MHD Nanofluid Flow Analysis in a Semi-Porous Channel by a Combined Series Solution Method

Trans. Phenom. Nano Micro Scales, 1(2): 124-137, Summer – Autumn 2013
DOI: 10.7508/tpnms.2013.02.006
ORIGINAL RESEARCH PAPER .
MHD Nanofluid Flow Analysis in a Semi-Porous Channel by a
Combined Series Solution Method
R. Nouri1
, D. D. Ganji1
, M. Hatami1,2,*
1. Babol University of Technology, Department of Mechanical Engineering, Babol, Iran.
2. Esfarayen Faculty of Industrial and Engineering, Department of Mechanical Engineering, Esfarayen, North Khorasan,
Iran
Abstract
In this paper, Least Square Method (LSM) and Differential Transformation Method (DTM) are used to solve
the problem of laminar nanofluid flow in a semi-porous channel in the presence of transverse magnetic field.
Due to existence some shortcomings in each method, a novel and efficient method named LS-DTM is
introduced which omitted those defects and has an excellent agreement with numerical solution. In the present
study, the effective thermal conductivity and viscosity of nanofluid are calculated by Maxwell–Garnetts (MG)
and Brinkman models, respectively. The influence of the three dimensionless numbers: the nanofluid volume
friction, Hartmann number and Reynolds number on non-dimensional velocity profile are considered. The
results show that velocity boundary layer thickness decrease with increase of Reynolds number and
nanoparticle volume friction and it increases as Hartmann number increases.
Keywords: Least Square Method (LSM); Nanofluid; Semi-porous channel; Uniform magnetic
1. Introduction
Most scientific problems in fluid mechanics and
heat transfer problems are inherently nonlinear. All
these problems and phenomena are modelled by
ordinary or partial nonlinear differential equations.
Most of these described physical and mechanical
problems are with a system of coupled nonlinear
differential equations. For an example heat transfer by
natural convection which frequently occurs in many
physical problems and engineering applications such
as geothermal systems, heat exchangers, chemical
catalytic reactors and nanofluid flow in a semi-porous
channel has a system of coupled nonlinear differential
________
equations for temperature or velocity distribution
equations.
The flow problem in porous tubes or channels has
been under considerable attention in recent years
because of its various applications in biomedical
engineering, for example, in the dialysis of blood in
artificial kidney, in the flow of blood in the capillaries,
in the flow in blood oxygenators as well as in many
other engineering areas such as the design of filters, in
transpiration cooling boundary layer control [1] and
gaseous diffusion [2]. In 1953, Berman [3] described
an exact solution of the Navier-Stokes equation for
steady two-dimensional laminar flow of a viscous,
incompressible fluid in a channel with parallel, rigid
*
Corresponding author
Email Address: m.hatami2010@gmail.com
124

