Application of differential transformation method (DTM) for heat and mass transfer in a porous channel Q2 Q1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3[.]
Trang 1H O S T E D B Y
ORIGINAL ARTICLE
Application of differential transformation method
(DTM) for heat and mass transfer in a porous
channel
Q2
S Sepasgozara,n, M Farajib, P Valipourc
Q1
a
Department of Civil Engineering, Babol University of Technology, Babol, Iran
b
Department of Mechanical Engineering, Babol University of Technology, Babol, Iran
c
Department of Textile and Apparel, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran
Received 10 April 2015; accepted 4 March 2016
KEYWORDS
Differential
transfor-mation method
(DTM);
Axisymmetric channel;
Rotating disk;
Porous media;
Non-Newtonianfluid
Abstract In the present paper a differential transformation method (DTM) is used to obtain the solution of momentum and heat transfer equations of non-Newtonian fluid flow in an axisymmetric channel with porous wall The comparison between the results from the differential transformation method and numerical method are in well agreement which proofs the capability of this method for solving such problems After this validity, results are investigated for the velocity and temperature for various values of Reynolds number, Prandtl number and power law index
& 2017 National Laboratory for Aeronautics and Astronautics Production and hosting by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license
( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).
1 Introduction
The problem of non-Newtonianfluid flow has been under a
lot of attention in recent years Various applications in different
fields of engineering specially the interest in heat transfer
problems of non-Newtonian fluid flow, such as hot rolling, lubrication, cooling problems and drag reduction was the main reason for this considerable attention Debruge and Han [1] Q3
studied a problem concerning heat transfer in channel flow, which can be considered as an application of the previous works reported by Yuan [2], White [3] and Treill [4] Q4
Increasing the resistance of the blades against the hot stream around the blades for cooling was interested However the cooling process gives rise to excess energy consumption which leads to largely decrease of turbine efficiency
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57 58 59 60 61 62 63 64 65 66 67 68 69 70
http://ppr.buaa.edu.cn/
www.sciencedirect.com
Propulsion and Power Research
http://dx.doi.org/10.1016/j.jppr.2017.01.001
2212-540X & 2017 National Laboratory for Aeronautics and Astronautics Production and hosting by Elsevier B.V This is an open access article under the
CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).
n Corresponding author.
E-mail address: s.sepasgozar@yahoo.com (S Sepasgozar).
Peer review under responsibility of National Laboratory for Aeronautics
and Astronautics, China.
Propulsion and Power Research ]]]];](]):]]]–]]]
Trang 2Most of phenomena in our world are essentially nonlinear and
are described by nonlinear equations Nonlinear differential
equations usually arise from mathematical modeling of many
physical systems Some of them are solved using numerical
methods and some are solved using analytic methods such as
perturbation Perturbation techniques are based on the existence
of small or large parameters, this is called perturbation quantity
Unfortunately, many nonlinear problems in science and
engi-neering do not contain such kind of perturbation quantities
Therefore, same as the HAM[5,6], HPM[7–10], ADM[11,12]
and OHAM[13,14]can overcome the foregoing restrictions and
limitations of perturbation methods One of the semi-exact
methods which does not need small parameters is the differential
transformation method This method constructs an analytical
solution in the form of a polynomial It is different from the
traditional higher-order Taylor series method The Taylor series
method is computationally expensive for large orders The
differential transform method is an alternative procedure for
obtaining an analytic Taylor series solution of differential
equations The main advantage of this method is that it can be
applied directly to nonlinear differential equations without
