Stagnation-Point Flow toward a Vertical, Nonlinearly ... · point flow towards a stretching surface in a non-Newtonian fluid with heat transfer,”International Journal of Thermal

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 528717 6 pageshttpdxdoiorg1011552013528717

Research ArticleStagnation-Point Flow toward a Vertical Nonlinearly StretchingSheet with Prescribed Surface Heat Flux

Sin Wei Wong1 M A Omar Awang1 and Anuar Ishak2

1 Institute of Mathematical Sciences University of Malaya 50603 Kuala Lumpur Malaysia2 School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia43600 Bangi Selangor Malaysia

Correspondence should be addressed to Anuar Ishak anuarishakyahoocom

Received 27 November 2012 Revised 31 January 2013 Accepted 31 January 2013

Academic Editor Chein-Shan Liu

Copyright copy 2013 Sin Wei Wong et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An analysis is carried out to study the steady two-dimensional stagnation-point flow of an incompressible viscous fluid towardsa stretching vertical sheet It is assumed that the sheet is stretched nonlinearly with prescribed surface heat flux This problemis governed by three parameters buoyancy velocity exponent and velocity ratio Both assisting and opposing buoyant flows areconsideredThe governing partial differential equations are transformed into a system of ordinary differential equations and solvednumerically by finite difference Keller-box method The flow and heat transfer characteristics for different values of the governingparameters are analyzed and discussed Dual solutions are found in the opposing buoyant flows while the solution is unique forthe assisting buoyant flows

1 Introduction

The study of fluid flow and heat transfer due to a stretchingsurface has significant application in the industrial processesfor example in polymer sheet extrusion from a die drawingof plastic films and manufacturing of glass fiber The qualityof the final product greatly depends on the heat transfer rate atthe stretching surface as explained by Karwe and Jaluria [1 2]

Different from the flow induced by a stretching horizontalplate (see Crane [3] Weidman and Magyari [4] and Weid-man and Ali [5]) the effect of the buoyancy force could notbe neglected for the vertical plateThere are several works thatreported the flow and heat transfer characteristics that arebrought about by the buoyancy force [6ndash11] Ramachandranet al [12] studied the effect of buoyancy force on thestagnation point flows past a vertically heated surface atrest and found that dual solutions exist in the buoyancyopposing flow region In the present paper in addition to theflow under the influence of buoyancy force as discussed byRamachandran et al [12] we discuss the consequent flow andheat transfer characteristics that are also brought about by

the stretching sheet with power-law velocity variation It isworth mentioning that the problems of the stagnation-pointflow toward a stretching sheet have been considered by manyauthors [13ndash27] by considering various flow configurationsas well as surface heating conditions

2 Problem Formulation

Consider amixed convection stagnation-point flow towards avertical nonlinearly stretching sheet immersed in an incom-pressible viscous fluid as shown in Figure 1 The Cartesiancoordinates (119909 119910) are taken such that the 119909-axis is measuredalong the sheet oriented in the upwards or downwardsdirection and the 119910-axis is normal to it It is assumed thatthe wall stretching velocity is given by 119880

119908= 119886119909119898 and the

far field inviscid velocity distribution in the neighborhoodof the stagnation point (0 0) is given by 119880

infin(119909) = 119887119909

119898119881infin(119910) = minus119887119910

119898 The surface heat flux is in the form of119902119908(119909) = 119888119909

(5119898minus3)2 (see Merkin and Mahmood [28]) where119886 119887 119888 and 119898 are constants This 119902

119908(119909) ensured that the

2 Journal of Applied Mathematics

119879infin119881infin

119880infin

119880119908

119902119908

119910

119892

119906

119907

119910

119902119908

119880119908

119880infin

119879infin

119881infin

(a) Assisting flow (120582 gt 0) (b) Opposing flow (120582 lt 0)

119909

119909

Figure 1 Physical model and coordinate system

minus10 minus8 minus6 minus4 minus2 0 2 4minus3

minus2

minus1

01

2

34

120582

119898 = 05

119898 = 2

Upper branchLower branch

119898 = 1

119891998400998400

(0)