Hatami et al./ TPNMS 1 (2013) 124-137
125
Nomenclature Greek Symbols
* *
,A B Constant parameter υ Kinematic viscosity
P Fluid pressure σ Electrical conductivity
q Mass transfer parameter ε Aspect ratio h/Lx
kx General coordinates µ Dynamic viscosity
f Velocity function υ Kinematic viscosity
k Fluid thermal conductivity ρ Fluid density
n Power law index in temperature
distribution
Re Reynolds number Subscripts
Ha Hartmann number ∞ Condition at infinity
u,v Dimensionless components velocity in
x and y directions, respectively
nf Nanofluid
u*,v* Velocity components in x and y
directions respectively
f Base fluid
x,y Dimensionless horizontal, vertical
coordinates respectively
s Nano-solid-particles
x*,y* Distance in x,y directions parallel to
the plates
porous walls driven by uniform, steady suction or
injection at the walls. This mass transfer is paramount
in some industrial processes. More recently,
Sheikholeslami et al. [4] analyzed the effects of a
magnetic field on the nanofluid flow in a porous
channel through weighted residual methods called
Galerkin method. Nanofluid, which is a mixture of
nano-sized particles (nanoparticles) suspended in a
base fluid, is used to enhance the rate of heat transfer
via its higher thermal conductivity compared to the
base fluid. Soleimani et al. [5] studied natural
convection heat transfer in a semi-annulus enclosure
filled with nanofluid using the Control Volume based
Finite Element Method. They found that the angle of
turn has an important effect on the streamlines,
isotherms and maximum or minimum values of local
Nusselt number. Natural convection of a non-
Newtonian copper-water nanofluid between two
infinite parallel vertical flat plates is investigated by
Domairry et al. [6]. They conclude that as the
nanoparticle volume fraction increases, the
momentum boundary layer thickness increases,
whereas the thermal boundary layer thickness
decreases. Sheikholeslami et al. [7] performed a
numerical analysis for natural convection heat transfer
of Cu-water nanofluid in a cold outer circular
enclosure containing a hot inner sinusoidal circular
cylinder in presence of horizontal magnetic field using
the Control Volume based Finite Element Method.
There are some simple and accurate approximation
techniques for solving differential equations called the
Weighted Residuals Methods (WRMs). Collocation,
Galerkin and Least Square are examples of the
WRMs. Stern and Rasmussen [8] used collocation
method for solving a third order linear differential
equation. Vaferi et al. [9] have studied the feasibility
of applying of Orthogonal Collocation method to
solve diffusivity equation in the radial transient flow
system. Hendi and Albugami [10] used Collocation
and Galerkin methods for solving Fredholm–Volterra
integral equation. Recently Least square method is
introduced by A. Aziz and M.N. Bouaziz [11] and is
applied for a predicting the performance of a
longitudinal fin [12]. They found that least squares
method is simple compared with other analytical
methods. Shaoqin and Huoyuan [13] developed and
analyzed least-squares approximations for the
incompressible magneto-hydrodynamic equations.
The concept of differential transformation method
(DTM) was first introduced by Zhou [14] in 1986 and
it was used to solve both linear and nonlinear initial
value problems in electric circuit analysis. This
method can be applied directly for linear and
nonlinear differential equation without requiring
linearization, discretization, or perturbation and this is

126
the main benefit of this method. S. Ghafoori et al. [15]
used the DTM for solving the nonlinear oscillation
equation.
The main aim of this paper is to investigate the
problem of laminar nanofluid flow in a semi-porous
channel in the presence of transverse magnetic field
using Least Square (LSM) and Differential
Transformation Methods (DTM). Also a novel and
combined method from these two methods is
introduced as LS-DTM which is very accurate and
efficient. The effects of the nanofluid volume friction,
Hartmann number and Reynolds number on velocity
profile are considered. Furthermore velocity profiles
for different structures of nanofluid (copper and silver
nanoparticles in water or ethylene glycol) are
investigated.
2. Problem Description
Consider the laminar two-dimensional stationary flow
of an electrically conducting incompressible viscous
fluid in a semi-porous channel made by a long
rectangular plate with length of xL in uniform
translation in x * direction and an infinite porous
plate.
The distance between the two plates is h . We observe
a normal velocity qon the porous wall. A uniform
magnetic field B is assumed to be applied towards
direction *y (Fig. 1).
In the case of a short circuit to neglect the electrical
field and perturbations to the basic normal field and
without any gravity forces, the governing equations
are:
* *
0,
* *
u v
x y
∂ ∂
+ =
∂ ∂
(1)
22 2
2 2
* * 1 *
* *
* * *
* *
* ,
* *
nf
nf nf
nf nf
u u P
u v
x y x
Bu u
u
x y
ρ
µ σ
ρ ρ
∂ ∂ ∂
+ = − +
∂ ∂ ∂
 ∂ ∂
+ − 
∂ ∂ 
(2)
2 2
2 2
* * 1 *
* *
* * *
* *
,
* *
nf
nf
nf
v v P
u v
x y y
v v
x y
ρ
µ
ρ
∂ ∂ ∂
+ = −
∂ ∂ ∂
 ∂ ∂
+ + 
∂ ∂ 
(3)
The appropriate boundary conditions for the velocity
are:
0* 0 : * *, * 0,y u u v= = = (4)
* : * 0, * ,y h u v q= = = − (5)
Calculating a mean velocity U by the relation:
0* 0 : * *, * 0,y u u v= = = (6)
We consider the following transformations:
* *
; ,
x
x y
x y
L h
= = (7))
2
* * *
; ,
.
y
f
u v P
u v P
U q qρ
= = = (8)
Then, we can consider two dimensionless numbers:
the Hartman number Ha for the description of
magnetic forces [16] and the Reynolds number Re
for dynamic forces:
,
.
f
f f
Ha Bh
σ
ρ υ
= (9)
Re .nf
nf
hq
ρ
µ
= (10)
where the effective density( nfρ ) is defined as [12]:
(1 )nf f sρ ρ φ ρ φ= − + (11)
Where φ is the solid volume fraction of nanoparticles.
The dynamic viscosity of the nanofluids given by
Brinkman [12] is
2.5
(1 )
f
nf
µ
µ
φ
=
−
(12)
the effective thermal conductivity of the nanofluid can
be approximated by the Maxwell–Garnetts (MG)
model as [12]:
2 2 ( )
2 ( )
n f s f f s
f s f f s
k k k k k
k k k k k
φ
φ
+ − −
=
+ + −
(13)
The effective electrical conductivity of nanofluid was
presented by Maxwell [17] as below

