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The ext ended hom ot opy pert urbat i on

m et hod and boundary l ayer f l ow due

t o condensat i on and nat ur al convect i on

on a porous vert i cal pl at e

Qasem M. Al - Mdal l al a , Muham m ed I . Syam a & P . Donal d Ar i el b

a Depar t m ent of Mat hem at i cal Sci ences, Col l ege of Sci ence,

Uni t ed Ar ab Em i r at es Uni ver si t y , PO Box 17551, Al - Ai n, Uni t ed

Ar ab Em i r at es

b Depar t m ent of Mat hem at i cal Sci ences, Tr i ni t y W est er n

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Avai l abl e onl i ne: 22 Aug 2011

To ci t e t hi s art i cl e: Qasem M. Al - Mdal l al , Muham m ed I . Syam & P . Donal d Ar i el ( 2011) : The

ext ended hom ot opy per t ur bat i on m et hod and boundar y l ayer f l ow due t o condensat i on and

nat ur al convect i on on a por ous ver t i cal pl at e, I nt er nat i onal Jour nal of Com put er Mat hem at i cs,

DOI : 10. 1080/00207160. 2011. 606905

To l i nk t o t hi s art i cl e: ht t p: //dx. doi . or g/10. 1080/00207160. 2011. 606905

PLEASE SCROLL DOW N FOR ARTICLE

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International Journal of Computer Mathematics

2011, 1–18, iFirst

The extended homotopy perturbation method and boundary

layer flow due to condensation and natural convection on a

porous vertical plate

Qasem M.Al-Mdallala, Muhammed I. Syama* and P. DonaldArielb

aDepartment of Mathematical Sciences, College of Science, United Arab Emirates University, PO Box

17551, Al-Ain, United Arab Emirates;bDepartment of Mathematical Sciences, Trinity Western University,

7600 Glover Road, Langley, BC, Canada V2Y 1Y1

(Received 13 April 2011; revised version received 14 July 2011; accepted 16 July 2011)

Theextendedhomotopyperturbationmethod(EHPM),whichisanextensiontothewell-knownhomotopy

perturbation method (HPM) due to He, is applied to derive the analytical solution of the boundary layer

flow of falling vapour due to the condensation on a cold, porous vertical plate. The EHPM calculates the

solution automatically by adjusting the scaling factor of the independent similarity variable normal to the

plate.TheresultsobtainedbytheEHPMareinexcellentagreementwiththeexactnumericalsolution.Also

an asymptotic solution, valid for a large suction parameter is developed which matches very well with the

exact solution even for moderate values of the suction velocity. Finally, it is shown that the EHPM solution

is also applicable to the moderate values of blowing across the plate.

Keywords: homotopy perturbation method; extended homotopy perturbation method; a porous vertical

plate;Ackroyd’s method; asymptotic solution

2010 AMS Subject Classifications: 62Z05; 65L99

Introduction

Homotopy is a highly interesting and useful concept in topology, an important branch of pure

mathematics.Essentially,itstandsforacontinuousdeformationofmathematicalobjectsfromone

state to another. The idea has been extensively utilized in deriving the solution of the nonlinear

boundaryvalueproblems(BVPs)occurringinvariousdisciplinesofscienceandtechnology.With

the advent of the digital computers, at first, extensive numerical algorithms were developed for

difficultnonlinearproblems,butlater,becauseoftherapidadvancesincomputeralgebrasystems,

homotopy provided a boon to deriving even the analytical solutions of the same problems.

The basic idea behind the homotopy solutions is to introduce a parameter, say, p which takes

continuouslythevaluesfrom0to1.Whenpequalszero,theBVPgoverningthenonlinearproblem

simplifies to a simple linear BVP whose solution can be obtained rather trivially. As p changes

from 0 to 1, continuous deformations take place in the BVP, and eventually when p becomes 1,

*Corresponding author. Email: m.syam@uaeu.ac.ae

ISSN 0020-7160 print/ISSN 1029-0265 online

© 2011 Taylor & Francis

DOI: 10.1080/00207160.2011.606905

http://www.informaworld.com

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2

Q.M. Al-Mdallal et al.

the solution of the given nonlinear problem is recovered. The basic idea narrated above is further

facilitated by assuming that the variables of interest can be expressed in a power series of p. By

equating various powers of p, one obtains BVP(s) at each stage of deformation, which, hopefully,

can be solved analytically.

In many cases, the homotopy solution is similar to the regular perturbation solution expressible

intermsofsomesmallphysicalparameterdescribingthenonlinearproblem.However,theregular

perturbation solutions in most cases are valid only within the radius of convergence of the cor-

responding physical parameter. The main task of the homotopy methods is to enlarge the radius

of convergence so that the solution would become applicable over a much larger range of the

physical parameters. This is done by introducing some parameters in the homotopy formulation

of the problem.