requiring linearization, discretization and therefore, it is not
affected by errors associated to discretization The concept of
DTM wasfirst introduced by Zhou[15], who solved linear and
nonlinear problems in electrical circuits Chen and Ho [16]
developed this method for partial differential equations and Ayaz
[17]applied it to the system of differential equations Jang et al
[18]applied the two-dimensional differential transform method
to the solution of partial differential equations Sheikholeslami
et al [19]used this method to solve the problem of nanofluid
flow between parallel plates considering Thermophoresis and
Brownian effects Sheikholeslami and Ganji[20]applied DTM
to solve the problem of nanofluid flow and heat transfer between
parallel plates considering Brownian motion They concluded
that Nusselt number increases with augment of nanoparticle
volume fraction, Hartmann number while it decreases with
increase of the squeeze number Natural convection of a
non-Newtonian copper–water nanofluid between two infinite parallel
verticalflat plates is investigated by Domairry et al.[21] They conclude that as the nanoparticle volume fraction increases, the momentum boundary layer thickness increases, whereas the thermal boundary layer thickness decreases New analytical and numerical method has been developed in recent year in different field of science[22–56]
In this study, the purpose is to solve nonlinear equations via DTM It can be seen that this method is strongly capable for solving a large class of coupled and nonlinear differ-ential equations without tangible restriction of sensitivity to the degree of the nonlinear term
2 Mathematical formulation
2.1 Flow analysis
This study is concerned with simultaneous development of flow and heat transfer for non-Newtonian viscoelastic fluid flow on the turbine disc for cooling purposes The problem to
be considered is depicted schematically inFigure 1 The r-axis
is parallel to the surface of disk and the z-axis is normal to it
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112
Nomenclature
A, B symmetric kinematic matrices
C specific heat
Cn blade-wall temperature coefficients
f velocity function
F transformation ofƒ
xk general coordinates
Kr rotation parameter
k fluid thermal conductivity
n power law index in temperature distribution
p fluid pressure
qnð Þη temperature function
Pr Prandtl number
Re injection Reynolds number
T temperature
ur; uz velocity components in r, z directions, respectively
V injection velocity r; θ; z cylindrical coordinate symbols
δv m
δx n velocity gradients
δa m
δx n acceleration gradients
Greek symbols
ρ fluid density
τij stress tensor component
ϕk viscosity coefficients
φ dissipation function
η dimension less coordinates in z direction
ψ stream function
Trang 3The porous disc of the channel is at z¼þL The wall that
coincides with the r-axis is heated externally and from the
other perforated wall non-Newtonian fluid is injected
uni-formly in order to cool the heated wall
As can observed inFigure 1the cooling problem of the disk
can be considered as a stagnation pointflow with injection For
a steady, ax symmetric, non-newtonianfluid flow the following
equations can be written in cylindrical coordinates
The continuity equation:
∂ðrurÞ
∂r þ
∂ðruzÞ
And the momentum equations:
ur
∂ðurÞ
∂r þ uz
∂ðurÞ
∂z
¼ 1ρ∂P∂rþ1ρ ∂τrr
∂r þ
1
rðτrrτθθÞ þ∂τrz
∂z
ð2Þ
ur∂ðuzÞ
∂r þ uz∂ðuzÞ
∂z
¼ 1
ρ
∂P
∂zþ
1 ρ
∂τzr
∂r þ
1
rτrzþ∂τzz
∂z
ð3Þ
Hereτrr; τrz; τzr; τzz are the components of stress matrix
The analytical model under consideration leads to the
following boundary conditions:
@z ¼ 0 ur¼ uz¼ 0 ð4Þ
@z ¼ L ur¼ 0; uz¼ V ð5Þ
Here ur; uz are the velocity components in the r and z
directions and V is the injection velocity;ρ; P are the density,
and pressure For particular class of viscoelastic and
viscoi-nelasticfluids Rivlin[36]showed that if the stress components
τij at a point xk ðk ¼ 1; 2; 3Þ and time t are assumed to be
polynomials in the velocity gradientδv m
δx n ðm; n ¼ 1; 2; 3Þ and the acceleration gradients δa m
δx n ðm; n ¼ 1; 2; 3Þ, and if in addition the medium is assumed to
Q5 be isotropic the stress
matrix can be expressed in the form
‖τij‖ ¼ ϕ0Iþ ϕ1A þ ϕ2Bþ ϕ3A2þ ::: ð6Þ