Figure 2 Variation of the skin friction coefficient 11989110158401015840(0) with 120582 forvarious values of119898 when 120576 = 05

minus

16 minus14 minus12 minus10 minus8 minus6 minus4 minus2 0 2 4minus4

minus3

minus2

minus1

0

1

23


120582

119898 = 05

119898 = 2

119898 = 1

119891998400998400

(0)


buoyancy parameter is independent of 119909 For the assistingflow as shown in Figure 1(a) the 119909-axis points upwards inthe same direction of the stretching surface such that theexternal flow and the stretching surface induce flow andheat transfer in the velocity and thermal boundary layers

minus10 minus8 minus6 minus4 minus2 005

1

15

2

25

3

35

120579(0

)

2


120582

119898 = 05

119898 = 2

119898 = 1

Figure 4 Variation of the wall temperature 120579(0) with 120582 for variousvalues of119898 when 120576 = 05

minus16 minus14 minus12 minus10 minus8 minus6 minus4 minus2 0 2040608

112141618

222

120579(0

)


119898 = 05

119898 = 2

119898 = 1

120582


respectively On the other hand for the opposing flow asshown in Figure 1(b) the 119909-axis points vertically downwardsin the same direction of the stretching surface such that theexternal flow and the stretching surface also induce flowand heat transfer respectively in the velocity and thermal

Journal of Applied Mathematics 3

0 1 2 3 4 5 6 7minus04

minus02

002040608

1

120578

119898 = 05 1 2

119891998400 (120578

)


Figure 6 Velocity profile1198911015840(120578) for various values of119898when 120576 = 05and 120582 = minus05

0 1 2 3 4 5 6 7012345678

119898 = 2 1 05


120578

120579(120578

)

Figure 7 Temperature profile 120579(120578) for various values of 119898 when120576 = 05 and 120582 = minus05

boundary layers The steady boundary layer equations withBoussinesq approximation are

120597119906

120597119909

+

120597119907

120597119910

= 0 (1)

119906

120597119906

120597119909

+ 119907

120597119906

120597119910

= 119880infin

119889119880infin

119889119909

+ 120584

1205972

119906

1205971199102+ 119892120573 (119879 minus 119879

infin)

(2)

119906

120597119879

120597119909

+ 119907

120597119879

120597119910

= 120572

1205972

119879

1205971199102 (3)

subject to the boundary conditions

119906 = 119880119908(119909)

119907 = 0

120597119879

120597119910

= minus

119902119908

119896

at 119910 = 0

119906 997888rarr 119880infin(119909) 119879 997888rarr 119879

infinas 119910 997888rarr infin

(4)

where 119906 and 119907 are the velocity components along the 119909- and119910-axes respectively 119892 is the acceleration due to gravity 120572 is

minus10 minus8 minus6 minus4 minus2 0 2 4 6 8 100

051

152

253

354

119909

120578

120595(119887120584)12 = 02 05 1 2 120595(119887120584)12 = 02 05 1 2

Figure 8 Streamlines for the upper branch solutions when119898 = 1 120576 = 1 and 120582 = minus2


1

2

3

4

5

119909

120578

120595(119887120584)12 = 02 05 1 2 120595(119887120584)12 = 02 05 1 2

Figure 9 Streamlines for the lower branch solutions when119898 = 1 120576 = 1 and 120582 = minus2

the thermal diffusivity of the fluid120584 is the kinematic viscosity120573 is the coefficient of thermal expansion 120588 is the fluid densityand 119879

infinis the far field ambient constant temperature

The continuity equation (1) can be satisfied automaticallyby introducing a stream function 120595 such that 119906 = 120597120595120597119910

and 119907 = minus120597120595120597119909 The momentum and energy equations aretransformed by the similarity variables

120578 = (

119880infin

120584119909

)

12

119910

120595 = [120584119909119880infin]12

119891 (120578)

120579 (120578) =

119896 (119879 minus 119879infin)

119902119908

(

119880infin

120584119909

)

