close

Вход

Забыли?

вход по аккаунту

?

Newapplications of approximate methods in fluid mechanics.

код для вставкиСкачать
ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2011; 6: 669–678
Published online 28 June 2010 in Wiley Online Library
(wileyonlinelibrary.com) DOI:10.1002/apj.472
Research article
New applications of approximate methods in fluid
mechanics
Z. Ziabakhsh,1 * G. Domairry,2 * Ammar Domiri2 and S. M. Moghimi3
1
IOR/EOR Research Institute, National Iranian Oil Company (NIOC), Tehran, Iran
Department of Mechanical Engineering, Babol University of Technology, PO Box 484, Babol, Iran
3
Department of Mechanical Engineering, Islamic Azad University, Qaemshahr branch, Qaemshahr, Iran
2
Received 22 February 2010; Revised 2 May 2010; Accepted 19 May 2010
ABSTRACT: In this paper, we have modeled boundary layer flows induced by continuous stretched surfaces by
implementing one of the newest analytical methods of solving nonlinear differential equations called homotopy analysis
method (HAM), which gives us a vast freedom to choose the answer type. We have used an iterating analytical method
to cope with the nonlinearity. A new adapting boundary condition is proposed in this work that is based on an initial
guess and then it is developed to the solution expression. The analytic results are compared with the numerical solution
(NS) and the comparison reveals that a good agreement exists between the NS and HAM solution. Also the convergence
of the obtained HAM solution is discussed explicitly. The obtained approximate solutions are valid for all values of
the dimensionless parameter β, as it is shown later in the paper.  2010 Curtin University of Technology and John
Wiley & Sons, Ltd.
KEYWORDS: homotopy analysis method; boundary layer flows; numerical solution; continuous stretched surfaces;
heat transfer; nonlinear problem
INTRODUCTION
The heat transfer over an unsteady stretching permeable
surface with prescribed wall temperature has considered
by many investigators throughout its applications in
engineering practice, particularly in chemical industries.
Some common examples are the cases of boundary layer
control, transpiration cooling and gaseous diffusion.
Crane[1] has presented an exact analytical solution
for the steady two-dimensional flow due to a stretching
surface in a quiescent fluid; additionally many authors
such as Ishak et al .,[2] Gupta et al .[3] and Grubka
et al .[4] have considered various aspects of this problem
and achieved similarity solutions.
It is necessary to mention that through above referred
studies which discussed stretching surfaces, the flows
were assumed to be steady. Unsteady flows due to
stretching surfaces have received less heed; a few of
them are those considered by Devi et al .,[5] Andersson et al .,[6] Nazar et al .,[7] and very recently Ali
*Correspondence to: Z. Ziabakhsh, IOR/EOR Research Institute,
National Iranian Oil Company (NIOC), Tehran, Iran.
E-mail: ziagandji@yahoo.com
G. Domairry, Department of Mechanical Engineering, Babol
University of Technology, PO Box 484, Babol, Iran.
E-mail: amirganga111@yahoo.com
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Curtin University is a trademark of Curtin University of Technology
and Mehmood[8 – 9] introduced a similar transformation
which was used by Williams and Rhyne,[10] in which it
transforms the governing partial differential equations
with three independent variables in two independent
variables, which are more convenient for numerical
computations. In addition, S. Nadeem et al .[11] considered unsteady shrinking sheet through porous medium
with variable viscosity.
In this paper, we explore the analytic solution of
the nonlinear heat transfer over an unsteady stretching
permeable surface with prescribed wall temperature and
will make a comparison between the homotopy analysis
method (HAM) and numerical results given by Magyari
et al .[12]
HAM which was recently developed by Liao[13] is
one of the most successful and efficient methods in
solving nonlinear equations. In comparison with previous analytic techniques, the following merits can be
mentioned for HAM:[14] first and foremost, unlike all
previous analytic techniques, the HAM provides us
great freedom to express solutions of a given nonlinear
problem by means of different base functions. Secondly, unlike all previous analytic techniques, the HAM
always provides us with a family of solution expressions in the auxiliary parameter h̄, even if a nonlinear
