Research ArticleOn a Homotopy Perturbation Treatment of SteadyLaminar Forced Convection Flow over a NonlinearlyStretching Porous Sheet

Nemat Dalir and Salman Nourazar

Department of Mechanical Engineering Amirkabir University of Technology (Tehran Polytechnic) 424 Hafez AvenuePO Box 15875-4413 Tehran Iran

Correspondence should be addressed to Nemat Dalir dalirautacir

Received 13 December 2013 Accepted 10 February 2014 Published 8 April 2014

Academic Editors K Ariyur and R Maceiras

Copyright copy 2014 N Dalir and S Nourazar This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The steady two-dimensional laminar forced convection boundary layer flow of an incompressible viscous Newtonian fluid overa nonlinearly stretching porous (permeable) sheet with suction is considered The sheetrsquos permeability is also considered tobe nonlinear The boundary layer equations are transformed by similarity transformations to a nonlinear ordinary differentialequation (ODE)Then the homotopy perturbationmethod (HPM) is used to solve the resultant nonlinear ODEThe dimensionlessentrainment parameter and the dimensionless sheet surface shear stress are obtained for various values of the suction parameterand the nonlinearity factor of sheet stretching and permeability The results indicate that the dimensionless sheet surface shearstress decreases with the increase of suction parameterThe results of present HPM solution are compared to the values obtained ina previous study by the homotopy analysis method (HAM)The HPM results show that they are in good agreement with the HAMresults within 2 error

1 Introduction

Boundary-layer flowof an incompressible fluid over a stretch-ing sheet has many applications in engineering such as inliquid film condensation process aerodynamic extrusion ofplastic sheets cooling process of metallic plate in a coolingbath and glass and polymer industries

In the last decade many semianalytical methods havebeen used to solve the boundary layer flow problems Forexample He [1] proposed a new perturbation techniquecoupled with the homotopy technique which requires nosmall parameters in the equations and can readily eliminatethe limitations of the traditional perturbation techniques Henamed this method as the homotopy perturbation method(HPM) Esmaeilpour and Ganji [2] presented the problem offorced convection over a horizontal flat plate and employedthe HPM to compute an approximation to the solutionof the system of nonlinear differential equations governingthe problem Xu [3] obtained an approximate solution of aboundary layer equation in unbounded domain by means

of Hersquos homotopy perturbation method (HPM) Fathizadehand Rashidi [4] solved the convective heat transfer equationsof boundary layer flow with pressure gradient over a flatplate using the HPM They studied the effects of Prandtlnumber and pressure gradient on both temperature andvelocity profiles in the boundary layer Raftari and Yildirim[5] obtained by means of the HPM an approximate analyticalsolution of the magnetohydrodynamic (MHD) boundarylayer flow of an upper-convected Maxwell (UCM) fluid overa permeable stretching sheet Raftari et al [6] obtainedby means of the HPM an approximate solution of themagnetohydrodynamic (MHD) boundary layer flow Liao [7]solved the boundary-layer flow over a stretched impermeablewall by means of another semianalytic technique namelythe homotopy analysis method (HAM) He [8] comparedthe HAM with the HPM and stated that the difference isclear just as the Taylor series method is different from theperturbation methods He also illustrated the effectivenessand convenience of HPM which can powerfully use themodern perturbation methods Liao [9] investigated the

Hindawi Publishing CorporationChinese Journal of EngineeringVolume 2014 Article ID 297163 8 pageshttpdxdoiorg1011552014297163

2 Chinese Journal of Engineering

steady-state boundary layer flows over a permeable stretchingsheet by the HAM He obtained two branches of solutionsin which one of them agrees well with the known numericalsolutions but the other is new and has not been reportedin general cases Mahgoub [10] investigated experimentallythe non-Darcian forced convection heat transfer over ahorizontal flat plate in a porousmediumof spherical particlesHe examined the effects of particle diameter and particlesmaterials of different thermal conductivities with air as theworking fluid Dinarvand et al [11] compared the HPM andHAMsolutions for a nonlinear ordinary differential equationarising from Bermanrsquos similarity problem for the steady two-dimensional flow of a viscous incompressible fluid through achannel with wall suction or injection

In the present study the steady boundary layer flow ofan incompressible viscous fluid past a nonlinearly stretchingand nonlinearly permeable flat horizontal sheet is consideredUsing the similarity method the boundary layer equationsare transformed to a nonlinear ordinary differential equation(ODE)The HPM solution of the governing ODE is obtainedand compared to the HAM results [9] The results reveal thatthe HPM is very effective such that the analytical solutionobtained by using only two terms from HPM solutioncoincides well with the HAM solution

2 Mathematical Formulation

The boundary layer flow over a nonlinearly stretching sheetis considered The governing equations of mass momentumand energy for the steady two-dimensional laminar forcedconvection boundary layer flow of an incompressible viscousNewtonian fluid over a stretching permeable sheet are asfollows [9]

120597119906

120597119909+120597V120597119910

= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= 120592

1205972119906

1205971199102 (2)

where 119906 and V are the fluid velocity components in 119909 and119910 directions respectively and 120592 is the kinematic viscosity ofthe fluid The boundary conditions for fluid velocity are asfollows

119906 = 119880119908= 119886(119909 + 119887)

120582 V = 119881

119908= 119860(119909 + 119887)

(120582minus1)2

at 119910 = 0

119906 997888rarr 0 as 119910 997888rarr infin

(3)

where 119880119908represents the stretching velocity of the sheet and

119881119908denotes the velocity of fluid through the porous sheet 119886

119887 120582 and 119860 are constants If 119860 gt 0 then there is injection(blowing) through the sheet and if 119860 lt 0 then there issuction of fluid through the sheet respectively If 119886 gt 0then the sheet stretches in the positive 119909-direction and if119886 lt 0 then the sheet stretches in the negative 119909-directionrespectively

The stream function 120595 is defined such that it satisfies thecontinuity equation (1)

119906 =120597120595

120597119910 V = minus

120597120595

120597119909 (4)

To use similarity method the following similarity trans-formations are applied to the problem

120578 = radic119886 (1 + 120582)

120592(119909 + 119887)

(120582minus1)2119910

119891 (120578) =120595

119886(119909 + 119887)(120582+1)2

radic119886 (1 + 120582)

120592

(5)

where 120578 is similarity variable and 119891 is the dimensionlessvelocity variable Using the similarity transformations (5)the boundary layer equations (1) and (2) transform to thefollowing third order nonlinear ordinary differential equation(ODE)

119891101584010158401015840(120578) +

1

2119891 (120578) 119891

10158401015840(120578) minus 120573119891

10158402(120578) = 0 (6)

and the boundary conditions transform to the followingforms

119891 (0) = 120574 1198911015840(0) = 1 119891

1015840(infin) = 0 (7)

where we have

120574 = minus2119860

radic119886 (120582 + 1) 120592 120573 =

120582

1 + 120582 (8)

Here 120574 is the injection (blowing) or suction parameter 120574 lt 0and 120574 gt 0 correspond to the lateral injection (119881

119908gt 0)

and suction (119881119908

lt 0) of fluid through the porous sheetrespectively and 120574 = 0 indicates the impermeable sheet120573 is the nonlinearity factor of the sheet permeability andstretching and when 120573 = 0 the sheet moves with a constantvelocity From (5) the fluid velocity components can beobtained as follows

119906 (119909 119910) = 119886(119909 + 119887)1205821198911015840(120578)

V (119909 119910) = minus1

2radic119886 (1 + 120582) 120592(119909 + 119887)

(120582minus1)2

times [119891 (120578) + (2120573 minus 1) 1205781198911015840(120578)]

(9)

The fluid vertical velocity far away from the sheet is called theentrainment velocity of the fluid and is obtained as

V (119909 +infin) = minus1

2radic119886 (1 + 120582) 120592(119909 + 119887)

(120582minus1)2119891 (+infin) (10)

The shear stress on the surface of sheet can be written as

120591119908= minus120583

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= minus119886120588radic119886 (1 + 120582) 120592(119909 + 119887)(3120582minus1)2

11989110158401015840(0)

(11)

Thus 119891(+infin) and 11989110158401015840(0) have physical meanings 119891(+infin) isthe dimensionless entrainment parameter and 11989110158401015840(0) is thedimensionless sheet surface shear stress respectively

Chinese Journal of Engineering 3

Table 1 Values of 119891(+infin) and 11989110158401015840(0) for various 120574 when 120573 = 1

120574 120572119891(+infin)

(present HPM solution)11989110158401015840(0)

(present HPM solution)119891(+infin)

(HAM [9])11989110158401015840(0)

(HAM [9])0 070711 117851 minus0942814 128077378 minus0906375511 100000 191666 minus116666 193111056 minus1175614092 136603 269936 minus148804 269794357 minus1513449325 268614 536798 minus274819 536649491 minus27733583210 509808 101955 minus513077 1019523827 minus514619724

3 Application of HPM

The main feature of the HPM is that it deforms a difficultproblem into a set of problems which are easier to solveHPM produces analytical expressions for the solution ofnonlinear differential equations [1] The obtained analyticalsolution by HPM is in the form of an infinite power seriesUsing HPM [1] the original nonlinear ODE which cannotbe solved easily is divided into some linear ODEs which aresolved easily in a recursivemanner bymathematical symbolicsoftware MATHEMATICA First (6) and (7) are written inthe following forms

119906101584010158401015840+1

211990611990610158401015840minus 12057311990610158402= 0 (12)

119906 (0) = 120574 1199061015840(0) = 1 119906

1015840(infin) = 0 (13)

where the dependent variable is changed to 119906 instead of 119891Then a homotopy is constructed in the following form

119906101584010158401015840minus 12057221199061015840+ 119901(

1

211990611990610158401015840minus 12057311990610158402+ 12057221199061015840) = 0 (14)

where 119901 is a small parameter (0 le 119901 le 1) and 120572 is a constantwhich is further to be determined It can be seen that when119901 = 0 (14) becomes 119906101584010158401015840 minus 12057221199061015840 = 0 and when 119901 = 1 (14)results in (12) According to HPM the following polynomialin 119901 is substituted in (14)

119906 = 1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot (15)

where 1199060 1199061 and so forth are the analytical first second

and so forth terms of the solution 119906 Substituting (15) in (14)performing some algebraic manipulation and equating theidentical powers of 119901 to zero give

1199010 119906101584010158401015840

0minus 12057221199061015840

0= 0 119906

0(0) = 120574

1199061015840

0(0) = 1 119906

1015840

0(infin) = 0

(16)

1199011 119906101584010158401015840

1minus 12057221199061015840

1+1

2119906011990610158401015840

0minus 12057311990610158402

0+ 12057221199061015840

0= 0

1199061(0) = 0 119906

1015840

1(0) = 0 119906

1015840

1(infin) = 0

(17)

The equation for 1199010 (16) has the following solution

1199060(120578) = 120574 +

1

120572(1 minus exp (minus120572120578)) (18)

Now if the solution for 1199060 (18) is substituted in the equation

for 1199011 (17) becomes

119906101584010158401015840

1minus 12057221199061015840

1= (

1

2120572120574 +

1

2minus 1205722) exp (minus120572120578)

