An Exponential Series Method for the Solution of Free Convection Boundary Layer Flow in a Saturated Porous Medium

doi:10.4236/ajcm.2011.12010

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American Journal of Computational Mathematics, 2011, 1, 104-110

doi:10.4236/ajcm.2011.12010 Published Online June 2011 (http://www.scirp.org/journal/ajcm)

An Exponential Series Method for the Solution of Free

Convection Boundary Layer Flow in a Saturated Porous

Medium

Vishwanath B. Awati1, N. M. Bujurke2, Ramesh B. Kudenatti3

1Department of Mat hem at i cs, Maharani’s Science College for Women, Bangalore, India

2Department of Mat hem at i cs, Karnatak University, Dharwad, India

3Department of Mat hem at i cs, Bangalore Uni versi t y, Bangalore, India

E-mail: [email protected]

Received March 30, 2011; revised May 6, 2011; acce pted May 15, 2011

Abstract

Third order nonlinear ordinary differential equation, subject to appropriate boundary conditions, arising in

fluid mechanics is solved exactly using more suggestive schemes-Dirichlet series and method of stretching

variables. These methods have advantages over pure numerical methods in obtaining derived quantities ac-

curately for various values of the parameters involved at a stretch and are valid in a much larger domain

compared with classical numerical schemes.

Keywords: Boundary Layer Equations, Stretching Surface, Dirichlet Series, Powell’s Method, Stretching of

Variables.

1. Introduction

In this article we consider the effect of blowing and suc-

tion along a vertical flat plate on free convection in air or

water which are of significant interest in recent years.

Earliest study on this topic was by Eichhorn (1960) who

investigated the effects of both wall temperature and the

blowing or suction velocity with prescribed power func-

tions of distance from the leading edge. Eichhorn (1960)

shows that the similarity solutions for the problem are

possible if the exponents in the prescribed power func-

tions are related in a particular manner. Sparrow and

Celss (1961) show a perturbation method for analyzing

more general problem with arbitrary values of exponents

and these results were confirmed by Mabuchi (1960) by

an integral method.

Our aim here is to present qualitative features of the

physical problems of interest. Majority of the problems

considered here are from fluid dynamics. The residual

warm water discharged from a geothermal power plant is

disposed of through subsurface re-injection wells which

can be idealized as vertical plane surface in porous me-

dium. The buoyancy flow past bodies immersed in a

saturated porous medium have been studied by Cheng

(1977, 1977). Merkin (1978) investigated the effect of

uniform mass flux on the fr ee convection boundar y layer

on a vertical wall in a saturated porous medium. Cheng

and Minkowycz (1977) have studied similarity solution

for the case of wall temperature and suction velocity

varying as powers of x, the longitudinal distance. In all

cases, the numerical solutions have been given for se-

lected values of para meter s involved.

The third order nonlinear ordinary differential equa-

tions over an infinite interval with suction/injection pa-

rameter w

appear in various branches of physics and

engineering and are of special interest. In very few cases,

they have analytical solution. Sakiadias (1961) investi-

gated the boundary layer flow on continuous solid sur-

face with a constant speed. Erickson et al. (1966) have

studied the problem of moving surface with suction or

injection. Since the surface is flexible the filament may

be stretched during the course of ejection and so only the

surface velocity deviates without being uniform. Samuel

and Hall (1973), who investigated, the similarity solution

for laminar boundary layer on a continuous moving po-

rous surface, obtain a series with exponential terms as

their solutions. The heat and mass transfer on stretching

sheet with suction or blowing was investigated by Gupta

and Gupta (1977). Ackroyed (1978) obtains the series

solution of steady two dimensional laminar boundary

V. B. AWATI ET AL.

105

layer flows in fluids of constant density and constant

viscosity. The stretching sheet may be considered either

as an impermeable or permeable. The two dimensional

steady boundary layer flow in a permeable surface with

stretching velocity in a quiescent fluid in the presence of

suction or injection and obtain exact solution fo r specific

parameters and express the smallest entrainment velocity

corresponding to a vanishing skin friction in a closed

form was studied (Magyari and Keller, 2000).