Here I is the unit matrices, A and B are symmetric
kinematic matrixes defined by:
A¼ δvi
δxj
þδvj
δxi
;
B¼ δai
δxj
þδaj
δxi
þ 2δvm
δxi
δvm
δxj
And ϕk ðk ¼ 0; 1; 2; 3Þ are polynomials in the invariants
of A, B, A2 This study is restricted to second orderfluids for
which ϕk ðk ¼ 0; 1; 2; 3Þ are constant and ϕk ðk ¼ 4; 5; …Þ
are zero So that the stress components are as follows:
τrr¼ ϕ1Arrþ ϕ2Arr2þ ϕ3Brr ð8Þ
τzz¼ ϕ1Azzþ ϕ2Azz2þ ϕ3Bzz ð9Þ
τθθ¼ ϕ1Aθθþ ϕ2Aθθ2þ ϕ3Bθθ ð10Þ
τrz¼ ϕ1Arzþ ϕ2Arz2þ ϕ3Brz ð11Þ For the solution of the problem depicted inFigure 1 in the case of axially symmetricflow it is convenient to define
a stream function so that the continuity equation is satisfied:
ψ ¼ Vr2fðηÞ ð12Þ Where η ¼ z
L and the velocity components can be derived as:
ur¼V r
L f
0
uz¼ 2Vf ðηÞ ð14Þ Using Eqs.(12)–(14)the equations of motion reduce to:
f022f f00¼ L2
ρV2r
∂P
∂rþ
ϕ1
ρVLf
000
þ ϕ2
ρL2ð f0022f0f000Þ þ ϕ3
ρL2ð f0022f fivÞ
ð15Þ 4f f0¼ L2
ρV2
∂P
∂z 2
ϕ1
ρVLf
00
þ 2 ϕ2
ρL2 14f0f00þr2
Lf
00
f000
þ 4 ϕ3
ρL2 11f0f00þ f f000þr2
Lf
00
f000
ð16Þ The pressure term can be eliminated by differentiating Eq.(15)
with respect to z and Eq.(16)with respect to r and subtracting the resulting equations This gives the following equations:
2f f000¼f iv
ReK1 ð4f00f000þ 2f0fivÞ
K2 ð4f00f000þ 2f0fivþ 2f fvÞ ð17Þ Where K1¼ Φ 2
ρL 2 is the injection Reynolds number For K2¼ 0,the equation turned to:
fivþ 2 Re f f000 K1 Reð4f00f000þ 2f0fivÞ ¼ 0 ð18Þ The boundary conditions are:
fð0Þ ¼ 0; f0ð0Þ ¼ 0;
fð1Þ ¼ 1; f0ð1Þ ¼ 0: ð19Þ
2.2 Heat transfer analysis
The energy equation for the present problem with viscous dissipation in non-dimensional form is given by:
ρC ur
∂T
∂r þ uz
∂T
∂z
¼ k∇2
Tþ φ ð20Þ
φ ¼ τrr∂ur
∂r þ τθθ
ur
r þ τzz∂uz
∂z þ τrz ∂ur
∂z þ
∂uz
∂r
ð21Þ
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112
Trang 4Here ur; uz are the velocity components in the r and z
directions and V is the injection velocity;ρ; P; T; C; k are the
density, pressure, temperature, specific heat, and heat conduction
coefficient of fluid, respectively φ is the dissipation function
Letting the blade wall (z¼0) temperature distribution be
Tw¼ T0þ P1
r L
n
And assuming thefluid temperature to have the form of
Ref.[1]
T¼ T0þX1
Cn
r L
n
qnðηÞ ð22Þ Where T0 is the temperature of the incoming coolant
(z¼L) and neglecting dissipation effect the following
equations and boundary conditions are obtained:
qn00 Pr Re ð f0
qn2f qn
0
Þ ¼ 0;
ðn ¼ 0; 2; 3; 4; :::Þ ð23Þ
qnð0Þ ¼ 1; qnð1Þ ¼ 0 ð24Þ
3 Fundamentals of differential transform
method [20]
Basic definitions and operations of differential
transfor-mation are introduced as follows Differential
transforma-tion of the functransforma-tion fð Þ is defined as follows:η
F kð Þ ¼ 1
k!
dkfð Þη
dηk
η ¼ η 0
ð25Þ
In Eq (6), fð Þ is the original function and F kη ð Þ is the
transformed function which is called the T-function (it is also
called the spectrum of the fð Þ at η ¼ ηη 0, in the k domain) The
differential inverse transformation of F kð Þ is defined as:
fð Þ ¼η X1
FðkÞðηη0Þk ð26Þ
by combining Eq.(25)and Eq.(26), fð Þ can be obtained:η
fð Þ ¼η X1
dkfð Þη
dηk
η ¼ η 0
ð ηη0Þk
k! ð27Þ
Eq.(27)implies that the concept of the differential transforma-tion is derived from Taylor's series expansion, but the method does not evaluate the derivatives symbolically However, relative derivatives are calculated by an iterative procedure that is described by the transformed equations of the original functions
From the definitions of Eqs.(25)and(26), it is easily proven that the transformed functions comply with the basic mathematical operations shown in below In real applications, the function fð Þη
in Eq.(27)is expressed by afinite series and can be written as:
fð Þ ¼η X
N
FðkÞðηη0Þk ð28Þ
Eq (9) implies that fð Þ ¼η P1k¼ Nþ1FðkÞðηη0Þk
is negligibly small, where N is series size
Theorems to be used in the transformation procedure, which can be evaluated from Eqs.(25)and(26), are given below (Table 1)
4 Solution with DTM
Now differential transformation method into governing equations has been applied Taking the differential trans-forms of Eqs (18) and (23) with respect to χ and considering H¼1 gives:
ðk þ 1Þðk þ 2Þðk þ 3Þðk þ 4ÞF½k þ 4
þ 2 Re X
k
F½k mðm þ 1Þ
ðm þ 2Þðm þ 3ÞF½m þ 3
!