+ (120573 minus1

2) exp (minus2120572120578)

(19)

The equation for 1199061 (19) can be solved in an unbounded

domain under the boundary conditions 1199061(0) = 0 1199061015840

1(0) = 0

1199061015840

1(infin) = 0 as it is shown in the Appendix which gives

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+minus121205722+ 6120574120572 + 8120573 + 2

241205723exp (minus120572120578)

(20)

in which 120572 = 1205744 + radic(1205744)2+ (12) It should be noted that

in fact 120572 = 1205744 plusmn radic(1205744)2+ (12) but here as 120572 should be

positive (120572 gt 0) thus 120572 = 1205744 + radic(1205744)2+ (12) Therefore

the first-order approximate solution by HPM that is 119891(120578) =119906(120578) = 119906

0(120578) + 119906

1(120578) is as follows

119891 (120578) = (120574 +1

120572+minus2 + 6120572

2minus 2120573 minus 3120572120574

121205723)

+1 minus 2120573

121205723exp (minus2120572120578)

+ (minus361205722+ 6120574120572 + 8120573 + 2

241205723) exp (minus120572120578)

(21)

According to (21) the dimensionless entrainment parameter119891(+infin) and the dimensionless sheet surface shear stress11989110158401015840(0) are obtained as follows

119891 (+infin) =121205741205723+ 18120572

2minus 3120574120572 minus 2 minus 2120573

121205723

11989110158401015840(0) =

minus361205722+ 6120574120572 + 10 minus 8120573

24120572

(22)

4 Results and Discussion

The boundary layer flow over a stretching permeable sheetwith suction is investigated using HPM The dimensionless

4 Chinese Journal of Engineering

Table 2 Values of 119891(+infin) and 11989110158401015840(0) for various 120574 when 120573 = 10

120574 120572119891(+infin)

(present HPM solution)11989110158401015840(0)

(present HPM solution)119891(+infin)

(HAM [9])11989110158401015840(0)

(HAM [9])0 070711 056407 minus268544 0658388 minus26081481 100000 141666 minus286666 1541900 minus28243452 136603 221091 minus318419 2457505 minus30678723 178078 328118 minus355903 3392704 minus33409115 268614 529059 minus386503 5300584 minus397365410 509808 101842 minus571923 10181382 minus5928155

Table 3 Values of 119891(+infin) and 11989110158401015840(0) for various 120573 when 120574 = 1 (120572 = 1)

120573119891(+infin)

(present HPM solution)11989110158401015840(0)

(present HPM solution)119891(+infin)

(HAM [9])11989110158401015840(0)

(HAM [9])minus0656192 219270 minus0414603 23 minus0381592minus0395769 214929 minus0601410 22 minus05691800 208333 minus0833333 2093662 minus07864001 191666 minus116666 1931110 minus11756143 162333 minus175333 1764465 minus16994315 145000 minus225000 1670900 minus2088843

entrainment parameter and the dimensionless sheet surfaceshear stress are obtained and compared to the values obtainedby the HAM solution [9] Table 1 shows the values of thedimensionless entrainment parameter 119891(+infin) and the val-ues of the dimensionless sheet surface shear stress 11989110158401015840(0)using the present HPM method and the HAM solution [9]for various values of 120574 when 120573 = 1 It can be seen that thepresent HPM solution agrees well with the HAM solution [9]for both 119891(+infin) and 11989110158401015840(0) within plusmn2 error It can also beseen that at 120573 = 1 when 120574 increases 119891(+infin) increases Thismeans that the increase of suction through the sheet causesaccording to (10) the increase of the fluid vertical downwardvelocity far away from the sheet It is also observed that at120573 = 1 11989110158401015840(0) decreases with the increase of 120574 that is theincrease of suction through the sheet causes the increase ofsheet surface shear stress

Table 2 demonstrates the values of 119891(+infin) and the valuesof 11989110158401015840(0) using the present HPM solution and the HAMsolution [9] for various values of 120574 when 120573 = 10 It can beseen that at 120573 = 10 when suction parameter 120574 increases119891(+infin) increases while 11989110158401015840(0) decreases which means thatthe suction increase causes the boost of fluid far-away verticalvelocity and sheet surface shear stress

Table 3 shows the values of 119891(+infin) and 11989110158401015840(0) using thepresent HPM solution and the HAM solution [9] for variousvalues of 120573 when 120574 = 1 It is observed that at 120574 = 1 theincrease of 120573 results in the reduction of 119891(+infin) This meansthat when there is suction through the sheet the increaseof nonlinearity factor of sheet permeability and stretchingcauses the fluid entrainment velocity to decrease It may alsobe seen that at 120574 = 1 the increase of120573 decreases11989110158401015840(0) that isthe increase of nonlinearity factor 120573 decreases fluid friction atthe sheet surface

0 2 4 6 8 100

2

4

6

8

10

12

f(+infin

)

120574

HAM and 120573 = 1HAM and 120573 = 10

HPM and 120573 = 1HPM and 120573 = 10

Figure 1 Variation of119891(+infin)with 120574 for various120573 (120573 = 1amp 10) usingpresent HPM solution and HAM [9]

Table 4 depicts 119891(+infin) and 11989110158401015840(0) values using the HPMsolution of the present paper and the HAM solution [9] forvarious values of 120573 when 120574 = 10 It is observable that 119891(+infin)

values reduce and 11989110158401015840(0) decreases with the increase of 120573 ata constant 120574 (120574 = 10) Thus it can be concluded that insituations where lower sheet surface friction is desired lowervalues of suction parameter 120574 and nonlinearity factor 120573 canbe helpful

Figure 1 indicates the variation of 119891(+infin) with 120574 forvarious 120573 using present HPM solution and HAM [9] The

Chinese Journal of Engineering 5

Table 4 Values of 119891(+infin) and 11989110158401015840(0) for various 120573 when 120574 = 10 (120572 = 509808)

120573119891(+infin)

(present HPM solution)11989110158401015840(0)

(present HPM solution)119891(+infin)

(HAM [9])11989110158401015840(0)

(HAM [9])minus5949297 102043 minus467640 1021 minus4413988minus1495596 101987 minus496760 102 minus48997760 101968 minus506539 10197086 minus50493511 101955 minus513077 10195238 minus51461973 101930 minus526154 10191759 minus53330355 101905 minus539231 10188534 minus551167010 101842 minus571923 10181382 minus592815520 101716 minus637307 10169894 minus666363430 101590 minus702691 10160896 minus730736140 101465 minus768075 10153542 minus7886163

0 1 2 3 4 50

05

1

15

2

25

3

f(+infin

)

HAMHPM

120574 = 1

120573

Figure 2 Variation of 119891(+infin) with 120573 by present HPM solution andHAM [9] for 120574 = 1

present HPM solution is observed to agree quite well with theHAM results [9] It is seen that 119891(+infin) augments somewhatlinearly with the increase of 120574 The reason is that when 120574

increases the suction through the porous sheet increasesHence the fluid vertical velocity on the sheet boosts whichresults in increases of the fluid vertical velocity far away fromthe sheet

Figure 2 demonstrates the variation of 119891(+infin) with 120573

using present HPM solution and HAM [9] for 120574 = 1 TheHPMresults are in quite good agreementwith theHAMonesIt is observed that 119891(+infin) reduces linearly with the increaseof 120573 This result is due to the fact that when 120573 increases thevelocity of the fluid in direction perpendicular to the sheetincreases

Figure 3 illustrates the variation of 119891(+infin) with 120573 usingpresent HPM solution and HAM [9] for 120574 = 10 Thedifference with Figure 2 is in the values of 119891(+infin) where120574 = 10 results in higher values of 119891(+infin) while the slopefor reduction of 119891(+infin) is lower in 120574 = 10 (Figure 3)

0 10 20 30 401014

1016

1018

102

1022

1024

f(+infin

)

HAMHPM

120574 = 10

120573

Figure 3 Variation of 119891(+infin) with 120573 by present HPM solution andHAM [9] for 120574 = 10

Figure 4 depicts the variation of 11989110158401015840(0) with 120574 for variousvalues of 120573 (120573 = 1 amp 10) using present HPM solution andHAM [9] For both 120573 = 1 and 120573 = 10 there is a good matchbetween the results of HPM with the ones of HAM It is alsoapparent that 11989110158401015840(0) diminishes with a growth in 120574 When 120574is increased the resultant higher suction leads to a reductionin the stickiness of fluid particles to the sheet Hence the fluidfriction at sheet surface experiences moderation

Figure 5 shows variation of 11989110158401015840(0) with 120573 using presentHPM solution and HAM [9] for 120574 = 1 The results of the twomethods HPM and HAM [9] indicate a quite good matchWith an increase in 120573 a decrease in 11989110158401015840(0) is observed whichcan be attributed to lower fluid-surface interaction by thehigher nonlinearity of sheet stretching and permeability

The variation of11989110158401015840(0)with120573using presentHPMsolutionandHAM [9] for 120574 = 10 is shown in Figure 6 In this case thesame conclusions made in Figure 5 are also quite acceptableHowever for 120574 = 10 (Figure 6) lower values of 11989110158401015840(0) witha lower slope of the curve can be observed

6 Chinese Journal of Engineering

0 2 4 6 8 10minus6

minus5

minus4

minus3

minus2

minus1

0

120574

HAM and 120573 = 1HAM and 120573 = 10

HPM and 120573 = 1HPM and 120573 = 10

f998400998400(0)

Figure 4 Variation of 11989110158401015840(0)with 120574 for various 120573 (120573 = 1amp 10) usingpresent HPM solution and HAM [9]

0 1 2 3 4 5minus3

minus25

minus2

minus15

minus1

minus05

0

f998400998400(0)

HAMHPM

120574 = 1

120573

Figure 5 Variation of 11989110158401015840(0) with 120573 by present HPM solution andHAM [9] for 120574 = 1

5 Conclusions

Thesteady 2-Dboundary layer flowof an incompressible fluidon a nonlinearly stretching permeable sheet is consideredTheHPMsolution of the boundary layer flow is obtainedThedimensionless entrainment parameter and the dimensionlesssheet surface shear stress are obtained and compared to thevalues of a previous HAM solution which show that thepresent HPM solution with only two terms agrees well withthe results of HAMThe results obtained are as follows

(1) The increase of suction through the sheet causes theincrease of fluid vertical downward velocity far awayfrom the sheet

0 10 20 30 40minus8

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

f998400998400(0)

HAMHPM

120574 = 10

120573

Figure 6 Variation of 11989110158401015840(0) with 120573 by present HPM solution andHAM [9] for 120574 = 10

(2) When there is suction through the sheet the increaseof nonlinearity factor of sheet permeability andstretching causes the fluid entrainment velocity todecrease

(3) The fluid friction at the sheet surface decreases withthe increase of nonlinearity factor of sheet stretchingand permeability

Appendix

The equation for 1199061(120578) (19) is solved using the symbolic

software MATHEMATICA under the boundary conditions1199061(0) = 0 1199061015840

1(0) = 0 as

119906101584010158401015840

1minus 12057221199061015840

1= (

1

2120572120574 +

1

2minus 1205722) exp (minus120572120578)