12

(5)

into the following nonlinear ordinary differential equations

119891101584010158401015840

+

119898 + 1

2

11989111989110158401015840

+ 119898(1 minus 11989110158402

) + 120582120579 = 0

1

Pr12057910158401015840

+

119898 + 1

2

1198911205791015840

minus (2119898 minus 1) 1198911015840

120579 = 0

(6)

The transformed boundary conditions are

119891 (0) = 0 1198911015840

(0) = 120576 1205791015840

(0) = minus1

1198911015840

(120578) 997888rarr 1 120579 (120578) 997888rarr 0 as 120578 997888rarr infin

(7)


Table 1 Comparison of the values of 11989110158401015840(0)with those ofWang [29]when119898 = 1 and 120582 = 0

120576 Wang [29] Present5 minus1026475 minus10264752 minus188731 minus1887301 0 005 071330 07132902 105113 10511301 114656 1146560 1232588 123259

where 120576 = 119886119887 Here primes denote differentiation withrespect to 120578 120582 = 119866119903

119909Re52119909

is the buoyancy or mixedconvection parameter Pr = 120584120572 is the Prandtl number119866119903119909

= 1198921205731199021199081199094

(1198961205842

) is the local Grashof number andRe119909= 119880infin119909120584 is the local Reynolds number We note that

120582 is a constant with 120582 gt 0 corresponds to the assistingflow and 120582 lt 0 denotes the opposing flow whilst 120582 = 0

is for forced convective flow For the forced convection flow(120582 = 0) the corresponding temperature problem possesses alarger similarity-solution domain than the mixed convection(120582 = 0) one The reason is that in the forced convection casethe existence of similarity solutions does not require therestriction of the applied wall heat flux to the special form119902119908(119909) = 119888119909

(5119898minus3)2 Namely in the forced convection casethe much weaker assumption 119902

119908(119909) = 119888119909

119899 suffices for thesimilarity reduction of the problem where the flux exponent119899 does not depend on the velocity exponent119898 in any way

The main physical quantities of interest are the values of11989110158401015840

(0) being a measure of the skin friction and the non-dimensional wall temperature 120579(0) Our main aim is to findhow the values of 11989110158401015840(0) and 120579(0) vary in terms of theparameters 120582 and119898

3 Results and Discussion

Equations (6) subject to the boundary conditions (7) are inte-grated numerically using a finite difference scheme known asthe Keller box method [30] Numerical results are presentedfor different physical parameters To conserve space weconsider the Prandtl number as unity throughout this paperThe results presented here are comparable very well withthose of Ramanchandran et al [12] For no buoyancy effects120582 = 0 and119898 = 1 comparison of the values of11989110158401015840(0)wasmadewith those of Wang [29] as presented Table 1 which shows afavourable agreement

Figures 2 and 3 show the skin friction coefficient 11989110158401015840(0)against buoyancy parameter 120582 for some values of velocityexponent parameter 119898 when velocity ratio parameter is 120576 =05 and 120576 = 1 Two branches of solutions are foundThe solidlines are the upper branch solutions and the dash lines arethe lower branch solutions With increasing119898 the range of 120582for which the solution exists increases Also from both figuresof the upper branch solutions the skin friction is higher forthe assisting flow (120582 gt 0) compared to the opposing flow(120582 lt 0) This implies that increasing the buoyancy parameter

120582 increases the skin friction coefficient11989110158401015840(0)whilst for 120576 = 1the values of11989110158401015840(0) as shown in Figure 3 are positive for 120582 gt 0and negatives for 120582 lt 0 Physically this means that positive11989110158401015840

(0) implies the fluid exerts a drag force on the sheet andnegative implies the reverse Similarly this also happens for120576 = 05 but at different values of 120582

As seen in Figures 2 and 3 there exists a critical valueof velocity ratio 120582

119888such that for 120582 lt 120582

119888there will be no

solutions for 120582119888lt 120582 lt 0 there will be dual solutions and

when 120582 gt 0 the solution is unique Our numerical compu-tations presented in Figure 2 show that for the velocity ratio120576 = 05 120582