670
Z. ZIABAKHSH et al.
problem has a unique solution. Thirdly, unlike perturbation techniques, the HAM is independent through any
small or large quantities. So, the HAM can be applied
no matter if governing equations and boundary/initial
conditions of a given nonlinear problem contain small
or large quantities or not. Finally, in the works of previous authors such as Hayat et al .[15] and Liao,[16] it has
also shown that the HAM method logically contains
some previous techniques such as Adomian’s decomposition method, Lyapunov’s artificial small parameter method and the δ-expansion method. Many authors
such as Liao[17] have applied HAM in solving permeable and impulsively stretched plate; Hayat et al .[18]
considered Magnetohydrodynamics (MHD) flow of a
micropolar fluid near a stagnation-point toward a nonlinear stretching surface by HAM. Unsteady boundary
layer flow adjacent to permeable stretching surface in a
porous medium by HAM was considered by Mehmood
et al .[19]
Domairry and Nadim,[20] Nadeem et al .,[21] Abbasbandy[22] and Ziabakhsh et al .[23 – 24] and others[25 – 33]
have successfully applied HAM in solving different
types of nonlinear problems i.e. coupled, decoupled,
homogeneous and nonhomogeneous equations arising
in different physical problems such as heat transfer,
fluid flow, oscillatory systems, etc.
Above all, there are no rigorous theories to direct
us to choose the initial approximations, auxiliary linear
operators, auxiliary functions and auxiliary parameter h̄.
From practical viewpoints, there are some fundamental
rules such as the rule of solution expression, the rule of
coefficient ergodicity and the rule of solution existence,
which play important roles within the HAM. Unfortunately, the rule of solution expression implies such an
assumption that we should have, more or less, some
knowledge about a given nonlinear problem a priori.
So, theoretically, this assumption impairs the HAM,
although we can always attempt some base functions
even if a given nonlinear problem is completely new
for us. Therefore in the present work, we reexamine
the heat transfer characteristics of boundary layer flows
induced by continuous stretched surfaces, and attempt
to obtain its solution using the HAM.
DESCRIPTION OF THE PROBLEM
As the stretching sheet interacts with the surrounding
fluid both thermally and mechanically, different sets
of prescribed boundary mechanical and thermal conditions can be considered. Consider the steady velocity
and thermal boundary layers induced by a continuous
(in general nonisothermal) stretching surface moving
through a quiescent incompressible fluid of constant
temperature T∞ (Magyari and Keller[12] ). As shown in
Fig. 1 with the usual boundary layer approximations,
the governing equations of momentum, energy balance
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pacific Journal of Chemical Engineering
Figure 1. Schematic diagram of the problem under
consideration.
and mass transfer are as follows:
∂u
∂v
+
=0
∂x
∂y
∂u
∂u
u
+v
=υ
∂x
∂y
u
(1)
∂ 2u
∂y 2
∂T
∂T
∂ 2T
+υ
=α 2
∂x
∂y
∂y
(2)
(3)
Accordingly to stretching velocity and surface temperature distribution (uw , Tw ), the boundary condition
