International Journal of Heat and Mass Transfer 54 (2011) 854–862
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
Radiation effects on MHD flow of Maxwell fluid in a channel with porous medium
T. Hayat a,b,⇑, R. Sajjad a, Z. Abbas c, M. Sajid d, Awatif A. Hendi e
a
Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
c
Department of Mathematics, FBAS, International Islamic University, Islamabad 44000, Pakistan
d
Theoretical Plasma Physics Division, PINSTECH, P.O. Nilore, Islamabad 44000, Pakistan
e
Physics Department, Faculty of Science, King Saud University, P.O. Box 1846, Riyadh 11321, Saudi Arabia
b
a r t i c l e
i n f o
Article history:
Received 23 April 2010
Received in revised form 22 September 2010
Accepted 27 September 2010
Available online 8 November 2010
Keywords:
Maxwell fluid
Heat transfer
Porous medium
Porous channel
a b s t r a c t
This paper describes the heat transfer analysis with thermal radiation on the two-dimensional magnetohydrodynamic (MHD) flow in a channel with porous walls. The upper-convected Maxwell (UCM) fluid
fills the porous space between the channel walls. The corresponding boundary layer equations are transformed into ordinary differential equations by means of similarity transformations. The resulting problems are solved by employing homotopy analysis method (HAM). Convergence of the derived series
solutions is ensured. The effects of embedded parameters on the dimensionless velocity components
and temperature are examined through plots. The variation of local Nusselt number is also analyzed.
Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction
The flows in porous channels/tubes are of special interest in the
several applications in biomedical and mechanical engineering.
Such flows appear in the blood dialysis in artificial kidney, flow
in the capillaries, flow in blood oxygenators, the design of filters
and design of porous pipe. Many fluids of industrial importance
are non-Newtonian. An extension of flow analysis from viscous
to the non-Newtonian fluids is not so straightforward. In fact, the
difficulties occur by the diversity of non-Newtonian fluids in their
constitutive relationship and simultaneous effects of viscosity and
elasticity. These viscoelastic effects add complexities in the resulting differential equations. Some interesting contributions on the
topic can be found in the studies [1–15] and several references
therein.
The non-Newtonian fluids are mainly classified into three types
namely differential, rate and integral. The simplest subclass of the
rate type fluids is the Maxwell model [16]. This fluid model can
very well describe the relaxation time effects. It is worthmentioning to point out that Maxwell did not developed his model for polymeric liquids, but instead for air, the methodology used by him has
been extended by Rajagopal and Srinivasa [17] to produce a plethora of rate type models [18]. Choi et al. [19] discussed the steady
hydrodynamic boundary layer flow of an incompressible Maxwell
⇑ Corresponding author at: Department of Mathematics, College of Science, King
Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia. Tel.: +92 51 90642172.
E-mail address: pensy_t@yahoo.com (T. Hayat).
0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijheatmasstransfer.2010.09.069
fluid in a porous channel. Abbas et al. [20] reported magnetohydrodynamic effects on the flow analysis presented in a study [19].
Hayat et al. [21,22] extended this analysis for the second grade
and Jeffery fluids. Then Hayat and Abbas [23] discussed boundary
layer flow of an incompressible Maxwell fluid in porous channel
with chemical reaction.
The aim of present attempt is to venture further in this regime.
For that we have an interest to examine the steady boundary layer
flow of an upper-convected Maxwell fluid in a porous channel with
heat transfer analysis when radiation effects are present. An
incompressible fluid saturates the porous medium. The resulting
nonlinear problem is treated for a series solution by homotopy
analysis method (HAM) [24–45]. Convergence of the HAM solution
is established and the variations of emerging parameters are highlighted on the velocity and temperature.
2. Definition of the problem
Let us examine the heat transfer characteristics on MHD twodimensional flow of an incompressible upper-convected Maxwell
fluid in a channel with porous walls at y = ±H/2 (see, Fig. 1). Porous
medium fills the space between the walls of the channel. Flow is
induced by suction/blowing. A constant magnetic field B0 is applied in the y-direction and there is no external electric field. The
induced magnetic field is neglected under the assumption of small
magnetic Reynolds number. Furthermore, symmetric nature of the
flow is taken into account and pressure gradient is neglected.