Liao [24] has systematically developed a homotopy method called the homotopy analysis

method (HAM), the main feature of which is the presence of an auxiliary parameter h, which

is primarily used to control the convergence of the power series representing various physical

quantities. Besides, there is also a provision of an auxiliary function that can supplement the

role of h. Liao’s pioneering work has attracted the attention of several researchers and during

the last few years a number of papers have appeared in the literature, which have attempted to

solve numerous nonlinear BVPs using the HAM.The interested reader is specially referred to the

studies of Liao, Hayat,Abbasbandy and their coworkers [1–4,14,16,17,25–29].

On the other hand, He [18] introduced the homotopy perturbation method (HPM). The HPM

does not have a standard auxiliary parameter h per se, but it strives to introduce other parameters

which promise even faster convergence in comparison with the HAM. While, typically, one

requires 20–50 terms in the power series of a HAM solution to obtain a reasonably accurate

solution, a judicious selection of the right kind of parameters in the HPM would require only 3–4

terms – sometimes only one correction term leads to a fully acceptable solution. He and others

[11,12,15,19–22]demonstratedtheversatilityoftheHPMbysolvingahostofimportantphysical

problems involving nonlinear differential equations.

The HPM has been found to be particularly effective in obtaining the analytical solutions of the

flow problems in fluid dynamics when the flow is caused by the moving boundaries. Ariel et al.

[13] demonstrated to a large extent the power of the HPM by computing the steady, axisymmetric

flow of a viscous, incompressible fluid past a linearly stretching sheet.They showed that only one

correction term in the HPM is able to generate a solution which is remarkably accurate. Further,

the presence of the physical parameters such as the suction or the magnetic field, which for large

values usually cause havoc with the numerical integration schemes because of the thin boundary

layers, in fact, improve the performance of the HPM solution. The attractive idea of one-term

correction in the HPM solution was consequently invoked by Ariel [7,8] to get the solution of

the three-dimensional flow past a stretching sheet, and the axisymmetric flow past a stretching

sheet when there is a partial velocity slip at the sheet. In both cases, again, the HPM solution was

fairly close to the exact solution and could have been easily accepted by a practicing engineer

or a scientist. The main criticism against the one-term correction solution of the HPM is that

it cannot be readily extended to the many-term solution without shedding the principle of the

solution which is free of the secular terms. The reason for insisting the solution being free of the

secular terms is well known in perturbation theory; the rational being that each perturbation term

is a correction to the solution obtained by terminating the perturbation solution at the preceding

term and if a secular term is included in any term of the perturbation solution then the limit of

this term compared to the rest of the solution would become infinite as the independent variable

tends to infinity. If one still sticks to the latter principle then there can be situations where one-

term correction solution of the HPM would not give a sufficiently accurate solution of a physical

problem. Thus, there arose the need to extend the HPM solution so that a solution, free of the

secularterms,couldbegeneratedtoanydesireddegreeofaccuracy.Ariel[9]proposedtherequired

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International Journal of Computer Mathematics

3

extension (EHPM) by expanding the independent variable occurring in the BVP describing the

axi-symmetricflowpastastretchingsheetalsoinapowerseriesofp.Thecoefficientsinthepower

series of both, the dependent and the independent variables could then be calculated by insisting

that the solution is free of the secular terms. The extended version of the HPM was put to test

to compute the flow due to a rotating disk [10]. For this problem, the one-term correction term

of the HPM solution gives quite a poor approximation, but the EHPM generates a fully accurate

solution which completely matches with the solution ofAckroyd [5] who needed nearly 50 terms

in his series solution.

The EHPM is an important extension of the HPM in that it allows solutions to be developed

to an arbitrary degree of accuracy. It can be applied in many different physical situations. In this

work, we reconsider an important classical problem that was first considered by Koh et al. [23],

withtheaimofderivingafullyanalyticalsolution.Theproblemconsistsofthemotionofavapour

falling under gravity near a colder vertical plate.As a result, a condensate layer is formed near the

plate. Koh et al. showed that the usual temperature term occurring in the momentum equation can

beneglectedatsufficientlyhighvaluesofthePrandtlnumbers,resultingintoasingleequationfor

the motion decoupled from the energy equation.The vertical plate is assumed to be porous which

can permit both, the suction and the injection across the plate. This introduces a new element in

the application of the EHPM, as the previous investigations involving the EHPM have assumed

the boundaries to be impermeable.

Equations of motion

We consider the two-dimensional motion of a film of vapour at a uniform temperature Tsfalling

under the gravity field near a vertical plate maintained at a uniform temperature Tw(Tw< Ts).

The schematic description of the flow is given in Figure 1. Because of the lower temperature

of the plate, condensation takes place which can be modelled by boundary layer equations

when the Prandtl number is greater than 10 [23]. The plate is taken along the x-axis. In the

following, we focus on the motion of the vapour layer only. Accordingly, we introduce the vari-

ables pertaining to the vapour layer, namely a stream function ?, and the similarity variable η,

as follows:

ψ = 4νvδx3/4(f?

