An approximate analytic solution of the Blasius problem

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Short communication

An approximate analytic solution of the Blasius problem

Faiz Ahmad

^a,*

, Wafaa H. Al-Barakati

^b

aCentre for Advanced Mathematics and Physics, National University of Science and Technology, EME Campus, Peshawar Road, Rawalpindi, Pakistan

bDepartment 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 2007

Abstract

The [4/3] Pade approximant for the derivative is modiﬁed so that the resulting expression has the required asymptotic behavior. This gives an analytical result which represents the solution of the classical Blasius problem on the whole domain.

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 nonlinear two-point boundary valueBlasius problem

f⁰⁰⁰ðgÞ þ1

2fðgÞf⁰⁰⁰ðgÞ ¼0; gP0 ð1:1aÞ

fð0Þ ¼0; f⁰ð0Þ ¼0; f⁰ð1Þ ¼1 ð1:1bÞ

wheregandfðgÞare, respectively, the dimensionless coordinate and the dimensionless stream function. Bla- sius[1]found the following analytic solution for the problem

fðgÞ ¼X¹

k¼0

1 2 k

A_kr^kþ1

ð3kþ2Þ!g^3kþ2; ð1:2Þ

whereA₀¼A₁¼1 and

doi:10.1016/j.cnsns.2007.12.010

* Corresponding author. Tel.: +92 3455334211.

E-mail address:[email protected](F. Ahmad).

Available online at www.sciencedirect.com

Communications in Nonlinear Science and Numerical Simulation xxx (2008) xxx–xxx

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Please cite this article in press as: Ahmad F, Al-Barakati WH, An approximate analytic solution of the Blasius problem, Commun Nonlinear Sci Numer Simul (2008), doi:10.1016/j.cnsns.2007.12.010

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A_k ¼X^k1

r¼0

3k1 3r

A_rA_kr1; kP2: ð1:3Þ

In(1.2)rdenotes the unknown f⁰⁰ð0Þ:In spite of the presence ofð3kþ2Þ!in the denominator, the above series converges only within a ﬁnite interval½0;g₀whereg₀^1:8894_r , thus making it impossible to use(1.2)to ﬁndrby applying the last condition in(1.1b)for a largeg. Howarth[2]solved the Blasius problem numerically and foundr0:33206: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 problems governed by a nonlinear differential equation in that although a solution exists over an unbounded domain, but a power series representation of the solution converges only within a finite interval. For such problems finding an analyticalsolution which is uniformly valid over the whole domain is of fundamental interest. 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], forgP5, only in the fourth decimal position. However, Liao’s solution contains a large number of terms so that an explicit expression for the 35th order solution will require several pages to write upon.

In this short note, we derive ashortanalytical expression for the derivative of the solution. The idea is to form a hybrid expression which takes care off⁰ðgÞnot only for smallgbut also whenggrows very large. The

½4=3Pade approximant forf⁰ðgÞis slightly modiﬁed to the eﬀect that the resulting expression represents the function on the entire domain½0;1Þwith 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 entire domain. For largegour 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 make no attempt toevaluater, but use its value as found numerically in[2,3]or analytically by Liao[5]or Ahmad [8]. Also we use the numerical resultfð7Þ ¼5:27924. Our emphasis is on ﬁnding a simple expression capable of producing accurate results over the entire domain½0;1Þ.

2. Uniformly valid analytic solution

We start with the analytical solution(1.2)with four terms fðgÞ ¼rg²

2 r²g⁵

240þ 11r³g⁸

161;280 5r⁴g¹¹

4;257;792þ ð2:1Þ

On diﬀerentiation, we get f⁰ðgÞ ¼rgr²g⁴

48 þ11r³g⁷

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

f⁰ðgÞ ’rgþ₅₆₀³ r²g⁴

1þ₄₂₀¹¹rg³ ð2:3Þ

In the above expression letr¼0:332057 and modify it by addingag⁵expð^g₄²1Þto the numerator as well as the denominator. Denote the new expression bygðgÞ.