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T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 854–862
Invoking the following non-dimensional parameters [19]
x ¼
x
;
H
hðy Þ ¼
y ¼
y
;
H
u ¼ Vx f 0 ðy Þ;
v ¼ Vf ðy Þ;
T
;
TH
ð9Þ
the similarity equations resulting from Eqs. (2) and (6) are
0
0
00 0
00
f 000 M Ref þ Deff Kf þ Reðf 02 ff Þ þ De 2ff f 00 f 2 f 000 ¼ 0;
ð10Þ
4
002
1 þ Rd h00 PrRef h0 þ PrEcf ¼ 0:
3
ð11Þ
Eqs. (7) and (8) now give
f ¼ 0;
f 00 ¼ 0;
h0 ¼ 0 at y ¼ 0;
1
;
2
f 0 ¼ 0;
h ¼ 1 at y ¼
Fig. 1. Physical model.
f ¼
Both the walls have same temperature TH. The temperature at
the centerline (y = 0) is @T
¼ 0. The boundary layer equations for
@y
the flow under consideration are
@u @ v
þ
¼ 0;
@x @y
qcp u
@T
@T
þv
@x
@y
¼ k0
2
@ 2 T @qr
@u
l
þ
@y2 @y
@y
ð2Þ
ð3Þ
in which u and v are the velocity components in the x and y-directions respectively, q is the fluid density, m is the kinematic viscosity,
r is the electrical conductivity, / is the porosity, k is the permeability of the porous medium, k is the relaxation time, cp is the specific
heat at constant temperature, k0 is the thermal conductivity of the
fluid, T is the temperature and qr is the radiative heat flux. Further, it
is pointed out that Eq. (2) is a correct version of the equation of motion in the previous studies [20,46–52].
Using the Rosseland approximation for radiation in an optically
thick layer [53] one obtains
qr ¼ 4
4r @T
;
3k @y
ð4Þ
M¼
rB20 H
H2 /
HV
kV 2
; Re ¼
; De ¼
;
; K¼
k
qV
m
m
Pr ¼
cp l
;
k0
Ec ¼
¼
k0 þ
16r T 3H
3k
!
2
@2T
@u
þ
l
:
@y2
@y
fy2nþ1 ; n P 0g;
ð16Þ
fy2n ; n P 0g;
ð17Þ
u ¼ 0;
v ¼ 0;
v¼
V
;
2
@T
¼ 0 at y ¼ 0;
@y
T ¼ TH
at y ¼
H
2
1
X
an y2nþ1 ;
ð18Þ
bn y2n ;
ð19Þ
n¼0
ð6Þ
hðyÞ ¼
1
X
n¼0
where an and bn are the coefficients to be determined. The initial
guesses f0(y) and h0(y) are chosen as
The relevant boundary conditions are
@u
¼ 0;
@y
ð15Þ
For HAM solutions, the set of base functions for f(y) and h (y) can
be expressed, respectively by
With the help of Eqs. (4) and (5), Eq. (3) becomes
ð14Þ
3. Homotopy analysis solutions
f ðyÞ ¼
@T
@T
þv
@x
@y
4r T 3H
k0 k
In next section, series solutions for problem given by Eqs. (10)–(13)
will be constructed by the homotopy analysis method.
in the form
Rd ¼
h
i
16r T 3
x k0 þ 3k H @T
@y
xqw
y¼1=2
Nux ¼
¼
k 0 ðT H Þ
k 0 ðT H Þ
4
¼ Re1=2 1 þ Rd h0 ð1=2Þ:
3
T 4 ffi 4T 3H T 3T 4H :
qcp u
V 2 x2
;
cp ðT H Þ
are respectively called the Hartman number M, the porosity parameter K, the Reynolds number Re, the Deborah number De, the Prandtl number Pr, the Eckert number Ec and radiation parameter Rd.
It is worth mentioning to note that Re > 0 corresponds to suction
case and Re < 0 for blowing. For Newtonian fluid De = 0.
The local Nusselt number Nux is defined as
where r* is the Stefan–Boltzmann constant and k* is the mean
absorption coefficient. By Taylors’ series about TH, T4 can be written
as
ð5Þ
ð13Þ
in which
ð1Þ
"
#
@u
@u
@2u
@2u
@2u
u
þv
þ k u2 2 þ v 2 2 þ 2uv
@x
@y
@x
@y
@x@y
2
2
@ u rB
@u
/m
¼ m 2 0 u þ kv
u;
@y
@y
q
k
1
2
ð12Þ
ð7Þ
ð8Þ
in which V > 0 corresponds to suction and V < 0 indicates
blowing.