+ (120573 minus1

2) exp (minus2120572120578)

1199061(0) = 0 119906

1015840

1(0) = 0

(A1)

which gives the following solution

1199061(120578) =

minus3 + 41205722+ 2120573 minus 2120572120574 + 8120572

2119862 (2)

41205723

+1 minus 2120573

121205723exp (minus2120572120578)

+3 minus 6120572

2+ 3120572120574 minus 8120572

2119862 (2)

81205723exp (minus120572120578)

+7 minus 6120572

2minus 8120573 + 3120574120572 minus 24120572

2119862 (2)

241205723exp (120572120578)

+minus21205722+ 1 + 120572120574

41205722120578 exp (minus120572120578)

(A2)

Chinese Journal of Engineering 7

where 119862(2) is the integration constant If the boundarycondition 1199061015840

1(infin) = 0 is applied to the solution (A2) it gives

119862(2) and 120572 as

7 minus 61205722minus 8120573 + 3120574120572 minus 24120572

2119862 (2)

241205722= 0

997888rarr 119862 (2) =7

241205722minus1

4minus

120573

31205722+

120574

8120572

minus21205722+ 1 + 120572120574

minus4120572= 0 997888rarr 120572 =

120574 plusmn radic1205742 + 8

4

(A3)

Now if119862(2) from (A3) is substituted in (A2) 1199061(120578) becomes

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+2 minus 12120572

2+ 6120572120574 + 8120573

241205723exp (minus120572120578)

+minus21205722+ 1 + 120572120574

41205722120578 exp (minus120572120578)

(A4)

Here it can be checked and seen that for (A4) 1199061(0) =

0 11990610158401(0) = 0 If the third boundary condition 119906

1015840

1(infin) =

0 is applied to (A4) it gives the value of 120572 = 1205744 plusmn

radic(1205744)2+ (12) This value of 120572 removes the secular term

from the ordinary differential equation (ODE) for 1199061(120578) If 120572

is substituted in the last term of (A4) the last term vanishesand 119906

1(120578) takes its final form as

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+2 minus 12120572

2+ 6120572120574 + 8120573

241205723exp (minus120572120578)

(A5)

Nomenclature

119886 Stretching rate (mdash)119860 Permeability rate (mdash)119887 Displacement value of the sheet stretching

and permeability (m)119891 Dimensionless velocity variable

(= (120595119886(119909 + 119887)(120582+1)2

)radic119886(1 + 120582)120592) (mdash)119901 Small parameter in (14) (mdash)119906 Velocity in x-direction (m sminus1) dependent

variable in (12) instead of 119891 (mdash)119880119908 Velocity of the sheet (m sminus1)

V Velocity in y-direction (m sminus1)119881119908 Velocity of fluid through the porous sheet(m sminus1)

119909 Horizontal coordinate (m)119910 Vertical coordinate (m)

Greek Symbols

120572 Constant in (14) (mdash)120573 Nonlinearity factor of the sheet

stretching and permeability(= 120582(1 + 120582)) (mdash)

120574 Suction or injection (blowing)parameter (= minus2119860radic119886(120582 + 1)120592) (mdash)

120578 Similarity variable(= (119909 + 119887)

(120582minus1)2119910radic119886(1 + 120582)120592) (mdash)

120582 Exponent of sheet stretching andpermeability velocity (mdash)

120583 Dynamic viscosity (N smminus2)120592 Kinematic viscosity (m2 sminus1)120588 Density (kgmminus3)120591 Shear stress (Nmminus2)120591119908 Shear stress on sheet surface (Nmminus2)

120595 Stream function (m2 sminus1)

Subscripts

infin Infinity119891 Fluid119908 Sheet surface

Conflict of Interests

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

References

[1] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[2] M Esmaeilpour andD D Ganji ldquoApplication of Hersquos homotopyperturbation method to boundary layer flow and convectionheat transfer over a flat platerdquo Physics Letters A vol 372 no 1pp 33ndash38 2007

[3] L Xu ldquoHersquos homotopy perturbation method for a boundarylayer equation in unbounded domainrdquo Computers and Math-ematics with Applications vol 54 no 7-8 pp 1067ndash1070 2007

[4] M Fathizadeh and F Rashidi ldquoBoundary layer convective heattransfer with pressure gradient using Homotopy PerturbationMethod (HPM) over a flat platerdquo Chaos Solitons and Fractalsvol 42 no 4 pp 2413ndash2419 2009

[5] B Raftari and A Yildirim ldquoThe application of homotopyperturbation method for MHD flows of UCM fluids aboveporous stretching sheetsrdquo Computers and Mathematics withApplications vol 59 no 10 pp 3328ndash3337 2010

[6] B Raftari S T Mohyud-Din and A Yildirim ldquoSolution tothe MHD flow over a non-linear stretching sheet by homotopyperturbation methodrdquo Science China Physics Mechanics andAstronomy vol 54 no 2 pp 342ndash345 2011

[7] S J Liao ldquoA new branch of solutions of boundary-layer flowsover an impermeable stretched platerdquo International Journal ofHeat and Mass Transfer vol 48 no 12 pp 2529ndash2539 2005

[8] J-H He ldquoComparison of homotopy perturbation methodand homotopy analysis methodrdquo Applied Mathematics andComputation vol 156 no 2 pp 527ndash539 2004

8 Chinese Journal of Engineering

[9] S-J Liao ldquoA new branch of solutions of boundary-layer flowsover a permeable stretching platerdquo International Journal of Non-Linear Mechanics vol 42 no 6 pp 819ndash830 2007

[10] S EMahgoub ldquoForced convection heat transfer over a flat platein a porous mediumrdquoAin Shams Engineering Journal vol 4 pp605ndash613 2013

[11] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoComparison ofHAMandHPMmethods for Bermanrsquosmodel of two-dimensional viscous flow in porous channel withwall suction or injectionrdquo Advances in Theoretical and AppliedMechanics vol 1 no 7 pp 337ndash347 2008

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

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

Antennas andPropagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Chinese Journal of Engineering 3

Table 1 Values of 119891(+infin) and 11989110158401015840(0) for various 120574 when 120573 = 1

120574 120572119891(+infin)

(present HPM solution)11989110158401015840(0)

(present HPM solution)119891(+infin)

(HAM [9])11989110158401015840(0)

(HAM [9])0 070711 117851 minus0942814 128077378 minus0906375511 100000 191666 minus116666 193111056 minus1175614092 136603 269936 minus148804 269794357 minus1513449325 268614 536798 minus274819 536649491 minus27733583210 509808 101955 minus513077 1019523827 minus514619724

3 Application of HPM

The main feature of the HPM is that it deforms a difficultproblem into a set of problems which are easier to solveHPM produces analytical expressions for the solution ofnonlinear differential equations [1] The obtained analyticalsolution by HPM is in the form of an infinite power seriesUsing HPM [1] the original nonlinear ODE which cannotbe solved easily is divided into some linear ODEs which aresolved easily in a recursivemanner bymathematical symbolicsoftware MATHEMATICA First (6) and (7) are written inthe following forms

119906101584010158401015840+1

211990611990610158401015840minus 12057311990610158402= 0 (12)

119906 (0) = 120574 1199061015840(0) = 1 119906

1015840(infin) = 0 (13)

where the dependent variable is changed to 119906 instead of 119891Then a homotopy is constructed in the following form

119906101584010158401015840minus 12057221199061015840+ 119901(

1

211990611990610158401015840minus 12057311990610158402+ 12057221199061015840) = 0 (14)

where 119901 is a small parameter (0 le 119901 le 1) and 120572 is a constantwhich is further to be determined It can be seen that when119901 = 0 (14) becomes 119906101584010158401015840 minus 12057221199061015840 = 0 and when 119901 = 1 (14)results in (12) According to HPM the following polynomialin 119901 is substituted in (14)

119906 = 1199060+ 1199011199061+ 11990121199062+ sdot sdot sdot (15)

where 1199060 1199061 and so forth are the analytical first second

and so forth terms of the solution 119906 Substituting (15) in (14)performing some algebraic manipulation and equating theidentical powers of 119901 to zero give

1199010 119906101584010158401015840

0minus 12057221199061015840

0= 0 119906

0(0) = 120574

1199061015840

0(0) = 1 119906

1015840

0(infin) = 0

(16)

1199011 119906101584010158401015840

1minus 12057221199061015840

1+1

2119906011990610158401015840

0minus 12057311990610158402

0+ 12057221199061015840

0= 0

1199061(0) = 0 119906

1015840

1(0) = 0 119906

1015840

1(infin) = 0

(17)

The equation for 1199010 (16) has the following solution

1199060(120578) = 120574 +

1

120572(1 minus exp (minus120572120578)) (18)

Now if the solution for 1199060 (18) is substituted in the equation

for 1199011 (17) becomes

119906101584010158401015840

1minus 12057221199061015840

1= (

1

2120572120574 +

1

2minus 1205722) exp (minus120572120578)

+ (120573 minus1

2) exp (minus2120572120578)

(19)

The equation for 1199061 (19) can be solved in an unbounded

domain under the boundary conditions 1199061(0) = 0 1199061015840

1(0) = 0

1199061015840

1(infin) = 0 as it is shown in the Appendix which gives

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+minus121205722+ 6120574120572 + 8120573 + 2

241205723exp (minus120572120578)

(20)

in which 120572 = 1205744 + radic(1205744)2+ (12) It should be noted that

in fact 120572 = 1205744 plusmn radic(1205744)2+ (12) but here as 120572 should be

positive (120572 gt 0) thus 120572 = 1205744 + radic(1205744)2+ (12) Therefore

the first-order approximate solution by HPM that is 119891(120578) =119906(120578) = 119906

0(120578) + 119906

1(120578) is as follows

119891 (120578) = (120574 +1

120572+minus2 + 6120572

2minus 2120573 minus 3120572120574

121205723)

+1 minus 2120573

121205723exp (minus2120572120578)

+ (minus361205722+ 6120574120572 + 8120573 + 2

241205723) exp (minus120572120578)

(21)

According to (21) the dimensionless entrainment parameter119891(+infin) and the dimensionless sheet surface shear stress11989110158401015840(0) are obtained as follows

119891 (+infin) =121205741205723+ 18120572

2minus 3120574120572 minus 2 minus 2120573

121205723

11989110158401015840(0) =

minus361205722+ 6120574120572 + 10 minus 8120573

24120572

(22)

4 Results and Discussion

The boundary layer flow over a stretching permeable sheetwith suction is investigated using HPM The dimensionless

4 Chinese Journal of Engineering

Table 2 Values of 119891(+infin) and 11989110158401015840(0) for various 120574 when 120573 = 10

120574 120572119891(+infin)

(present HPM solution)11989110158401015840(0)

(present HPM solution)119891(+infin)

(HAM [9])11989110158401015840(0)

(HAM [9])0 070711 056407 minus268544 0658388 minus26081481 100000 141666 minus286666 1541900 minus28243452 136603 221091 minus318419 2457505 minus30678723 178078 328118 minus355903 3392704 minus33409115 268614 529059 minus386503 5300584 minus397365410 509808 101842 minus571923 10181382 minus5928155