is
u(x , 0) = uw (x ), v (x , 0) = 0, u(x , ∞) = 0 (4)
T (x , 0) = Tw (x ), T (x , ∞) = T∞ = constant (5)
Equations 1–3 lead to an independent flow boundary value problem and a forced thermal convection
problem. In terms of the stream function ψ = ψ(x , y)
defined by u = ∂ψ/∂y, v = −∂ψ/∂x . Equations 1–3
is reduced to:
∂ψ ∂ 2 ψ
∂ 3ψ
∂ψ ∂ 2 ψ
−
=
v
∂y ∂x ∂y
∂x ∂y 2
∂y 3
(6)
∂ψ ∂T
∂ψ ∂T
∂ 2T
−
=α 2
∂y ∂x
∂x ∂y
∂y
(7)
By using the transformation as described by Schlichting and Gersten[34]
ψ(x , y) = A(x )f (x ), η = B (x )y, T (x , y) = T∞
+ C (x )θ (η)
(8)
Asia-Pac. J. Chem. Eng. 2011; 6: 669–678
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
FLUID MECHANICS
where A(x ) = const and B (x ) > 0, and by using similarity solution
A(x ) = A0 x m+1/2 , B (x ) = B0 x m−1/2 ,
m = −1, C (x ) = C0 .x n
(9)
where A0 , B0 , n and m are constant. With the power
low similarity (Eqn 9) and the boundary conditions
(Eqns 4 and 5), the quantities of physical become:
ψ(x , y) = A0 .x
m+1/2
f (η), η = B0 x
in which ci (i = 1 − 5) are constants. Let P ∈ [0, 1]
denotes the embedding parameter and h̄ indicates
nonzero auxiliary parameters. We then construct the following equations:
m−1/2
The zeroth-order deformation equations
(1 − P )L1 [f (η; p) − f0 (η)] = p h̄ 1 N1 [f (η; p), θ (η; p)]
y,
(21)
m = −1, B0 > 0, T (x , y) = T∞ + C0 x θ (η)
n
f (0; p) = 0;
(10)
f (0; p) = 1;
f (∞; p) = 0
(22)
(1 − P )L2 [θ (η; p) − θ0 (η)] = p h̄ 2 N2 [θ (η; p), f (η; p)]
m u(x , y) = A0 B0 x f (η)
m −1 m−1/2 m + 1
v (x , y) = −A0 x
f (η) +
ηf (η)
2
2
(11)
(23)
θ (0; p) = 1;
N1 [f (η; p), θ (η; p)] =
m uw (x ) = A0 B0 x f (0), v (x , ∞)
m + 1 m−1/2
x
f∞
= −A0
2
Tw (x ) = T∞ + C0 x n θ (0)
(12)
(13)
The balance Eqns 6–7 are reduced to the ordinary
differential equations for f (η) and θ (η). Then we
have
bf (η) + f (η)f (η) − βf (η)2 = 0
b θ (η) + f (η)θ (η) − γ f (η)θ (η) = 0
Pr
(14)
(15)
2vB0 , β = 2m , γ = 2n and
where b =
m +1
m +1
(m + 1)A0
Pr = v /α is the Prandtl number.
The boundary conditions are
f (0) = 0, f (0) = 1, f (∞) = 0
(16)
θ (0) = 1, θ (∞) = 0
(17)
θ (∞; p) = 0
−β
df (η; p)
dη
L1 (f ) = f
+f ,
=0
L2 (θ ) = θ + θ L2 (c4 exp(−η) + c5 ) = 0
When p increases from 0 to 1, then f (η; p) and
θ (η; p) vary from f0 (η) and θ0 (η) to f (η) and θ (η).
By Taylor’s theorem and using Eqns 27 and 28, f (η; p)
and θ (η; p) can be expanded in a power series of p as
follows:
∞
f (η; p) = f0 (η) +
fm (η)p m , fm (η) =
m−1
1 ∂ m (f (η; p))
m!
∂p m
(29)
∞
θm (η)p m , θm (η) =
1 ∂ m (θ (η; p))
m!
∂p m
(30)
where h̄ is chosen in such a way that these two series are
convergent at p = 1, therefore we have through Eqns 29
and 30 that
∞
m−1
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
(26)
(28)
f (η) = f0 (η) +
(20)
1 d 2 θ (η; p)
dθ (η; p)
+ f (η; p)
Pr dη2
dη
θ (η; 0) = θ0 (η) θ (η; 1) = θ (η)
(19)
L1 (c3 exp(−η) + c2 η + c3 ) = 0,
(25)
(27)
m−1
θ0 (η) = exp(−η) (18)
dη2
f (η; 0) = f0 (η) f (η; 1) = f (η)
APPLICATIONS OF HAM
d 2 f (η; p)
For p = 0 and p = 1, we have
θ (η; p) = θ0 (η) +
f0 (η) = 1 − exp(−η),
+ f (η; p)
df (η; p)
=0
dη
For special case we take b = 1 in Eqns14 and 15 and
solve them by HAM.