η = δx−1/4(f?

c)1/2F(η), (1)

c)1/2(y − ys), (2)

Figure 1. Schematic diagram.

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Q.M. Al-Mdallal et al.

where

δ =

?

g

4ν2

V

?1/4

. (3)

Here μ is the dynamic viscosity, ν represents kinematic viscosity, ρ represents the density and g

thegravitationalacceleration.Thesubscriptscandvrefertophysicalquantitiesincondensateand

vapour, respectively. Further, the suffix s is used to denote conditions at the condensate surface.

Moreover, fcand f?

layer variables at the condensate surface. Consequently, the momentum equation for the vapour

boundary layers becomes

care, respectively, the stream function and its derivative in terms of condensate

f???+ 3ff??− 2f?2= 0,

f(0) = A,

(4)

f?(0) = 1,

f?(∞) = 0, (5)

where

A =

?(ρμ)c

(ρμ)v

?

fc(f?

c)−1/2. (6)

It may be noted that when A > 0, the vapour boundary layer has suction applied to it at the

condensate surface. Whereas, in the case A < 0 blowing takes place at the surface.

Extended homotopy perturbation solution

The extended HPM is based on stretching the independent variable ζ by means of a scaling

parameter, say, α. We thus introduce

η = αζ.(7)

The BVP (4) and (5) transforms to

αd3f

dη3+ 3fd2f

dη2− 2

αdf(0)

?df

= 1,

dη

?2

= 0,

df(∞)

dη

(8)

f(0) = A,

dη

= 0.(9)

We also introduce a new dependent variable F as follows:

F = αf.(10)

Consequently, the BVP (8) and (9) can be rewritten as

α2d3F

dη3+ 3Fd2F

dη2− 2

dF(0)

dη

?dF

= 1,

dη

?2

= 0,

dF(∞)

dη

(11)

F(0) = αA,

= 0.(12)

We further introduce the parameter Z defined by

Z = αA. (13)

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International Journal of Computer Mathematics

5

As a result the problem is now inverted in that Z would be prescribed a priori and A would be

determined in the solution process. Now we set up the homotopy equation

?

α2

?d3F

dη3−dF

dη

?

+ p

3Fd2F

dη2− 2

?dF

dη

?2

+ α2dF

dη

?

= 0.(14)

Expanding F and α2in power series of p, one obtains

F = F0+ F1p + F2p2+ ··· =

∞

?

∞

?

n=0

Fnpn,(15)

α2= b0+ b1p + b2p2+ ··· =

n=0

bnpn.(16)

Substituting for F and α2in Equations (12)–(14), and equating like powers of p on both sides, we

obtain the following system of equations: For n = 0:

?d3F0

F0(0) = Z,

dη

b0

dη3−dF0

dη

dF0(0)

?

= 0, (17)

= 1,

dF0(∞)

dη

= 0,(18)

and for (n ≥ 1):

b0

?d3Fn

dη3−dFn

dη

?

= −

n−1

?

m=0

bm+1

??d3Fn−m−1

− 2dFm

dη

dη3

−dFn−m−1

dη

?

+ 3Fmd2Fn−m−1

?

dη2

dFn−m−1

dη

dFn(∞)

dη

+ bmdFn−m−1

dη

,(19)

Fn(0) = 0,

dFn(0)

dη

= 0,

= 0.(20)

It can be easily seen that the solution for F0is given as

F0= 1 + Z − e−η,

dF0

dη

= e−η.(21)

Substituting for F0in Equation (19) for n = 1, we obtain

?d3F1

The solution of which, subject to boundary conditions (20) for n = 1, is

?3(1 + Z)

Assuming that the solution has to be free of the secular terms, the coefficient of ηe−ηin

Equation (23) must be zero; this leads to

b0

dη3−dF1

dη

?

= [3(1 + Z) − b0]e−η− e−2η. (22)

F1=

2b0

−1

2

?

ηe−η+

?7 + 9Z

6b0

−1

2

?

e−η+

1

6b0

e−2η+1

2−

4

3b0. (23)

b0= 3(1 + Z)

(24)

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6

Q.M. Al-Mdallal et al.

and

F1=

1

18(1 + Z)(1 − e−η)2,

dF1

dη

=

1

9(1 + Z)e−η(1 − e−η). (25)

For the sake of simplicity, we use the notation

c =

1

1 + Z, (26)

so that the solution developed so far can be rewritten as

F0=1

c− e−η,

c

18(1 − e−η)2,

dF0

dη

= e−η,

dF1

dη

b0=3

c,(27)

F1=

=c

9e−η(1 − e−η)2.(28)

The second-order system (n = 2) is

?d3F2

b0

dη3−dF2

dη

?