3
2y2 ;
f0 ðyÞ ¼ y
2
ð20Þ
h0 ðyÞ ¼ 1:
ð21Þ
The auxiliary linear operators Lf and Lh with their properties are given below
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T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 854–862
Lf ðf Þ ¼ f 000 ;
ð22Þ
Lh ðf Þ ¼ f 00 ;
Lf C 0þ C 1 y þ C 2 y2 ¼ 0;
ð23Þ
ð24Þ
Lh ½C 3 þ C 4 y ¼ 0;
ð25Þ
where Ci(i = 0 4) are the arbitrary constants.
Now the zeroth-order deformation problems are
Finally the mth order deformation problems are
Lf ½fm ðyÞ vm fm1 ðyÞ ¼ hf Rfm ðyÞ;
ð39Þ
Lh ½hm ðyÞ vm hm1 ðyÞ ¼ hh Rhm ðyÞ;
ð40Þ
fm ð0Þ ¼ fm00 ð0Þ ¼ fm
1
1
¼ fm0 ð Þ ¼ 0;
2
2
ð1 qÞLf f ðy; qÞ f0 ðyÞ ¼ qhf N f f ðy; qÞ ;
ð26Þ
qÞ h0 ðyÞ ¼ qhh N h f ðy; qÞ; hðy; qÞ ;
ð1 qÞLh hðy;
ð27Þ
h0m ð0Þ ¼ hm
ð28Þ
0 0
000
Rfm ðyÞ ¼ fm1
M Re fm1
þ Defm1k fk00 Kf m1
0
2
3
0
00
Re fm1k fk fm1k fk
m1
X6
7
k 5;
þ
4
P
0
00
000
þDef
2f
f
f
f
m1k
kl
kl l
l
k¼0
f ð0; qÞ ¼ 0;
f 00 ð0; qÞ ¼ 0;
f 1 ; q ¼ 1 ;
2
2
f 0 1 ; q ¼ 0;
2
h 1 ; q ¼ 1;
2
h0 ð0; qÞ ¼ 0;
ð29Þ
!
!
2
@ 3 f ðy; qÞ
@ f ðy; qÞ
f ðy; qÞ @ f ðy; qÞ
þ
De
N f ½f ðy; qÞ ¼
M
Re
@y3
@y2
@y
0
1
!2
@f ðy; qÞ
@f ðy; qÞ
@ 2f ðy; qÞA
@
þ Re
f ðy; qÞ
K
@y2
@y
@y
!
3
@f ðy; qÞ @ 2f ðy; qÞ
2 @ f ðy; qÞ
;
ðf ðy; qÞÞ
þ De 2f ðy; qÞ
@y2
@y3
@y
ð30Þ
f ðy; 1Þ ¼ f ðyÞ;
hðy; 0Þ ¼ h0 ðyÞ;
hðy; 1Þ ¼ hðyÞ:
ð31Þ
ð32Þ
when q varies from 0 to 1, f ðy; qÞvaries from f0(y) to f(y) and
hðy; qÞvaries from h0(y) to h(y). In view of Taylors’ series
f ðy; qÞ ¼ f0 ðyÞ þ
1
X
fm ðyÞqm ;
Rhm ðyÞ ¼
vm ¼
m1
X
4
1 þ Rd h00m1 PrRefk h0m1k PrEcfm1k fk00 ;
3
k¼0
0;
m 6 1;
1;
m > 1:
1
X
ð34Þ
hm ðyÞ ¼ hm ðyÞ þ C 3 þ C 4 y;
ð46Þ
hm ðyÞ
where
and
denote the particular solutions. Now Eqs. (39)
and (40) can be solved by Mathematica one after the other in the
order m = 1 ,2, 3, . . .
4. Convergence of the HAM solutions
Convergence of derived series solutions (37) and (38) strongly
depend upon the auxiliary parameters hf and hh . For this purpose,
we plot h
f and hh -curves in the Figs. 2 and 3. From these figures, it
is clear that admissible ranges of hf and hh are 1:65 6 hf 6 0:8
and 1:3 6 hh 6 0:35, respectively. However all forthcoming
computations are made when hf ¼ 1 ¼ hh . Convergence of the derived series solutions is ensured and shown in Table 1.