119888= minus8331 minus2677 and minus07411 for119898 = 2 1 and 05

respectively On the other hand for the velocity ratio 120576 = 1

shown in Figure 3 120582119888= minus1498 minus4764 and minus1301 for119898 = 2

1 and 05 respectively The dual solutions exhibit the normalforward flowbehavior and also the reverse flowwhere1198911015840(120578) lt0 From these two results it seems that an increase in velocityratio parameter 120576 leads to an increase of the critical values of|120582119888| This increases the dual solutions range of (6)-(7)Figures 4 and 5 display the variations of the wall temper-

ature 120579(0) against the buoyancy parameter 120582 for some valuesof119898when 120576 = 05 and 120576 = 1 respectively Both figures clearlyshow that the wall temperature increases as 119898 increases forthe upper branch solutions For the lower branch solutionsthe wall temperature becomes unbounded as 120582 rarr 0

minusThe velocity and temperature profiles for 120576 = 05 when

120582 = minus05 are presented in Figures 6 and 7 respectivelyFigure 6 shows that the velocity increases as 119898 increases forthe upper branch solutions while opposite trend is observedfor the lower branch solutions In Figure 7 for the upperbranch solutions it is seen that an increase in 119898 tends todecrease the temperature Besides that the temperature ishigher for the lower branch solution than the upper branchsolution at all points near and away from the solid surface Itcan be seen from Figures 6 and 7 that all profiles approachthe far field boundary conditions (7) asymptotically thussupporting the numerical results obtained Finally the typicalstreamlines for 119898 = 1 120576 = 1 and 120582 = minus2 for both solutionbranches are shown in Figures 8 and 9

4 Conclusions

The problem of mixed convection stagnation-point flowtowards a nonlinearly stretching vertical sheet immersed inan incompressible viscous fluid was investigated numericallyThe effects of the velocity exponent parameter 119898 buoyancyparameter 120582 and velocity ratio parameter 120576 on the fluidflow and heat transfer characteristics were discussed It wasfound that for the assisting flow the solution is unique whiledual solutions were found to exist for the opposing flowup to a certain critical value 120582

119888 Moreover increasing the

velocity exponent parameter119898 is to increase the range of thebuoyancy parameter 120582 for which the solution exists

Nomenclature

119886 119887 119888 Constants119891 Dimensionless stream function119892 Acceleration due to gravity


119866119903119909 Local Grashof number

119896 Thermal conductivity119898 Velocity exponent parameterPr Prandtl number119902119908 Surface heat flux

Re119909 Local Reynolds number

119879 Fluid temperature119879infin Ambient temperature

119906 119907 Velocity components along the 119909- and119910-directions respectively

119880infin Free stream velocity

119880119908 Stretching velocity

119909 119910 Cartesian coordinates along the surfaceand normal to it respectively

Greek Letters

120572 Thermal diffusivity120573 Thermal expansion coefficient120576 Velocity ratio parameter120578 Similarity variable120579 Dimensionless temperature120582 Buoyancy parameter120584 Kinematic viscosity120588 Fluid densityΨ Stream function

Subscripts

119908 Condition at the solid surfaceinfin Condition far away from the solid surface

Superscript

1015840 Differentiation with respect to

Acknowledgments

The authors wish to thank the anonymous reviewers for theirvaluable comments and suggestions The third author wouldlike to acknowledge the financial supports received from theUniversiti Kebangsaan Malaysia under the incentive grantsUKM-GUP-2011-202 and DIP-2012-31

References

[1] M V Karwe and Y Jaluria ldquoFluid flow and mixed convectiontransport from a moving plate in rolling and extrusion pro-cessesrdquo Journal ofHeat Transfer vol 110 no 3 pp 655ndash661 1988

[2] M V Karwe and Y Jaluria ldquoNumerical simulation of thermaltransport associated with a continuously moving flat sheet inmaterials processingrdquo ASME Journal of Heat Transfer vol 113no 3 pp 612ndash619 1991

[3] L J Crane ldquoFlow past a stretching platerdquo Journal of AppliedMathematics and Physics vol 21 no 4 pp 645ndash647 1970

[4] P D Weidman and Magyari ldquoGeneralized Crane flow inducedby continuous surfaces stretchingwith arbitrary velocitiesrdquoActaMechanica vol 209 no 3-4 pp 353ndash362 2010

