ldquoMHDvisco-elastic fluid flow over a non-isothermal stretching sheetrdquoInternational Journal of Engineering Science vol 42 no 8-9 pp935ndash946 2004

[8] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[9] M Subhas Abel S K Khan and K V Prasad ldquoStudy ofvisco-elastic fluid flow and heat transfer over a stretching sheetwith variable viscosityrdquo International Journal of Non-LinearMechanics vol 37 no 1 pp 81ndash88 2002

[10] I Anwar N Amin and I Pop ldquoMixed convection boundarylayer flow of a viscoelastic fluid over a horizontal circular cyli-nderrdquo International Journal ofNon-LinearMechanics vol 43 no9 pp 814ndash821 2008

[11] R M Sonth S K KhanM S Abel and K V Prasad ldquoHeat andmass transfer in a visco-elastic fluid flow over an acceleratingsurface with heat sourcesink and viscous dissipationrdquoHeat andMass Transfer vol 38 no 3 pp 213ndash220 2002

[12] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[13] K-L Hsiao ldquoHeat and mass mixed convection for MHD visco-elastic fluid past a stretching sheet with ohmic dissipationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 7 pp 1803ndash1812 2010

[14] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012

[15] A J Chamkha M Mujtaba A Quadri and C Issa ldquoThermalradiation effects on MHD forced convection flow adjacent to anon-isothermal wedge in the presence of a heat source or sinkrdquoHeat and Mass Transfer vol 39 no 4 pp 305ndash312 2003

[16] K-L Hsiao ldquoMHDmixed convection for viscoelastic fluid pasta porouswedgerdquo International Journal of Non-LinearMechanicsvol 46 no 1 pp 1ndash8 2011

[17] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010

[18] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuously moving stretching surface by HomotopyAnalysis Method with two auxiliary parametersrdquo Journal ofApplied Mathematics vol 2012 Article ID 780415 19 pages2012

[19] T R Mahapatra S Dholey and A S Gupta ldquoOblique stag-nation-point flow of an incompressible visco-elastic fluid towa-rds a stretching surfacerdquo International Journal of Non-LinearMechanics vol 42 no 3 pp 484ndash499 2007

[20] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC New York NY USA2004

[21] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[22] MMustafa T Hayat I Pop S Asghar and S Obaidat ldquoStagna-tion-point flow of a nanofluid towards a stretching sheetrdquo Inter-national Journal of Heat and Mass Transfer vol 54 no 25-26pp 5588ndash5594 2011

[23] M M Rashidi T Hayat E Erfani S A Mohimanian Pour andA A Hendi ldquoSimultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steadyMHDconvective flow

10 Advances in Mechanical Engineering

due to a rotating diskrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[24] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[25] 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

[26] M M Rashidi M Ali N Freidoonimehr and F Nazari ldquoPara-metric analysis and optimization of entropy generation inunsteady MHD flow over a stretching rotating disk using arti-ficial neural network and particle swarm optimization algori-thmrdquo Energy vol 55 pp 497ndash510 2013

[27] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[28] S Abbasbandy E Magyari and E Shivanian ldquoThe homotopyanalysis method for multiple solutions of nonlinear boundaryvalue problemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 9-10 pp 3530ndash3536 2009

[29] H Shahmohamadi M M Rashidi and S Dinarvand ldquoAna-lytic approximate solutions for unsteady two-dimensional andaxisymmetric squeezing flows between parallel platesrdquo Mathe-matical Problems in Engineering vol 2008 Article ID 93509512 pages 2008

[30] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica 2013

[31] X Su L Zheng X Zhang and J Zhang ldquoMHD mixed conve-ctive heat transfer over a permeable stretching wedge withthermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[32] M Massoudi ldquoLocal non-similarity solutions for the flow ofa non-Newtonian fluid over a wedgerdquo International Journal ofNon-Linear Mechanics vol 36 no 6 pp 961ndash976 2001

[33] D Pal ldquoCombined effects of non-uniform heat sourcesink andthermal radiation on heat transfer over an unsteady stretchingpermeable surfacerdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 4 pp 1890ndash1904 2011

[34] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[35] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[36] M M Rashidi S A Mohimanian pour and S AbbasbandyldquoAnalytic approximate solutions for heat transfer of a microp-olar fluid through a porous mediumwith radiationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 16no 4 pp 1874ndash1889 2011