= −b1

+ 4dF0

?d3F1

dη3−dF1

dF1

dη

dη

?

− 3F0d2F1

− b1dF0

dη2− 3F1d2F0

dη2

dη

− b0dF1

dη

dη, (29)

and in view of Equations (27) and (28), it takes the form

d3F2

dη3−dF2

dη

=

c

54[(3c − 18b1)e−η− (6 + 4c − 6cb1)e−2η+ 7ce−3η].(30)

For the solution of F2to be free of the secular terms, we must have

b1=c

6.

(31)

Substituting for b1in Equation (30), we obtain

d3F2

dη3−dF2

dη

= −c

54[(6 + 4c − c2)e−2η− 7ce−3η].(32)

The solution for F2is given by

F2=

dF2

dη

c

1296(1 − e−η)2(24 + 2c − 4c2− 7ce−η),

c

1296e−η(1 − e−η)(48 + 11c − 8c2− 21ce−η).

=

(33)

The value of b2can be found in the next step by assuming that the solution of F2to be free of

the secular terms; we obtain

c

216(12 + c − 2c2).

We observe from the foregoing that the value of bnis calculated only at the (n + 1)st step. In fact

the expressions for Fn+1and dFn+1/dη calculated at the (n + 1)st step involve bn, which is then

b2=

(34)

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International Journal of Computer Mathematics

7

evaluated on applying the condition that the solution is free of the secular terms. In fact, it can be

seen that bnis related to the value of Fnat infinity through the relation

bn= 3 lim

η→∞Fn(η).(35)

This can be shown using the mathematical induction on n, as was done by Ariel [9] for the

axisymmetric flow past a stretching sheet.

Tables 1 and 2 show the evaluated values of bnand d2Fn(0)/dη2for the first few terms of

expansion for arbitrary value of Z. It is clearly seen that every new term after n = 1 in the

perturbation solution leads to two extra terms of c both in bnand d2Fn(0)/dη2. Note that the

expressions for these two important parameters, bnand d2Fn(0)/dη2, are simple and elegant.

A natural question arises regarding the number of terms after which the perturbation solution

must be terminated. Once a tolerance is decided, a number of criteria are plausible. We decided

to terminate the solution when the coefficient of the highest degree term in c in both bnand

d2Fn(0)/dη2met the prescribed tolerance criterion. Selecting a 10-digit accuracy criterion, it was

found that the perturbation solution could be terminated after 12 terms. It may be mentioned here

that at the present state of investigation no formal proof has been derived for the convergence of

the EHPM. As can be observed with Ackroyd’s method [6], the convergence of the perturbation

Table 1. Listing of the values of bnfor n = 0,1,...,8.

Nbn

0

2

c

c

6

1

2

c

18+

?1

?

?

⎛

⎜

⎛

⎜

⎜

⎛

⎜

⎜

c2

216−

864−257c2

c3

108

3

c

54+

5c

38880−

c3

1296+

c4

972

?

4

c

1

162+

125c

31104−

257c2

777600−

353c3

279936+

389c4

349920+

5c5

34992−

5c6

34992

?

5

c

1

486+

823c

373248−

60851c2

46656000−

271c3

233280+

30881c4

41990400+

121c5

419904−

43c6

209952−

35c7

1259712+

7c8

314928

?

6

c

⎜

⎝

1

1458+

1591c

1492992+

3291511c2

8398080000−

3773c7

56687040+

3503473c2

55987200000−

1434683c7

16325867520−

813239c3

1007769600−

2233c8

56687040+

14156309c3

30233088000−

192311c8

5101833600+

11c12

17006112

27745483c2

1119744000000−

170267176043c6

6399740067840000−

414997c10

48977602560−

1527251c4

4408992000+

7c9

1259712−

766879711c4

7407106560000+

1309c9

85030560−

2288773c5

7054387200−

7c10

1889568

83983c6

503884800

+

⎞

⎟

⎟

⎠

7

c

⎜

⎜

⎜

⎝

⎜

1

4374+

689527859c6

7618738176000−

25583c

53747712+

177088277c5

658409472000

+

1057c10

136048896−

77c11

68024448

+

⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎟

8

c

⎜

⎜

⎜

⎝

⎜

1

13122+

2709687427963c5

14932726824960000+

13268221c9

571405363200+

129997c

644972544+

864295541c3

3627970560000−

34570330259c7

411411861504000−

3179c12

2040733440+

6699920263c4

1555492377600000

587507819c8

25713241344000

143c13

612220032−

+

+

17413c11

4897760256+

143c14

1224440064

⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎟

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Page 10

8

Q.M. Al-Mdallal et al.

Table 2.Listing of the values of d2Fn(0)/dη2for n = 0,1,...,8.