[5] P D Weidman and M E Ali ldquoAligned and nonaligned radialstagnation flow on a stretching cylinderrdquo European Journal ofMechanics BFluids vol 30 no 1 pp 120ndash128 2011

[6] C H Chen ldquoLaminar mixed convection adjacent to verticalcontinuously stretching sheetsrdquo Heat and Mass Transfe vol 33no 5-6 pp 471ndash476 1998

[7] M Ali and F Al-Yousef ldquoLaminar mixed convection boundarylayers induced by a linearly stretching permeable surfacerdquoInternational Journal of Heat and Mass Transfer vol 45 no 21pp 4241ndash4250 2002

[8] A Ishak R Nazar and I Pop ldquoMixed convection boundarylayers in the stagnation-point flow toward a stretching verticalsheetrdquoMeccanica vol 41 no 5 pp 509ndash518 2006

[9] A Ishak R Nazar and I Pop ldquoHydromagnetic flow and heattransfer adjacent to a stretching vertical sheetrdquo Heat and MassTransfer vol 44 no 8 pp 921ndash927 2008

[10] F M Ali R Nazar N M Arifin and I Pop ldquoEffect of Hallcurrent on MHDmixed convection boundary layer flow over astretched vertical flat platerdquoMeccanica vol 46 no 5 pp 1103ndash1112 2011

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

[12] N Ramachandran T S Chen and B F Armaly ldquoMixedconvection in stagnation flows adjacent to vertical surfacesrdquoASME Journal of Heat Transfer vol 110 no 2 pp 373ndash377 1988

[13] M Ayub H Zaman M Sajid and T Hayat ldquoAnalyticalsolution of stagnation-point flow of a viscoelastic fluid towardsa stretching surfacerdquo Communications in Nonlinear Science andNumerical Simulation vol 13 no 9 pp 1822ndash1835 2008

[14] N Bachok A Ishak and I Pop ldquoOn the stagnation-point flowtowards a stretching sheet with homogeneous-heterogeneousreactions effectsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4296ndash4302 2011

[15] N Bachok A Ishak and I Pop ldquoBoundary layer stagnation-point flow and heat transfer over an exponentially stretch-ingshrinking sheet in a nanofluidrdquo International Journal ofHeat and Mass Transfer vol 55 pp 8122ndash8128 2012

[16] N Bachok A Ishak and I Pop ldquoMelting heat transferin boundary layer stagnation-point flow towards a stretch-ingshrinking sheetrdquo Physics Letters A vol 374 no 40 pp4075ndash4079 2010

[17] K Bhattacharyya ldquoDual solutions in boundary layerstagnation-point flow and mass transfer with chemicalreaction past a stretchingshrinking sheetrdquo InternationalCommunications in Heat and Mass Transfer vol 38 no 7 pp917ndash922 2011

[18] T Hayat Z Abbas and M Sajid ldquoMHD stagnation-point flowof an upper-convected Maxwell fluid over a stretching surfacerdquoChaos Solitons and Fractals vol 39 no 2 pp 840ndash848 2009

[19] W Ibrahim B Shankar and M M Nandeppanavar ldquoMHDstagnation point flow andheat transfer due to nanofluid towardsa stretching sheetrdquo International Journal of Heat and MassTransfer vol 56 pp 1ndash9 2013

[20] A Ishak K Jafar R Nazar and I Pop ldquoMHD stagnation pointflow towards a stretching sheetrdquo Physica A vol 388 no 17 pp3377ndash3383 2009


[21] F Labropulu D Li and I Pop ldquoNon-orthogonal stagnation-point flow towards a stretching surface in a non-Newtonianfluid with heat transferrdquo International Journal of ThermalSciences vol 49 no 6 pp 1042ndash1050 2010

[22] G C Layek S Mukhopadhyay and Sk A Samad ldquoHeatand mass transfer analysis for boundary layer stagnation-point flow towards a heated porous stretching sheet withheat absorptiongeneration and suctionblowingrdquo InternationalCommunications in Heat and Mass Transfer vol 34 no 3 pp347ndash356 2007