;
ð35Þ
m¼1
in which
fm ðyÞ ¼
hm ðyÞ ¼
1 @ mf ðy; qÞ
m! @qm
1 @ m hðy; qÞ
m! @qm
q¼0
ð36Þ
q¼0
and auxiliary parameters h
f and hh are so properly chosen that the
series (33) and (34) converge at q = 1 and so
f ðyÞ ¼ f0 ðyÞ þ
1
X
fm ðyÞ;
ð37Þ
m¼1
hðyÞ ¼ h0 ðyÞ þ
1
X
m¼1
hm ðyÞ:
ð44Þ
ð45Þ
ð33Þ
hm ðyÞqm
ð43Þ
fm ðyÞ ¼ fm ðyÞ þ C 0 þ C 1 y þ C 2 y2 ;
m¼1
hðy; qÞ ¼ h0 ðyÞ þ
ð42Þ
The general solution of Eqs. (39)–(41) are
fm ðyÞ
Here hf and hh indicate the non-zero auxiliary parameters and
q 2 [0, 1] is an embedding parameter such that
f ðy; 0Þ ¼ f0 ðyÞ;
ð41Þ
l¼0
where the nonlinear operators N f and N h are given by
2
4
@ hðy; qÞ
@ hðy; qÞ
N h ½hðy; qÞ ¼ 1 þ Rd
PrRef ðy; qÞ
@y2
3
@y
!2
@ 2 f ðy; qÞ
þ PrEc
:
@y2
1
¼ 0;
2
ð38Þ
Fig. 2. ⁄f-curve of f000 (1/2) for 20th-order of approximation.
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T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 854–862
Re = - 20, De = 0, K = 0.5
2
M
M
M
M
f (y) and f ' (y)
1.5
=
=
=
=
0
1
2
4
f'
1
f
0.5
0
0
0.1
0.2
0.3
0.4
0.5
y
Fig. 5. Effects of M on f and f0 in case of blowing Re < 0.
Fig. 3. ⁄h-curve of h0 (1/2) for 20th-order of approximation.
Table 1
Convergence of the HAM solution for different order of approximations.
M = 0.5, Re = 20, De = 0
f000 (1/2)
h0 (1/2)
01
05
10
15
20
25
30
18.45714286
20.83493238
20.84666717
20.84668110
20.84668112
20.84668112
20.84668112
12.0000000
9.90965002
9.90738003
9.90737855
9.90737856
9.90737856
9.90737856
5. Results and discussion
f (y) and f ' (y)
1.5
Order of approximation
In order to see the physical insight of the considered problem,
the influence of emerging parameters on the dimensionless velocity components, temperature and the local Nusselt number both
for suction and blowing are analyzed in this section. Sections 5.1
and 5.2 describe the respective results of Newtonian (De = 0) and
viscoelastic (non-Newtonian De – 0) fluids. Figs. 4–27 have been
plotted in this regard.
1
f'
=
=
=
=
0
4
8
16
f
0.5
0
0
K
K
K
K
0.1
0.2
0.3
0.4
0.5
y
Fig. 6. Effects of K on f and f0 in case of suction Re > 0.
5.1. Newtonian fluid (De = 0)
M = 0.5, Re = - 20, De = 0
1.6
Figs. 4–15 are plotted in order to illustrate the influences of M,
K, Re, Rd, Pr and Ec on the dimensionless velocity components and
K
K
K
K
1.4
1.2
1.5
f (y) and f ' (y)
M
M
M
M
1
=
=
=
=
f (y) and f ' (y)
Re = 20, De = 0, K = 0.5
0
1
2
4
f'
1
=
=
=
=
0
4
8
16
f'
0.8
0.6
f
0.4
0.2
f
0.5
0
0
0.1
0.2
0.3
0.4
0.5
y
Fig. 7. Effects of K on f and f0 in case of blowing Re < 0.
0
0
0.1
0.2
0.3
0.4
y
0
Fig. 4. Effects of M on f and f in case of suction Re > 0.
0.5
temperature f, f0 and h for the Newtonian fluid (De = 0) with suction
and blowing.
Fig. 4 shows that by increasing M for suction Reynolds number
(Re > 0) f decreases and f0 decreases for y < 0.23 whereas it
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T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 854–862
Fig. 8. Effects of Re on h for the suction case.
Fig. 11. Effects of Rd on h, for the blowing case.
Fig. 9. Effects of Re on h for the blowing case.
Fig. 12. Effects of Pr on h for the suction case.