For HAM solutions, we choose the initial guesses and
auxiliary linear operators in the following form:
dη3
2
N2 [f (η; p), θ (η; p)] =
− γ θ (η; p)
d 3 f (η; p)
(24)
fm (η), θ (η) = θ0 (η) +
∞
θm (η)
m−1
(31)
Asia-Pac. J. Chem. Eng. 2011; 6: 669–678
DOI: 10.1002/apj
671
672
Z. ZIABAKHSH et al.
Asia-Pacific Journal of Chemical Engineering
The mth-order deformation equations
L1 [fm (η) − χm fm−1 (η)] = h̄ 1 Rmf (η)
fm (0) = 0;
fm (0)
= 0;
L2 [θm (η) − χm θm−1 (η)]
θm (0) = 0;
fm (∞) =
= h̄ 2 Rmθ (η)
(32)
0
(33)
(34)
θm (∞) = 0
Rmf (η) = f m−1 +
m−1
n=0
fn fm−1−n
−β
(35)
m−1
fn fm−1−n
n=0
(36)
Rmθ (η) =
m−1
m−1
1 θm−1 +
fm−1−n θn − γ
fm−1−n
θn
Pr
n=0
n=0
(37)
We have found the solution by Maple analytic solution device.
Figure 2. The h̄1 -validity for f (0) when Pr = 1, γ = 0
and β = 3.
CONVERGENCE OF THE HAM SOLUTION
As mentioned by Liao,[14] HAM provides us with great
freedom in choosing the solution of a nonlinear problem
by different base functions. This has a great effect on
the convergence region because the convergence region
and rate of a series are chiefly determined by the base
functions used to express the solution. Therefore, we
can approximate a nonlinear problem more efficiently
by choosing a proper set of base functions and ensure
its convergency. On the other hand, as pointed out by
Liao, the convergence and rate of approximation for
the HAM solution strongly depends on the value of
auxiliary parameter h̄. Even if the initial approximation
f0 (η) and θ0 (η) and the auxiliary linear operator L, are
given, we still have great freedom to choose the value
of the auxiliary parameter h̄. So, the auxiliary parameter
h̄ provides us with an additional way to conveniently
adjust and control the convergence region and rate of
solution series. By means of the so-called h̄-curves, it is
easy to find out the so-called valid regions of h̄ to gain a
convergent solution series. When the valid region of h̄ is
a horizontal line segment then the solution (converges)
is converged. In our case study, according to Figs 2
and 3, the convergence and rate of approximation for
the HAM solution strongly depend on the values of
auxiliary parameters h̄ 1 and h̄ 2 .
According to Figs 2 and 3 for Pr = 1, γ = 0 and β =
3 the ranges for values of f (0) and θ (0) are −0.4 <
h̄ 1 < −0.15 and −1.6 < h̄ 2 < −0.4, respectively.
Figures 4 and 5 show how auxiliary parameters h̄ 1
and h̄ 2 varies with changing β. If β increases, then the
range of convergency of solution is restricted and then
decreased. As well as according to Figs 6 and 7 the
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Figure 3. The h̄2 -validity for θ (0) when Pr = 1, γ = 0
and β = 3.
range of convergency is restricted with increasing in
Pr and β for θ (0). Finally, Figs 8 and 9 illustrate the
effect of parameter γ on convergency range; at these
figures by increasing of γ the range of convergency is
decreased.
NUMERICAL METHOD (NM)
The best approximate for solving Eqns 14 and 15 that
can be used is fourth order Runge–Kutta method.
Asia-Pac. J. Chem. Eng. 2011; 6: 669–678
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
Figure 4. The h̄1 -validity of f (0) for various value of β
when γ = 1, Pr = 1 and 11th order of approximation.
FLUID MECHANICS
Figure 6. The h̄2 -validity of θ (0) for various value of β
when γ = 5, Pr = 1.2 and 11th order of approximation.
Figure 5. The h̄2 -validity of θ (0) for various value of β
when γ = 1, Pr = 1 and 11th order of approximation.
It is often utilized to solve differential equation systems. Third-order differential equations can be usually changed into second-order equations and then first
order. After that, it can be solved through Runge–Kutta
method.
RESULT AND DISCUSSION
Figures 10–12 illustrate effect of β on the nondimensional boundary layer flow f (η), its derivative f (η)
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Figure 7. The h̄2 -validity of θ (0) for various value of Pr
when γ = 1, β = 2 and 11th order of approximation.
and temperature θ (η) for Pr = 1 and γ = 0 as well.
Figure 13 shows the result of θ (η) for various β when
Pr = 1 and γ = 1. Figures 14 and 15 show the result of
θ (η) and f (η) for various β when Pr = 1.2 and γ = 5,
respectively. Figures 13 and 14 illustrate the decrement
of the velocity and temperature profiles as β increases.