For specific type of boundary conditions i.e ()f







Dirichlet series solution is more efficient and pro-

vides a uniformly valid solution throughout the boundary

layer flow caused by saturated porous medium at high

Raleigh numbers. Kravchenko and Yablonskii (1965)

were the first to use Dirichlet series for the solution of

third order nonlinear boundary value problem over infi-

nite range. A general discussion of the convergence of

the Dirichlet series may be found in (Riesz, 1957). The

accuracy as well as uniqueness of the solution can be

confirmed using other powerful semi-numerical schemes.

Semi-numerical methods in this category require intro-

duction of new variables, thus converting third order

equations into second order equations, whose solution

may be obtained by using power series. We also find the

approximate analytical solution by the method of

stretching variables. Sachdev et al. (2005) have analyzed

various problems from fluid dynamics of stretching sheet

using this approach and analyzed governing equations,

solution obtained are more accurate compared with ear-

lier numerical findings.

The paper is organized as follows: In Section 2 the

mathematical formulation of the proposed problem with

relevant boundary conditions is given, its exact solution

for 1

 and is also presented. Section 3 is

devoted to semi-numerical method for the solution of the

problem using Dirichlet series and in Section 4 the

method of stretching variables is used. In Section 5 de-

tailed results obtained by the nov el procedures explained

here are compared with the corresponding numerical

solutions.

1B

2. Mathematical For mulation

Case I: The boundary layer equations of momentu m and

energy corresponding to flow past a vertical plate em-

bedded in a saturated porous medium can be reduced to

the form (Cheng, 1977)



 



(1)

2ff









 











(2)

where the plate temperature and suction or injection ve-

locity are given by

and n

TTAx vax





 (3)

where





1/2







The relevant b oundary conditions are

at 0: 1,

as : 0, 0









 (4)

and w

is the non-dimensional form which is positive

for the withdrawal of fluid (suction) and negative for the

discharge of fluid (injection).

Eliminating



from (1) and (2), we get

ffff







 







 (5)

and the boundary conditions become







, , (6)



01f



0f

Case II: The momentum equation for the vapour

boundary layer derived by (Ackryod, 1978) is

32ffff

 0



 (7)

with the boundar y conditions











01f, (8)



0f

This problem describes the vapour boundary layer in-

duced by the falling motion of the condensate layer on a

cold vertical plate. The vapour is at saturation tempera-

ture and far from the plate at rest. Koh et al (1961) who

first investigated the problem have shown that the con-

densate motion is restrained somewhat by the shear

stress produced in the vapour bo undary layer.

Case III: We also consider the self-similar two-di-

mensional steady boundary layer flow induced by a per-

meable surface stretching with velocity





uUx in quiescent fluid in the presence of suc-

tion or injection with velocity



vx ax



. For

and by (Magyari and Keller, 2000)

f1m

20fff f



 





, where 2



 (9)

satisfying boundary conditions











01f, (10)



0f

The equation describing the above boundary value

problems can be put conveniently in more general third

order nonlinear differential equation of the type

20fAff f



 







 (11)

satisfying boundary con ditions are











0f1



and (12)



0f

where

, are constants, B

is always positive and

B may be positive or negative.

106 V. B. AWATI ET AL.

Under the transformation

 

1/2









 and

1/2



Equation (11) becomes

 

where

FFFF





 





，



(13)

and the boundary conditions (12) become



1/2w



, , (14)



01F



0F

For the particular case with 1

 and (

1B1





)

integrating (13) twice with respect to



, using



  , and subjected to the boundary conditions

(14), we get

 

(15)

the Equation (15) is a Riccati typ e equation, whose solu-

tion is given as



 

(/2) (/2)

fe fe

Ffe fe

 



 

















(16)

where 2



 . For Equation (13) be-

comes 0

f



22tanh/F





3. Dirichlet Series Approach

We seek Dirichlet series solution for Equation (11) sat-

isfying in the form (Kravchenko and

Yablonskii, 1965)



0f

fba













 (17)

where 0



 and 1a. Substituting (17) into (11),

we get

+ ()0

ii ii

iii

kik

iii

kik

ibae ibae

Akbbae

Bkikbbae

 









 





 













(18)

For , we have 11 . We assume 1i0ba ba 11b



and is any arbitrary parameter. We rewrite (18) for

recurrence relation to obtain coefficients as



(1)

bAkBkik











k





(19)

for . If 2, 3, 4,i1a and 11b, then the se-

ries (17) converges absolutely for any 0



 and





 , where

ln 0















(20)

and 0



 is a sufficiently small number depending on

and



. The series (17) converges absolutely and

uniformly on the half axis





 .