[37] A Ishak R Nazar and I Pop ldquoFalkner-Skan equation for flowpast a moving wedge with suction or injectionrdquo Journal ofApplied Mathematics and Computing vol 25 no 1-2 pp 67ndash832007

[38] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mechanical Engineering

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Advances inAcoustics ampVibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

thinspJournalthinspofthinsp

Sensors

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Antennas andPropagation

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Advances in Mechanical Engineering 7

0 1 2 3 4 5 60

02

04

06

08

1

M = 0

M = 1

M = 2

M = 3

M = 4

M = 5

120578

120579(120578)

Figure 8 Effect of magnetic parameter on the temperature distri-bution when 119891

119908= Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

k1 = 00

k1 = 05

k1 = 10

k1 = 15

k1 = 20

k1 = 30

120578

f998400 (120578)

Figure 9 Effect of viscoelastic parameter on the velocity profilewhen119872 = 119891

119908= 1 and 120573 = 13

0 1 2 3 4 5 60

02

04

06

08

1

k1 = 00

k1 = 05

k1 = 10

k1 = 15

k1 = 20

k1 = 30

120578

120579(120578)

Figure 10 Effect of viscoelastic parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 119873119903 = 1 and 120573 = 13

0 1 2 3 4 5 60

02

04

06

08

1

120578

f998400 (120578)

fw = 00

fw = 05

fw = 10

fw = 20

fw = 30

Figure 11 Effect of suction parameter on the velocity profile when119872 = 1 120573 = 13 and 119896

1= 05

8 Advances in Mechanical Engineering

0 1 2 3 4 5 60

02

04

06

08

1

120578

fw = 00

fw = 05

fw = 10

fw = 20

fw = 30

120579(120578)

Figure 12 Effect of suction parameter on the temperature distribu-tion when119872 = Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05

Applying the suction at the wedge surface causes the amountof the fluid to draw into the surface and consequently thehydrodynamic boundary-layer becomes thinner In additionthe thermal boundary-layer gets depressed by increasing thesuction parameter

The effects of the thermal radiation parameter and thePrandtl number on the temperature distribution are shownin Figures 13 and 14 when 119872 = 119891

119908= 119873119903 = 1 120573 = 13

and 1198961= 05 The rate of energy transport to the fluid

increases by increasing the thermal radiation parameterThus the temperature of the fluid increases On the otherhand the increase of radiation parameter leads to overcom-ing the effect of convective heat transfer Based on the Prandtlnumber definition (Pr = ]120572) this parameter is definedas the ratio between the momentum diffusion to thermaldiffusion Thus with the increase of Prandtl number thethermal diffusion decreases and so the thermal boundary-layer becomes thinner as seen in Figure 14 It physicallymeans that the flow with large Prandtl number dissipates theheat faster to the fluid as the temperature gradient gets steeperand hence increasing the heat transfer coefficient between thesurface and the fluid

5 Conclusions

In this paper the semi-analyticalnumerical techniqueknown as HAM has been implemented to solve the trans-formed differential equations describing the MHD mixedconvective heat transfer for an incompressible laminarand electrically conducting viscoelastic fluid flow over a

0 1 2 3 4 5 60

02

04

06

08

1

120578

f998400 (120578)

Nr = 00

Nr = 05

Nr = 10

Nr = 20

Nr = 30

Figure 13 Effect of thermal radiation parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 1 120573 = 13 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

120578

120579(120578)

Pr = 071

Pr = 100

Pr = 200

Pr = 300

Pr = 500

Figure 14 Effect of Prandtl number on the temperature distributionwhen119872 = 119891

119908= 119873119903 = 1 120573 = 13 and 119896

1= 05

porous wedge in the presence of the thermal radiation effectThe present semi-numericalanalytical simulations agreeclosely with the previous studies for some special casesHAM has been shown to be a very strong and efficient

Advances in Mechanical Engineering 9

technique in finding analytical solutions for nonlineardifferential equations HAM is displayed to illustrate exce-llent convergence and accuracy and is currently beingemployed to extend the present study to mixed convectiveheat transfer simulations The effects of different physicalkey parameters such as wedge angle parameter magneticparameter viscoelastic parameter suction parameterthermal radiation parameter and Prandtl number are plottedand discussedThe results show that as the wedge angle incre-ases the heat transfer to the fluid increases for other constantspecified parameter and for Pr = 1 The magnetic field has aweak effect on the thermal boundary thickness however thesuction has a remarkable effect on it Increasing the thermalradiation parameter reduces the heat transfer coefficientbetween the wedge and the fluid however increasing Prandtlnumber increases it