According to Figs 16 and 17 by increasing γ ,
the nondimensional temperature θ (η) decreases. By
Asia-Pac. J. Chem. Eng. 2011; 6: 669–678
DOI: 10.1002/apj
673
674
Z. ZIABAKHSH et al.
Figure 8. The h̄2 -validity of θ (0) for various value of γ
when Pr = 0.2, β = 5 and 11th order of approximation.
Asia-Pacific Journal of Chemical Engineering
Figure 10. The result of f (η) for various β when Pr = 1
and γ = 0.
Figure 11. The result of f (η) for various β when Pr = 1
and γ = 0.
Figure 9. The h̄2 -validity of θ (0) for various value of γ
when Pr = 1, β = 1.5 and 11th order of approximation.
increasing Pr in Figs 18 and 19, the nondimensional
temperature θ (η) decreases.
As shown in Figs 20 and 21, it has been attempted
to show the accuracy, capabilities and wide-range
applications of the HAM in comparison with the
numerical solution (NS) of heat transfer characteristics
of boundary layer flows induced by continuous stretched
surfaces. Figure 20 shows the boundary layer flow
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
f (η), and Fig. 21 indicates the comparison of the
nondimensional temperature θ (η) and NS for known
values of the parameters Pr = 1, γ = 1 and β = 0.
According to the Tables 1–3 these approximate solutions are in excellent agreement with the corresponding NSs.
CONCLUSIONS
In the present work, we have applied the HAM to
compute the heat transfer characteristics of boundary
Asia-Pac. J. Chem. Eng. 2011; 6: 669–678
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
Figure 12. The result of θ(η) for various β when Pr = 1
and γ = 0.
Figure 13. The result of θ(η) for various β when Pr = 1
and γ = 1.
layer flows induced by continuous stretched surfaces.
The convergency of the solution series and the power
of HAM in controlling and adjusting the convergence
region and rate of solution series was discussed. When
compared with other analytic methods, it is clear that
HAM provides highly accurate analytic solutions for
nonlinear problems.
The results show that:
1. HAM can give much better approximations for
nonlinear differential equations than do the previous solutions including perturbation method in the
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
FLUID MECHANICS
Figure 14. The result of θ(η) for various β when Pr = 1.2
and γ = 5.
Figure 15. The result of f (η) for various β when Pr = 1.2
and γ = 5.
heat transfer characteristics of boundary layer flows
induced by continuously stretched surfaces problem.
2. The proper range of the auxiliary parameters h̄ 1
and h̄ 2 to ensure the convergency of the solution series were obtained through the so-called
h̄-curves. In HAM, the auxiliary parameter h̄ provides us with a convenient way to adjust and
control the convergence and its rate for the solutions
series.
Asia-Pac. J. Chem. Eng. 2011; 6: 669–678
DOI: 10.1002/apj
675
676
Z. ZIABAKHSH et al.
Figure 16. The result of θ(η) for various γ when Pr = 0.2
and β = 5.
Figure 17. The result of θ(η) for various γ when Pr = 1 and
β = 1.5.
3. Solutions of HAM can be expressed with different
functions and therefore they can be originated from
the nature of the problems.
4. The boundary conditions can be applied not only
on the initial guess but also on other terms as
well.
5. In analytic approach, the phenomena with same
physical manners have the same solution
expressions.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pacific Journal of Chemical Engineering
Figure 18. The result of θ(η) for various Pr when β = 2 and
γ = 1.
Figure 19. The result of θ(η) for various Pr when β = 3 and
γ = 3.
6. It is necessary to determine the validity space of
auxiliary parameters to the convergence of the results
and the assumption must be alike for all coupled
equations.
7. The solutions, though restricted to the first-order
expansion, are quite elegant and fully acceptable in
accuracy. Governing equations are easily solved by
the approximate method.
