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Wall properties and slip effects on the magnetohydrodynamic peristaltic motion of a viscous fluid with heat transfer and porous space.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2011; 6: 649–658
Published online 28 June 2010 in Wiley Online Library
(wileyonlinelibrary.com) DOI:10.1002/apj.470
Research article
Wall properties and slip effects on the
magnetohydrodynamic peristaltic motion
of a viscous fluid with heat transfer and porous
space
T. Hayat1,2 * and Maryiam Javed1
1
2
Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan
Department of Mathematics, College of Sciences, King Saud University, PO Box 2455, Riyadh 11451, Saudi Arabia
Received 8 March 2010; Revised 29 April 2010; Accepted 23 May 2010
ABSTRACT: Peristaltic flows are important for understanding the transport of fluids, especially blood flow, in
physiological systems and in biomedical instruments. Several studies have considered the peristaltic flows of Newtonian
and non-Newtonian fluids with one or more simplifying assumptions, and for peristaltic flows in biomechanics
applications such as magnetohydrodynamic (MHD) pumps and blood flow. This paper presents a theoretical study
of the effects of slip conditions and wall properties on the MHD peristaltic flow of a viscous fluid filling the porous
space in a channel. The heat transfer analysis is also presented. The analytical solutions of the stream function,
velocity and temperature distributions are constructed in closed form under long wavelength and low Reynolds number
approximations. The effects of various influential key parameters and groups are reported and discussed.  2010 Curtin
University of Technology and John Wiley & Sons, Ltd.
KEYWORDS: magnetohydrodynamics; peristaltic flow; slip effects; wall properties; porous media; heat transfer
INTRODUCTION
During the past few decades, the peristaltic flows
have drawn intensive attention from many mathematicians, engineers, modelers and computer scientists. Areas of applications of peristaltic flows cover
widely the transport of fluids in physiological systems, in biomedical instruments, in industry, etc.
There are several attempts dealing with the peristaltic flows of Newtonian and non-Newtonian fluids
under one or more simplified assumptions.[1 – 15] Jaffrin and Shapiro[15] and Rath[16] have made a complete survey of peristaltic flows in the domain of
biomechanics.
Little has been said about the influence of heat
transfer on the peristaltic transport of fluids. Vajravelu
et al .[17] discussed the heat transfer effects on peristaltic motion in a vertical porous annulus. Mekheimer
and Elmaboud[18] studied the heat transfer and MHD
effects on the peristaltic transport of a viscous fluid
in a vertical annulus. Srinivas and Kothandapani[19]
*Correspondence to: T. Hayat, Department of Mathematics, Quaidi-Azam University, Islamabad 44000, Pakistan.
E-mail: pensy t@yahoo.com
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Curtin University is a trademark of Curtin University of Technology
examined the heat transfer analysis for MHD peristaltic flow of a viscous fluid in an asymmetric
channel. Very recently, Radhakrishnamacharya and
Srinivasulu[20] discussed the influence of wall properties on the peristaltic transport with heat transfer in a
uniform channel. Kothandapani and Srinivas[21] studied the effect of wall properties on the MHD peristaltic
transport with heat transfer and porous medium. Note
that in Refs [18–21] no-slip condition is taken into
account.
To date the slip, heat transfer and wall effects on
the MHD peristaltic transport in a porous space has
not been analyzed to the author’s knowledge. The
goal is to develop a model that discusses the slip
and heat transfer effects on the peristaltic flow in a
channel with compliant walls. It is worth mentioning
to note that such analysis for hydrodynamic fluid is
even not available yet in the existing literature. The
velocity slip condition is imposed in terms of the shear
stress. The MHD consideration is important especially
in MHD pumps and blood flow. In the present study, the
analytic solutions to the axial velocity, stream function,
temperature and heat transfer coefficient are presented.
Moreover, some flow quantities of interest are analyzed
through graphical results.