127
3 1
1
2 1
s
fnf
f s s
f f
σ
φ
σσ
σ σ σ
φ
σ σ
 
−  
 = +
   
+ − −      
   
(14)
The thermo physical properties of the nanofluid are
given in Table 1.
Introducing Eqs. (6) and (10) into Eqs. (1) and (3)
leads to the dimensionless equations:
0,
u v
x y
∂ ∂
+ =
∂ ∂ (15)
2
2 2 2 *
2
2 2 *
1
,
Re
y
nf
nf
Pu u
u v
x y x
u u Ha B
u
hq x y A
ε
µ
ε
ρ
∂∂ ∂
+ = − +
∂ ∂ ∂
 ∂ ∂
+ − 
∂ ∂ 
(16)
2 2
2
2 2
1
.
y nf
nf
Pv v v
u v
x y x hq x y
µυ
ε
ρ
∂  ∂ ∂ ∂ ∂
+ = − + + 
∂ ∂ ∂ ∂ ∂  (17)
where *
A and *
B are constant parameters:
*
3 1
(1 ) , * 1
2 1
s
fs
f s s
f f
A B
σ
φ
σρ
φ φ
ρ σ σ
φ
σ σ
 
−  
 = − + = +
   
+ − −      
   
(18)
Quantity of ε is defined as the aspect ratio between
distance h and a characteristic length xL of the slider.
This ratio is normally small. Berman’s similarity
transformation is used to be free from the aspect ratio
ofε :
( ) ( )0
*
; .
u dV
v V y u u U y x
U dy
= − = = +
(19)
Introducing Eq. (19) in the second momentum
equation (17) shows that quantity yP y∂ ∂ does not
depend on the longitudinal variable x . With the first
momentum equation, we also observe that
2 2
yP x∂ ∂ is independent of x .
We omit asterisks for simplicity. Then a separation of
variables leads to [16]:
( )
'2 '' '''
2.5*
22 *
' 2 2
* 2
1 1
Re 1
1
,
Re
y y
V VV V
A
P PHa B
V
A x x x
φ
ε ε
− −
−
∂ ∂
+ = =
∂ ∂
(20)
( )
( )
' '
2.5*
2.5'' 2 *
1 1
Re 1
1 .
UV VU
A
U Ha B U
φ
φ
− =
−
 × − −
 
(21)
The right-hand side of Eq. (20) is constant. So, we
derive this equation with respect to x . This gives:
( )
( )
2.52 * ''
2.5* ' '' '''
1
Re 1 ,
IV
V Ha B V
A V V VV
φ
φ
= − +
 − − 
(22)
Where primes denote differentiation with respect to y
and asterisks have been omitted for simplicity. The
dynamic boundary conditions are:
'
'
0: 1; 0; 0,
1: 0; 1; 0.
y U V V
y U V V
 = = = =

= = = =
(23)
3. Analytical Methods
3. 1 Least Square Method (LSM)
Suppose a differential operator D is acted on a
function u to produce a function p:
( ( )) ( )D u x p x= (24)
It is considered that u is approximated by a functionu
, which is a linear combination of basic functions
chosen from a linearly independent set. That is,
1
n
i i
i
u u c ϕ
=
≅ = ∑ (25)
Now, when substituted into the differential
operator, D, the result of the operations generally isn’t
p(x). Hence an error or residual will exist:

128
( ) ( ( )) ( ) 0R x D u x p x= − ≠ (26)
The notion in WRMs is to force the residual to zero in
some average sense over the domain. That is:
( ) ( ) 0 1,2,...,iX
R x W x i n= =∫ (27)
Where the number of weight functions Wi is exactly
equal the number of unknown constants ci inu . The
result is a set of n algebraic equations for the unknown
constants ci. If the continuous summation of all the
squared residuals is minimized, the rationale behind
the name can be seen. In other words, a minimum of
2
( ) ( ) ( )
X X
S R x R x dx R x dx= =∫ ∫ (28)
In order to achieve a minimum of this scalar
function, the derivatives of S with respect to all the
unknown parameters must be zero. That is,
2 ( ) 0
i iX
S R
R x dx
c c
∂ ∂
= =
∂ ∂∫ (29)
Comparing with Eq. (27), the weight functions are
seen to be
2i
i
R
W
c
∂
=
∂
(30)
However, the “2” coefficient can be dropped, since
it cancels out in the equation. Therefore the weight
functions for the Least Squares Method are just the
derivatives of the residual with respect to the
unknown constants
i
i
R
W
c
∂
=
∂
(31)
Because trial functions must satisfy the boundary
conditions in Eq. (23), so they will be considered as,
2
1
3
2
2 3
3
2 4 2 5
4 5
( ) 1 ( )
( )
( ) ( )
2 3
( ) ( )
2 4 2 5
U y y c y y
c y y
y y
V y c
y y y y
c c
 = − + −

+ −

 = −


+ − + −

(32)
In this problem, we have two coupled equations (Eqs.
(21) and (22)), so two residual functions will be
appeared as,
2 3 2 4
2 3 2 3 4
1 1 2 3 4 5 1 2 3 4 5 3 4
2
1 2
2 5
2
5 1 2
( , , , , , ) (1 ( ) ( ))( ( ) ( ) ( )) (
2 3 2 4
3 1
2 6 1
2 1
)( 1 (1 2 ) (1 3 )
2 5
s
f
s s
f f
y y y y
R c c c c c y y c y y c y y c y y c y y c y y c c
c c y Ha
y y
c c y c y
σ
φ
σ
σ σ
φ
σ σ
   
= − + − + − − + − + − − − + − +   
   
 
− 
 − − − +
   
+ − −   
     
− − + − + − − 
 
2.5 2 3
1 2
2.5
2 2.5 2 3
2 1 2 3 4 5 4 5 3 4 5
(1 ) (1 ( ) ( ))
Re 1 (1 )
3 1
( , , , , , ) 6 24 1 (1 ) ( (1 2 ) (1 3 ) (1 4 ))
2 1
s
f
s
f
s s
f f
y c y y c y y
R c c c c c y c c y Ha c y c y c y
φ
ρ φ
φ φ
ρ
σ
φ
σ
φ
σ σ
φ
σ σ
 
 
  − − + − + −
 
  
 
 
− + − 
 
  
−  
  = − − − + − − + − + − +
    
+ − −     
    
2.5
2 3 2 4 2 5
2 3 4 2 3 2
3 4 5 3 4 5 3 4 5 3 4 5
Re 1 (1 )
( ( ) ( ) ( ))( (1 2 ) (1 3 ) (1 4 )) ( 2 6 12 )
2 3 2 4 2 5
s
f
y y y y y y
c y y c y y c y y c y c y c y c c c c c y c y
ρ φ
φ φ
ρ















 
− + − 
 


      
− + − + − − + − + − − − + − + − − − −      
      
(33)
By substituting the residual functions,
R1(c1,c2,c3,c4,c5, y) and R2(c1,c2,c3,c4,c5, y), into Eq.
(29), a set of equation with five equations will
appear and by solving this system of equations,
coefficients c1-c5 will be determined. For example,
Using Least Square Method for a water-copper
nanofluid with Re=0.5, Ha=0.5 and φ =0.05. U(y)
and V(y) are as follows:

126
2 3
2 3
4 5
( ) 1 1.334953917
0.3461783819 0.01122446534
( ) 1.8703229 3.1584693
6.9279074 2.8991152
U y y
y y
V y y y
y y
= − +

−

= + −
 +
(34)
3.2 Differential Transformation Method
(DTM)
In this section the fundamental basic of the
Differential Transformation Method is introduced.
For understanding method’s concept, suppose that
x(t) is an analytic function in domain D, and t=ti
represents any point in the domain. The function
x(t) is then represented by one power series whose
center is located at ti . The Taylor series expansion
function of x(t) is in form of:
0
( ) ( )
( )
!
t ti
k k
i
k
k
t t d x t
x t t D
k dt
=
∞
=
 −
= ∀ ∈ 
 
∑ (35)
The Maclaurin series of x(t) can be obtained by
taking ti=0 in Eq. (35) expressed as:
0
0
( )
( )
!
t
k k
k
k
t d x t
x t t D
k dt
=
∞
=
 