Asia-Pac. J. Chem. Eng. 2011; 6: 669–678
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
FLUID MECHANICS
Table 1. The results of HAM and NS for f (0) and θ (0) when γ = 1 and Pr = 1.
f (0)
β
0.0
0.5
1.0
1.5
2.0
θ (0)
h̄ 1
h̄ 2
NM
HAM
NM
HAM
−0.5
−0.5
−0.5
−0.5
−0.4
−0.6
−0.6
−0.6
−0.6
−0.5
−0.627563
−0.829954
−1.000000
−1.148601
−1.281816
−0.627553
−0.829941
−1.000000
−1.148594
−1.281746
−1.062324
−1.028183
−1.000000
−0.975917
−0.954811
−1.062326
−1.028180
−1.000000
−0.975902
−0.954762
Figure 20. The comparison between HAM solution and NS
for f (η) and f (η) that solid lines are HAM solutions and
symbols are NS when Pr = 1, h̄1 = −0.5, β = 0 and γ = 1.
Table 2. The results of HAM and NS for θ (0) when
γ = 1 and β = 0.
θ (0)
Figure 21. The comparison between HAM solution and NS
for θ(η) that solid line is HAM solution and symbol is NS
when Pr = 1, h̄1 = −0.5, h̄2 = −0.6, β = 0 and γ = 1.
Table 3. The results of HAM and NS for θ (0) when Pr
= 1 and β = 0.
θ (0)
γ
Pr
h̄ 1
h̄ 2
NM
HAM
0.5
1.5
2.0
3.0
5.0
−0.4
−0.4
−0.4
−0.4
−0.4
−0.5
−0.5
−0.6
−0.8
−1.6
−0.682933
−1.351373
−1.593483
−1.997453
−2.634798
−0.682150
−1.351726
−1.593503
−1.994660
−2.633785
Consequently, these equations are solved by utilizing
MAPLE 10, mathematical software, whose results are
given in following figures and tables.
Acknowledgements
The authors are grateful to the reviewers for their
helpful and useful comments and suggestions.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
0.0
1.0
2.0
3.0
4.0
5.0
h̄ 1
h̄ 2
NM
HAM
−0.5
−0.5
−0.5
−0.5
−0.5
−0.5
−0.6
−0.6
−0.6
−0.6
−0.6
−0.6
−0.627563
−1.062324
−1.403498
−1.689876
−1.940194
−2.164833
−0.627559
−1.062324
−1.403471
−1.689899
−1.940223
−2.164841
NOMENCLATURE
A(x )
A0
B (x )
B0
b
C0
f
Scaling function
Constant parameter
Scaling function
Constant parameter
Constant parameter
Constant parameter
Dimensionless stream function
Asia-Pac. J. Chem. Eng. 2011; 6: 669–678
DOI: 10.1002/apj
677
678
Z. ZIABAKHSH et al.
h̄ 1 , h̄ 2
k
L1 , L2
m
n
N1 , N2
Pr
T
u
v
x
y
Asia-Pacific Journal of Chemical Engineering
Auxiliary parameters
Thermal conductivity
Linear operator of HAM
Stretching exponent
Temperature exponent
Nonlinear operator
Prandtl parameter
Temperature
Dimensional longitudinal velocity
Dimensional transversal velocity
Dimensional wall coordinate
Dimensional transversal coordinate
Greek symbols
α
β
γ
η
θ
υ
ψ
Thermal diffusivity
Constant parameter
Constant parameter
Dimensionless similarity variable
Dimensionless temperature
Kinematic viscosity
Stream function
Subscripts
w
∞
Wall conditions
Conditions at infinity
REFERENCES
[1] L.J. Crane. Z. Angew. Math. Phys., 1970; 21, 645–647.
[2] A. Ishak, R. Nazar, I. Pop. Nonlinear Anal. Real World Appl.,
2909; 10(5), 2909–2913.
[3] P.S. Gupta, A.S. Gupta. Can. J. Chem. Eng., 1977; 55,
744–749.
[4] L.J. Grubka, K.M. Bobba. ASME J. Heat Transf., 1985; 107,
248–250.
[5] C.D.S. Devi, H.S. Takhar, G. Nath. J. Heat Mass Transf.,
1991; 26, 71–79.
[6] H.I. Andersson, J.B. Aarseth, B.S. Dandapat. Int. J. Heat Mass
Transf., 2000; 43, 69–74.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
[7] R. Nazar, N. Amin, I. Pop. Mech. Res. Commun., 2004; 31,
121–128.