The series (17) contains two free parameters namely a

and



. These unkno wn parameters are d etermined from

the remaining boundary conditions (12) at 0





(0) i

ba f







 

 (21)

and 2

(0)( )1

fib







a





 (22)

The solution of transcendental Equations (21) and (22)

yield constants and a



. The solution of these tran-

scendental equations is equivalent to the unconstrained

minimization of the functional

() 1

iw i

bafi ba









 





 



 



(23)

We use Powell’s method of conjugate directions

(Press et al 1987) which is one of the most efficient

techniques for solving unconstrained optimization prob-

lems. This helps in fixing the unknowns and a



uniquely for different values of the parameters

and w

. Alternatively, Newton method is also used to

determine the unknown parameters accurately for dif-

ferent value of w

For the shear stress at the surface for the problem (11)

it is given by

(0)( )

ba i







 

 (24)

4. Method of Stretching Variables

Most of the boundary value problems over infinite inter-

vals are not amenable in obtaining analytical solution. In

such situations, it is possible to obtain approximate solu-

tion of these problems. As the governing equation is

sometimes too difficult to solve exactly, one has to ap-

proach the approximate analysis. Many approximate

methods are ad-hoc and often provide solutions for major

engineering problems and physical insight into the prob-

lems. The approximate solution considered here to these

problems, are based on the idea of stretching the vari-

ables of the flow problems. We have to choose suitable

trial velocity profile



satisfying the boundary condi-

tions automatically and later integrate

 which will

satisfy the remaining boundary conditions. Substitution

his resulting function into the given equation gives the t

V. B. AWATI ET AL.

107

Table 1. Comparison of the values of f





f0 0





of Equation (5) obtained by the dirichlet series method, method of

stretching variable and pure numerical method for different values of .

