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Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1021–1024

www.elsevier.com/locate/cnsns

Short communication

An approximate analytic solution of the Blasius problem

Faiz Ahmad a,*, Wafaa H. Al-Barakati b

a Centre for Advanced Mathematics and Physics, National University of Science and Technology, EME Campus, Peshawar Road,

Rawalpindi, Pakistanb Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 15905 Jeddah 21454, Saudi Arabia

Received 19 September 2007; received in revised form 26 December 2007; accepted 31 December 2007Available online 10 January 2008

Abstract

The [4/3] Pade approximant for the derivative is modified so that the resulting expression has the required asymptoticbehavior. This gives an analytical result which represents the solution of the classical Blasius problem on the wholedomain.� 2008 Elsevier B.V. All rights reserved.

PACS: 47.15.Cb; 02.30.Mv

Keywords: Viscous flow; Blasius problem; Analytical solution; Pade approximation

1. Introduction

The two dimensional steady state laminar viscous flow over a semi-infinite flat plate is modeled by the non-linear two-point boundary value Blasius problem

1007-5

doi:10.

* CoE-m

f 000ðgÞ þ 1

2f ðgÞf 000ðgÞ ¼ 0; g P 0 ð1:1aÞ

f ð0Þ ¼ 0; f 0ð0Þ ¼ 0; f 0ð1Þ ¼ 1 ð1:1bÞ

where g and f ðgÞ are, respectively, the dimensionless coordinate and the dimensionless stream function. Bla-sius [1] found the following analytic solution for the problem

f ðgÞ ¼X1k¼0

� 1

2

� �k Akrkþ1

ð3k þ 2Þ! g3kþ2; ð1:2Þ

where A0 ¼ A1 ¼ 1 and

704/$ - see front matter � 2008 Elsevier B.V. All rights reserved.

1016/j.cnsns.2007.12.010

rresponding author. Tel.: +92 3455334211.ail address: faizmath@hotmail.com (F. Ahmad).

1022 F. Ahmad, W.H. Al-Barakati / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1021–1024

Ak ¼Xk�1

r¼0

3k � 1

3r

� �ArAk�r�1; k P 2: ð1:3Þ

In (1.2) r denotes the unknown f 00ð0Þ: In spite of the presence of ð3k þ 2Þ! in the denominator, the aboveseries converges only within a finite interval ½0; g0� where g0 � 1:8894

r , thus making it impossible to use (1.2) tofind r by applying the last condition in (1.1b) for a large g. Howarth [2] solved the Blasius problem numericallyand found r � 0:33206: Asaithambi [3] found this number correct to nine decimal positions as 0.332057336.Although the Blasius problem is almost a century old, it is still a topic of active current research [4–11].

The series (1.2) fails to converge outside a finite interval. This difficulty is inherent in several physical prob-lems governed by a nonlinear differential equation in that although a solution exists over an unboundeddomain, but a power series representation of the solution converges only within a finite interval. For suchproblems finding an analytical solution which is uniformly valid over the whole domain is of fundamentalinterest. For the Blasius problem such a solution did not exist until 1999, when Liao, in a land mark paper,published a solution by using the homotopy analysis method [5]. His 35th order solution differs from How-ath’s numerical solution [2], for g P 5, only in the fourth decimal position. However, Liao’s solution containsa large number of terms so that an explicit expression for the 35th order solution will require several pages towrite upon.

In this short note, we derive a short analytical expression for the derivative of the solution. The idea is toform a hybrid expression which takes care of f 0ðgÞ not only for small g but also when g grows very large. The½4=3� Pade approximant for f 0ðgÞ is slightly modified to the effect that the resulting expression represents thefunction on the entire domain ½0;1Þ with remarkable accuracy. The actual solution is found by a simple quad-rature. A comparison with the numerical results shows that our solution gives accurate results over the entiredomain. For large g our expression yields results in agreement with the numerical up to four decimal positions.

We remark that whereas Liao’s solution is completely analytical, ours is partly numerical in that we makeno attempt to evaluate r, but use its value as found numerically in [2,3] or analytically by Liao [5] or Ahmad[8]. Also we use the numerical result f ð7Þ ¼ 5:27924. Our emphasis is on finding a simple expression capable ofproducing accurate results over the entire domain ½0;1Þ.