N

d2Fn(0)/dη2

0

−1

c

9

?1

?1

?

⎛

⎜

⎛

⎜

⎛

⎜

⎛

⎜

⎜

1

2

c

27−

5c

648−

c2

162

?

3

c

81−

19c

7776−127c2

29160+

c3

1458+

c4

1458

?

4

c

1

243−

c

10368−1523c2

583200+

1459c3

4199040+

67c4

87480−

5c5

69984−

5c6

52488

?

5

c

⎜

⎝

1

729+

421c

1119744−

47101c2

34992000−

127c3

41990400+

7c7

944784+

58367c4

94478400−

7c8

472392

2216519c3

15116544000+

91c8

3149280−

16975841c3

113374080000+

117449c8

3401222400+

205294801c2

2519424000000−

176981396261c6

2399902525440000−

590527c10

73466403840−

1481c5

37791360−

307c6

2099520

+

⎞

⎟

⎞

⎟

⎟

⎠

6

c

⎜

⎝

1

2187+

4079c

13436928−

66557c6

453496320+

9199c

53747712−

164259493c6

1428513408000−

1

19683+

137048054393c5

2133246689280000−

3061243c9

685686435840−

3792761c2

6298560000−

49c7

30233088+

29816669c2

125971200000−

747487c7

63489484800+

160615c

1934917632−

2682863c4

6613488000+

7c9

11337408−

3723055577c4

16665989760000+

401c9

408146688−

5693776361c3

54419558400000+

5329798837c7

239990252544000+

7271c12

6122200320+

8051c5

362797056

−

7c10

2834352

⎟

⎠

7

c

⎜

⎝

1

6561+

451197839c5

7618738176000

11c12

25509168

5673757781c4

54685278900000

305266787c8

9795520512000

143c13

7346640384−

−

11c10

1889568+

⎞

⎟

⎟

⎠

8

c

⎜

⎜

⎜

⎝

⎜

+

+

3509c11

7346640384+

143c14

1836660096

⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎟

solution is very much problem-dependent. For the axi-symmetric flow past a stretching sheet, it

has been demonstrated byAriel [9] that the convergence consistently improves as more terms are

taken into the perturbation solution. The same remarks hold for the present problem whenever

there is a convergence to a solution. The analytical solution for α2and d2F(0)/dη2are given by

α2=

12

?

− 2.0838593048 × 10−2c3+ 4.8851052662 × 10−3c4+ 3.2219199832 × 10−3c5

+ 1.3933764722 × 10−3c6− 5.6995605798 × 10−4c7+ 4.0436021835 × 10−4c8

+ 1.0438389045 × 10−4c9+ 1.1440731723 × 10−4c10− 2.0513512839 × 10−5c11

+ 2.9797831397 × 10−5c12+ 4.7097654765 × 10−6c13+ 6.644102936 × 10−6c14

− 1.1618398403 × 10−6c15+ 1.1713886924 × 10−6c16+ 2.4667444745 × 10−7c17

+ 1.4918191976 × 10−7c18− 3.7669640248 × 10−8c19+ 1.2026549238 × 10−8c20

+ 3.5389750441 × 10−9c21+ 4.5735138803 × 10−10c22

− 1.5244918389 × 10−10c23,

n=0

bn=3

c+ 2.4999952958 × 10−1c + 1.8515665098 × 10−2c2

(36)

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Page 11

International Journal of Computer Mathematics

9

d2F(0)

dη2

=

12

?

− 1.5433479413 × 10−2c3+ 5.1814804637 × 10−4c4+ 2.8522279394 × 10−3c5

+ 1.4775443509 × 10−4c6− 6.3608681255 × 10−4c7+ 9.7210095795 × 10−5c8

+ 1.5232853346 × 10−4c9+ 3.6690921152 × 10−5c10− 3.6984503971 × 10−5c11

+ 1.0608993132 × 10−5c12+ 8.5583057744 × 10−6c13+ 2.3876936595 × 10−6c14

− 1.7461063878 × 10−6c15+ 4.0424697798 × 10−7c16+ 2.882796244 × 10−7c17

+ 4.8273180225 × 10−8c18− 3.5003261072 × 10−8c19+ 3.6125011969 × 10−9c20

+ 2.7295663402 × 10−9c21+ 1.2704098658 × 10−10c22

− 1.0163278926 × 10−10c23.

n=0

d2Fn(0)

dη2

= −1 + 1.6666635305 × 10−1c − 9.2608059603 × 10−3c2

(37)

It is well known that in order to obtain a higher accuracy solution in any type of series solution

method, one should increase the number of terms in the series. It requires a costly execution time

thatgrowsassomepowerofnumberoftermsintheseries.Inthispaper,weappliedShanks’trans-

formation[30],forimprovingtheconvergenceoftheseriesofα2andd2F(0)/dη2.Basically,ifthe

sequence {Sk} denotes the values of either α2=?k

k= Sk+2−(?Sk)2

n=0bnor d2F(0)/dη2=?k

n=0d2Fn(0)/dη2,

with k even, we compute ?Skand ?2Sk, then we apply the following formula:

S?