Dirichlet series









 Numerical





 Method of stretching

–4

–3

–2

–1

–0.8

–0.6

–0.4

–0.2

0.2

0.4

0.6

0.8

---

–5.81754

–4.98883

–4.26616

–3.63959

–3.09976

–2.63759

–2.24441

–1.91159

–1.63128

–1.39596

–1.19881

–0.59957

–0.33854

–0.21181

---

0.68320

0.70035

0.72103

0.74565

0.77457

0.80806

0.84637

0.88943

0.93732

0.98986

1.04683

1.38705

1.79089

2.23011

---

–0.20404

–0.24291

–0.28633

–0.33431

–0.38682

–0.44375

–0.50499

–0.57006

–0.63888

–0.71111

–0.78640

–1.19824

–1.64747

–2.11606

–0.00305

–0.01823

–0.097213

–0.20404

–0.24291

–0.28633

–0.33431

–0.38682

–0.44375

–0.50490

–0.57004

–0.63888

–0.71110

–0.78640

–1.19824

–1.64747

–2.11606

–0.08012

–0.10391

–0.14549

–0.22871

–0.25461

–0.28493

–0.32032

–0.36129

–0.40825

–0.46129

–0.52032

–0.58493

–0.65461

–0.72871

–1.14549

–1.60391

–2.08012

Table 2. Comparison of the values of for





f01 /3





of Equation (5) obtained by the Dirichlet series method, Method

of stretching variable and pure numerical method for different values of w

Dirichlet series









 Numerical







 Method of stretching

–4

–3

–2

–1

–0.8

–0.6

–0.4

–0.2

0.2

0.4

0.6

0.8

---

–7.86898

–3.96721

–3.37044

–2.85267

–2.40691

–2.02624

–1.70376

–1.43260

–1.20609

–1.01785

–0.86197

–0.73313

–0.35388

–0.19604

–0.12140

---

0.56238

0.64512

0.67926

0.71814

0.76225

0.81201

0.86775

0.92965

0.99773

1.07187

1.15179

1.23715

1.72986

2.29516

2.89895

---

–0.24571

–0.39700

–0.44153

–0.49164

–0.54760

–0.60957

–0.67765

–0.75172

–0.83161

–0.91701

–1.00753

–1.10272

–1.63357

–2.22255

–2.84146

–0.12499

–0.16640

–0.24372

–0.39700

–0.44153

–0.49164

–0.54759

–0.60958

–0.67765

–0.75172

–0.83161

–0.91701

–1.00753

–1.10274

–1.63357

–2.22255

–2.84145

–0.15738

–0.20185

–2.27614

–0.41202

–0.45136

–0.49602

–0.54654

–0.60333

–0.66667

–0.73667

–0.81320

–0.89602

–0.98469

–1.07869

–1.60948

–2.20185

–2.82405

residual of the form









which is called defect

function. Using least squares method, the residual of the

defect function can be minimized. For details see (Afzal,

1982; Ariel, 1994; Mamaloukas, 2002).

where is an amplification factor. In view of (27),

the system (25-26) are transformed to the form

0a





22 0

HAf HHBH



 



, 'd



 (28)

Using the transformation w



 into the system

(11), we get with the boundary conditions

(0) 0H



, (0) 1H



, (29) () 0H



20,

FAfFFBF

 





 (25) We choose velocity profile for general A, B and

to be of the form

and the boundary conditions become



00F,



01F



, (26)



0F exp( )H







 (30)

We introduce a stretching parameter



for both F and



in the form which satisfies the derivative conditions in (29) at 0





and



. Integrating (30) with respect to



be-

ween the limits 0 to

() ()HF





 and





 (27)



using conditions (29), we get t

V. B. AWATI ET AL.

108

Table 3. Comparison of the values of for





f01





of Equation (5) obtained by the Dirichlet series method, method of

stretching variable and pure numerical method for different values of w

Dirichlet series









 Numerical





0f Method of stretching

–4

–3

–2

–1

–0.8

–0.6

–0.4

–0.2

0.2

0.4

0.6

0.8

–17.94427

–10.90833

–5.82843

–2.61803

–2.18163

–1.80642

–1.48792

–1.22099

–1.00000

–0.81900

–0.67208

–0.55358

–0.45837

–0.38197

–0.17157

–0.09167

–0.05573

0.23607

0.30278

0.41421

0.61803

0.67703

0.74403

0.81980

0.90499

1.00000

1.10499

1.21980

1.34403

1.47703

1.61803

2.41421

3.30277

4.23607

–0.23607

–0.30278

–0.41421

–0.61803

–0.67703

–0.74403

–0.81980

–0.90499

–1.00000

–1.10499

–1.21980

–1.34403

–1.47703

–1.61803

–2.41421

–3.30277

–4.23607

–0.2360

–0.3027

–0.4142

–0.6180

–0.6770

–0.7440

–0.8198

–0.9049

–1.0000

–1.104

–1.219

–1.344

–1.477

–1.618

–2.414

–3.302

–4.236

–0.23607

–0.30277

–0.41421

–0.61803

–0.67703

–0.74403

–0.81980

–0.90499

–1.00000

–1.10499

–1.21980

–1.34403

–1.47703

–1.61803

–2.41421

–3.30278

–4.23607

Table 4. Comparison of the values of obtained by the Dirichlet series method, Method of stretching variable and

pure numerical method for different values of





Dirichlet series









 Numerical







 Method of stretching

–0.2

–0.4

–0.6

–0.8

–1.0

–0.01157

–0.02420

–0.07161

–0.36212

0.51828

–0.73429

–1.02209

–1.37560

–1.65643

9.32272

6.46670

3.80275

1.80184

1.55188

1.35370

1.20031

1.10092

1.13688

–9.269193

–6.389695

–3.672835

–1.540735

–1.256436

–1.026515

–0.847661

–0.744998

–0.590908

–9.269193

–6.389695

–3.672835

–1.540735

–1.256433

–1.026417

–0.845381

–0.745377

–0.597831

–9.25219

–6.36650

–3.64087

–1.52753

–1.25671

–1.04114

–0.87295

–0.74251

–0.64087

1 exp()H



 . (31)