4 Re K1
Xk
ðk m þ 1Þðk m þ 2Þ
F½k m þ 2ðm þ 1Þ
ðm þ 2Þðm þ 3ÞF½m þ 3
0 B
1 C
2 Re K1
Xk
ðk m þ 1ÞF½k m þ 1
ðm þ 1Þðm þ 2Þðm þ 3Þ
ðm þ 4ÞF½m þ 4
0 B
1 C
A ¼ 0
ð29Þ
F½0 ¼ 0; F½1 ¼ 0; F½2 ¼ α; F½3 ¼ β ð30Þ
ðk þ 1Þðk þ 2ÞQn½k þ 2
nPrReX
k
ðm þ 1ÞF½m þ 1Qn½k m
þ 2PrReX
k
ðm þ 1Þ Qn½m þ 1F½k m
ð31Þ
Qn½0 ¼ 1; Qn½1 ¼ λ ð32Þ where FðkÞ; QnðkÞ are the differential transforms of
fðηÞ; qnðηÞ and α; β; λ are constants which can be obtained through boundary condition, Eqs (19) and (24) This problem can be solved as followed:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112
method.
Original function Transformed function
f ðηÞ ¼ αgðηÞ7βhðηÞ F ½k ¼ αG½k7βH½k
f ðηÞ ¼ d n g ðηÞ
dη n F ½k ¼ ðkþnÞ!
k! G½k þ n
f ðηÞ ¼ gðηÞhðηÞ F ½k ¼Pk
m 0F½mH½k m
f ðτÞ ¼ sin ðϖη þ αÞ F ½k ¼ ϖ k
k! sinð πk
2 þ αÞ
f ðτÞ ¼ cos ðϖη þ αÞ F½k ¼ϖ k
k ! cosð πk
2 þ αÞ
f ðηÞ ¼ e λη F½k ¼ λ k
k !
FðηÞ ¼ 1 þ η ð Þ m
F ½k ¼ m ðm 1Þ:::ðm kþ1Þ
k!
f ðηÞ ¼ η m
F ½k ¼ δðk mÞ ¼ 10 ; k ¼ m; k am
(
Trang 5F½0 ¼ 0
F½1 ¼ 0
F½2 ¼ α
F½3 ¼ β
F½4 ¼ 2 Re K1αβ
F½5 ¼ 6
F½6 ¼ 1
þ 64
F½7 ¼ 1
105α2Re2K1β
þ 132
þ 265
F½8 ¼ 11
35Re3K1 α3β
þ 1752
þ 768
ð33Þ
Qn½0 ¼ 1
Qn½1 ¼ λ
Qn½2 ¼ 0
Qn½3 ¼ 1
3PrRenα
Qn½4 ¼ 1
6PrRenαλ þ1
4PrRenβ1
Qn½5 ¼ 3
20PrRenβλ þ2
1
Qn½6 ¼ 1
45Pr2Re2n2α2
þ 4
15PrRe2nK1αβλ þ1
þ 4
2
15Pr2Re2α2n
Qn½7 ¼ 1
126Pr2Re2n2α2λ þ1
28Pr2Re2n2βα
5
126Pr2Re2nα2λ þ1
7PrRe2nλK1β2
þ 4
þ 44
2
2
21Pr2Re2βnα þ 2
63Pr2Re2α2λ
Qn½8 ¼ 1
70Pr2Re2n2αβλ þ13
210Pr2Re3n2K1α2β
53
840Pr2Re2nαβλ þ 3
224Pr2Re2n2β2
1
280PrRe2nλαβ þ33
þ 48
11
þ 204
þ 1
16
7Pr2Re3K1α2βn
1
ð34Þ
The above process is continuous By substituting Eqs
(33)and (34)into the main equation based on DTM, it can
be obtained that the closed form of the solutions is:
FðηÞ ¼ 1 þ α η2þ βη3þ 2 Re Kð 1αβÞ η4
þ 6
η5
þ
1
þ64
!