Conflict of Interests

On behalf of all the authors there is no conflict of interests toreport

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[2] K Hiemenz Die Grenzschicht an einem in den gleichformigenFlussigkeitsstrom eingetauchten geraden Kreiszylinder WeberBerlin Germany 1911

[3] E RG EckertDie Berechnung desWarmeubergangs in der lam-inaren Grenzschicht umstromter Korper VDI Berlin Germany1942

[4] P D Ariel ldquoA numerical algorithm for computing the stagna-tion point flow of a second grade fluid withwithout suctionrdquoJournal of Computational and Applied Mathematics vol 59 no1 pp 9ndash24 1995

[5] M S Abel P G Siddheshwar andMMNandeppanavar ldquoHeattransfer in a viscoelastic boundary layer flow over a stretchingsheet with viscous dissipation and non-uniform heat sourcerdquoInternational Journal of Heat andMass Transfer vol 50 no 5-6pp 960ndash966 2007

[6] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[7] P S Datti K V Prasad M S Abel and A Joshi ldquoMHDvisco-elastic fluid flow over a non-isothermal stretching sheetrdquoInternational Journal of Engineering Science vol 42 no 8-9 pp935ndash946 2004

[8] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[9] M Subhas Abel S K Khan and K V Prasad ldquoStudy ofvisco-elastic fluid flow and heat transfer over a stretching sheetwith variable viscosityrdquo International Journal of Non-LinearMechanics vol 37 no 1 pp 81ndash88 2002

[10] I Anwar N Amin and I Pop ldquoMixed convection boundarylayer flow of a viscoelastic fluid over a horizontal circular cyli-nderrdquo International Journal ofNon-LinearMechanics vol 43 no9 pp 814ndash821 2008

[11] R M Sonth S K KhanM S Abel and K V Prasad ldquoHeat andmass transfer in a visco-elastic fluid flow over an acceleratingsurface with heat sourcesink and viscous dissipationrdquoHeat andMass Transfer vol 38 no 3 pp 213ndash220 2002

[12] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[13] K-L Hsiao ldquoHeat and mass mixed convection for MHD visco-elastic fluid past a stretching sheet with ohmic dissipationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 7 pp 1803ndash1812 2010

[14] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012

[15] A J Chamkha M Mujtaba A Quadri and C Issa ldquoThermalradiation effects on MHD forced convection flow adjacent to anon-isothermal wedge in the presence of a heat source or sinkrdquoHeat and Mass Transfer vol 39 no 4 pp 305ndash312 2003

[16] K-L Hsiao ldquoMHDmixed convection for viscoelastic fluid pasta porouswedgerdquo International Journal of Non-LinearMechanicsvol 46 no 1 pp 1ndash8 2011

[17] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010

[18] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuously moving stretching surface by HomotopyAnalysis Method with two auxiliary parametersrdquo Journal ofApplied Mathematics vol 2012 Article ID 780415 19 pages2012

[19] T R Mahapatra S Dholey and A S Gupta ldquoOblique stag-nation-point flow of an incompressible visco-elastic fluid towa-rds a stretching surfacerdquo International Journal of Non-LinearMechanics vol 42 no 3 pp 484ndash499 2007

[20] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC New York NY USA2004

[21] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[22] MMustafa T Hayat I Pop S Asghar and S Obaidat ldquoStagna-tion-point flow of a nanofluid towards a stretching sheetrdquo Inter-national Journal of Heat and Mass Transfer vol 54 no 25-26pp 5588ndash5594 2011

[23] M M Rashidi T Hayat E Erfani S A Mohimanian Pour andA A Hendi ldquoSimultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steadyMHDconvective flow

10 Advances in Mechanical Engineering

due to a rotating diskrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[24] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[25] 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

[26] M M Rashidi M Ali N Freidoonimehr and F Nazari ldquoPara-metric analysis and optimization of entropy generation inunsteady MHD flow over a stretching rotating disk using arti-ficial neural network and particle swarm optimization algori-thmrdquo Energy vol 55 pp 497ndash510 2013

[27] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[28] S Abbasbandy E Magyari and E Shivanian ldquoThe homotopyanalysis method for multiple solutions of nonlinear boundaryvalue problemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 9-10 pp 3530ndash3536 2009