Substituting (31) into (28) we get the residual defect

function







 







(,) exp2exp

1exp1 w

 



 





(32)

By using the least square s method for minimization of

error and using Euler-Lagrange equation which is sim-

plified to minimization of error in the form



,dR

 







 0

. (33)

Substituting (32) into Equation (33) and solving for



, we get

13483

23 ww

fABAf















 (34)

Thus, the final form of the solution becomes



11exp





 







. (35)

The expression (35) gives the solution of Equation (9)

for all

, and w

. It is striking that the Equation

(35) also admits analytical solutions for 1

, 1B





and w0f



; 1





 , for 1

, 1B



and

w0f



; 2ta/2nh(



)f. It is of interest to note

that, the former exact solution may also be recovered

from the method of stretching variable.

V. B. AWATI ET AL.

109

Table 5. Comparison of the values of obtained by the Dirichlet series method, Method of stretching variable and

pure numerical method for different values of







and w

Dirichlet series











 Numerical



0f Method of stretching

2 1.0

0.5

0.0

–0.33757

–0.51732

–0.81027

1.58166

1.22111

0.90564

–1.84998

–1.54028

–1.28181

–1.84989

–1.54047

–1.28215

–1.88444

–1.56498

–1.29099

1 1.0

0.5

0.0

–0.38197

–0.60961

–1.00000

1.61803

1.28078

0.99999

–1.61803

–1.28078

–1.00000

–1.61803

–1.28078

–1.00000

–1.61803

–1.28078

–1.00000

–1 1.0

0.5

0.0

–0.53591

–1.00705

–1.30794

1.73207

1.51099

1.61320

–1.00002

–0.50747

–0.41993

–1.00009

–0.50845

---

–2 1.0

0.5

0.0

–0.91332

–0.88355

–0.83355

2.12093

1.72753

1.42219

–0.37867

–0.23406

–0.15829

---

5. Numerical Results

In the present paper, we have given exact analytic solu-

tion of nonlinear boundary value problem (11) and (12)

in the form of Dirichlet series (17) and approximate so-

lution by using method of stretching variable (11). The

calculated values of representing the shear stress

at the surface associated with different parameters



0f

and w

for different sets of values of and a



are give n i n Tables 1-5.

The problem explained in (5) corresponds to





, B



 and for different values of w

. An

analytic solution in terms of series is obtained using

Dirichlet series method and also by an approximate solu-

tion method of stretching variable. Comparison of the

solution obtained is made with existing numerical solu-

tion by (Cheng, 1977) for 1

0, , 1



 and these are

given in Table 1-3. An excellent agreement between the

present computation and numerical values is achieved.

Also, the exact analytical solution for 1

 and 1B



have been recovered.

The problem mentioned in (7) corresponds to 3A



and and different values of w

2B

. The results

obtained using Dirichlet series and method of stretching

variables is given in Table 4. These results agree very

well with the pure numerical solutions.

The problem (9) corresponds to 1



and



. For specific values of 1

0, 3





and

, for which exact analytic solution of the problem

subjected to the boundary conditions (12) is given by

(Magyari and Keller, 2000). For corresponds to

1m



, 1B







for different w

, an excellent

agreement between Dirichlet series and method of

stretching variable with exact solution is achieved and

these are given in Table 3 . Also, for 1

mwhich cor-

responds to 1



, 1B







, we recover an exact

analytical solution (16).

6. Conclusions

In this article, a class of nonlinear ordinary differential

uations with relevant boundary conditions arising in

boundary layer theory has been solved using Dirichlet

series and method of stretching variables. These two

methods give a simple and efficient way to solve the

boundary value problems, particularly, when





0f



.

All the results thus obtained have been compared with

that of direct numerical solution and are rather remark-

able.

7. Acknowledgements

Authors are thankful to the referee for the valuable

comments on the earlier draft of the paper.

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