[23] L Y Yian N Amin and I Pop ldquoMixed convection flow near anon-orthogonal stagnation point towards a stretching verticalplaterdquo International Journal of Heat and Mass Transfer vol 50no 23-24 pp 4855ndash4863 2007

[24] T R Mahapatra S Dholey and A S Gupta ldquoObliquestagnation-point flow of an incompressible visco-elastic fluidtowards a stretching surfacerdquo International Journal of Non-Linear Mechanics vol 42 no 3 pp 484ndash499 2007

[25] J Paullet and P Weidman ldquoAnalysis of stagnation point flowtoward a stretching sheetrdquo International Journal of Non-LinearMechanics vol 42 no 9 pp 1084ndash1091 2007

[26] H Rosali A Ishak and I Pop ldquoStagnation point flow and heattransfer over a stretchingshrinking sheet in a porous mediumrdquoInternational Communications in Heat and Mass Transfer vol38 no 8 pp 1029ndash1032 2011

[27] N A Yacob A Ishak and I Pop ldquoMelting heat transferin boundary layer stagnation-point flow towards a stretch-ingshrinking sheet in a micropolar fluidrdquo Computers andFluids vol 47 no 1 pp 16ndash21 2011

[28] J H Merkin and T Mahmood ldquoMixed convection boundarylayer similarity solutions prescribed wall heat fluxrdquo Journal ofApplied Mathematics and Physics vol 40 no 1 pp 51ndash68 1989

[29] C Y Wang ldquoStagnation flow towards a shrinking sheetrdquoInternational Journal of Non-LinearMechanics vol 43 no 5 pp377ndash382 2008

[30] H B Keller ldquoA new difference scheme for parabolic problemsrdquoinNumerical Solution of Partial-Differential Equations J Bram-ble Ed vol 2 pp 327ndash350 Academic Press New York NYUSA 1970

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014


Stochastic AnalysisInternational Journal of


119879infin119881infin

119880infin

119880119908

119902119908

119910

119892

119906

119907

119910

119902119908

119880119908

119880infin

119879infin

119881infin

(a) Assisting flow (120582 gt 0) (b) Opposing flow (120582 lt 0)

119909

119909

Figure 1 Physical model and coordinate system

minus10 minus8 minus6 minus4 minus2 0 2 4minus3

minus2

minus1

01

2

34

120582

119898 = 05

119898 = 2


119898 = 1

119891998400998400

(0)


minus

16 minus14 minus12 minus10 minus8 minus6 minus4 minus2 0 2 4minus4

minus3

minus2

minus1

0

1

23


120582

119898 = 05

119898 = 2

119898 = 1

119891998400998400

(0)


buoyancy parameter is independent of 119909 For the assistingflow as shown in Figure 1(a) the 119909-axis points upwards inthe same direction of the stretching surface such that theexternal flow and the stretching surface induce flow andheat transfer in the velocity and thermal boundary layers

minus10 minus8 minus6 minus4 minus2 005

1

15

2

25

3

35

120579(0

)

2


120582

119898 = 05

119898 = 2

119898 = 1


minus16 minus14 minus12 minus10 minus8 minus6 minus4 minus2 0 2040608

112141618

222

120579(0

)


119898 = 05

119898 = 2

119898 = 1

120582


respectively On the other hand for the opposing flow asshown in Figure 1(b) the 119909-axis points vertically downwardsin the same direction of the stretching surface such that theexternal flow and the stretching surface also induce flowand heat transfer respectively in the velocity and thermal


0 1 2 3 4 5 6 7minus04

minus02

002040608

1

120578

119898 = 05 1 2

119891998400 (120578

)



0 1 2 3 4 5 6 7012345678

119898 = 2 1 05


120578

120579(120578

)



120597119906

120597119909

+

120597119907

120597119910

= 0 (1)

119906

120597119906

120597119909

+ 119907

120597119906

120597119910

= 119880infin

119889119880infin

119889119909

+ 120584

1205972

119906

1205971199102+ 119892120573 (119879 minus 119879

infin)