Table 3 Values of 119891(+infin) and 11989110158401015840(0) for various 120573 when 120574 = 1 (120572 = 1)

120573119891(+infin)

(present HPM solution)11989110158401015840(0)

(present HPM solution)119891(+infin)

(HAM [9])11989110158401015840(0)

(HAM [9])minus0656192 219270 minus0414603 23 minus0381592minus0395769 214929 minus0601410 22 minus05691800 208333 minus0833333 2093662 minus07864001 191666 minus116666 1931110 minus11756143 162333 minus175333 1764465 minus16994315 145000 minus225000 1670900 minus2088843

entrainment parameter and the dimensionless sheet surfaceshear stress are obtained and compared to the values obtainedby the HAM solution [9] Table 1 shows the values of thedimensionless entrainment parameter 119891(+infin) and the val-ues of the dimensionless sheet surface shear stress 11989110158401015840(0)using the present HPM method and the HAM solution [9]for various values of 120574 when 120573 = 1 It can be seen that thepresent HPM solution agrees well with the HAM solution [9]for both 119891(+infin) and 11989110158401015840(0) within plusmn2 error It can also beseen that at 120573 = 1 when 120574 increases 119891(+infin) increases Thismeans that the increase of suction through the sheet causesaccording to (10) the increase of the fluid vertical downwardvelocity far away from the sheet It is also observed that at120573 = 1 11989110158401015840(0) decreases with the increase of 120574 that is theincrease of suction through the sheet causes the increase ofsheet surface shear stress

Table 2 demonstrates the values of 119891(+infin) and the valuesof 11989110158401015840(0) using the present HPM solution and the HAMsolution [9] for various values of 120574 when 120573 = 10 It can beseen that at 120573 = 10 when suction parameter 120574 increases119891(+infin) increases while 11989110158401015840(0) decreases which means thatthe suction increase causes the boost of fluid far-away verticalvelocity and sheet surface shear stress

Table 3 shows the values of 119891(+infin) and 11989110158401015840(0) using thepresent HPM solution and the HAM solution [9] for variousvalues of 120573 when 120574 = 1 It is observed that at 120574 = 1 theincrease of 120573 results in the reduction of 119891(+infin) This meansthat when there is suction through the sheet the increaseof nonlinearity factor of sheet permeability and stretchingcauses the fluid entrainment velocity to decrease It may alsobe seen that at 120574 = 1 the increase of120573 decreases11989110158401015840(0) that isthe increase of nonlinearity factor 120573 decreases fluid friction atthe sheet surface

0 2 4 6 8 100

2

4

6

8

10

12

f(+infin

)

120574

HAM and 120573 = 1HAM and 120573 = 10

HPM and 120573 = 1HPM and 120573 = 10

Figure 1 Variation of119891(+infin)with 120574 for various120573 (120573 = 1amp 10) usingpresent HPM solution and HAM [9]

Table 4 depicts 119891(+infin) and 11989110158401015840(0) values using the HPMsolution of the present paper and the HAM solution [9] forvarious values of 120573 when 120574 = 10 It is observable that 119891(+infin)

values reduce and 11989110158401015840(0) decreases with the increase of 120573 ata constant 120574 (120574 = 10) Thus it can be concluded that insituations where lower sheet surface friction is desired lowervalues of suction parameter 120574 and nonlinearity factor 120573 canbe helpful

Figure 1 indicates the variation of 119891(+infin) with 120574 forvarious 120573 using present HPM solution and HAM [9] The

Chinese Journal of Engineering 5

Table 4 Values of 119891(+infin) and 11989110158401015840(0) for various 120573 when 120574 = 10 (120572 = 509808)

120573119891(+infin)

(present HPM solution)11989110158401015840(0)

(present HPM solution)119891(+infin)

(HAM [9])11989110158401015840(0)

(HAM [9])minus5949297 102043 minus467640 1021 minus4413988minus1495596 101987 minus496760 102 minus48997760 101968 minus506539 10197086 minus50493511 101955 minus513077 10195238 minus51461973 101930 minus526154 10191759 minus53330355 101905 minus539231 10188534 minus551167010 101842 minus571923 10181382 minus592815520 101716 minus637307 10169894 minus666363430 101590 minus702691 10160896 minus730736140 101465 minus768075 10153542 minus7886163

0 1 2 3 4 50

05

1

15

2

25

3

f(+infin

)

HAMHPM

120574 = 1

120573

Figure 2 Variation of 119891(+infin) with 120573 by present HPM solution andHAM [9] for 120574 = 1

present HPM solution is observed to agree quite well with theHAM results [9] It is seen that 119891(+infin) augments somewhatlinearly with the increase of 120574 The reason is that when 120574

increases the suction through the porous sheet increasesHence the fluid vertical velocity on the sheet boosts whichresults in increases of the fluid vertical velocity far away fromthe sheet

Figure 2 demonstrates the variation of 119891(+infin) with 120573

using present HPM solution and HAM [9] for 120574 = 1 TheHPMresults are in quite good agreementwith theHAMonesIt is observed that 119891(+infin) reduces linearly with the increaseof 120573 This result is due to the fact that when 120573 increases thevelocity of the fluid in direction perpendicular to the sheetincreases

Figure 3 illustrates the variation of 119891(+infin) with 120573 usingpresent HPM solution and HAM [9] for 120574 = 10 Thedifference with Figure 2 is in the values of 119891(+infin) where120574 = 10 results in higher values of 119891(+infin) while the slopefor reduction of 119891(+infin) is lower in 120574 = 10 (Figure 3)

0 10 20 30 401014

1016

1018

102

1022

1024

f(+infin

)

HAMHPM

120574 = 10

120573

Figure 3 Variation of 119891(+infin) with 120573 by present HPM solution andHAM [9] for 120574 = 10

Figure 4 depicts the variation of 11989110158401015840(0) with 120574 for variousvalues of 120573 (120573 = 1 amp 10) using present HPM solution andHAM [9] For both 120573 = 1 and 120573 = 10 there is a good matchbetween the results of HPM with the ones of HAM It is alsoapparent that 11989110158401015840(0) diminishes with a growth in 120574 When 120574is increased the resultant higher suction leads to a reductionin the stickiness of fluid particles to the sheet Hence the fluidfriction at sheet surface experiences moderation

Figure 5 shows variation of 11989110158401015840(0) with 120573 using presentHPM solution and HAM [9] for 120574 = 1 The results of the twomethods HPM and HAM [9] indicate a quite good matchWith an increase in 120573 a decrease in 11989110158401015840(0) is observed whichcan be attributed to lower fluid-surface interaction by thehigher nonlinearity of sheet stretching and permeability

The variation of11989110158401015840(0)with120573using presentHPMsolutionandHAM [9] for 120574 = 10 is shown in Figure 6 In this case thesame conclusions made in Figure 5 are also quite acceptableHowever for 120574 = 10 (Figure 6) lower values of 11989110158401015840(0) witha lower slope of the curve can be observed

6 Chinese Journal of Engineering

0 2 4 6 8 10minus6

minus5

minus4

minus3

minus2

minus1

0

120574

HAM and 120573 = 1HAM and 120573 = 10

HPM and 120573 = 1HPM and 120573 = 10

f998400998400(0)

Figure 4 Variation of 11989110158401015840(0)with 120574 for various 120573 (120573 = 1amp 10) usingpresent HPM solution and HAM [9]

0 1 2 3 4 5minus3

minus25

minus2

minus15

minus1

minus05

0

f998400998400(0)

HAMHPM

120574 = 1

120573

Figure 5 Variation of 11989110158401015840(0) with 120573 by present HPM solution andHAM [9] for 120574 = 1

5 Conclusions

Thesteady 2-Dboundary layer flowof an incompressible fluidon a nonlinearly stretching permeable sheet is consideredTheHPMsolution of the boundary layer flow is obtainedThedimensionless entrainment parameter and the dimensionlesssheet surface shear stress are obtained and compared to thevalues of a previous HAM solution which show that thepresent HPM solution with only two terms agrees well withthe results of HAMThe results obtained are as follows

(1) The increase of suction through the sheet causes theincrease of fluid vertical downward velocity far awayfrom the sheet

0 10 20 30 40minus8

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

f998400998400(0)

HAMHPM

120574 = 10

120573

Figure 6 Variation of 11989110158401015840(0) with 120573 by present HPM solution andHAM [9] for 120574 = 10

(2) When there is suction through the sheet the increaseof nonlinearity factor of sheet permeability andstretching causes the fluid entrainment velocity todecrease

(3) The fluid friction at the sheet surface decreases withthe increase of nonlinearity factor of sheet stretchingand permeability

Appendix

The equation for 1199061(120578) (19) is solved using the symbolic

software MATHEMATICA under the boundary conditions1199061(0) = 0 1199061015840

1(0) = 0 as

119906101584010158401015840

1minus 12057221199061015840

1= (

1

2120572120574 +

1

2minus 1205722) exp (minus120572120578)

+ (120573 minus1

2) exp (minus2120572120578)

1199061(0) = 0 119906

1015840

1(0) = 0

(A1)

which gives the following solution

1199061(120578) =

minus3 + 41205722+ 2120573 minus 2120572120574 + 8120572

2119862 (2)

41205723

+1 minus 2120573

121205723exp (minus2120572120578)

+3 minus 6120572

2+ 3120572120574 minus 8120572

2119862 (2)

81205723exp (minus120572120578)

+7 minus 6120572

2minus 8120573 + 3120574120572 minus 24120572

2119862 (2)

241205723exp (120572120578)

+minus21205722+ 1 + 120572120574

41205722120578 exp (minus120572120578)

(A2)

Chinese Journal of Engineering 7

where 119862(2) is the integration constant If the boundarycondition 1199061015840

1(infin) = 0 is applied to the solution (A2) it gives

119862(2) and 120572 as

7 minus 61205722minus 8120573 + 3120574120572 minus 24120572

2119862 (2)

241205722= 0

997888rarr 119862 (2) =7

241205722minus1

4minus

120573

31205722+

120574

8120572

minus21205722+ 1 + 120572120574

minus4120572= 0 997888rarr 120572 =

120574 plusmn radic1205742 + 8

4

(A3)

Now if119862(2) from (A3) is substituted in (A2) 1199061(120578) becomes

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+2 minus 12120572

2+ 6120572120574 + 8120573

241205723exp (minus120572120578)

+minus21205722+ 1 + 120572120574

41205722120578 exp (minus120572120578)

(A4)

Here it can be checked and seen that for (A4) 1199061(0) =

0 11990610158401(0) = 0 If the third boundary condition 119906

1015840

1(infin) =

0 is applied to (A4) it gives the value of 120572 = 1205744 plusmn

radic(1205744)2+ (12) This value of 120572 removes the secular term

from the ordinary differential equation (ODE) for 1199061(120578) If 120572

is substituted in the last term of (A4) the last term vanishesand 119906

1(120578) takes its final form as

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+2 minus 12120572

2+ 6120572120574 + 8120573

241205723exp (minus120572120578)

(A5)