[29] H Shahmohamadi M M Rashidi and S Dinarvand ldquoAna-lytic approximate solutions for unsteady two-dimensional andaxisymmetric squeezing flows between parallel platesrdquo Mathe-matical Problems in Engineering vol 2008 Article ID 93509512 pages 2008

[30] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica 2013

[31] X Su L Zheng X Zhang and J Zhang ldquoMHD mixed conve-ctive heat transfer over a permeable stretching wedge withthermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[32] M Massoudi ldquoLocal non-similarity solutions for the flow ofa non-Newtonian fluid over a wedgerdquo International Journal ofNon-Linear Mechanics vol 36 no 6 pp 961ndash976 2001

[33] D Pal ldquoCombined effects of non-uniform heat sourcesink andthermal radiation on heat transfer over an unsteady stretchingpermeable surfacerdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 4 pp 1890ndash1904 2011

[34] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[35] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[36] M M Rashidi S A Mohimanian pour and S AbbasbandyldquoAnalytic approximate solutions for heat transfer of a microp-olar fluid through a porous mediumwith radiationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 16no 4 pp 1874ndash1889 2011

[37] A Ishak R Nazar and I Pop ldquoFalkner-Skan equation for flowpast a moving wedge with suction or injectionrdquo Journal ofApplied Mathematics and Computing vol 25 no 1-2 pp 67ndash832007

[38] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mechanical Engineering

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Advances inAcoustics ampVibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

thinspJournalthinspofthinsp

Sensors

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Antennas andPropagation

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

8 Advances in Mechanical Engineering

0 1 2 3 4 5 60

02

04

06

08

1

120578

fw = 00

fw = 05

fw = 10

fw = 20

fw = 30

120579(120578)

Figure 12 Effect of suction parameter on the temperature distribu-tion when119872 = Pr = 119873119903 = 1 120573 = 13 and 119896

1= 05

Applying the suction at the wedge surface causes the amountof the fluid to draw into the surface and consequently thehydrodynamic boundary-layer becomes thinner In additionthe thermal boundary-layer gets depressed by increasing thesuction parameter

The effects of the thermal radiation parameter and thePrandtl number on the temperature distribution are shownin Figures 13 and 14 when 119872 = 119891

119908= 119873119903 = 1 120573 = 13

and 1198961= 05 The rate of energy transport to the fluid

increases by increasing the thermal radiation parameterThus the temperature of the fluid increases On the otherhand the increase of radiation parameter leads to overcom-ing the effect of convective heat transfer Based on the Prandtlnumber definition (Pr = ]120572) this parameter is definedas the ratio between the momentum diffusion to thermaldiffusion Thus with the increase of Prandtl number thethermal diffusion decreases and so the thermal boundary-layer becomes thinner as seen in Figure 14 It physicallymeans that the flow with large Prandtl number dissipates theheat faster to the fluid as the temperature gradient gets steeperand hence increasing the heat transfer coefficient between thesurface and the fluid

5 Conclusions

In this paper the semi-analyticalnumerical techniqueknown as HAM has been implemented to solve the trans-formed differential equations describing the MHD mixedconvective heat transfer for an incompressible laminarand electrically conducting viscoelastic fluid flow over a

0 1 2 3 4 5 60

02

04

06

08

1

120578

f998400 (120578)

Nr = 00

Nr = 05

Nr = 10

Nr = 20

Nr = 30

Figure 13 Effect of thermal radiation parameter on the temperaturedistribution when119872 = 119891

119908= Pr = 1 120573 = 13 and 119896

1= 05

0 1 2 3 4 5 60

02

04

06

08

1

120578

120579(120578)

Pr = 071

Pr = 100

Pr = 200

Pr = 300

Pr = 500

Figure 14 Effect of Prandtl number on the temperature distributionwhen119872 = 119891

119908= 119873119903 = 1 120573 = 13 and 119896

1= 05

porous wedge in the presence of the thermal radiation effectThe present semi-numericalanalytical simulations agreeclosely with the previous studies for some special casesHAM has been shown to be a very strong and efficient

Advances in Mechanical Engineering 9

technique in finding analytical solutions for nonlineardifferential equations HAM is displayed to illustrate exce-llent convergence and accuracy and is currently beingemployed to extend the present study to mixed convectiveheat transfer simulations The effects of different physicalkey parameters such as wedge angle parameter magneticparameter viscoelastic parameter suction parameterthermal radiation parameter and Prandtl number are plottedand discussedThe results show that as the wedge angle incre-ases the heat transfer to the fluid increases for other constantspecified parameter and for Pr = 1 The magnetic field has aweak effect on the thermal boundary thickness however thesuction has a remarkable effect on it Increasing the thermalradiation parameter reduces the heat transfer coefficientbetween the wedge and the fluid however increasing Prandtlnumber increases it