η6þ :::
ð35Þ
Qnð Þ ¼ 1 þ λη þη 1
3Pr Re nα
η3
þ 1
6Pr Re nαλ þ1
4Pr Re nβ1
6Pr Reαλ
η4
þ
3
20Pr Re nβλ þ2
5Pr Re2nK1αβ
1
10Pr Reβλ
!
η5
þ
1
15Pr Re2nK1αβλ
þ1
5Pr Re2n K1β2
þ4
5Pr Re3nK1 α2β 2
15Pr Re2K1αβλ
1
!
η6
by substituting the boundary condition from Eqs.(19)and
(24) into Eqs (35) and (36) in point η ¼ 1, it can be obtained the values ofα; β; λ
Fð1Þ ¼ 1 þ α þ β þ 2 Re Kð 1αβÞ
þ 6
þ 1
F0ð1Þ ¼ 1 þ 2α þ 3β þ 4 2 Re Kð 1αβÞ η4
þ 5 6
þ 6
1
þ64
!
Qnð Þ ¼ 1 þ λ þ1 1
3Pr Re nα
þ 1
6Pr Re nαλ þ1
4Pr Re nβ1
6Pr Reαλ
þ 3
20Pr Re nβλ þ2
5Pr Re2nK1αβ 1
10Pr Reβλ
þ 1
15Pr Re2nK1αβλ
þ 1
5Pr Re
5Pr Re
2
15Pr Re
15Pr
2Re2α2n
By solving Eqs.(37)–(39)gives the values ofα; β; λ By substituting obtainedα; β; λ into Eqs (35)and (36), it can
be obtained the expression of FðηÞ; QnðηÞ
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112
Trang 65 Results and discussion
The objective of the present study was to apply the differential
transformation method to obtain an explicit analytic solution of
heat transfer equation of a non-Newtonian fluid flow in an
axisymmetric channel with a porous wall for turbine cooling
applications (Figure 1) The results that obtained by the
differential transformation method were well matched with the
results carried out by the numerical solution obtained by a
four-1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112
η N ¼4 Error N ¼6 Error
0.1 0.031767 0.00214 0.031767 0.002141
0.2 0.116853 0.007411 0.116874 0.007432
0.3 0.239782 0.013885 0.239975 0.014078
0.4 0.384867 0.01927 0.385834 0.020237
0.5 0.536216 0.020762 0.539633 0.024179
0.6 0.677728 0.014878 0.68739 0.024541
0.7 0.793091 0.002704 0.816482 0.020687
0.8 0.865789 0.037278 0.916281 0.013214
0.9 0.879096 0.095201 0.978892 0.004596
η N ¼8 Error N ¼10 Error
0.1 0.029687 0.000059 0.02955 0.000076
0.2 0.109737 0.000295 0.109271 0.000171
0.3 0.226588 0.000692 0.225724 0.000173
0.4 0.366737 0.00114 0.365524 0.000073
0.5 0.516936 0.001481 0.515532 0.000077
0.6 0.664418 0.001569 0.66306 0.000211
0.7 0.797125 0.00133 0.796061 0.000267
0.8 0.903891 0.000823 0.903288 0.00022
0.9 0.974563 0.000266 0.974391 0.000094
Trang 7
order Runge–Kutta method as shown inTable 2 In these tables,
the error is introduced as followed:
Error¼ f ðηÞ NMf ðηÞDTM ð40Þ
As shown in Figures 2 and 3 for constant value of K1,
velocity values increase as a result of Reynolds number
increase At low Reynolds numbers the velocity profile exhibit
center line symmetry indicating a Poiseuille flow for
non-Newtonianfluids At higher Reynolds numbers the maximum
velocity point is shifted to the solid wall where shear stress
becomes larger as the Reynolds number grows Since f00ð0Þ is
measure of friction force, it is advisable to use viscoinelastic
fluids as a coolant fluid for industrial gas turbine engines
In Figure 4 temperature profiles for different values of
power law index (n) are shown It shows that for constant
value of η temperature increase if power law index
decrease Increasing Reynolds number leads to increasing
the curve of temperature profile and decreasing of qnðηÞ
values as shown in Figure 5
6 Conclusion
In this paper, non-Newtonianfluid flow in an axisymmetric
channel with a porous wall for turbine cooling applications
problem (Figure 1) has been solved via a sort of analytical
method, differential transformation method (DTM), and this
problem has been also solved by a numerical method (the
Runge–Kutta method of order 4) The observed good
agree-ment between the present method and numerical results shows
that differential transformation method is a powerful approach
for solving nonlinear differential equation such as this problem