Fig. 10. Effects of Rd on h for the suction case.
Fig. 13. Effects of Pr on h for the blowing case.
increases when y > 0.23. Fig. 5 depicts that by increasing M for
blowing Reynolds number (Re < 0) f increases and f0 increases for
y < 0.22 whereas it decreases when y > 0.22. It is noticed that for
suction and blowing f and f0 show opposite behavior when M
varies.
Figs. 6 and 7 indicate that by increasing K (both for suction and
blowing Reynolds number Re(?0)) f decreases and f0 decreases for
y < 0.22 whereas it increases when y > 0.22. It is noticed that suction and blowing do not alter f and f0 when K varies. Figs. 8–15 have
been prepared in order to check the variations of Re, Rd, Pr and Ec
on temperature profile h in suction and blowing cases. It is evident
from Figs. 8 and 9 that h increases by increasing suction Reynolds
number Re > 0 whereas it decreases for blowing Reynolds number
Re < 0. The effect of Rd for Re ? 0 on h are plotted in Figs. 10 and 11.
Here h decreases by increasing Rd both for suction and blowing
Reynolds number Re ? 0 (Re = ±20). Thus it is observed that suction
and blowing do not alter h when Rd varies. The influence of Pr both
859
T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 854–862
M = 0.5, Re = 20, De = 0.5
f (y) and f ' (y)
1.5
K
K
K
K
f'
1
=
=
=
=
0
4
8
16
f
0.5
0
0
0.1
0.2
0.3
0.4
0.5
y
Fig. 14. Effects of Ec on h for the suction case.
Fig. 17. Effects of K on f and f0 in case of suction Re > 0, (when De > 0).
Fig. 15. Effects of Ec on h for the blowing case.
Fig. 18. Effects of Re on h for the suction case (when De > 0).
Re = 20, De = 0.5, K = 0.5
1.5
f (y) and f ' (y)
M
M
M
M
1
0
1
2
4
f'
f
0.5
0
=
=
=
=
0
0.1
0.2
0.3
0.4
0.5
y
Fig. 16. Effects of M on f and f0 in case of suction Re > 0 (when De > 0).
for suction and blowing Reynolds number Re ? 0 on h are plotted
in the Figs. 12 and 13. Here h increases by increasing Pr both for
Re ? 0 (Re = ±20). So the effect of Pr on h both for suction and blowing is quite similar. For the Newtonian case, the behavior of Ec for
Re ? 0 on temperature h are plotted in Figs. 14 and 15. This shows
that h increases by increasing Ec both for suction and blowing
cases.
Fig. 19. Effects of Rd on h for the suction case (when De > 0).
5.2. Viscoelastic (non-Newtonian) fluid ðDe – 0Þ
5.2.1. Suction case (Re P 0, De > 0)
Figs. 16–21 are plotted in order to see the variations of K, Re, Rd,
Pr and Ec on the dimensionless velocity components and temperature f, f0 and h for non-Newtonian fluid with suction flow when
De > 0.
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T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 854–862
M = 0.5, Re = - 20, De = - 0.5
f (y) and f ' (y)
1.5
f'
1
=
=
=
=
0
4
8
16
f
0.5
0
0
Fig. 20. Effects of Pr on h for the suction case (when De > 0).
K
K
K
K
0.1
0.2
0.3
0.4
0.5
y
Fig. 23. Effects of K on f and f0 in case of blowing Re < 0 (when De < 0).
Fig. 21. Effects of Ec on h for the suction case (when De > 0).
Fig. 24. Effects of Re on h for the blowing case (when De < 0).
Re = - 20, De = - 0.5, K = 0.5
2
M
M
M
M
f (y) and f ' (y)
1.5
=
=
=
=
0
1
2
4
f'
1
f
0.5
0
0
0.1
0.2
0.3
0.4
0.5
y
Fig. 22. Effects of M on f and f0 in case of blowing Re < 0 (when De < 0).
Fig. 16 illustrates that by increasing M for suction Reynolds
number (Re > 0) when De > 0, f decreases and f0 decreases for
y < 0.23 whereas it increases when y > 0.23.
Fig. 17 shows the influence of K on f and f0 for Re > 0 and De > 0,
which shows qualitative similar behavior as noted for the Newtonian fluid (Fig. 6).
