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
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
f
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.
0
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
A
 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)
0f

  
(1)
10
2ff




  


 
w
(2)
where the plate temperature and suction or injection ve-
locity are given by
and n
w
TTAx vax


 (3)
where


1/2  n


The relevant b oundary conditions are
at 0: 1,
as : 0, 0
w
f
f
f





 (4)
and w
f
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
2
10
2
ffff
 

 



 (5)
and the boundary conditions become


0w
f
f

, , (6)

01f

0f
Case II: The momentum equation for the vapour
boundary layer derived by (Ackryod, 1978) is
2
32ffff
 0

 (7)
with the boundar y conditions


0w
f
f

,


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


ww
uUx in quiescent fluid in the presence of suc-
tion or injection with velocity


1
2
m
w
vx ax

. For
and by (Magyari and Keller, 2000)
0
w
f1m
20fff f

 

 , where 2
1
m
m
  (9)
satisfying boundary conditions


0w
f
f

,


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

 

 'd
d

 (11)
satisfying boundary con ditions are


0w
f
f

,


0f1

and (12)

0f
where
A
, are constants, B
A
is always positive and
B may be positive or negative.

106 V. B. AWATI ET AL.
Under the transformation
 
1/2
1
fF
A




 and
1/2
=
A
 Equation (11) becomes
 
2
0
where
FFFF
B
A


 


， 
(13)
and the boundary conditions (12) become

1/2w
F
Af  , , (14)

01F

0F
For the particular case with 1
A
 and ( 1B1


)
integrating (13) twice with respect to

, using

0w
f
f
  , and subjected to the boundary conditions
(14), we get
22
11
22
w
F
Ff
 
(15)
the Equation (15) is a Riccati typ e equation, whose solu-
tion is given as

 
 
(/2) (/2)
(/2) (/2)
ww
ww
fe fe
Ffe fe
 
 

 







 
(16)
where 2 2w
f
  . For Equation (13) be-
comes 0
w
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
1
ii
i
i
fba
Ae






 (17)
where 0

 and 1a. Substituting (17) into (11),
we get
32
11
12
21
1
21
+ ()0
ii ii
ii
ii
iii
kik
ik
iii
kik
ik
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
a

12
21
1
(1)
i
i
k
bAkBkik
ii






k
ik
bb


(19)
for . If 2, 3, 4,i1a and 11b, then the se-
ries (17) converges absolutely for any 0

 and

  , where
ln 0 a







(20)
and 0

 is a sufficiently small number depending on
and a

. 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


1
(0) i
i
iw
f
ba f
A



 
 (21)
and 2
1
(0)( )1
i
i
i
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
22
2
11
() 1
ii
iw i
ii
bafi ba
A




 


 

 

(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
A
,
and w
B
f
. Alternatively, Newton method is also used to
determine the unknown parameters accurately for dif-
ferent value of w
f
.
For the shear stress at the surface for the problem (11)
it is given by
2
1
(0)( )
i
i
i
F
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
f

satisfying the boundary condi-
tions automatically and later integrate
f
 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 or


f0 0


of Equation (5) obtained by the dirichlet series method, method of
stretching variable and pure numerical method for different values of .
w
f
Dirichlet series
w
f
a



0f

 Numerical
 0f

 Method of stretching
–4
–3
–2
–1
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
1
2
3
4
---
---
---
–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
f
.
Dirichlet series
w
f
A



0f

 Numerical


0f

 Method of stretching
–4
–3
–2
–1
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
1
2
3
4
---
---
–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

,R


 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
w
HAf HHBH 
 

, 'd
d

 (28)
Using the transformation w
f
fF

 into the system
(11), we get with the boundary conditions
(0) 0H

, (0) 1H

, (29) () 0H

20,
w
FAfFFBF
  
'd
d

 (25) We choose velocity profile for general A, B and
f
w
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
f
.
Dirichlet series
w
f
a



0f

 Numerical


0f Method of stretching
–4
–3
–2
–1
–0.8
–0.6
–0.4
–0.2
0
0.2
0.4
0.6
0.8
1
2
3
4
–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


f0
w
f
.
Dirichlet series
w
f
a



0f

 Numerical


0f

 Method of stretching
3
2
1
0
–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

,R


 

  

2
(,) exp2exp
1exp1 w
RB
Af
 

 


(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

2
0
,dR  



 0
. (33)
Substituting (32) into Equation (33) and solving for

, we get
22
13483
23 ww
A
fABAf 






 (34)
Thus, the final form of the solution becomes

11exp
w
ff


 



. (35)
The expression (35) gives the solution of Equation (9)
for all
A
, and w B
f
. It is striking that the Equation
(35) also admits analytical solutions for 1
A
, 1B


and w0f

; 1
f
e


 , for 1
A
, 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


f0

and w
f
.
Dirichlet series

w
f
a



0f

 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
A
,
and w
B
f
for different sets of values of and a

are give n i n Tables 1-5.
The problem explained in (5) corresponds to
1
2
A


, B

 and for different values of w
f
. 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
3
  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
A
 and 1B

have been recovered.
The problem mentioned in (7) corresponds to 3A

and and different values of w 2B
f
. 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
A

and
2
1
m
Bm
 . For specific values of 1
0, 3
 
and
1
2
, 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
1
A

, 1B


 for different w
f
, 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
3
mwhich cor-
responds to 1
A

, 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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