= ∀ ∈ 
 
∑ (36)
As explained in [14] the differential transformation
of the function x(t) is defined as follows:
0
0
( )
( )
!
t
k k
k
k
H d x t
X k
k dt
=
∞
=
 
=  
 
∑ (37)
Where X(k) represents the transformed function and
x(t) is the original function. The differential
spectrum of X(k) is confined within the interval
[0, ]t H∈ , where H is a constant value. The
differential inverse transform of X(k) is defined as
follows:
0
( ) ( ) ( )k
k
t
x t X k
H
∞
=
= ∑ (38)
It is clear that the concept of differential
transformation is based upon the Taylor series
expansion. The values of function X(k) at values of
argument k are referred to as discrete, i.e. X(0) is
known as the zero discrete, X(1) as the first discrete,
etc. The more discrete available, the more precise it
is possible to restore the unknown function. The
function x(t) consists of the T-function X(k), and its
value is given by the sum of the T-function with
(t/H)k as its coefficient. In real applications, at the
right choice of constant H, the larger values of
argument k the discrete of spectrum reduce rapidly.
The function x(t) is expressed by a finite series and
Eq. (38) can be written as:
0
( ) ( ) ( )
n
k
k
t
x t X k
H=
= ∑ (39)
Some important mathematical operations performed
by differential transform method are listed in Table
2. Now we apply Differential Transformation
Method (DTM) from Table 2 in to Eqs. (21) and
(22) for finding ( ) and ( )U y V y .
( )
0
2.5
0
2 2.5
( 1 ). ( ). ( 1 )
1 1
( 1 ). ( ). ( 1 ) .
Re .(1 )
( 1).( 2). ( 2) . .(1 ) . ( ) 0
k
l
k
l
k l U l V k l
k l V l U k l
A
k k U k Ha B U k
φ
φ
=
=
+ − + −
− + − + − −
−
× + + + − − =
∑
∑
(40)
2 2.5
2.5
0
0
( 1)( 2)( 3)( 4) ( 4)
. .(1 ) ( 1)( 2) ( 2)
Re. .(1 ) ( 1) ( 1)( )( 1 )
( )( 1 )( 2 )( 3 ) 0
k
l
k
l
k k k k V k
Ha B k k V k
A l V l k l k l
V k k l k l k l
φ
φ
=
=
+ + + + + −
− + + + −

− × + + − + −


− + − + − + − = 

∑
∑
(41)
Where and VU represent the DTM transformed
form of U and V respectively. The transformed
form of boundary conditions can be written as:
(0) 0, (1) 0, (2) , (3)
(0) 1, (1) .
V V V a V b
U U c
 = = = =

= =
(42)

127
Using transformed boundary condition and Eq. we
have,
2
2 2 2
2
2 2 2
2
2 2
2 2
(2) 0.5 1
1 0.5 1
(4) 0.0833 1
0.1667 1 0.0833 1
(3) 0.333 Re 1 0
.667 Re 1 0.333 Re 1
0.1667 1 0.333 1
0.1667 1
(5) 0.05
U Ha B
Ha B Ha B
V Ha B a
Ha B a Ha B a
U a A
a A a A
Ha B c Ha B c
Ha B c
V Ha
φ
φφ φφ
φ
φ φ φ φ
φ
φφ φφ
φ φ φ
φ φ
 = − −

− + −

= − −

− + −
= − −
− + −
+ − − −
+ −
= 2
2 2 2
1
0.1 1 0.05 1
B b
Ha B b Ha B b
φ
φ φ φ φ








− −

− + −
(43)
Where a, b, c are unknown coefficients that after
specifying ( ) and ( )U y V y and applying
boundary condition (Eq. (42)) into it, will be
determined. For water-copper nanofluid with
Re=0.5, Ha=0.5 and φ =0.05 following values
were determined for a, b and c coefficients.
a= 3.011719150, b= - 2.049532443,
c= - 1.673547080
(44)
Finally, U(y) and V(y) are as follows,
2
3
2 3
4 5
( ) 1 1.673547080y 0.1273175011y
+0.5462295787y
( ) 3.011719150y 2.049532443y
0.06390742601y 0.02609413491y
U y
V y
 = − −