Conflict of Interests

On behalf of all the authors there is no conflict of interests toreport

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[2] K Hiemenz Die Grenzschicht an einem in den gleichformigenFlussigkeitsstrom eingetauchten geraden Kreiszylinder WeberBerlin Germany 1911

[3] E RG EckertDie Berechnung desWarmeubergangs in der lam-inaren Grenzschicht umstromter Korper VDI Berlin Germany1942

[4] P D Ariel ldquoA numerical algorithm for computing the stagna-tion point flow of a second grade fluid withwithout suctionrdquoJournal of Computational and Applied Mathematics vol 59 no1 pp 9ndash24 1995

[5] M S Abel P G Siddheshwar andMMNandeppanavar ldquoHeattransfer in a viscoelastic boundary layer flow over a stretchingsheet with viscous dissipation and non-uniform heat sourcerdquoInternational Journal of Heat andMass Transfer vol 50 no 5-6pp 960ndash966 2007

[6] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[7] P S Datti K V Prasad M S Abel and A Joshi ldquoMHDvisco-elastic fluid flow over a non-isothermal stretching sheetrdquoInternational Journal of Engineering Science vol 42 no 8-9 pp935ndash946 2004

[8] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[9] M Subhas Abel S K Khan and K V Prasad ldquoStudy ofvisco-elastic fluid flow and heat transfer over a stretching sheetwith variable viscosityrdquo International Journal of Non-LinearMechanics vol 37 no 1 pp 81ndash88 2002

[10] I Anwar N Amin and I Pop ldquoMixed convection boundarylayer flow of a viscoelastic fluid over a horizontal circular cyli-nderrdquo International Journal ofNon-LinearMechanics vol 43 no9 pp 814ndash821 2008

[11] R M Sonth S K KhanM S Abel and K V Prasad ldquoHeat andmass transfer in a visco-elastic fluid flow over an acceleratingsurface with heat sourcesink and viscous dissipationrdquoHeat andMass Transfer vol 38 no 3 pp 213ndash220 2002

[12] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[13] K-L Hsiao ldquoHeat and mass mixed convection for MHD visco-elastic fluid past a stretching sheet with ohmic dissipationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 7 pp 1803ndash1812 2010

[14] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012

[15] A J Chamkha M Mujtaba A Quadri and C Issa ldquoThermalradiation effects on MHD forced convection flow adjacent to anon-isothermal wedge in the presence of a heat source or sinkrdquoHeat and Mass Transfer vol 39 no 4 pp 305ndash312 2003

[16] K-L Hsiao ldquoMHDmixed convection for viscoelastic fluid pasta porouswedgerdquo International Journal of Non-LinearMechanicsvol 46 no 1 pp 1ndash8 2011

[17] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010

[18] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuously moving stretching surface by HomotopyAnalysis Method with two auxiliary parametersrdquo Journal ofApplied Mathematics vol 2012 Article ID 780415 19 pages2012

[19] T R Mahapatra S Dholey and A S Gupta ldquoOblique stag-nation-point flow of an incompressible visco-elastic fluid towa-rds a stretching surfacerdquo International Journal of Non-LinearMechanics vol 42 no 3 pp 484ndash499 2007

[20] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC New York NY USA2004

[21] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[22] MMustafa T Hayat I Pop S Asghar and S Obaidat ldquoStagna-tion-point flow of a nanofluid towards a stretching sheetrdquo Inter-national Journal of Heat and Mass Transfer vol 54 no 25-26pp 5588ndash5594 2011

[23] M M Rashidi T Hayat E Erfani S A Mohimanian Pour andA A Hendi ldquoSimultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steadyMHDconvective flow

10 Advances in Mechanical Engineering

due to a rotating diskrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[24] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[25] 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

[26] M M Rashidi M Ali N Freidoonimehr and F Nazari ldquoPara-metric analysis and optimization of entropy generation inunsteady MHD flow over a stretching rotating disk using arti-ficial neural network and particle swarm optimization algori-thmrdquo Energy vol 55 pp 497ndash510 2013