2. Uniformly valid analytic solution

We start with the analytical solution (1.2) with four terms

f ðgÞ ¼ rg2

2� r2g5

240þ 11r3g8

161; 280� 5r4g11

4; 257; 792þ � � � ð2:1Þ

On differentiation, we get

f 0ðgÞ ¼ rg� r2g4

48þ 11r3g7

20; 160� 5r4g10

387; 072þ � � � ð2:2Þ

We shall use a Pade approximant to represent the above expression [12]. A Pade [4/3] approximant of (2.2)gives

f 0ðgÞ ’rgþ 3

560r2g4

1þ 11420

rg3ð2:3Þ

In the above expression let r ¼ 0:332057 and modify it by adding ag5 expðg2

4� 1Þ to the numerator as well as

the denominator. Denote the new expression by gðgÞ.We get

gðgÞ ¼0:332057gþ 0:00059069g4 þ ag5 expðg2

4� 1Þ

1þ 0:00869674g3 þ ag5 expðg2

4� 1Þ

ð2:4Þ

The above step is motivated by the idea that the function g will represent f 0 with fair amount of accuracy forsmall g and g will quickly approach unity as g becomes large. The parameter a is to be chosen in such a

F. Ahmad, W.H. Al-Barakati / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1021–1024 1023

manner that we get close agreement in sofar as possible in the transition from small g to large g. Also thechoice of the exponential function is dictated by the following heuristic argument. Eq. (1.1a) can be writtenin the form

TableCompa

g

00.40.81.21.62.02.42.83.23.64.04.44.64.85.05.25.45.65.86.06.46.87.07.48.0

1020

100

f 000ðgÞf 00ðgÞ ¼ �

1

2f ðgÞ

An integration from 0 to g gives

f 00ðgÞ ¼ r exp � 1

2

Z g

0

f ðuÞdu� �

ð2:5Þ

For large u, f 0ðuÞ ! 1, therefore, as an approximation we choose f ðuÞ ¼ u in (2.5) which givesf 00ðgÞ � r expð� 1

2g2Þ: An integration from g to 1 will give

1� f 0ðgÞ � rZ 1

ge�

12u2

du ð2:6Þ

If we express the right side in the form of an asymptotic series and keep only the first term, we obtain

f 0ðgÞ � 1þ rexpð� 1

2g2Þ

gð2:7Þ

An inspection of (2.4) shows that, for large g, gðgÞ ¼ f 0ðgÞ behaves as required by (2.7).If we choose a so that

R b0 gðgÞdg is close to the value of the exact solution at b such that gðbÞ � 1, the inte-

gral will continue to be close to the exact solution in ½b;1Þ. We arbitrarily select b ¼ 7 and find that witha ¼ 2:88� 10�6

1rison of analytical and numerical results

Approximate solution Numerical solution

0 00.0266 0.02660.1061 0.10610.2379 0.23790.4203 0.42030.6500 0.65000.9223 0.92231.2311 1.23101.5693 1.56911.9297 1.92952.3058 2.30572.6922 2.69242.8879 2.88823.0848 3.08533.2827 3.28333.4813 3.48193.6805 3.68093.8799 3.88034.0796 4.07994.2794 4.27964.6793 4.67945.0792 5.07935.2792 5.27925.6792 5.67926.2792 6.27928.2792 8.2792

18.2792 18.279298.2792 98.2792

1024 F. Ahmad, W.H. Al-Barakati / Communications in Nonlinear Science and Numerical Simulation 14 (2009) 1021–1024

Z 7

0

gðgÞdg ¼ 5:27923

while the exact value of the solution, found numerically, is 5.27924. We substitute the above value of a in (2.4)and expect that the resulting expression will provide an approximate analytical expression for the derivative ofthe solution on the entire domain ½0;1Þ. A comparison of the analytical results with the numerical in Table 1justifies this expectation.

With a ¼ 2:88� 10�6; gðgÞ becomes

gðgÞ ¼0:332057gþ 0:00059069g4 þ 0:00000288g5 expðg2

4� 1Þ

1þ 0:00869674g3 þ 0:00000288g5 expðg2

4� 1Þ

: ð2:8Þ

In Table 1, we compare the approximate resultsR g

0gðuÞdu with the ones obtained by a numerical solution of

the Blasius problem. We observe that the two results match to four decimal positions for 0 6 g 6 2:8: In thetransition stage 2:8 6 g 6 7:0 the two results agree to three decimal positions and even in this interval the max-imum error is less than two parts in ten thousand. For g P 7:0 the two results again match to four decimalpositions. In view of this, we can claim that the function

R g0

gðuÞdu represents the solution of the Blasius prob-lem on the whole domain ½0;1Þ with maximum error less than one part in five thousand.

Acknowledgement

Part of this work was done while the first author was at the King Abdulaziz University. He wishes to thankthe University for its hospitality.

References

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[10] Wang L. A new algorithm for solving classical Blasius equation. Appl Math Comput 2004;157:1–9.[11] Ahmad F. Degeneracy in the Blasius problem. Electron J Differ Equations 2007;2007(92):1–8.[12] Baker Jr GA, Graves Morris P. Pade approximants. Cambridge University Press; 1996.