(2)

119906

120597119879

120597119909

+ 119907

120597119879

120597119910

= 120572

1205972

119879

1205971199102 (3)


119906 = 119880119908(119909)

119907 = 0

120597119879

120597119910

= minus

119902119908

119896

at 119910 = 0

119906 997888rarr 119880infin(119909) 119879 997888rarr 119879


(4)



051

152

253

354

119909

120578

120595(119887120584)12 = 02 05 1 2 120595(119887120584)12 = 02 05 1 2



1

2

3

4

5

119909

120578

120595(119887120584)12 = 02 05 1 2 120595(119887120584)12 = 02 05 1 2






120578 = (

119880infin

120584119909

)

12

119910

120595 = [120584119909119880infin]12

119891 (120578)

120579 (120578) =

119896 (119879 minus 119879infin)

119902119908

(

119880infin

120584119909

)

12

(5)


119891101584010158401015840

+

119898 + 1

2

11989111989110158401015840

+ 119898(1 minus 11989110158402

) + 120582120579 = 0

1

Pr12057910158401015840

+

119898 + 1

2

1198911205791015840

minus (2119898 minus 1) 1198911015840

120579 = 0

(6)


119891 (0) = 0 1198911015840

(0) = 120576 1205791015840

(0) = minus1

1198911015840


(7)





119909Re52119909


= 1198921205731199021199081199094

(1198961205842





119908(119909) = 119888119909










119888such that for 120582 lt 120582











4 Conclusions




Nomenclature











Greek Letters


Subscripts


Superscript


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




0 1 2 3 4 5 6 7minus04

minus02

002040608

1

120578

119898 = 05 1 2

119891998400 (120578

)



0 1 2 3 4 5 6 7012345678

119898 = 2 1 05


120578

120579(120578

)



120597119906

120597119909

+

120597119907

120597119910

= 0 (1)

119906

120597119906

120597119909

+ 119907

120597119906

120597119910

= 119880infin

119889119880infin

119889119909

+ 120584

1205972

119906

1205971199102+ 119892120573 (119879 minus 119879

infin)

(2)

119906

120597119879

120597119909

+ 119907

120597119879

120597119910

= 120572

1205972

119879

1205971199102 (3)


119906 = 119880119908(119909)

119907 = 0

120597119879

120597119910

= minus

119902119908

119896

at 119910 = 0

119906 997888rarr 119880infin(119909) 119879 997888rarr 119879


(4)



051

152

253

354

119909

120578

120595(119887120584)12 = 02 05 1 2 120595(119887120584)12 = 02 05 1 2



1

2

3

4

5

119909

120578

120595(119887120584)12 = 02 05 1 2 120595(119887120584)12 = 02 05 1 2






120578 = (

119880infin

120584119909

)

12

119910

120595 = [120584119909119880infin]12

119891 (120578)

120579 (120578) =

119896 (119879 minus 119879infin)

119902119908

(

119880infin

120584119909

)

12

(5)


119891101584010158401015840

+

119898 + 1

2

11989111989110158401015840

+ 119898(1 minus 11989110158402

) + 120582120579 = 0

1

Pr12057910158401015840

+

119898 + 1

2

1198911205791015840

minus (2119898 minus 1) 1198911015840

120579 = 0

(6)


119891 (0) = 0 1198911015840

(0) = 120576 1205791015840

(0) = minus1

1198911015840


(7)





119909Re52119909


= 1198921205731199021199081199094

(1198961205842





119908(119909) = 119888119909










119888such that for 120582 lt 120582











4 Conclusions




Nomenclature











Greek Letters


Subscripts


Superscript


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014







119909Re52119909


= 1198921205731199021199081199094

(1198961205842





119908(119909) = 119888119909










119888such that for 120582 lt 120582











4 Conclusions




Nomenclature











Greek Letters


Subscripts


Superscript


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014












Greek Letters


Subscripts


Superscript


Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014










Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



Documents

Stagnation-Point Flow toward a Vertical, Nonlinearly ... · point flow towards a stretching surface in a non-Newtonian fluid with heat transfer,”International Journal of Thermal