Some facts were observed through the results Friction force
increased as a result of Reynolds numbers increase Increasing
power law index leads to decreasing temperature value
between two plates Nusselt number increases as a result of
Reynolds number increase Prandtl number increase leads to
Nusselt number increase as well Also Nusselt number increase
as an effect made by power law increase
References
[1] L.L Debruge, L.S Han, Heat transfer in a channel with a
porous wall for turbine cooling application, J Heat Transf
Trans ASME (1972) 385–390
[2] S.W Yuan, A.B Finkelstein, Laminar pipe flow with
injection and suction through a porous wall, Trans ASME
78 (1956) 719–724
[3] C Kurtcebe, M.Z Erim, Heat transfer of a non-Newtonian
viscoinelasticfluid in an axisymmetric channel with a porous
wall for turbine cooling application, Int Commun Heat
Mass Transf 29 (2002) 971–982
[4] J.L White, A.B Metzner, Constitutive equations for
viscoe-lasticfluids with application to rapid external flows, AIChE J
11 (1965) 324–330
[5] M Sheikholeslami, R Ellahi, H.R Ashorynejad, G
Domairry, T Hayat, Effects of heat transfer in flow of
nanofluids over a permeable stretching wall in a porous
medium, J Comput Theor Nanosci 11 (2014) 1–11
[6] M Sheikholeslami, H.R Ashorynejad, G Domairry, I
Hashim, Flow and heat transfer of Cu–water nanofluid Q6
between a stretching sheet and a porous surface in a rotating system, J Appl Math (2012), http://dx.doi.org/10.1155/
2012/421320 [7] M Sheikholeslami, D.D Ganji, Heat transfer of Cu–water nanofluid flow between parallel plates, Powder Technol 235 (2013) 873–879
[8] M Sheikholeslami, H.R Ashorynejad, D.D Ganji, A Kolahdooz, Investigation of rotating MHD viscous flow and heat transfer between stretching and porous surfaces using analytical method, Math Probl Eng (2011),http://dx.doi.org/10.1155/2011/258734 [9] M Sheikholeslami, H.R Ashorynejad, D.D Ganji, A
Yıldırım, Homotopy perturbation method for three-dimensional problem of condensation film on inclined rotating disk, Sci Iran B 19 (3) (2012) 437–442
[10] M Sheikholeslami, D.D Ganji, H.B Rokni, Nanofluid flow
in a semi-porous channel in the presence of uniform magnetic field, IJE Trans C: Asp 26 (6) (2013) 653–662
[11] M Sheikholeslami, D.D Ganji, H.R Ashorynejad, Investi-gation of squeezing unsteady nanofluid flow using ADM, Powder Technol 239 (2013) 259–265
[12] M Sheikholeslami, D.D Ganji, H.R Ashorynejad, H.B
Rokni, Analytical investigation of Jeffery–Hamel flow with high magnetic field and nano particle by adomian decom-position method, Appl Math Mech -Engl Ed 33 (1) (2012) 1553–1564
[13] M Sheikholeslami, D.D Ganji, Magnetohydrodynamicflow
in a permeable channelfilled with nanofluid, Sci Iran B 21 (1) (2014) 203–212
[14] M Sheikholeslami, H.R Ashorynejad, D Domairry, I
Hashim, Investigation of the laminar viscous flow in a semi-porous channel in the presence of uniform magnetic field using optimal homotopy asymptotic method, Sains Malays 41 (10) (2012) 1177–1229
[15] C.K Chen, S.H Ho, Solving partial differential equations by two dimensional differential transform method, Appl Math
Comput 106 (1999) 171–179
[16] F Ayaz, Solutions of the systems of differential equations by differential transform method, Appl Math Comput 147 (2004) 547–567
[17] I.H Abdel, H Hassan, Comparison differential transforma-tion technique with Adomian decompositransforma-tion method for linear and nonlinear initial value problems, Chaos Solitons Fractals 36 (2008) 53–65
[18] M.J Jang, C.L Chen, Y.C Liu, Two-dimensional differen-tial transform for pardifferen-tial differendifferen-tial equations, Appl Math