Nomenclature

119886 Stretching rate (mdash)119860 Permeability rate (mdash)119887 Displacement value of the sheet stretching

and permeability (m)119891 Dimensionless velocity variable

(= (120595119886(119909 + 119887)(120582+1)2

)radic119886(1 + 120582)120592) (mdash)119901 Small parameter in (14) (mdash)119906 Velocity in x-direction (m sminus1) dependent

variable in (12) instead of 119891 (mdash)119880119908 Velocity of the sheet (m sminus1)

V Velocity in y-direction (m sminus1)119881119908 Velocity of fluid through the porous sheet(m sminus1)

119909 Horizontal coordinate (m)119910 Vertical coordinate (m)

Greek Symbols

120572 Constant in (14) (mdash)120573 Nonlinearity factor of the sheet

stretching and permeability(= 120582(1 + 120582)) (mdash)

120574 Suction or injection (blowing)parameter (= minus2119860radic119886(120582 + 1)120592) (mdash)

120578 Similarity variable(= (119909 + 119887)

(120582minus1)2119910radic119886(1 + 120582)120592) (mdash)

120582 Exponent of sheet stretching andpermeability velocity (mdash)

120583 Dynamic viscosity (N smminus2)120592 Kinematic viscosity (m2 sminus1)120588 Density (kgmminus3)120591 Shear stress (Nmminus2)120591119908 Shear stress on sheet surface (Nmminus2)

120595 Stream function (m2 sminus1)

Subscripts

infin Infinity119891 Fluid119908 Sheet surface

Conflict of Interests

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

References

[1] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[2] M Esmaeilpour andD D Ganji ldquoApplication of Hersquos homotopyperturbation method to boundary layer flow and convectionheat transfer over a flat platerdquo Physics Letters A vol 372 no 1pp 33ndash38 2007

[3] L Xu ldquoHersquos homotopy perturbation method for a boundarylayer equation in unbounded domainrdquo Computers and Math-ematics with Applications vol 54 no 7-8 pp 1067ndash1070 2007

[4] M Fathizadeh and F Rashidi ldquoBoundary layer convective heattransfer with pressure gradient using Homotopy PerturbationMethod (HPM) over a flat platerdquo Chaos Solitons and Fractalsvol 42 no 4 pp 2413ndash2419 2009

[5] B Raftari and A Yildirim ldquoThe application of homotopyperturbation method for MHD flows of UCM fluids aboveporous stretching sheetsrdquo Computers and Mathematics withApplications vol 59 no 10 pp 3328ndash3337 2010

[6] B Raftari S T Mohyud-Din and A Yildirim ldquoSolution tothe MHD flow over a non-linear stretching sheet by homotopyperturbation methodrdquo Science China Physics Mechanics andAstronomy vol 54 no 2 pp 342ndash345 2011

[7] S J Liao ldquoA new branch of solutions of boundary-layer flowsover an impermeable stretched platerdquo International Journal ofHeat and Mass Transfer vol 48 no 12 pp 2529ndash2539 2005

[8] J-H He ldquoComparison of homotopy perturbation methodand homotopy analysis methodrdquo Applied Mathematics andComputation vol 156 no 2 pp 527ndash539 2004

8 Chinese Journal of Engineering

[9] S-J Liao ldquoA new branch of solutions of boundary-layer flowsover a permeable stretching platerdquo International Journal of Non-Linear Mechanics vol 42 no 6 pp 819ndash830 2007

[10] S EMahgoub ldquoForced convection heat transfer over a flat platein a porous mediumrdquoAin Shams Engineering Journal vol 4 pp605ndash613 2013

[11] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoComparison ofHAMandHPMmethods for Bermanrsquosmodel of two-dimensional viscous flow in porous channel withwall suction or injectionrdquo Advances in Theoretical and AppliedMechanics vol 1 no 7 pp 337ndash347 2008

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

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

Antennas andPropagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

4 Chinese Journal of Engineering

Table 2 Values of 119891(+infin) and 11989110158401015840(0) for various 120574 when 120573 = 10

120574 120572119891(+infin)

(present HPM solution)11989110158401015840(0)

(present HPM solution)119891(+infin)

(HAM [9])11989110158401015840(0)

(HAM [9])0 070711 056407 minus268544 0658388 minus26081481 100000 141666 minus286666 1541900 minus28243452 136603 221091 minus318419 2457505 minus30678723 178078 328118 minus355903 3392704 minus33409115 268614 529059 minus386503 5300584 minus397365410 509808 101842 minus571923 10181382 minus5928155

Table 3 Values of 119891(+infin) and 11989110158401015840(0) for various 120573 when 120574 = 1 (120572 = 1)

120573119891(+infin)

(present HPM solution)11989110158401015840(0)

(present HPM solution)119891(+infin)

(HAM [9])11989110158401015840(0)

(HAM [9])minus0656192 219270 minus0414603 23 minus0381592minus0395769 214929 minus0601410 22 minus05691800 208333 minus0833333 2093662 minus07864001 191666 minus116666 1931110 minus11756143 162333 minus175333 1764465 minus16994315 145000 minus225000 1670900 minus2088843

entrainment parameter and the dimensionless sheet surfaceshear stress are obtained and compared to the values obtainedby the HAM solution [9] Table 1 shows the values of thedimensionless entrainment parameter 119891(+infin) and the val-ues of the dimensionless sheet surface shear stress 11989110158401015840(0)using the present HPM method and the HAM solution [9]for various values of 120574 when 120573 = 1 It can be seen that thepresent HPM solution agrees well with the HAM solution [9]for both 119891(+infin) and 11989110158401015840(0) within plusmn2 error It can also beseen that at 120573 = 1 when 120574 increases 119891(+infin) increases Thismeans that the increase of suction through the sheet causesaccording to (10) the increase of the fluid vertical downwardvelocity far away from the sheet It is also observed that at120573 = 1 11989110158401015840(0) decreases with the increase of 120574 that is theincrease of suction through the sheet causes the increase ofsheet surface shear stress

Table 2 demonstrates the values of 119891(+infin) and the valuesof 11989110158401015840(0) using the present HPM solution and the HAMsolution [9] for various values of 120574 when 120573 = 10 It can beseen that at 120573 = 10 when suction parameter 120574 increases119891(+infin) increases while 11989110158401015840(0) decreases which means thatthe suction increase causes the boost of fluid far-away verticalvelocity and sheet surface shear stress

Table 3 shows the values of 119891(+infin) and 11989110158401015840(0) using thepresent HPM solution and the HAM solution [9] for variousvalues of 120573 when 120574 = 1 It is observed that at 120574 = 1 theincrease of 120573 results in the reduction of 119891(+infin) This meansthat when there is suction through the sheet the increaseof nonlinearity factor of sheet permeability and stretchingcauses the fluid entrainment velocity to decrease It may alsobe seen that at 120574 = 1 the increase of120573 decreases11989110158401015840(0) that isthe increase of nonlinearity factor 120573 decreases fluid friction atthe sheet surface

0 2 4 6 8 100

2

4

6

8

10

12

f(+infin

)

120574

HAM and 120573 = 1HAM and 120573 = 10

HPM and 120573 = 1HPM and 120573 = 10

Figure 1 Variation of119891(+infin)with 120574 for various120573 (120573 = 1amp 10) usingpresent HPM solution and HAM [9]

Table 4 depicts 119891(+infin) and 11989110158401015840(0) values using the HPMsolution of the present paper and the HAM solution [9] forvarious values of 120573 when 120574 = 10 It is observable that 119891(+infin)

values reduce and 11989110158401015840(0) decreases with the increase of 120573 ata constant 120574 (120574 = 10) Thus it can be concluded that insituations where lower sheet surface friction is desired lowervalues of suction parameter 120574 and nonlinearity factor 120573 canbe helpful

Figure 1 indicates the variation of 119891(+infin) with 120574 forvarious 120573 using present HPM solution and HAM [9] The

Chinese Journal of Engineering 5

Table 4 Values of 119891(+infin) and 11989110158401015840(0) for various 120573 when 120574 = 10 (120572 = 509808)

120573119891(+infin)

(present HPM solution)11989110158401015840(0)

(present HPM solution)119891(+infin)

(HAM [9])11989110158401015840(0)

(HAM [9])minus5949297 102043 minus467640 1021 minus4413988minus1495596 101987 minus496760 102 minus48997760 101968 minus506539 10197086 minus50493511 101955 minus513077 10195238 minus51461973 101930 minus526154 10191759 minus53330355 101905 minus539231 10188534 minus551167010 101842 minus571923 10181382 minus592815520 101716 minus637307 10169894 minus666363430 101590 minus702691 10160896 minus730736140 101465 minus768075 10153542 minus7886163

0 1 2 3 4 50

05

1

15

2

25

3

f(+infin

)

HAMHPM

120574 = 1

120573

Figure 2 Variation of 119891(+infin) with 120573 by present HPM solution andHAM [9] for 120574 = 1

present HPM solution is observed to agree quite well with theHAM results [9] It is seen that 119891(+infin) augments somewhatlinearly with the increase of 120574 The reason is that when 120574

increases the suction through the porous sheet increasesHence the fluid vertical velocity on the sheet boosts whichresults in increases of the fluid vertical velocity far away fromthe sheet

Figure 2 demonstrates the variation of 119891(+infin) with 120573

using present HPM solution and HAM [9] for 120574 = 1 TheHPMresults are in quite good agreementwith theHAMonesIt is observed that 119891(+infin) reduces linearly with the increaseof 120573 This result is due to the fact that when 120573 increases thevelocity of the fluid in direction perpendicular to the sheetincreases

Figure 3 illustrates the variation of 119891(+infin) with 120573 usingpresent HPM solution and HAM [9] for 120574 = 10 Thedifference with Figure 2 is in the values of 119891(+infin) where120574 = 10 results in higher values of 119891(+infin) while the slopefor reduction of 119891(+infin) is lower in 120574 = 10 (Figure 3)

0 10 20 30 401014

1016

1018

102

1022

1024

f(+infin

)

HAMHPM

120574 = 10

120573

Figure 3 Variation of 119891(+infin) with 120573 by present HPM solution andHAM [9] for 120574 = 10

Figure 4 depicts the variation of 11989110158401015840(0) with 120574 for variousvalues of 120573 (120573 = 1 amp 10) using present HPM solution andHAM [9] For both 120573 = 1 and 120573 = 10 there is a good matchbetween the results of HPM with the ones of HAM It is alsoapparent that 11989110158401015840(0) diminishes with a growth in 120574 When 120574is increased the resultant higher suction leads to a reductionin the stickiness of fluid particles to the sheet Hence the fluidfriction at sheet surface experiences moderation

Figure 5 shows variation of 11989110158401015840(0) with 120573 using presentHPM solution and HAM [9] for 120574 = 1 The results of the twomethods HPM and HAM [9] indicate a quite good matchWith an increase in 120573 a decrease in 11989110158401015840(0) is observed whichcan be attributed to lower fluid-surface interaction by thehigher nonlinearity of sheet stretching and permeability

The variation of11989110158401015840(0)with120573using presentHPMsolutionandHAM [9] for 120574 = 10 is shown in Figure 6 In this case thesame conclusions made in Figure 5 are also quite acceptableHowever for 120574 = 10 (Figure 6) lower values of 11989110158401015840(0) witha lower slope of the curve can be observed