T. HAYAT AND M. JAVED
Asia-Pacific Journal of Chemical Engineering
(b)
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
M = 1.5
M = 2.0
M = 2.5
M = 3.0
0
1
y
y
(a)
2
3
4
5
6
7
8
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
9
K = 0.2
K = 0.5
K = 2.0
K = inf
0
1
2
3
u
4
5
7
6
u
(c)
(d)
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-0.5
β = 0.0
β = 0.1
β = 0.2
β = 0.3
0
y
y
0.5
u
1
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
1.5
ε = 0.10
ε = 0.15
ε = 0.20
ε = 0.25
0
1
2
3
4
5
6
7
8
u
(e)
y
650
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
E1 = 1, E2 = 0.5, E3 = 0.1
E1 = 2, E2 = 0.3, E3 = 0.2
E1 = 1, E2 = 1, E3 = 0.1
E1 = 2, E2 = 0.5, E3 = 0
0
2
4
6
u
8
10
12
Figure 1. Variations of the longitudinal velocity u for different values of Hartman number M, porosity parameter
K, velocity slip parameter β, amplitude ratio ε and wall parameters. The other parameters chosen are E1 = 0.5,
E2 = 0.2, E3 = 0.1, ε = 0.2, K = 5, β = 0.2, x = 0.2 and t = 0.2 (a); E1 = 1.0, E2 = 0.5, E3 = 0.1, ε = 0.1, M = 2,
β = 0.2, x = 0.2 and t = 0.2 (b); E1 = 1, E2 = 0.5, E3 = 0.5, ε = 0.1, K = 5, M = 2, x = 0.3 and t = 0.1 (c); E1 = 0.5,
E2 = 0.2, E3 = 0.1, M = 2, K = 5, β = 0.2, x = 0.2 and t = 0.2 (d) and ε = 0.1, M = 2, K = 5, β = 0.2, x = 0.2 and
t = 0.2 (e).
MATHEMATICAL MODEL
We consider a channel of width 2d. The channel is
filled with an incompressible and magnetohydrodynamic (MHD) viscous fluid. A uniform magnetic field
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
B = (0, B0 , 0) is applied in the y-direction. The induced
magnetic field is negligible for small magnetic Reynolds
number. The fluid fills the porous space. The temperatures of the upper and lower walls of the channel are
maintained at T0 and T1 , respectively. The imposed
Asia-Pac. J. Chem. Eng. 2011; 6: 649–658
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
MAGNETOHYDRODYNAMIC PERISTALTIC MOTION OF A VISCOUS FLUID
(a)
(b)
1
1
0.5
M = 1.0
M = 1.5
M = 2.0
M = 2.5
0
y
y
0.5
-0.5
-1
-1
0.5
1
θ
1.5
2
0
1
2
3
θ
4
5
6
(d)
(c)
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
1
0.5
β = 0.0
γ = 0.0
γ = 0.02
γ = 0.05
γ = 0.10
β = 0.1
β = 0.2
y
y
0
-0.5
0
K = 0.2
K = 0.5
K = 2.0
K = inf
β = 0.3
0
-0.5
-1
0
0.5
1
θ
1.5
0
2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
θ
Figure 2. Variations of temperature distribution for different values of Hartman number M, porosity parameter
K, velocity slip parameter β, thermal slip parameter γ , Brinkman number Br , amplitude ratio ε and wall parameters.
The other parameters chosen are E1 = 0.4, E2 = 0.2, E3 = 0.1, ε = 0.1, K = 5, β = 0.2, γ = 0.1, x = 0.4, Br = 1
and t = 0.1 (a); E1 = 1, E2 = 0.2, E3 = 0.1, ε = 0.2, M = 2, β = 0.2, γ = 0.1, x = 0.4, Br = 2 and t = 0.2 (b);
E1 = 0.8, E2 = 0.2, E3 = 0.1, ε = 0.1, M = 2, K = 5, γ = 0.1, x = 0.3, Br = 2 and t = 0.1 (c); E1 = 0.8, E2 = 0.2,
E3 = 0.1, ε = 0.2, M = 2, K = 5, x = 0.3, Br = 2 and t = 0.08 (d); E1 = 0.5, E2 = 0.2, E3 = 0.1, ε = 0.2, M = 2,
K = 5, β = 0.2, γ = 0.1, x = 0.3 and t = 0.1 (e); E1 = 0.5, E2 = 0.2, E3 = 0.1, M = 2, K = 5, β = 0.2, γ = 0.1,
Br = 2, x = 0.1 and t = 0.3 (f) and ε = 0.1, M = 2, K = 5, β = 0.2, γ = 0.1, Br = 2, x = 0.3 and t = 0.1 (g).