Comput 121 (2001) 261–270
[19] M Sheikholeslami, M.M Rashidi, D.M.A Saad, F Firouzi, H.B Rokni, G Domairry, Steady nanofluid flow between parallel plates considering Thermophoresis and Brownian effects, J King Saud Univ Sci (2015), http://dx.doi.org/
10.1016/j.jksus.2015.06.003 [20] M Sheikholeslami, D.D Ganji, Nanofluid flow and heat transfer between parallel plates considering Brownian motion using DTM, Comput Methods Appl Mech Eng 283 (2015) 651–663
[21] D Domairry, M Sheikholeslami, H.R Ashorynejad, R.S.R
Gorla, M Khani, Natural convection flow of a non-Newtonian nanofluid between two vertical flat plates, Proc
IMechE Part N: J Nanoeng Nanosyst 225 (3) (2012) 115–
122,http://dx.doi.org/10.1177/1740349911433468
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112
Trang 8[22] H.R Ashorynejad, A.A Mohamad, M Sheikholeslami,
Magnetic field effects on natural convection flow of a
nanofluid in a horizontal cylindrical annulus using Lattice
Boltzmann method, Int J Therm Sci 64 (2013) 240–250
[23] P.V Satya Narayana, Effects of variable permeability and
radiation absorption on magnetohydrodynamic (MHD) mixed
convective flow in a vertical wavy channel with traveling
thermal waves, Propuls Power Res 4 (3) (2015) 150–160
[24] F Shakeri Aski, S.J Nasirkhani, E Mohammadian, A
Asgari, Application of adomian decomposition method for
micropolarflow in a porous channel, Propuls Power Res 3
(1) (2014) 15–21
[25] M Sheikholeslami, H.R Ashorynejad, P Rana, Lattice
Boltzmann simulation of nanofluid heat transfer enhancement
and entropy generation, J Mol Liq 214 (2016) 86–95
[26] M Sheikholeslami, S Soleimani, D.D Ganji, Effect of
electric field on hydrothermal behavior of nanofluid in a
complex geometry, J Mol Liq 213 (2016) 153–161
[27] M Sheikholeslami, M.M Rashidi, D.D Ganji, Numerical
investigation of magnetic nanofluid forced convective heat
transfer in existence of variable magnetic field using two
phase model, J Mol Liq 212 (2015) 117–126
[28] M Sheikholeslami, K Vajravelu, M.M Rashidi, Forced
convection heat transfer in a semi annulus under the influence
of a variable magnetic field, Int J Heat Mass Transf 92
(2016) 339–348
[29] M Sheikholeslami, M Hatami, D.D Ganji, Numerical
investigation of nanofluid spraying on an inclined rotating
disk for cooling process, J Mol Liq 211 (2015) 577–583
[30] M Sheikholeslami, M.M Rashidi, D.D Ganji, Effect of
non-uniform magneticfield on forced convection heat transfer of
Fe3O4–water nanofluid, Comput Methods Appl Mech Eng
294 (2015) 299–312
[31] M Sheikholeslami, M.M Rashidi, Ferrofluid heat transfer
treatment in the presence of variable magnetic field, Eur
Phys J 130 (2015) 115
[32] M Sheikholeslami, R Ellahi, Three dimensional mesoscopic
simulation of magneticfield effect on natural convection of
nanofluid, Int J Heat Mass Transf 89 (2015) 799–808
[33] M Sheikholeslami, S Abelman, Two phase simulation of
nanofluid flow and heat transfer in an annulus in the presence
of an axial magneticfield, IEEE Trans Nanotechnol 14 (3)
(2015) 561–569
[34] M Sheikholeslami, M.M Rashidi, Effect of space dependent
magneticfield on free convection of Fe3O4–water nanofluid,
J Taiwan Inst Chem Eng 56 (2015) 6–15
[35] U Khan, N Ahmed, M Asadullah, S.T Mohyud-Din, Effects
of viscous dissipation and slip velocity on two-dimensional
and axisymmetric squeezing flow of Cu–water and Cu–
kerosene nanofluids, Propuls Power Res 4 (1) (2015) 40–49
[36] U Khan, N Ahmed, S.I.U Khan, S.T Mohyud-Din,
Thermo-diffusion effects on MHD stagnation point flow
towards a stretching sheet in a nanofluid, Propuls Power
Res 3 (3) (2014) 151–158