6 Chinese Journal of Engineering

0 2 4 6 8 10minus6

minus5

minus4

minus3

minus2

minus1

0

120574

HAM and 120573 = 1HAM and 120573 = 10

HPM and 120573 = 1HPM and 120573 = 10

f998400998400(0)

Figure 4 Variation of 11989110158401015840(0)with 120574 for various 120573 (120573 = 1amp 10) usingpresent HPM solution and HAM [9]

0 1 2 3 4 5minus3

minus25

minus2

minus15

minus1

minus05

0

f998400998400(0)

HAMHPM

120574 = 1

120573

Figure 5 Variation of 11989110158401015840(0) with 120573 by present HPM solution andHAM [9] for 120574 = 1

5 Conclusions

Thesteady 2-Dboundary layer flowof an incompressible fluidon a nonlinearly stretching permeable sheet is consideredTheHPMsolution of the boundary layer flow is obtainedThedimensionless entrainment parameter and the dimensionlesssheet surface shear stress are obtained and compared to thevalues of a previous HAM solution which show that thepresent HPM solution with only two terms agrees well withthe results of HAMThe results obtained are as follows

(1) The increase of suction through the sheet causes theincrease of fluid vertical downward velocity far awayfrom the sheet

0 10 20 30 40minus8

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

f998400998400(0)

HAMHPM

120574 = 10

120573

Figure 6 Variation of 11989110158401015840(0) with 120573 by present HPM solution andHAM [9] for 120574 = 10

(2) When there is suction through the sheet the increaseof nonlinearity factor of sheet permeability andstretching causes the fluid entrainment velocity todecrease

(3) The fluid friction at the sheet surface decreases withthe increase of nonlinearity factor of sheet stretchingand permeability

Appendix

The equation for 1199061(120578) (19) is solved using the symbolic

software MATHEMATICA under the boundary conditions1199061(0) = 0 1199061015840

1(0) = 0 as

119906101584010158401015840

1minus 12057221199061015840

1= (

1

2120572120574 +

1

2minus 1205722) exp (minus120572120578)

+ (120573 minus1

2) exp (minus2120572120578)

1199061(0) = 0 119906

1015840

1(0) = 0

(A1)

which gives the following solution

1199061(120578) =

minus3 + 41205722+ 2120573 minus 2120572120574 + 8120572

2119862 (2)

41205723

+1 minus 2120573

121205723exp (minus2120572120578)

+3 minus 6120572

2+ 3120572120574 minus 8120572

2119862 (2)

81205723exp (minus120572120578)

+7 minus 6120572

2minus 8120573 + 3120574120572 minus 24120572

2119862 (2)

241205723exp (120572120578)

+minus21205722+ 1 + 120572120574

41205722120578 exp (minus120572120578)

(A2)

Chinese Journal of Engineering 7

where 119862(2) is the integration constant If the boundarycondition 1199061015840

1(infin) = 0 is applied to the solution (A2) it gives

119862(2) and 120572 as

7 minus 61205722minus 8120573 + 3120574120572 minus 24120572

2119862 (2)

241205722= 0

997888rarr 119862 (2) =7

241205722minus1

4minus

120573

31205722+

120574

8120572

minus21205722+ 1 + 120572120574

minus4120572= 0 997888rarr 120572 =

120574 plusmn radic1205742 + 8

4

(A3)

Now if119862(2) from (A3) is substituted in (A2) 1199061(120578) becomes

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+2 minus 12120572

2+ 6120572120574 + 8120573

241205723exp (minus120572120578)

+minus21205722+ 1 + 120572120574

41205722120578 exp (minus120572120578)

(A4)

Here it can be checked and seen that for (A4) 1199061(0) =

0 11990610158401(0) = 0 If the third boundary condition 119906

1015840

1(infin) =

0 is applied to (A4) it gives the value of 120572 = 1205744 plusmn

radic(1205744)2+ (12) This value of 120572 removes the secular term

from the ordinary differential equation (ODE) for 1199061(120578) If 120572

is substituted in the last term of (A4) the last term vanishesand 119906

1(120578) takes its final form as

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+2 minus 12120572

2+ 6120572120574 + 8120573

241205723exp (minus120572120578)

(A5)

Nomenclature

119886 Stretching rate (mdash)119860 Permeability rate (mdash)119887 Displacement value of the sheet stretching

and permeability (m)119891 Dimensionless velocity variable

(= (120595119886(119909 + 119887)(120582+1)2

)radic119886(1 + 120582)120592) (mdash)119901 Small parameter in (14) (mdash)119906 Velocity in x-direction (m sminus1) dependent

variable in (12) instead of 119891 (mdash)119880119908 Velocity of the sheet (m sminus1)

V Velocity in y-direction (m sminus1)119881119908 Velocity of fluid through the porous sheet(m sminus1)

119909 Horizontal coordinate (m)119910 Vertical coordinate (m)

Greek Symbols

120572 Constant in (14) (mdash)120573 Nonlinearity factor of the sheet

stretching and permeability(= 120582(1 + 120582)) (mdash)

120574 Suction or injection (blowing)parameter (= minus2119860radic119886(120582 + 1)120592) (mdash)

120578 Similarity variable(= (119909 + 119887)

(120582minus1)2119910radic119886(1 + 120582)120592) (mdash)

120582 Exponent of sheet stretching andpermeability velocity (mdash)

120583 Dynamic viscosity (N smminus2)120592 Kinematic viscosity (m2 sminus1)120588 Density (kgmminus3)120591 Shear stress (Nmminus2)120591119908 Shear stress on sheet surface (Nmminus2)

120595 Stream function (m2 sminus1)

Subscripts

infin Infinity119891 Fluid119908 Sheet surface

Conflict of Interests

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

References

[1] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[2] M Esmaeilpour andD D Ganji ldquoApplication of Hersquos homotopyperturbation method to boundary layer flow and convectionheat transfer over a flat platerdquo Physics Letters A vol 372 no 1pp 33ndash38 2007

[3] L Xu ldquoHersquos homotopy perturbation method for a boundarylayer equation in unbounded domainrdquo Computers and Math-ematics with Applications vol 54 no 7-8 pp 1067ndash1070 2007

[4] M Fathizadeh and F Rashidi ldquoBoundary layer convective heattransfer with pressure gradient using Homotopy PerturbationMethod (HPM) over a flat platerdquo Chaos Solitons and Fractalsvol 42 no 4 pp 2413ndash2419 2009

[5] B Raftari and A Yildirim ldquoThe application of homotopyperturbation method for MHD flows of UCM fluids aboveporous stretching sheetsrdquo Computers and Mathematics withApplications vol 59 no 10 pp 3328ndash3337 2010

[6] B Raftari S T Mohyud-Din and A Yildirim ldquoSolution tothe MHD flow over a non-linear stretching sheet by homotopyperturbation methodrdquo Science China Physics Mechanics andAstronomy vol 54 no 2 pp 342ndash345 2011

[7] S J Liao ldquoA new branch of solutions of boundary-layer flowsover an impermeable stretched platerdquo International Journal ofHeat and Mass Transfer vol 48 no 12 pp 2529ndash2539 2005

[8] J-H He ldquoComparison of homotopy perturbation methodand homotopy analysis methodrdquo Applied Mathematics andComputation vol 156 no 2 pp 527ndash539 2004

8 Chinese Journal of Engineering

[9] S-J Liao ldquoA new branch of solutions of boundary-layer flowsover a permeable stretching platerdquo International Journal of Non-Linear Mechanics vol 42 no 6 pp 819ndash830 2007

[10] S EMahgoub ldquoForced convection heat transfer over a flat platein a porous mediumrdquoAin Shams Engineering Journal vol 4 pp605ndash613 2013

[11] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoComparison ofHAMandHPMmethods for Bermanrsquosmodel of two-dimensional viscous flow in porous channel withwall suction or injectionrdquo Advances in Theoretical and AppliedMechanics vol 1 no 7 pp 337ndash347 2008

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Antennas andPropagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Chinese Journal of Engineering 5

Table 4 Values of 119891(+infin) and 11989110158401015840(0) for various 120573 when 120574 = 10 (120572 = 509808)

120573119891(+infin)

(present HPM solution)11989110158401015840(0)

(present HPM solution)119891(+infin)

(HAM [9])11989110158401015840(0)

(HAM [9])minus5949297 102043 minus467640 1021 minus4413988minus1495596 101987 minus496760 102 minus48997760 101968 minus506539 10197086 minus50493511 101955 minus513077 10195238 minus51461973 101930 minus526154 10191759 minus53330355 101905 minus539231 10188534 minus551167010 101842 minus571923 10181382 minus592815520 101716 minus637307 10169894 minus666363430 101590 minus702691 10160896 minus730736140 101465 minus768075 10153542 minus7886163

0 1 2 3 4 50

05

1

15

2

25

3

f(+infin

)

HAMHPM

120574 = 1

120573

Figure 2 Variation of 119891(+infin) with 120573 by present HPM solution andHAM [9] for 120574 = 1

present HPM solution is observed to agree quite well with theHAM results [9] It is seen that 119891(+infin) augments somewhatlinearly with the increase of 120574 The reason is that when 120574

increases the suction through the porous sheet increasesHence the fluid vertical velocity on the sheet boosts whichresults in increases of the fluid vertical velocity far away fromthe sheet

Figure 2 demonstrates the variation of 119891(+infin) with 120573

using present HPM solution and HAM [9] for 120574 = 1 TheHPMresults are in quite good agreementwith theHAMonesIt is observed that 119891(+infin) reduces linearly with the increaseof 120573 This result is due to the fact that when 120573 increases thevelocity of the fluid in direction perpendicular to the sheetincreases

Figure 3 illustrates the variation of 119891(+infin) with 120573 usingpresent HPM solution and HAM [9] for 120574 = 10 Thedifference with Figure 2 is in the values of 119891(+infin) where120574 = 10 results in higher values of 119891(+infin) while the slopefor reduction of 119891(+infin) is lower in 120574 = 10 (Figure 3)

0 10 20 30 401014

1016

1018

102

1022

1024

f(+infin

)

HAMHPM

120574 = 10

120573

Figure 3 Variation of 119891(+infin) with 120573 by present HPM solution andHAM [9] for 120574 = 10

Figure 4 depicts the variation of 11989110158401015840(0) with 120574 for variousvalues of 120573 (120573 = 1 amp 10) using present HPM solution andHAM [9] For both 120573 = 1 and 120573 = 10 there is a good matchbetween the results of HPM with the ones of HAM It is alsoapparent that 11989110158401015840(0) diminishes with a growth in 120574 When 120574is increased the resultant higher suction leads to a reductionin the stickiness of fluid particles to the sheet Hence the fluidfriction at sheet surface experiences moderation

Figure 5 shows variation of 11989110158401015840(0) with 120573 using presentHPM solution and HAM [9] for 120574 = 1 The results of the twomethods HPM and HAM [9] indicate a quite good matchWith an increase in 120573 a decrease in 11989110158401015840(0) is observed whichcan be attributed to lower fluid-surface interaction by thehigher nonlinearity of sheet stretching and permeability