[27] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[28] S Abbasbandy E Magyari and E Shivanian ldquoThe homotopyanalysis method for multiple solutions of nonlinear boundaryvalue problemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 9-10 pp 3530ndash3536 2009

[29] H Shahmohamadi M M Rashidi and S Dinarvand ldquoAna-lytic approximate solutions for unsteady two-dimensional andaxisymmetric squeezing flows between parallel platesrdquo Mathe-matical Problems in Engineering vol 2008 Article ID 93509512 pages 2008

[30] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica 2013

[31] X Su L Zheng X Zhang and J Zhang ldquoMHD mixed conve-ctive heat transfer over a permeable stretching wedge withthermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[32] M Massoudi ldquoLocal non-similarity solutions for the flow ofa non-Newtonian fluid over a wedgerdquo International Journal ofNon-Linear Mechanics vol 36 no 6 pp 961ndash976 2001

[33] D Pal ldquoCombined effects of non-uniform heat sourcesink andthermal radiation on heat transfer over an unsteady stretchingpermeable surfacerdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 4 pp 1890ndash1904 2011

[34] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[35] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[36] M M Rashidi S A Mohimanian pour and S AbbasbandyldquoAnalytic approximate solutions for heat transfer of a microp-olar fluid through a porous mediumwith radiationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 16no 4 pp 1874ndash1889 2011

[37] A Ishak R Nazar and I Pop ldquoFalkner-Skan equation for flowpast a moving wedge with suction or injectionrdquo Journal ofApplied Mathematics and Computing vol 25 no 1-2 pp 67ndash832007

[38] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mechanical Engineering

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Advances inAcoustics ampVibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

thinspJournalthinspofthinsp

Sensors

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Antennas andPropagation

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Advances in Mechanical Engineering 9

technique in finding analytical solutions for nonlineardifferential equations HAM is displayed to illustrate exce-llent convergence and accuracy and is currently beingemployed to extend the present study to mixed convectiveheat transfer simulations The effects of different physicalkey parameters such as wedge angle parameter magneticparameter viscoelastic parameter suction parameterthermal radiation parameter and Prandtl number are plottedand discussedThe results show that as the wedge angle incre-ases the heat transfer to the fluid increases for other constantspecified parameter and for Pr = 1 The magnetic field has aweak effect on the thermal boundary thickness however thesuction has a remarkable effect on it Increasing the thermalradiation parameter reduces the heat transfer coefficientbetween the wedge and the fluid however increasing Prandtlnumber increases it

Conflict of Interests

On behalf of all the authors there is no conflict of interests toreport

Acknowledgments

Theauthors express their gratitude to the anonymous refereesfor their constructive reviews of the paper and for helpfulcomments The authors extend their appreciation to theDeanship of Scientific Research at King Saud University forfunding this work through the research group Project noRGP-VPP-080

References

[1] M M Rashidi and E Erfani ldquoA new analytical study of MHDstagnation-point flow in porous media with heat transferrdquoComputers and Fluids vol 40 no 1 pp 172ndash178 2011

[2] K Hiemenz Die Grenzschicht an einem in den gleichformigenFlussigkeitsstrom eingetauchten geraden Kreiszylinder WeberBerlin Germany 1911

[3] E RG EckertDie Berechnung desWarmeubergangs in der lam-inaren Grenzschicht umstromter Korper VDI Berlin Germany1942

[4] P D Ariel ldquoA numerical algorithm for computing the stagna-tion point flow of a second grade fluid withwithout suctionrdquoJournal of Computational and Applied Mathematics vol 59 no1 pp 9ndash24 1995

[5] M S Abel P G Siddheshwar andMMNandeppanavar ldquoHeattransfer in a viscoelastic boundary layer flow over a stretchingsheet with viscous dissipation and non-uniform heat sourcerdquoInternational Journal of Heat andMass Transfer vol 50 no 5-6pp 960ndash966 2007

[6] K V Prasad D Pal V Umesh and N S P Rao ldquoThe effectof variable viscosity on MHD viscoelastic fluid flow and heattransfer over a stretching sheetrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 2 pp 331ndash3442010

[7] P S Datti K V Prasad M S Abel and A Joshi ldquoMHDvisco-elastic fluid flow over a non-isothermal stretching sheetrdquoInternational Journal of Engineering Science vol 42 no 8-9 pp935ndash946 2004