[8] A. Ali, A. Mehmood. Commun. Nonlinear Sci. Numer. Simul.,
2008; 13(2), 340–349.
[9] A. Ali, A. Mehmood, T. Shah. Commun. Nonlinear Sci.
Numer. Simul., 2008; 13(5), 902–912.
[10] J.C. Williams, T.B. Rhyne. SIAM J. Appl. Math., 1980; 38,
215–224.
[11] S. Nadeem, M. Awais. Phys. Lett. A, 2008; 372(30),
4965–4972.
[12] E. Magyari, B. Keller. Heat Mass Transf., 2006; 42, 679–687.
[13] S.J. Liao. Proposed homotopy analysis techniques for the
solution of nonlinear problems, PhD dissertation, Shanghai
Jiao Tong University, China, 1992.
[14] S.J. Liao. Beyond Perturbation: Introduction to Homotopy
Analysis Method, Chapman & Hall/CRC Press: Boca Raton,
2003.
[15] T. Hayat, Z. Abbas, M. Sajid. Comput. Fluid Dyn., 2006; 20,
229–238.
[16] S.J. Liao. Int. J. Heat Mass Transf., 2005; 48, 2529–2539.
[17] S.J. Liao. Commun. Nonlinear Sci. Numer. Simul., 2006; 11(3),
326–339.
[18] T. Hayat, T. Javed, Z. Abbas. Nonlinear Anal. Real World
Appl., 2009; 10(3), 1514–1526.
[19] A. Mehmood, A. Ali, T. Shah. Can. J. Phys., 2008; 86,
1079–1082.
[20] N. Nadim, G. Domairry. Energy Conver. Manage., 2009;
50(4), 1056–1061.
[21] S. Nadeem, T. Hayat, S. Abbasbandy, M. Ali. Nonlinear Anal.
Real World Appl., 2010; 11(2), 856–868.
[22] S. Abbasbandy. Int. Commun. Heat Mass Transf., 2007; 34,
380–387.
[23] Z. Ziabakhsh, G. Domairry, H.R. Ghazizadeh. J. Taiwan Inst.
Chem. Eng., 2009; 40(1), 91–97.
[24] Z. Ziabakhsh, G. Domairry, H. Bararnia, H. Babazadeh.
J. Taiwan Inst. Chem. Eng., 2010; 41, 22–28.
[25] A.A. Joneidi, G. Domairry, M. Babaelahi, M. Mozaffari. Int.
J. Numer. Meth. Fluids, 2010; 63(5), 548–563.
[26] S. Nadeem, A. Hussain, M. Khan. Commun. Nonlinear Sci.
Numer. Simul., 2010; 15(3), 475–481.
[27] A.A. Joneidi, G. Domairry, M. Babaelahi. J. Taiwan Inst. of
Chem. Eng., 2010; 41(1), 35–43.
[28] A.R. Sohouli, M. Famouri, A. Kimiaeifar, G. Domairry. Commun. Nonlinear Sci. Numer. Simul., 2010; 15(7), 1691–1699.
[29] S. Srinivas, R. Muthuraj. Commun. Nonlinear Sci. Numer.
Simul., 2010; 15(8), 2098–2108.
[30] A.A. Joneidi, G. Domairry, M. Babaelahi. Meccanica,; DOI:
10.1007/s11012-010-9295-y.
[31] M. Omidvar, A. Barari, M. Momeni, D.D. Ganji. Geomech.
Geoeng., 2010; (in press).
[32] M.o. Miansari, M.e. Miansari, A. Barari, D.D. Ganji. Int.
J. Comput. Meth. Eng. Sci. Mech., 2010; (in press).
[33] F. Fouladi, E. Hosseinzadeh, A. Barari, G. Domairry. J. Heat
Transf. Res., 41(2), 1–11.
[34] H. Schlichting, K. Gersten. Grenzschicht-Theorie, SpringerVerlag: Berlin, 1997.
Asia-Pac. J. Chem. Eng. 2011; 6: 669–678
DOI: 10.1002/apj
Документ
Категория
Без категории
Просмотров
0
Размер файла
523 Кб
Теги
newapplications, approximate, method, mechanics, fluid
1/--страниц
Пожаловаться на содержимое документа