[37] M Sheikholeslami, R Ellahi, M Hassan, S Soleimani, A
study of natural convection heat transfer in a nanofluid filled
enclosure with elliptic inner cylinder, Int J Numer Methods
Heat Fluid Flow 24 (8) (2014) 1906–1927
[38] M.S Kandelousi, Effect of spatially variable magneticfield on
ferrofluid flow and heat transfer considering constant heat flux
boundary condition, Eur Phys J 129 (11) (2014) 129–248
[39] M.S Kandelousi, KKL correlation for simulation of nano-fluid flow and heat transfer in a permeable channel, Phys
Lett A 378 (45) (2014) 3331–3339
[40] M Sheikholeslami, D.D Ganji, Entropy generation of nanofluid in presence of magnetic field using Lattice Boltz-mann MethodPhysica A 417 (2015) 273–286
[41] M.T Darvishi, F Khani, F.G Awad, A.A Khidir, P
Sibanda, Numerical investigation of theflow of a micropolar fluid through a porous channel with expanding or contracting walls, Propuls Power Res 3 (3) (2014) 133–142
[42] T.I.-P Shih, S.G Ramachandran, M.K Chyu, Time-accurate CFD conjugate analysis of transient measurements of the heat-transfer coefficient in a channel with pin fins, Propuls
Power Res 2 (1) (2013) 10–19
[43] Z.F Yang, H Hu, An experimental investigation on the trailing edge cooling of turbine blades, Propuls Power Res 1 (1) (2012) 36–47
[44] M Sheikholeslami, M Gorji-Bandpy, K Vajravelu, Lattice Boltzmann simulation of magnetohydrodynamic natural con-vection heat transfer of Al2O3–water Nanofluid in a hor-izontal cylindrical enclosure with an inner triangular cylinder, Int J Heat Mass Transf 80 (2015) 16–25
[45] M Sheikholeslami, Effect of uniform suction on nanofluid flow and heat transfer over a cylinder, J Braz Soc Mech
Sci Eng 37 (2015) 1623–1633
[46] M Sheikholeslami, D.D Ganji, M.Y Javed, R Ellahi, Effect
of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of two phase model, J
Magn Magn Mater 374 (2015) 36–43
[47] M Sheikholeslami, S Abelman, D.D Ganji, Numerical simula-tion of MHD nanofluid flow and heat transfer considering viscous dissipation, Int J Heat Mass Transf 79 (2014) 212–222
[48] M Sheikholeslami, D.D Ganji, Ferrohydrodynamic and magnetohydrodynamic effects on ferrofluid flow and con-vective heat transfer, Energy 75 (2014) 400–410
[49] M Jafaryar, F Farkhadnia, E Mohammadian, M Hosseini, A.M Khazaee, Analytical investigation of laminar flow through expanding or contracting gaps with porous walls, Propuls Power Res 3 (4) (2014) 222–229
[50] G Lin, K Kusterer, D Bohn, T Sugimoto, R Tanaka, M
Kazari, Investigation on heat transfer enhancement and pressure loss of double swirl chambers cooling, Propuls
Power Res 2 (3) (2013) 177–187
[51] M Sheikholeslami, D.D Ganji, Unsteady nanofluid flow and heat transfer in presence of magnetic field considering thermal radiation, J Braz Soc Mech Sci Eng 37 (3) (2015) 895–902
[52] M Sheikholeslami, D.D Ganji, Heated permeable stretching surface in a porous medium using Nanofluids, J Appl Fluid Mech 7 (3) (2014) 535–542
[53] M Sheikholeslami, D.D Ganji, Numerical investigation for two phase modeling of nanofluid in a rotating system with permeable sheet, J Mol Liq 194 (2014) 13–19
[54] M Sheikholeslami, D.D Ganji, Three dimensional heat and mass transfer in a rotating system using nanofluid, Powder Technol 253 (2014) 789–796
[55] M Sheikholeslami, F.B Sheykholeslami, S Khoshhal, H
Mola-Abasi, D.D Ganji, H.B Rokni, Effect of magnetic field on Cu–water nanofluid heat transfer using GMDH-type neural network, Neural Comput Appl 25 (2014) 171–178
[56] R.S Rivlin, J.L Ericksen, Stress-deformation relations for isotropic materials, J Ration Mech Anal 4 (1955) 323–425
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112