[8] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[9] M Subhas Abel S K Khan and K V Prasad ldquoStudy ofvisco-elastic fluid flow and heat transfer over a stretching sheetwith variable viscosityrdquo International Journal of Non-LinearMechanics vol 37 no 1 pp 81ndash88 2002

[10] I Anwar N Amin and I Pop ldquoMixed convection boundarylayer flow of a viscoelastic fluid over a horizontal circular cyli-nderrdquo International Journal ofNon-LinearMechanics vol 43 no9 pp 814ndash821 2008

[11] R M Sonth S K KhanM S Abel and K V Prasad ldquoHeat andmass transfer in a visco-elastic fluid flow over an acceleratingsurface with heat sourcesink and viscous dissipationrdquoHeat andMass Transfer vol 38 no 3 pp 213ndash220 2002

[12] D Pal ldquoHeat andmass transfer in stagnation-point flow towardsa stretching surface in the presence of buoyancy force andthermal radiationrdquoMeccanica vol 44 no 2 pp 145ndash158 2009

[13] K-L Hsiao ldquoHeat and mass mixed convection for MHD visco-elastic fluid past a stretching sheet with ohmic dissipationrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 7 pp 1803ndash1812 2010

[14] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012

[15] A J Chamkha M Mujtaba A Quadri and C Issa ldquoThermalradiation effects on MHD forced convection flow adjacent to anon-isothermal wedge in the presence of a heat source or sinkrdquoHeat and Mass Transfer vol 39 no 4 pp 305ndash312 2003

[16] K-L Hsiao ldquoMHDmixed convection for viscoelastic fluid pasta porouswedgerdquo International Journal of Non-LinearMechanicsvol 46 no 1 pp 1ndash8 2011

[17] E Erfani M M Rashidi and A B Parsa ldquoThe modifieddifferential transform method for solving off-centered stag-nation flow toward a rotating discrdquo International Journal ofComputational Methods vol 7 no 4 pp 655ndash670 2010

[18] MM Rashidi E Momoniat and B Rostami ldquoAnalytic approx-imate solutions forMHD boundary-layer viscoelastic fluid flowover continuously moving stretching surface by HomotopyAnalysis Method with two auxiliary parametersrdquo Journal ofApplied Mathematics vol 2012 Article ID 780415 19 pages2012

[19] T R Mahapatra S Dholey and A S Gupta ldquoOblique stag-nation-point flow of an incompressible visco-elastic fluid towa-rds a stretching surfacerdquo International Journal of Non-LinearMechanics vol 42 no 3 pp 484ndash499 2007

[20] S J Liao Beyond Perturbation Introduction to the HomotopyAnalysis Method Chapman amp HallCRC New York NY USA2004

[21] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004

[22] MMustafa T Hayat I Pop S Asghar and S Obaidat ldquoStagna-tion-point flow of a nanofluid towards a stretching sheetrdquo Inter-national Journal of Heat and Mass Transfer vol 54 no 25-26pp 5588ndash5594 2011

[23] M M Rashidi T Hayat E Erfani S A Mohimanian Pour andA A Hendi ldquoSimultaneous effects of partial slip and thermal-diffusion and diffusion-thermo on steadyMHDconvective flow

10 Advances in Mechanical Engineering

due to a rotating diskrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[24] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[25] 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

[26] M M Rashidi M Ali N Freidoonimehr and F Nazari ldquoPara-metric analysis and optimization of entropy generation inunsteady MHD flow over a stretching rotating disk using arti-ficial neural network and particle swarm optimization algori-thmrdquo Energy vol 55 pp 497ndash510 2013

[27] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[28] S Abbasbandy E Magyari and E Shivanian ldquoThe homotopyanalysis method for multiple solutions of nonlinear boundaryvalue problemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 9-10 pp 3530ndash3536 2009

[29] H Shahmohamadi M M Rashidi and S Dinarvand ldquoAna-lytic approximate solutions for unsteady two-dimensional andaxisymmetric squeezing flows between parallel platesrdquo Mathe-matical Problems in Engineering vol 2008 Article ID 93509512 pages 2008

[30] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica 2013

[31] X Su L Zheng X Zhang and J Zhang ldquoMHD mixed conve-ctive heat transfer over a permeable stretching wedge withthermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[32] M Massoudi ldquoLocal non-similarity solutions for the flow ofa non-Newtonian fluid over a wedgerdquo International Journal ofNon-Linear Mechanics vol 36 no 6 pp 961ndash976 2001