traveling waves of small amplitudes on the insulating
walls of the channel are:
2π
(x − ct)
y = ±η = ± d + a sin
λ
+µ
(2)
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
∂2
∂x
2
+
∂2
∂y
2
u − σ B02 u −
∂
∂
∂p
∂
+u
+v
v =−
ρ
∂t
∂x
∂y
∂y
2
2
µ
∂
∂
+ 2 v− v
+µ
2
k
∂x
∂y
∂
∂
κ
∂
+u
+v
T = ∇ 2T
ζ
∂t
∂x
∂y
ρ
  2 2 
 ∂u + ∂v 
∂x
∂y

+ ν 2
 + ∂u + ∂v 2 
∂x
∂y
(1)
where λ is the wavelength, c is the wave speed, a
is the wave amplitude and x and y are the Cartesian
coordinates with x measured in the direction of the wave
propagation and y measured in the direction normal to
the mean position of the channel walls.
The equations which can govern the motion for the
present flow are:
∂v
∂u
+
=0
∂x
∂y
∂
∂
∂
∂p
ρ
+u
+v
u=−
∂t
∂x
∂y
∂x
µ
u
k
(3)
(4)
(5)
subjected to the boundary conditions
u ±α
∂u
= 0 aty = ±η
∂y
Asia-Pac. J. Chem. Eng. 2011; 6: 649–658
DOI: 10.1002/apj
651
T. HAYAT AND M. JAVED
(e)
Asia-Pacific Journal of Chemical Engineering
(f)
Br = 0.0
1
Br = 1.0
0.4
Br = 3.0
0.5
0.8
0.6
Br = 2.0
ε = 0.10
ε = 0.15
ε = 0.18
ε = 0.20
0.2
0
y
y
0
-0.2
-0.5
-0.4
-0.6
-1
-0.8
0
0.5
1
1.5
θ
2
2.5
0
3
0.5
1
θ
1.5
2
(g)
y
652
E1 = 0.5, E2 = 0.1, E3 = 0.3
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
E1 = 1, E2 = 0.1, E3 = 0.3
E1 = 0.6, E2 = 0.1, E3 = 0
E1 = 0.7, E2 = 0.3, E3 = 0.3
0
0.5
1
1.5
θ
Figure 2. (Continued).
2π
= ± d + a sin
(x − ct)
(6)
λ
∂p
∂ 2u
∂ 2u
∂
L(η) =
=µ
+ 2
∂x
∂x
∂y 2
∂y
∂u
∂u
∂u
+u
+v
−ρ
∂t
∂x
∂y
µ
− σ B02 u − u at y = ±η
(7)
k
∂T
T −ξ
= T0 on y = −η
(8)
∂y
∂T
= T1 on y = η
(9)
T +ξ
∂y
where u and v are the velocity components in x and
y directions, respectively, and ρ, t, p, µ, ν, σ , ζ , κ
and T are the fluid density, the time, the pressure, the
dynamic viscosity, the kinematic viscosity, the electrical conductivity, the specific heat, the thermal conductivity and the temperature,
respectively.