?2Sk

,

where ?Sk= Sk+2− Sk+1and ?2Sk= Sk+2− 2Sk+1+ Sk, to obtain the new improved sequence

{S?

the order of convergence of the new sequence {S?

the original one {Sk}. In the following, we limit ourselves to an eight-digit accuracy solution. In

Tables 3–8, we list the values of α,−F??(0) and A for different non-zero values of Z when the

perturbation series is terminated after k terms. Note that when Z is given then α can be found

using Equation (36) and, therefore, A can be calculated using Equation (13), i.e. A = Z/α. Also

k} but with k − 2 terms. Since Shanks’ transformation is a special kind of the extrapolation,

k} is higher than the order of convergence of

Table 3.

(i) without using Shanks’transformation, and (ii) using Shanks’transformation.

Illustrating the variation of α, −f??(0) and A for Z = 0 with N, the number of terms in the perturbation solution

α

−d2f(0)/dη2

A

N

Without Shanks With ShanksWithout ShanksWith Shanks Without ShanksWith Shanks

0

1

2

3

4

5

6

7

8

9

1.73205081

1.77951304

1.79376492

1.79876215

1.80063285

1.80135741

1.80164369

1.80175823

1.80180445

1.80182321

1.80183086

1.80183398

1.80183527

1.73205081 1.73205081

1.58178937

1.55293537

1.54481654

1.54218451

1.54126808

1.54093552

1.54081165

1.54076471

1.54074671

1.54073974

1.54073702

1.54073596

1.732050810

0

0

0

0

0

0

0

0

0

0

0

0

0

1.79988097 1.546077860

1.801818201.540784390

1.801836011.540735590

1.80183616 1.540735270

10

11

12

1.80183616 1.540735270

1.80183616 1.540735270

Downloaded by [Qasem M. Al-Mdallal] at 20:56 06 October 2011

Page 12

10

Q.M. Al-Mdallal et al.

Table 4.

(i) without using Shanks’transformation, and (ii) using Shanks’transformation.

Illustrating the variation of α, −f??(0) and A for Z = 2 with N, the number of terms in the perturbation solution

α

−d2f(0)/dη2

A

N

Without Shanks With ShanksWithout Shanks With ShanksWithout ShanksWith Shanks

0

1

2

3

4

5

6

7

8

9

3.00000000

3.00924501

3.01234884

3.01343845

3.01383178

3.01397637

3.01403017

3.01405037

3.01405800

3.01406089

3.01406199

3.01406242

3.01406258

3.000000003.00000000

2.89779150

2.86686217

2.85676836

2.85330884

2.85208475

2.85164237

2.85148016

2.85142008

2.85139766

2.85138925

2.85138607

2.85138487

3.00000000 0.66666667

0.66461853

0.66393373

0.66369366

0.66360704

0.66357521

0.66356336

0.66355892

0.66355724

0.66355660

0.66355636

0.66355627

0.66355623

0.66666667

3.013917552.853441380.66358975

3.014062072.851387530.66355637

3.014062672.851383520.66355621

3.01406268 2.851384140.66355621

10

11

12

3.014062682.851384140.66355621

3.014062682.851384140.66355621

Table 5.

(i) without using Shanks’transformation, and (ii) using Shanks’transformation.

Illustratingthevariationofα,−f??(0)andAforZ = 10withN,thenumberoftermsintheperturbationsolution

α

−d2f(0)/dη2

A

N

Without Shanks With ShanksWithout Shanks With ShanksWithout ShanksWith Shanks

0

1

2

3

4

5

6

7

8

9

5.74456265

5.74588127

5.74632346

5.74647367

5.74652516

5.74654293

5.74654909

5.74655124

5.74655199

5.74655225

5.74655234

5.74655237

5.74655238

5.74456265 5.74456265

5.68784206

5.66932499

5.66315827

5.66107487

5.66036369

5.66011911

5.66003454

5.66000519

5.65999497

5.65999140

5.65999015

5.65998971

5.74456265 1.74077656

1.74037707

1.74024314

1.74019765

1.74018206

1.74017668

1.74017481

1.74017416

1.74017394

1.74017386

1.74017383

1.74017382

1.74017382

1.74077656

5.74654657 5.660349841.74017560

5.74655238 5.659989631.74017382

5.746552395.659989351.74017381

5.746552395.659989481.74017382

10

11

12

5.74655239 5.65998948 1.74017382

5.746552395.65998948 1.74017382

Table 6.

solution (i) without using Shanks’transformation, and (ii) using Shanks’transformation.