[33] D Pal ldquoCombined effects of non-uniform heat sourcesink andthermal radiation on heat transfer over an unsteady stretchingpermeable surfacerdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 4 pp 1890ndash1904 2011

[34] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[35] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[36] M M Rashidi S A Mohimanian pour and S AbbasbandyldquoAnalytic approximate solutions for heat transfer of a microp-olar fluid through a porous mediumwith radiationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 16no 4 pp 1874ndash1889 2011

[37] A Ishak R Nazar and I Pop ldquoFalkner-Skan equation for flowpast a moving wedge with suction or injectionrdquo Journal ofApplied Mathematics and Computing vol 25 no 1-2 pp 67ndash832007

[38] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mechanical Engineering

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Advances inAcoustics ampVibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

thinspJournalthinspofthinsp

Sensors

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Antennas andPropagation

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

10 Advances in Mechanical Engineering

due to a rotating diskrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 11 pp 4303ndash4317 2011

[24] Z Abbas Y Wang T Hayat and M Oberlack ldquoMixed convec-tion in the stagnation-point flow of a Maxwell fluid towardsa vertical stretching surfacerdquo Nonlinear Analysis Real WorldApplications vol 11 no 4 pp 3218ndash3228 2010

[25] 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

[26] M M Rashidi M Ali N Freidoonimehr and F Nazari ldquoPara-metric analysis and optimization of entropy generation inunsteady MHD flow over a stretching rotating disk using arti-ficial neural network and particle swarm optimization algori-thmrdquo Energy vol 55 pp 497ndash510 2013

[27] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoSeries solutions for unsteady laminar MHD flownear forward stagnation point of an impulsively rotating andtranslating sphere in presence of buoyancy forcesrdquo NonlinearAnalysis Real World Applications vol 11 no 2 pp 1159ndash11692010

[28] S Abbasbandy E Magyari and E Shivanian ldquoThe homotopyanalysis method for multiple solutions of nonlinear boundaryvalue problemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 14 no 9-10 pp 3530ndash3536 2009

[29] H Shahmohamadi M M Rashidi and S Dinarvand ldquoAna-lytic approximate solutions for unsteady two-dimensional andaxisymmetric squeezing flows between parallel platesrdquo Mathe-matical Problems in Engineering vol 2008 Article ID 93509512 pages 2008

[30] M M Rashidi N Freidoonimehr A Hosseini O A Begand T K Hung ldquoHomotopy simulation of nanofluid dynamicsfrom a non-linearly stretching isothermal permeable sheet withtranspirationrdquoMeccanica 2013

[31] X Su L Zheng X Zhang and J Zhang ldquoMHD mixed conve-ctive heat transfer over a permeable stretching wedge withthermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[32] M Massoudi ldquoLocal non-similarity solutions for the flow ofa non-Newtonian fluid over a wedgerdquo International Journal ofNon-Linear Mechanics vol 36 no 6 pp 961ndash976 2001

[33] D Pal ldquoCombined effects of non-uniform heat sourcesink andthermal radiation on heat transfer over an unsteady stretchingpermeable surfacerdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 4 pp 1890ndash1904 2011

[34] T Hayat M Mustafa and I Pop ldquoHeat and mass transfer forSoret and Dufourrsquos effect on mixed convection boundary layerflow over a stretching vertical surface in a porous medium filledwith a viscoelastic fluidrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 5 pp 1183ndash1196 2010

[35] S-J Liao ldquoAn explicit totally analytic approximate solution forBlasiusrsquo viscous flow problemsrdquo International Journal of Non-Linear Mechanics vol 34 no 4 pp 759ndash778 1999

[36] M M Rashidi S A Mohimanian pour and S AbbasbandyldquoAnalytic approximate solutions for heat transfer of a microp-olar fluid through a porous mediumwith radiationrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 16no 4 pp 1874ndash1889 2011

[37] A Ishak R Nazar and I Pop ldquoFalkner-Skan equation for flowpast a moving wedge with suction or injectionrdquo Journal ofApplied Mathematics and Computing vol 25 no 1-2 pp 67ndash832007

[38] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mechanical Engineering

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Advances inAcoustics ampVibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

thinspJournalthinspofthinsp

Sensors

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Antennas andPropagation

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mechanical Engineering

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Advances inAcoustics ampVibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

thinspJournalthinspofthinsp

Sensors

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Antennas andPropagation

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of