We get

gðgÞ ¼0:332057gþ0:00059069g⁴þag⁵expð^g₄²1Þ

1þ0:00869674g³þag⁵expð^g₄²1Þ ð2:4Þ

The above step is motivated by the idea that the functiongwill representf⁰with fair amount of accuracy for smallg and g will quickly approach unity as g becomes large. The parameter a is to be chosen in such a

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

f⁰⁰⁰ðgÞ f⁰⁰ðgÞ ¼ 1

2fðgÞ

An integration from 0 tog gives f⁰⁰ðgÞ ¼rexp 1

2 Z g

0

fðuÞdu

ð2:5Þ For large u, f⁰ðuÞ !1, therefore, as an approximation we choose fðuÞ ¼u in (2.5) which gives f⁰⁰ðgÞ rexpð¹₂g²Þ:An integration fromg to1will give

1f⁰ðgÞ r Z 1

g

e¹²^u²du ð2:6Þ

If we express the right side in the form of an asymptotic series and keep only the ﬁrst term, we obtain f⁰ðgÞ 1þrexpð¹₂g²Þ

g ð2:7Þ

An inspection of(2.4)shows that, for largeg,gðgÞ ¼f⁰ðgÞbehaves as required by(2.7).

If we chooseaso thatRb

0gðgÞdgis close to the value of the exact solution atbsuch thatgðbÞ 1, the inte- gral will continue to be close to the exact solution in ½b;1Þ. We arbitrarily selectb¼7 and ﬁnd that with a¼2:8810⁶

Table 1

Comparison of analytical and numerical results

g Approximate solution Numerical solution

0 0 0

0.4 0.0266 0.0266

0.8 0.1061 0.1061

1.2 0.2379 0.2379

1.6 0.4203 0.4203

2.0 0.6500 0.6500

2.4 0.9223 0.9223

2.8 1.2311 1.2310

3.2 1.5693 1.5691

3.6 1.9297 1.9295

4.0 2.3058 2.3057

4.4 2.6922 2.6924

4.6 2.8879 2.8882

4.8 3.0848 3.0853

5.0 3.2827 3.2833

5.2 3.4813 3.4819

5.4 3.6805 3.6809

5.6 3.8799 3.8803

5.8 4.0796 4.0799

6.0 4.2794 4.2796

6.4 4.6793 4.6794

6.8 5.0792 5.0793

7.0 5.2792 5.2792

7.4 5.6792 5.6792

8.0 6.2792 6.2792

10 8.2792 8.2792

20 18.2792 18.2792

100 98.2792 98.2792

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Z 7 0

gðgÞdg¼5:27923

while the exact value of the solution, found numerically, is 5.27924. We substitute the above value ofain(2.4) and expect that the resulting expression will provide an approximate analytical expression for the derivative of the solution on theentiredomain½0;1Þ. A comparison of the analytical results with the numerical inTable 1 justiﬁes this expectation.

Witha¼2:8810⁶;gðgÞbecomes

gðgÞ ¼0:332057gþ0:00059069g⁴þ0:00000288g⁵expð^g₄²1Þ

1þ0:00869674g³þ0:00000288g⁵expð^g₄²1Þ : ð2:8Þ

InTable 1, we compare the approximate resultsRg

0 gðuÞduwith the ones obtained by a numerical solution of the Blasius problem. We observe that the two results match to four decimal positions for 06g62:8:In the transition stage2:86g67:0 the two results agree to three decimal positions and even in this interval the maximum error is less than two parts in ten thousand. ForgP7:0 the two results again match to four decimal positions. In view of this, we can claim that the functionRg

0gðuÞdurepresents the solution of the Blasius problem on thewhole domain½0;1Þwith 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 thank the University for its hospitality.

References

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[12] Baker Jr GA, Graves Morris P. Pade approximants. Cambridge University Press; 1996.

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