The variation of11989110158401015840(0)with120573using presentHPMsolutionandHAM [9] for 120574 = 10 is shown in Figure 6 In this case thesame conclusions made in Figure 5 are also quite acceptableHowever for 120574 = 10 (Figure 6) lower values of 11989110158401015840(0) witha lower slope of the curve can be observed

6 Chinese Journal of Engineering

0 2 4 6 8 10minus6

minus5

minus4

minus3

minus2

minus1

0

120574

HAM and 120573 = 1HAM and 120573 = 10

HPM and 120573 = 1HPM and 120573 = 10

f998400998400(0)

Figure 4 Variation of 11989110158401015840(0)with 120574 for various 120573 (120573 = 1amp 10) usingpresent HPM solution and HAM [9]

0 1 2 3 4 5minus3

minus25

minus2

minus15

minus1

minus05

0

f998400998400(0)

HAMHPM

120574 = 1

120573

Figure 5 Variation of 11989110158401015840(0) with 120573 by present HPM solution andHAM [9] for 120574 = 1

5 Conclusions

Thesteady 2-Dboundary layer flowof an incompressible fluidon a nonlinearly stretching permeable sheet is consideredTheHPMsolution of the boundary layer flow is obtainedThedimensionless entrainment parameter and the dimensionlesssheet surface shear stress are obtained and compared to thevalues of a previous HAM solution which show that thepresent HPM solution with only two terms agrees well withthe results of HAMThe results obtained are as follows

(1) The increase of suction through the sheet causes theincrease of fluid vertical downward velocity far awayfrom the sheet

0 10 20 30 40minus8

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

f998400998400(0)

HAMHPM

120574 = 10

120573

Figure 6 Variation of 11989110158401015840(0) with 120573 by present HPM solution andHAM [9] for 120574 = 10

(2) When there is suction through the sheet the increaseof nonlinearity factor of sheet permeability andstretching causes the fluid entrainment velocity todecrease

(3) The fluid friction at the sheet surface decreases withthe increase of nonlinearity factor of sheet stretchingand permeability

Appendix

The equation for 1199061(120578) (19) is solved using the symbolic

software MATHEMATICA under the boundary conditions1199061(0) = 0 1199061015840

1(0) = 0 as

119906101584010158401015840

1minus 12057221199061015840

1= (

1

2120572120574 +

1

2minus 1205722) exp (minus120572120578)

+ (120573 minus1

2) exp (minus2120572120578)

1199061(0) = 0 119906

1015840

1(0) = 0

(A1)

which gives the following solution

1199061(120578) =

minus3 + 41205722+ 2120573 minus 2120572120574 + 8120572

2119862 (2)

41205723

+1 minus 2120573

121205723exp (minus2120572120578)

+3 minus 6120572

2+ 3120572120574 minus 8120572

2119862 (2)

81205723exp (minus120572120578)

+7 minus 6120572

2minus 8120573 + 3120574120572 minus 24120572

2119862 (2)

241205723exp (120572120578)

+minus21205722+ 1 + 120572120574

41205722120578 exp (minus120572120578)

(A2)

Chinese Journal of Engineering 7

where 119862(2) is the integration constant If the boundarycondition 1199061015840

1(infin) = 0 is applied to the solution (A2) it gives

119862(2) and 120572 as

7 minus 61205722minus 8120573 + 3120574120572 minus 24120572

2119862 (2)

241205722= 0

997888rarr 119862 (2) =7

241205722minus1

4minus

120573

31205722+

120574

8120572

minus21205722+ 1 + 120572120574

minus4120572= 0 997888rarr 120572 =

120574 plusmn radic1205742 + 8

4

(A3)

Now if119862(2) from (A3) is substituted in (A2) 1199061(120578) becomes

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+2 minus 12120572

2+ 6120572120574 + 8120573

241205723exp (minus120572120578)

+minus21205722+ 1 + 120572120574

41205722120578 exp (minus120572120578)

(A4)

Here it can be checked and seen that for (A4) 1199061(0) =

0 11990610158401(0) = 0 If the third boundary condition 119906

1015840

1(infin) =

0 is applied to (A4) it gives the value of 120572 = 1205744 plusmn

radic(1205744)2+ (12) This value of 120572 removes the secular term

from the ordinary differential equation (ODE) for 1199061(120578) If 120572

is substituted in the last term of (A4) the last term vanishesand 119906

1(120578) takes its final form as

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+2 minus 12120572

2+ 6120572120574 + 8120573

241205723exp (minus120572120578)

(A5)

Nomenclature

119886 Stretching rate (mdash)119860 Permeability rate (mdash)119887 Displacement value of the sheet stretching

and permeability (m)119891 Dimensionless velocity variable

(= (120595119886(119909 + 119887)(120582+1)2

)radic119886(1 + 120582)120592) (mdash)119901 Small parameter in (14) (mdash)119906 Velocity in x-direction (m sminus1) dependent

variable in (12) instead of 119891 (mdash)119880119908 Velocity of the sheet (m sminus1)

V Velocity in y-direction (m sminus1)119881119908 Velocity of fluid through the porous sheet(m sminus1)

119909 Horizontal coordinate (m)119910 Vertical coordinate (m)

Greek Symbols

120572 Constant in (14) (mdash)120573 Nonlinearity factor of the sheet

stretching and permeability(= 120582(1 + 120582)) (mdash)

120574 Suction or injection (blowing)parameter (= minus2119860radic119886(120582 + 1)120592) (mdash)

120578 Similarity variable(= (119909 + 119887)

(120582minus1)2119910radic119886(1 + 120582)120592) (mdash)

120582 Exponent of sheet stretching andpermeability velocity (mdash)

120583 Dynamic viscosity (N smminus2)120592 Kinematic viscosity (m2 sminus1)120588 Density (kgmminus3)120591 Shear stress (Nmminus2)120591119908 Shear stress on sheet surface (Nmminus2)

120595 Stream function (m2 sminus1)

Subscripts

infin Infinity119891 Fluid119908 Sheet surface

Conflict of Interests

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

References

[1] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[2] M Esmaeilpour andD D Ganji ldquoApplication of Hersquos homotopyperturbation method to boundary layer flow and convectionheat transfer over a flat platerdquo Physics Letters A vol 372 no 1pp 33ndash38 2007

[3] L Xu ldquoHersquos homotopy perturbation method for a boundarylayer equation in unbounded domainrdquo Computers and Math-ematics with Applications vol 54 no 7-8 pp 1067ndash1070 2007

[4] M Fathizadeh and F Rashidi ldquoBoundary layer convective heattransfer with pressure gradient using Homotopy PerturbationMethod (HPM) over a flat platerdquo Chaos Solitons and Fractalsvol 42 no 4 pp 2413ndash2419 2009

[5] B Raftari and A Yildirim ldquoThe application of homotopyperturbation method for MHD flows of UCM fluids aboveporous stretching sheetsrdquo Computers and Mathematics withApplications vol 59 no 10 pp 3328ndash3337 2010

[6] B Raftari S T Mohyud-Din and A Yildirim ldquoSolution tothe MHD flow over a non-linear stretching sheet by homotopyperturbation methodrdquo Science China Physics Mechanics andAstronomy vol 54 no 2 pp 342ndash345 2011

[7] S J Liao ldquoA new branch of solutions of boundary-layer flowsover an impermeable stretched platerdquo International Journal ofHeat and Mass Transfer vol 48 no 12 pp 2529ndash2539 2005

[8] J-H He ldquoComparison of homotopy perturbation methodand homotopy analysis methodrdquo Applied Mathematics andComputation vol 156 no 2 pp 527ndash539 2004

8 Chinese Journal of Engineering

[9] S-J Liao ldquoA new branch of solutions of boundary-layer flowsover a permeable stretching platerdquo International Journal of Non-Linear Mechanics vol 42 no 6 pp 819ndash830 2007

[10] S EMahgoub ldquoForced convection heat transfer over a flat platein a porous mediumrdquoAin Shams Engineering Journal vol 4 pp605ndash613 2013

[11] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoComparison ofHAMandHPMmethods for Bermanrsquosmodel of two-dimensional viscous flow in porous channel withwall suction or injectionrdquo Advances in Theoretical and AppliedMechanics vol 1 no 7 pp 337ndash347 2008

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

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

Antennas andPropagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

6 Chinese Journal of Engineering

0 2 4 6 8 10minus6

minus5

minus4

minus3

minus2

minus1

0

120574

HAM and 120573 = 1HAM and 120573 = 10

HPM and 120573 = 1HPM and 120573 = 10

f998400998400(0)

Figure 4 Variation of 11989110158401015840(0)with 120574 for various 120573 (120573 = 1amp 10) usingpresent HPM solution and HAM [9]

0 1 2 3 4 5minus3

minus25

minus2

minus15

minus1

minus05

0

f998400998400(0)

HAMHPM

120574 = 1

120573

Figure 5 Variation of 11989110158401015840(0) with 120573 by present HPM solution andHAM [9] for 120574 = 1

5 Conclusions

Thesteady 2-Dboundary layer flowof an incompressible fluidon a nonlinearly stretching permeable sheet is consideredTheHPMsolution of the boundary layer flow is obtainedThedimensionless entrainment parameter and the dimensionlesssheet surface shear stress are obtained and compared to thevalues of a previous HAM solution which show that thepresent HPM solution with only two terms agrees well withthe results of HAMThe results obtained are as follows

(1) The increase of suction through the sheet causes theincrease of fluid vertical downward velocity far awayfrom the sheet

0 10 20 30 40minus8

minus75

minus7

minus65

minus6

minus55

minus5

minus45

minus4

f998400998400(0)

HAMHPM

120574 = 10

120573

Figure 6 Variation of 11989110158401015840(0) with 120573 by present HPM solution andHAM [9] for 120574 = 10

(2) When there is suction through the sheet the increaseof nonlinearity factor of sheet permeability andstretching causes the fluid entrainment velocity todecrease

(3) The fluid friction at the sheet surface decreases withthe increase of nonlinearity factor of sheet stretchingand permeability

Appendix

The equation for 1199061(120578) (19) is solved using the symbolic

software MATHEMATICA under the boundary conditions1199061(0) = 0 1199061015840

1(0) = 0 as

119906101584010158401015840

1minus 12057221199061015840

1= (

1

2120572120574 +

1

2minus 1205722) exp (minus120572120578)

+ (120573 minus1

2) exp (minus2120572120578)

1199061(0) = 0 119906

1015840

1(0) = 0

(A1)

which gives the following solution

1199061(120578) =

minus3 + 41205722+ 2120573 minus 2120572120574 + 8120572

2119862 (2)

41205723

+1 minus 2120573

121205723exp (minus2120572120578)

+3 minus 6120572

2+ 3120572120574 minus 8120572

2119862 (2)

81205723exp (minus120572120578)

+7 minus 6120572

2minus 8120573 + 3120574120572 minus 24120572

2119862 (2)

241205723exp (120572120578)

+minus21205722+ 1 + 120572120574

41205722120578 exp (minus120572120578)

(A2)

Chinese Journal of Engineering 7

where 119862(2) is the integration constant If the boundarycondition 1199061015840

1(infin) = 0 is applied to the solution (A2) it gives

119862(2) and 120572 as

7 minus 61205722minus 8120573 + 3120574120572 minus 24120572

2119862 (2)