Now for suction flow (Re > 0) when De > 0, Figs. 18, 20 and 21
indicate that the dimensionless temperature h increases by
Fig. 25. Effects of Rd on h for the blowing case (when De < 0).
increasing Re, Pr and Ec, respectively while Fig. 19 depicts that h
decreases by increasing Rd.
5.2.2. Blowing case (Re < 0, De < 0)
Figs. 22–27 are plotted in order to see the influences of K, Re, Rd,
Pr and Ec on the dimensionless velocities and temperature f, f0 and
h for the non-Newtonian fluid in blowing case when De < 0.
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T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 854–862
Table 2
Numerical values of local Nusselt number for the suction case.
Re
M
0
5
10
20
5
1.0
0
2
4
8
1
K
1.0
0
4
8
16
1
Fig. 26. Effects of Pr on h for the blowing case (when De < 0).
De
Rd
Pr
Ec
Re1/2Nux
1.0
0.5
0.5
0.5
1.500753
1.635831
1.786391
2.182340
1.632181
1.651119
1.701858
1.835777
1.634007
1.643906
1.659531
1.701858
1.632262
1.635831
1.639995
1.647842
1.721740
1.659672
1.626706
1.606267
0.625678
1.289103
1.993054
2.740532
1.635831
3.271662
4.907493
6.543324
0.0
1.0
1.5
2.0
1.0
0.0
0.3
0.6
0.9
0.5
0.2
0.4
0.6
0.8
0.5
0.5
1.0
1.5
2.0
Table 3
Numerical values of local Nusselt number for the blowing case.
Fig. 27. Effects of Ec on h for the blowing case (when De > 0).
Fig. 22 depicts that by increasing M for blowing Reynolds number (Re < 0) when De < 0, f increases and f0 increases for y < 0.22
whereas it decreases when y > 0.22.
Fig. 23 plots the variation of K on f and f0 for Re < 0 and De < 0
which shows similar behavior as that for the Newtonian fluid with
suction and blowing cases (Figs. 6 and 7) and also for the non-Newtonian fluid with suction case (Fig. 17).
For blowing case (Re < 0) when De < 0, Figs. 24 and 25 depict
that h decreases by increasing blowing Reynolds number Re(<0)
and Rd, respectively whereas Figs. 26 and 27 indicate that it increases by increasing Pr and Ec.
Tables 2 and 3 present the numerical values of local Nusselt
number (Eq. (15)) for different values of embedding parameters
both for suction and blowing cases, respectively. The numerical
values of local Nusselt number increases for large suction Reynolds
number and it decreases for large blowing Reynolds number. The
numerical values of local Nusselt number increases by increasing
M, K, De, Pr and Ec while it decreases by increasing Rd for the suction case (Table 2). However the numerical values of local Nusselt
number increases by increasing M, K, De, RdPr and Ec in the blowing
case (Table 3).
6. Concluding remarks
The flow and heat transfer in a channel with porous medium in
the presence of radiation are discussed. The nonlinear analysis is
computed and results for velocity and temperature distributions
are obtained using the homotopy analysis method. The conver-
Re
M
5
10
20
5
1
2
4
6
8
1
K
1
0
4
8
16
1
De
Rd
Pr
Ec
Re1/2Nux
1
0.5
0.5
0.5
1.373870
1.252644
1.029914
1.377779
1.427874
1.578043
1.933996
1.373832
1.376067
1.383122
1.407723
1.373870
1.374013
1.374365
1.375137
1.295588
1.335939
1.363268
1.382998
0.580176
1.118977
1.619599
2.084982
1.373870
2.747741
4.121611
5.495482
1.0
1.5
2.0
2.5
1.0
0.0
0.2
0.4
0.6
0.5
0.2
0.4
0.6
0.8
0.5
0.5
1.0
1.5
2.0
gence of the results is explicitly shown. The main findings are
listed below.
For suction and blowing, the influence of M on the dimensionless velocity components f and f0 is qualitatively opposite.
The variation of K on the dimensionless velocity components f
and f0 is qualitative similar in suction and blowing.
The dimensionless temperature h shows opposite behavior in
suction and blowing cases for the variation of Re.
862
T. Hayat et al. / International Journal of Heat and Mass Transfer 54 (2011) 854–862
The effect of Rd, Pr and Ec on temperature are qualitatively similar in blowing and suction cases.
The numerical values of local Nusselt number for the suction
and blowing cases show qualitatively similar results for the
variations of M, K, De, Pr and Ec, and opposite results for the
variations of Re and Rd.
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