The extended homotopy perturbation method and boundary layer flow due to condensation and natural convection on a porous vertical plate.
ABSTRACT The extended homotopy perturbation method (EHPM), which is an extension to the wellknown homotopy 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. The results obtained by the EHPM are in excellent agreement with the exact numerical solution. 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.
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Article: The application of homotopy analysis method to nonlinear equations arising in heat transfer
[Show abstract] [Hide abstract]
ABSTRACT: Here, the homotopy analysis method (HAM), which is a powerful and easytouse analytic tool for nonlinear problems and dose not need small parameters in the equations, is compared with the perturbation and numerical and homotopy perturbation method (HPM) in the heat transfer filed. The homotopy analysis method contains the auxiliary parameter ℏ, which provides us with a simple way to adjust and control the convergence region of solution series.Physics Letters A 12/2006; · 1.63 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The purpose of the present paper is to introduce a method, probably for the first time, to predict the multiplicity of the solutions of nonlinear boundary value problems. This procedure can be easily applied on nonlinear ordinary differential equations with boundary conditions. This method, as will be seen, besides anticipating of multiplicity of the solutions of the nonlinear differential equations, calculates effectively the all branches of the solutions (on the condition that, there exist such solutions for the problem) analytically at the same time. In this manner, for practical use in science and engineering, this method might give new unfamiliar class of solutions which is of fundamental interest and furthermore, the proposed approach convinces to apply it on nonlinear equations by today’s powerful software programs so that it does not need tedious stages of evaluation and can be used without studying the whole theory. In fact, this technique has new point of view to wellknown powerful analytical method for nonlinear differential equations namely homotopy analysis method (HAM). Everyone familiar to HAM knows that the convergencecontroller parameter plays important role to guarantee the convergence of the solutions of nonlinear differential equations. It is shown that the convergencecontroller parameter plays a fundamental role in the prediction of multiplicity of solutions and all branches of solutions are obtained simultaneously by one initial approximation guess, one auxiliary linear operator and one auxiliary function. The validity and reliability of the method is tested by its application to some nonlinear exactly solvable differential equations which is practical in science and engineering.Communications in Nonlinear Science and Numerical Simulation 12/2010; · 2.57 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, the homotopy analysis method (HAM) is applied to numerically approximate the eigenvalues of the second and fourthorder Sturm–Liouville problems. These eigenvalues are calculated by starting the HAM algorithm with one initial guess. In this paper, it can be observed that the auxiliary parameter ℏ, which controls the convergence of the HAM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more important qualitative difference in analysis between HAM and other methods.Communications in Nonlinear Science and Numerical Simulation 01/2011; 16(1):112126. · 2.57 Impact Factor
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Thi s art i cl e w as dow nl oaded by: [Qasem M . Al M dal l al ]
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ht t p: //www. t andf onl i ne. com /l oi /gcom 20
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
Uni ver si t y , 7600 Gl over Road, Langl ey , BC, Canada, V2Y 1Y1
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
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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.AlMdallala, Muhammed I. Syama* and P. DonaldArielb
aDepartment of Mathematical Sciences, College of Science, United Arab Emirates University, PO Box
17551, AlAin, 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),whichisanextensiontothewellknownhomotopy
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 00207160 print/ISSN 10290265 online
© 2011 Taylor & Francis
DOI: 10.1080/00207160.2011.606905
http://www.informaworld.com
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2
Q.M. AlMdallal 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 oneterm
correction in the HPM solution was consequently invoked by Ariel [7,8] to get the solution of
the threedimensional 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 oneterm correction solution of the HPM is that
it cannot be readily extended to the manyterm 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
axisymmetricflowpastastretchingsheetalsoinapowerseriesofp.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 oneterm 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 twodimensional 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 xaxis. 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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4
Q.M. AlMdallal 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. AlMdallal 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 secondorder 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 10digit 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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8
Q.M. AlMdallal 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 problemdependent. For the axisymmetric 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 eightdigit accuracy solution. In
Tables 3–8, we list the values of α,−F??(0) and A for different nonzero 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 Shanks With 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
1.799880971.546077860
1.80181820 1.540784390
1.80183601 1.540735590
1.801836161.540735270
10
11
12
1.801836161.540735270
1.801836161.540735270
Downloaded by [Qasem M. AlMdallal] at 20:56 06 October 2011
Page 12
10
Q.M. AlMdallal 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 ShanksWith Shanks Without ShanksWith 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
3.01406207 2.851387530.66355637
3.014062672.851383520.66355621
3.014062682.851384140.66355621
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 ShanksWith Shanks Without 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
5.746546575.66034984 1.74017560
5.74655238 5.659989631.74017382
5.746552395.659989351.74017381
5.746552395.659989481.74017382
10
11
12
5.74655239 5.659989481.74017382
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 Shanks Without Shanks With 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
17.4069663117.37827176 5.74482642
17.40696633 17.378258035.74482641
17.4069663317.378258035.74482641
17.4069663417.378258035.74482641
10
11
12
17.4069663417.37825803 5.74482641
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 Shanks With ShanksWithout ShanksWith Shanks Without 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 Shanks Without ShanksWith Shanks Without 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
1.28104530 0.97011303
1.275092970.96685039
1.297509720.96664002
1.29696173 0.96663691
10
11
12
1.30260786 0.96662486
1.302711200.96675705
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 eightdecimal 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. AlMdallal 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)
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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 skinfriction 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. AlMdallal] at 20:56 06 October 2011