241205722= 0

997888rarr 119862 (2) =7

241205722minus1

4minus

120573

31205722+

120574

8120572

minus21205722+ 1 + 120572120574

minus4120572= 0 997888rarr 120572 =

120574 plusmn radic1205742 + 8

4

(A3)

Now if119862(2) from (A3) is substituted in (A2) 1199061(120578) becomes

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+2 minus 12120572

2+ 6120572120574 + 8120573

241205723exp (minus120572120578)

+minus21205722+ 1 + 120572120574

41205722120578 exp (minus120572120578)

(A4)

Here it can be checked and seen that for (A4) 1199061(0) =

0 11990610158401(0) = 0 If the third boundary condition 119906

1015840

1(infin) =

0 is applied to (A4) it gives the value of 120572 = 1205744 plusmn

radic(1205744)2+ (12) This value of 120572 removes the secular term

from the ordinary differential equation (ODE) for 1199061(120578) If 120572

is substituted in the last term of (A4) the last term vanishesand 119906

1(120578) takes its final form as

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+2 minus 12120572

2+ 6120572120574 + 8120573

241205723exp (minus120572120578)

(A5)

Nomenclature

119886 Stretching rate (mdash)119860 Permeability rate (mdash)119887 Displacement value of the sheet stretching

and permeability (m)119891 Dimensionless velocity variable

(= (120595119886(119909 + 119887)(120582+1)2

)radic119886(1 + 120582)120592) (mdash)119901 Small parameter in (14) (mdash)119906 Velocity in x-direction (m sminus1) dependent

variable in (12) instead of 119891 (mdash)119880119908 Velocity of the sheet (m sminus1)

V Velocity in y-direction (m sminus1)119881119908 Velocity of fluid through the porous sheet(m sminus1)

119909 Horizontal coordinate (m)119910 Vertical coordinate (m)

Greek Symbols

120572 Constant in (14) (mdash)120573 Nonlinearity factor of the sheet

stretching and permeability(= 120582(1 + 120582)) (mdash)

120574 Suction or injection (blowing)parameter (= minus2119860radic119886(120582 + 1)120592) (mdash)

120578 Similarity variable(= (119909 + 119887)

(120582minus1)2119910radic119886(1 + 120582)120592) (mdash)

120582 Exponent of sheet stretching andpermeability velocity (mdash)

120583 Dynamic viscosity (N smminus2)120592 Kinematic viscosity (m2 sminus1)120588 Density (kgmminus3)120591 Shear stress (Nmminus2)120591119908 Shear stress on sheet surface (Nmminus2)

120595 Stream function (m2 sminus1)

Subscripts

infin Infinity119891 Fluid119908 Sheet surface

Conflict of Interests

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

References

[1] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[2] M Esmaeilpour andD D Ganji ldquoApplication of Hersquos homotopyperturbation method to boundary layer flow and convectionheat transfer over a flat platerdquo Physics Letters A vol 372 no 1pp 33ndash38 2007

[3] L Xu ldquoHersquos homotopy perturbation method for a boundarylayer equation in unbounded domainrdquo Computers and Math-ematics with Applications vol 54 no 7-8 pp 1067ndash1070 2007

[4] M Fathizadeh and F Rashidi ldquoBoundary layer convective heattransfer with pressure gradient using Homotopy PerturbationMethod (HPM) over a flat platerdquo Chaos Solitons and Fractalsvol 42 no 4 pp 2413ndash2419 2009

[5] B Raftari and A Yildirim ldquoThe application of homotopyperturbation method for MHD flows of UCM fluids aboveporous stretching sheetsrdquo Computers and Mathematics withApplications vol 59 no 10 pp 3328ndash3337 2010

[6] B Raftari S T Mohyud-Din and A Yildirim ldquoSolution tothe MHD flow over a non-linear stretching sheet by homotopyperturbation methodrdquo Science China Physics Mechanics andAstronomy vol 54 no 2 pp 342ndash345 2011

[7] S J Liao ldquoA new branch of solutions of boundary-layer flowsover an impermeable stretched platerdquo International Journal ofHeat and Mass Transfer vol 48 no 12 pp 2529ndash2539 2005

[8] J-H He ldquoComparison of homotopy perturbation methodand homotopy analysis methodrdquo Applied Mathematics andComputation vol 156 no 2 pp 527ndash539 2004

8 Chinese Journal of Engineering

[9] S-J Liao ldquoA new branch of solutions of boundary-layer flowsover a permeable stretching platerdquo International Journal of Non-Linear Mechanics vol 42 no 6 pp 819ndash830 2007

[10] S EMahgoub ldquoForced convection heat transfer over a flat platein a porous mediumrdquoAin Shams Engineering Journal vol 4 pp605ndash613 2013

[11] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoComparison ofHAMandHPMmethods for Bermanrsquosmodel of two-dimensional viscous flow in porous channel withwall suction or injectionrdquo Advances in Theoretical and AppliedMechanics vol 1 no 7 pp 337ndash347 2008

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

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

Antennas andPropagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Chinese Journal of Engineering 7

where 119862(2) is the integration constant If the boundarycondition 1199061015840

1(infin) = 0 is applied to the solution (A2) it gives

119862(2) and 120572 as

7 minus 61205722minus 8120573 + 3120574120572 minus 24120572

2119862 (2)

241205722= 0

997888rarr 119862 (2) =7

241205722minus1

4minus

120573

31205722+

120574

8120572

minus21205722+ 1 + 120572120574

minus4120572= 0 997888rarr 120572 =

120574 plusmn radic1205742 + 8

4

(A3)

Now if119862(2) from (A3) is substituted in (A2) 1199061(120578) becomes

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+2 minus 12120572

2+ 6120572120574 + 8120573

241205723exp (minus120572120578)

+minus21205722+ 1 + 120572120574

41205722120578 exp (minus120572120578)

(A4)

Here it can be checked and seen that for (A4) 1199061(0) =

0 11990610158401(0) = 0 If the third boundary condition 119906

1015840

1(infin) =

0 is applied to (A4) it gives the value of 120572 = 1205744 plusmn

radic(1205744)2+ (12) This value of 120572 removes the secular term

from the ordinary differential equation (ODE) for 1199061(120578) If 120572

is substituted in the last term of (A4) the last term vanishesand 119906

1(120578) takes its final form as

1199061(120578) =

minus2 + 61205722minus 2120573 minus 3120572120574

121205723+1 minus 2120573

121205723exp (minus2120572120578)

+2 minus 12120572

2+ 6120572120574 + 8120573

241205723exp (minus120572120578)

(A5)

Nomenclature

119886 Stretching rate (mdash)119860 Permeability rate (mdash)119887 Displacement value of the sheet stretching

and permeability (m)119891 Dimensionless velocity variable

(= (120595119886(119909 + 119887)(120582+1)2

)radic119886(1 + 120582)120592) (mdash)119901 Small parameter in (14) (mdash)119906 Velocity in x-direction (m sminus1) dependent

variable in (12) instead of 119891 (mdash)119880119908 Velocity of the sheet (m sminus1)

V Velocity in y-direction (m sminus1)119881119908 Velocity of fluid through the porous sheet(m sminus1)

119909 Horizontal coordinate (m)119910 Vertical coordinate (m)

Greek Symbols

120572 Constant in (14) (mdash)120573 Nonlinearity factor of the sheet

stretching and permeability(= 120582(1 + 120582)) (mdash)

120574 Suction or injection (blowing)parameter (= minus2119860radic119886(120582 + 1)120592) (mdash)

120578 Similarity variable(= (119909 + 119887)

(120582minus1)2119910radic119886(1 + 120582)120592) (mdash)

120582 Exponent of sheet stretching andpermeability velocity (mdash)

120583 Dynamic viscosity (N smminus2)120592 Kinematic viscosity (m2 sminus1)120588 Density (kgmminus3)120591 Shear stress (Nmminus2)120591119908 Shear stress on sheet surface (Nmminus2)

120595 Stream function (m2 sminus1)

Subscripts

infin Infinity119891 Fluid119908 Sheet surface

Conflict of Interests

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

References

[1] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[2] M Esmaeilpour andD D Ganji ldquoApplication of Hersquos homotopyperturbation method to boundary layer flow and convectionheat transfer over a flat platerdquo Physics Letters A vol 372 no 1pp 33ndash38 2007

[3] L Xu ldquoHersquos homotopy perturbation method for a boundarylayer equation in unbounded domainrdquo Computers and Math-ematics with Applications vol 54 no 7-8 pp 1067ndash1070 2007

[4] M Fathizadeh and F Rashidi ldquoBoundary layer convective heattransfer with pressure gradient using Homotopy PerturbationMethod (HPM) over a flat platerdquo Chaos Solitons and Fractalsvol 42 no 4 pp 2413ndash2419 2009

[5] B Raftari and A Yildirim ldquoThe application of homotopyperturbation method for MHD flows of UCM fluids aboveporous stretching sheetsrdquo Computers and Mathematics withApplications vol 59 no 10 pp 3328ndash3337 2010

[6] B Raftari S T Mohyud-Din and A Yildirim ldquoSolution tothe MHD flow over a non-linear stretching sheet by homotopyperturbation methodrdquo Science China Physics Mechanics andAstronomy vol 54 no 2 pp 342ndash345 2011

[7] S J Liao ldquoA new branch of solutions of boundary-layer flowsover an impermeable stretched platerdquo International Journal ofHeat and Mass Transfer vol 48 no 12 pp 2529ndash2539 2005

[8] J-H He ldquoComparison of homotopy perturbation methodand homotopy analysis methodrdquo Applied Mathematics andComputation vol 156 no 2 pp 527ndash539 2004

8 Chinese Journal of Engineering

[9] S-J Liao ldquoA new branch of solutions of boundary-layer flowsover a permeable stretching platerdquo International Journal of Non-Linear Mechanics vol 42 no 6 pp 819ndash830 2007

[10] S EMahgoub ldquoForced convection heat transfer over a flat platein a porous mediumrdquoAin Shams Engineering Journal vol 4 pp605ndash613 2013

[11] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoComparison ofHAMandHPMmethods for Bermanrsquosmodel of two-dimensional viscous flow in porous channel withwall suction or injectionrdquo Advances in Theoretical and AppliedMechanics vol 1 no 7 pp 337ndash347 2008

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

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

Antennas andPropagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

8 Chinese Journal of Engineering

[9] S-J Liao ldquoA new branch of solutions of boundary-layer flowsover a permeable stretching platerdquo International Journal of Non-Linear Mechanics vol 42 no 6 pp 819ndash830 2007

[10] S EMahgoub ldquoForced convection heat transfer over a flat platein a porous mediumrdquoAin Shams Engineering Journal vol 4 pp605ndash613 2013

[11] S Dinarvand A Doosthoseini E Doosthoseini and M MRashidi ldquoComparison ofHAMandHPMmethods for Bermanrsquosmodel of two-dimensional viscous flow in porous channel withwall suction or injectionrdquo Advances in Theoretical and AppliedMechanics vol 1 no 7 pp 337ndash347 2008

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

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

Antennas andPropagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom 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

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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

Antennas andPropagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of