Illustrating the variation of α, −f??(0) and A for Z = 100 with N, the number of terms in the perturbation

α

−d2f(0)/dη2

A

N

Without ShanksWith ShanksWithout ShanksWith ShanksWithout ShanksWith Shanks

0

1

2

3

4

5

6

7

8

9

17.40689519

17.40694259

17.40695840

17.40696368

17.40696545

17.40696604

17.40696624

17.40696630

17.40696633

17.40696633

17.40696634

17.40696634

17.40696634

17.4068951917.40689519

17.38779303

17.38143891

17.37932070

17.37861343

17.37837698

17.37829787

17.37827138

17.37826250

17.37825953

17.37825853

17.37825820

17.37825809

17.406895195.74484990

5.74483425

5.74482903

5.74482729

5.74482671

5.74482651

5.74482645

5.74482643

5.74482642

5.74482642

5.74482641

5.74482641

5.74482641

5.74484990

17.4069663117.378271765.74482642

17.4069663317.37825803 5.74482641

17.40696633 17.378258035.74482641

17.4069663417.378258035.74482641

10

11

12

17.40696634 17.378258035.74482641

17.4069663417.378258035.74482641

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International Journal of Computer Mathematics

11

Table 7.

solution (i) without using Shanks’transformation, and (ii) using Shanks’transformation.

Illustrating the variation of α, −f??(0) and A for Z = 0.2 with N, the number of terms in the perturbation

α

−d2f(0)/dη2

A

N

Without ShanksWith ShanksWithout ShanksWith ShanksWithout ShanksWith Shanks

0

1

2

3

4

5

6

7

8

9

1.54919334

1.61503354

1.63307290

1.63934835

1.64169833

1.64261200

1.64297485

1.64312086

1.64318013

1.64320433

1.64321425

1.64321833

1.64322002

1.549193341.54919334

1.39072333

1.37002971

1.36402310

1.36211769

1.36146492

1.36123159

1.36114590

1.36111386

1.36110173

1.36109709

1.36109531

1.36109462

1.54919334

−0.12909944

−0.12383644

−0.12246851

−0.12199970

−0.12182506

−0.12175730

−0.12173041

−0.12171959

−0.12171520

−0.12171341

−0.12171267

−0.12171237

−0.12171225

−0.12909944

−0.12198810

−0.12171563

−0.12171222

−0.12171216

−0.12171216

−0.12171216

1.639880691.36692159

1.643174711.36121021

1.643220471.36109251

1.643221201.36109423

10

11

12

1.64322121 1.36109418

1.643221211.36109418

Table 8.

solution (i) without using Shanks’transformation, and (ii) using Shanks’transformation.

Illustrating the variation of α, −f??(0) and A for Z = 0.6 with N, the number of terms in the perturbation

α

−d2f(0)/dη2

A

N

Without ShanksWith ShanksWithout Shanks With ShanksWithout ShanksWith Shanks

0

1

2

3

4

5

6

7

8

9

1.09544512

1.27148207

1.28055254

1.29974079

1.29813369

1.30302712

1.30126736

1.30319140

1.30204239

1.30303022

1.30231396

1.30289737

1.30244129

1.09544512 1.09544512

0.91829261

0.99153895

0.95270031

0.97532590

0.96079134

0.97082789

0.96367538

0.96899323

0.96496990

0.96808803

0.96564049

0.96759003

1.09544512

−0.54772256

−0.47189025

−0.46854774

−0.46163051

−0.46220201

−0.46046624

−0.46108895

−0.46040819

−0.46081449

−0.46046515

−0.46071840

−0.46051210

−0.46067335

−0.54772256

−0.46839362

−0.47064479

−0.46242776

−0.46262549

−0.46061439

−0.46057625

1.281045300.97011303

1.27509297 0.96685039

1.297509720.96664002

1.296961730.96663691

10

11

12

1.302607860.96662486

1.30271120 0.96675705

presented in the tables are the same values when Shanks’transformation is applied to accelerate

the convergence. The efficiency of Shanks’transformation can readily be observed.

From Tables 3–6, we observe that as Z, a measure of the suction parameter, is increased

the number of terms required to obtain the solution within the desired accuracy decreases. For

example, when there is no suction, about eight terms are required for eight-decimal accuracy,

provided the Shanks transformation is used. For Z = 100, the number of terms for a similar

accuracy is reduced to four. This is indeed what is to be expected because as Z is increased a

suctionboundarylayersetsinanditiswellknownthatonlyafewtermsareneededtodescribethe

said boundary layer behaviour (see also the section on asymptotic solution.) On the other hand,

from Tables 7 and 8 it becomes evident that the performance of the EHPM starts deteriorating

when there is an increased injection across the surface. In fact, the EHPM solution becomes

divergent for values of Z less than −0.6. Again this is not unexpected since in an analogous

situation Ackroyd [6] found that the number of terms required in his series solution to get the

same accuracy increases dramatically as the value of −Z increases and approaches the value 1

(see the section on numerical solution.) This, as can be seen, is a serious deficiency of the EHPM

and needs to be addressed. Perhaps the inclusion of another parameter can obviate the problem

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Page 14

12

Q.M. Al-Mdallal et al.

of divergence. This is an issue that is being looked into and the developments relating to it will

be reported in due course of time.