= −
+ −
(45)
3.3 LS- DTM Combined Method
Since LSM and DTM have a little shortcoming
in some areas for predicting the V(y) and U(y) (See
results section), we combined these two methods
as LS-DTM combined method which eliminated
those defects and for all areas has an excellent
agreement with numerical procedure. For this
purpose we selected U(y) from Eq. (32) and V(y)
from Eq. (41). By using these two equations four
unknown coefficients will be existed: a, b, c1 and
c2. For finding these coefficients, four equations
are needed; two of them are obtained from Eq. (29)
for c1 and c2, and other two equations are selected
from boundary condition for V(y) in Eq. (23). For
water-copper nanofluid with Re=0.5, Ha=0.5 and
φ =0.05 following formula are calculated for U(y)
and V(y) by this efficient and novel method,
2
3
2 3
4 5
( ) 1 1.332674596 0.3491297634
0.01645516732
( ) 3.011719150y 2.049532443y
0.06390742601y 0.02609413491y
U y y y
y
V y
 = − +

−

= −
+ −
(46)
4. Results and discussion
In the present study LSM and DTM methods are
applied to obtain an explicit analytic solution of the
laminar nanofluid flow in a semi-porous channel in
the presence of uniform magnetic field (Fig. 1).
Fig. 1. Schematic of the problem (Nanofluid in a porous
media between parallel plates and magnetic field)
For this aim Eqs. (21) and (22) are solved for
different nanofluid structures (see Table 1) and
comparison between described methods is
demonstrated in Figs. 2 and 3. These figures explain
that each of these two methods has a shortcoming
which cannot completely predict the velocity
distribution in a special area. As seen in Figs. 2 and
3, LSM has a defect in predicting the V(y) and
DTM has a defect in demonstrating the U(y). For
eliminating this shortcoming and achieving to a
reliable result, these two methods were combined
and introduced as a novel method called LS-DTM
combined method.

128
Fig. 2. Comparison of the DTM and LSM results for a)
U(y) and b) V(y) for Cu-Water nanofluid when Re=0.5,
Ha=0.5 and φ =0.05
Fig. 3 Comparison of the DTM and LSM results for a)
U(y) and b) V(y) for Silver-Ethylene glycol nanofluid
when Re=1, Ha=1 and φ =0.01
Table 1
Thermo physical properties of nanofluids and
nanoparticles
Material
Density(kg /
m3
)
Electrical
conductivit
y,σ(( .m)-1
)
Silver 10500 6.30×107
Copper 8933 5.96×107
Ethylene glycol 1113.2 1.07×10-4
Drinking water 997.1 0.05
Silver 10500 6.30×107
Copper 8933 5.96×107
Ethylene glycol 1113.2 1.07×10-4

129
Fig. 4 is improved form of Fig. 3 using LS-DTM
combined method. This figure reveals that LS-DTM
has an excellent agreement with numerical method
in whole areas, and so, it is an accurate and
convenient method for solving these kinds of
problems.
Fig. 4 LS-DTM results compared with numerical
procedure for Fig. 3 state.
Effect of Hartman number (Ha) on dimensionless
velocities for water with copper is shown in Figs. 5.
Generally, when the magnetic field is imposed on
the enclosure, the velocity field suppressed owing to
the retarding effect of the Lorenz force. For low
Reynolds number, as Hartmann number increases
( )V y decreases for my y> but opposite trend is
observed for my y< , my is a meeting point that all
curves joint together at this point. When Reynolds
number increases this meeting point shifts to the
solid wall and it can be seen that ( )V y decreases
with increase of Hartmann number. Fig. 6-a and b
show the effect of nanoparticle volume fraction on
U(y) and V(y) for water with copper nanoparticles
when Re=1 and Ha=1. Velocity boundary layer
thickness decreases with increase of nanoparticle
volume fraction.
Effect of Reynolds number (Re) on dimensionless
velocities is shown in Figs. 6-c and d. It is worth to
mention that the Reynolds number indicates the
relative significance of the inertia effect compared
to the viscous effect. Thus, velocity profile
decreases as Re increases and in turn increasing Re
leads to increase in the magnitude of the skin
friction coefficient. Contour plots of the effect of
Hartman number and nanoparticles volume fraction
in wide range of data are depicted in Figs. 7.
Table 2
Some fundamental operations of the differential
transform method
Origin function Transformed function
( ) ( ) ( )x t f x g tα β= ± ( ) ( ) ( )X k F k G kα β= ±
( )
( )
m
m
d f t
x t
dt
=
( )! ( )
( )
!
k m F k m
X k
k
+ +
=
( ) ( ) ( )x t f t g t=
0
( ) ( ) ( )
k
l
X k F l G k l
=
= −∑
( ) m
x t t= 1, ifk=m,
( ) ( )
0, ifk m.
X k k mδ

= − =
≠
( ) exp( )x t t= 1
( )
!
X k
k
=
( ) sin( )x t tω α= +
( ) sin( )
! 2
k
k
X k
k
ω π
α= +
( ) cos( )x t tω α= +
( ) cos( )
! 2
k
k
X k
k
ω π
α= +
The effects of the nanoparticle and liquid phase
material on velocity’s profiles are shown in Tables
3 and 4 for U(y) and V(y), respectively. These

130
tables reveal that when nanofluid includes copper
(as nanoparticles) or ethylene glycol (as fluid phase)
in its structure, the U(y) and V(y) values are greater
than the other structures.
Table 3
U(y) variations for different types of nanofluids and nanoparticles, Re=1, Ha=1 and φ =0.04.
y Water-Copper Water-Silver Ethylene glycol-Copper Ethylene glycol-Silver
0.0 1.0 1.0 1.0 1.0
0.1 0.83570944 0.83422691 0.8366021 0.8352604
0.2 0.68526977 0.68251844 0.6869270 0.6844364
0.3 0.55072934 0.54706012 0.5529409 0.5496176
0.4 0.43267586 0.42850606 0.4351915 0.4314120
0.5 0.33056756 0.32632363 0.3331308 0.3292806
0.6 0.24305045 0.23912555 0.2454244 0.2418595
0.7 0.16826652 0.16499404 0.1702490 0.1672728
0.8 0.10414284 0.10178554 0.1055735 0.1034265
0.9 0.04864722 0.04739956 0.0494059 0.0482677
1.0 0.0 0.0 0.0 0.0
Table 4
V(y) variations for different types of nanofluids and nanoparticles, Re=1, Ha=1 and φ =0.04.
y Water-Copper Water-Silver Ethylene glycol-Copper Ethylene glycol-Silver
0.0 0.0 0.0 0.0 0.0
0.1 0.02610404 0.025997188 0.026167546 0.026071871
0.2 0.097561353 0.097211791 0.097768963 0.097456128
0.3 0.204165243 0.203542825 0.204534660 0.203977939
0.4 0.335581162 0.334740930 0.336079488 0.335328393
0.5 0.48112359 0.480180063 0.481682731 0.480839846
0.6 0.62958117 0.628678803 0.630115463 0.629309909
0.7 0.769057186 0.768337156 0.769483116 0.768840824
0.8 0.88678803 0.886349813 0.887046994 0.886656410
0.9 0.96889373 0.96874723 0.968980208 0.968849751
1.0 1.0 1.0 1.0 1.0

38
Fig. 5. Effect of Hartman number (Ha) on dimensionless velocities for water with copper nanoparticles, φ =0.04, a)
U(y), Re=1, b) V(y), Re=1, c) U(y), Re=5 and d) V(y), Re=5.

Hatami
Fig. 6. Effect of nanoparticle volume fraction,
Effect of Reynolds number (Re) on c) U(y) and d) V(y) when Ha=10
135
Effect of nanoparticle volume fraction,φ , on a) U(y) and b) V(y), for Cu-Water when Re=1 and Ha=1.
Effect of Reynolds number (Re) on c) U(y) and d) V(y) when Ha=10φ =0.04.
n Re=1 and Ha=1.

Hatami
Fig. 7 Contour plot for showing the effect of Hartman number (Left) and nanoparticles fraction (Right) on V(y)
6. Conclusion
In this paper, Least Square and Differe
Transformation Methods are combined to
eliminate the shortcoming of each method for
solving the problem of laminar nanofluid flow in
a semi-porous channel in the presence of
uniform magnetic field. Outcomes reveal that
this method is an accurate, pow
convenient approach compared by numerical
method for solving this problem. The results
indicate that velocity boundary layer thickness
decrease with increase of Reynolds number and
nanoparticles volume fraction and it increases as
Hartmann number increases. Choosing copper
(as nanoparticles) and ethylene glycol (as fluid
phase) leads to maximum increment in velocity.
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136
Contour plot for showing the effect of Hartman number (Left) and nanoparticles fraction (Right) on V(y)
distribution
In this paper, Least Square and Differential
Transformation Methods are combined to
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. The results
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