A numerical solution

Following the footsteps of Ackroyd [6], we look for a solution of the BVP (11) and (12) in the

form of a series involving exponential functions, i.e.

F(η) =

∞

?

n=0

ane−nη,

a0?= 0.(38)

SubstitutionforF anditsderivativesinEquation(11)resultsintothefollowingrecurrencerelation

for anwhen the like powers of exp(−nη) are equated on both sides

1

α2n2(n − 1)

From Equation (39), we obtain

an=

n−1

?

m=1

m(5m − 2n)aman−m,

n ≥ 2. (39)

a0=α2

3,

a2=

a2

4α2,

1

a3=

7a3

72α4,

1

a4=

a4

1

24α6,

a5=

7a5

384α8,

1

a6=

413a5

51840α8,··· .

1

(40)

However, the boundary conditions (12) give

∞

?

n=0

an= Z,

∞

?

n=1

nan= −1.(41)

Itisclearlyseenthatallthecoefficientsan’sareexpressedonlyintermsofa1andα2(unknowns).

ThoseunknownscanbedeterminedusingEquation(41).Thus,foragivenZ allthean’sareknown,

which completes the solution (38). For recovering the solution of the original BVP (4) and (5) the

reverse transformations (7) and (10) can be applied. The value of A is, however, determined post

priori only upon making use of Equation (13).The values of the various parameters of interest for

the present problem are presented in Table 9 using different techniques including the numerical

scheme described above. Excellent agreement between those techniques is obviously apparent.

An asymptotic solution for large suction

In this section, we take Ackroyd’s method to its logical conclusion by deducing an asymptotic

solution for large Z. To this end, we find it convenient to introduce the parameter

γ =a1

α2. (42)

Equation (41) can then be rewritten in terms of γ as

?1

?

a1

3γ+ 1 +1

1 +1

4γ +

7

24γ2+1

7

72γ2+

1

24γ3+

35

384γ4+

7

384γ4+

413

8640γ5+ ···

413

51840γ5+ ···

?

?

= Z, (43)

a1

2γ +

6γ3+ = −1.(44)

Downloaded by [Qasem M. Al-Mdallal] at 20:56 06 October 2011

Page 15

International Journal of Computer Mathematics

13

Table 9.

characterizing the flow.

Illustrating the variation of A, the suction parameter, α, the scaling factor of the flow, and −f??(0), a dimensionless measure of the skin-friction at the wall with Z, a parameter

Exact numericalExtended HPMAsymptotic for large Z

Za

α

−f??(0)

0.96672358

1.06850725

1.16873699

1.26643377

1.36109418

1.45252827

1.54073527

2.27918542

2.85138414

4.13100102

5.65998948

7.87518615

12.32913753

17.37825803

A

α

−f??(0)

0.96672358

1.06850725

1.16873696

1.26643377

1.36109418

1.45252827

1.54073527

2.27918542

2.85138414

4.13100102

5.65998948

7.87518615

12.32913753

17.37825803

NA

α

−f??(0)

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

1

2

5

10

20

50

100

Z → ∞

−0.46060039

−0.35978883

−0.27101669

−0.19223726

−0.12171216

−0.05801592

0

0.40400045

0.66355621

1.17713612

1.74017382

2.51952432

4.04219557

5.74482641

1.30264761

1.38970406

1.47592387

1.56057159

1.64322121

1.72366484

1.80183616

2.47524474

3.01406268

4.24759712

5.74655239

7.93800633

12.36951531

17.40696634

−0.46059972

−0.35978883

−0.27101669

−0.19223726

−0.12171216

−0.05801592

0

0.40400045

0.66355621

1.17713612

1.74017382

2.51952432

4.04219557

5.74482641

1.30264955

1.38970406

1.47592387

1.56057159

1.64322121

1.72366484

1.80183616

2.47524474

3.01406268

4.24759712

5.74655239

7.93800633

12.36951531

17.40696634

20

18

14

12

10

10

8

8

8

8

8

8

4

4

0.38593598

0.66326940

1.17713515

1.74017380

2.51952432

4.04219557

5.74482641

√(Z/3)

2.49979973

3.01439690

4.24759811

5.74655240

7.93800633

12.36951531

17.40696634

√(3Z)

2.11225346

2.84744939

4.13097433

5.65998888

7.87518613

12.32913753

17.37825803

√(3Z)

Downloaded by [Qasem M. Al-Mdallal] at 20:56 06 October 2011