Furthermore,
L = −τ ∂ 2 ∂x 2 + m∂ 2 ∂t 2 + C ∂ ∂t, τ is the elastic
tension in the membrane, m is the mass per unit area
and C is the coefficient of viscous damping.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
On setting
u=
∂
∂
, v =−
∂y
∂x
(10)
and defining the following nondimensional variables
and parameters:
x
y
, y ∗ = , ∗ =
,
λ
d
cd
ct
η
d 2p
t ∗ = , η∗ = , p ∗ =
λ
d
µcλ
d
k
a
K∗ = 2, ε = , δ = , M2
d
λ
d
σ B02 d 2
τd3
, E1 = − 3
=
µ
λ µc
x∗ =
E2 =
mcd 3
λ3 µ
, E3 =
Cd 3
λ2 µ
(12)
, Pr
c2
ρνζ
, E=
κ
ζ (T1 − T0 )
T − T0
cd
, θ=
R=
ν
T1 − T0
=
(11)
(13)
(14)
Asia-Pac. J. Chem. Eng. 2011; 6: 649–658
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
(a) 6
(b)
M=0
M=2
M=4
4
3
K = 1.0
2
1
0
0
-2
-1
-4
-2
-6
-3
0
0.2
K = 0.2
2
Z
Z
MAGNETOHYDRODYNAMIC PERISTALTIC MOTION OF A VISCOUS FLUID
0.4
0.6
0.8
K = inf
0
1
0.2
0.4
(c)
(d) 0.5
β = 0.0
β = 0.1
β = 0.2
6
0.6
0.8
1
0.6
0.8
1
x
x
γ = 0.0
γ = 0.1
γ = 0.2
0.4
0.3
4
0.2
0.1
0
0
Z
Z
2
-0.1
-2
-0.2
-4
-0.3
-6
-0.4
-0.5
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
x
x
(e) 2.5
Br = 1
2
Br = 2
(f) 6
4
Br = 3
1.5
1
2
Z
Z
0.5
0
-0.5
0
-2
-1
E1 = 0.5, E2 = 0.1, E3 = 0.5
E1 = 1, E2 = 0.1, E3 = 0.5
-1.5
-4
E1 = 0.5, E2 = 0.5, E3 = 0.5
-2
E1 = 1.0, E2 = 0.1, E3 = 0.0
-6
-2.5
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
x
x
Figure 3. Variations of heat transfer coefficient Z for different values of Hartman number M, porosity parameter K,
velocity slip parameter β, thermal slip parameter γ , Brinkman number Br and wall parameters. The other parameters
chosen are E1 = 1, E2 = 0.2, E3 = 0.1, ε = 0.2, K = 0.05, β = 0.2, γ = 0.1, Br = 1 and t = 0.2 (a); E1 = 0.3,
E2 = 0.2, E3 = 0.1, ε = 0.2, M = 4, β = 0.2, γ = 0.1, Br = 2 and t = 0.1 (b); E1 = 0.3, E2 = 0.2, E3 = 0.1, ε = 0.2,
M = 4, K = 0.2, γ = 0.1, Br = 2 and t = 0.2 (c); E1 = 0.8, E2 = 0.1, E3 = 0.1, ε = 0.2, M = 2, K = 0.02, β = 0.2,
Br = 2 and t = 0.2 (d); E1 = 0.3, E2 = 0.2, E3 = 0.1, ε = 0.2, M = 4, K = 0.2, β = 0.2, γ = 0.1 and t = 0.2 (e) and
ε = 0.15, M = 1, K = 0.05, Br = 3, β = 0.2, γ = 0.1 and t = 0.25 (f).
Equation (2) is automatically satisfied and Eqns (3)–
(9) under long wavelength and low Reynolds number
are reduced to:
∂ 3
∂p
−
=0
−N
3
∂y
∂x
∂y
2 ∂
(15)
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
1
Pr
∂y 2
∂ 3
∂y
∂ 2θ
3
− N2
+E
∂ 2
∂y 2
2
=0
(16)
∂
∂y
Asia-Pac. J. Chem. Eng. 2011; 6: 649–658
DOI: 10.1002/apj
653
654
T. HAYAT AND M. JAVED
Asia-Pacific Journal of Chemical Engineering
(a)
(b)
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
(c)
1.5
1
0.5
0
-0.5
-1
-1.5
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Figure 4. Streamlines for (a) M = 0, (b) M = 4, (c) M = 6. The other parameters chosen are (ε = 0.2, β = 0.1, E1 = 0.3,
E2 = 0.1, E3 = 0.1, K = 0.04 and t = 0).
∂3
∂3
∂2
= E1 3 + E2
+
E
3
∂x ∂t
∂x
∂x ∂t 2
η at y = ±η
SOLUTION OF THE PROBLEM
The closed form solutions of Eqns (15)–(20) are
(17)
2
=L
∂ ∂
± β 2 = 0 at y = ±η
∂y
∂y
= ±[1 + ε sin 2π(x − t)]
∂θ
θ −γ
= 0 at y = −η
∂y
∂θ
θ +γ
= 1 at y = η
∂y
2
(18)
(19)
(20)
in which asterisks have been suppressed for
simplicity,
∂p/∂y = 0, is a stream function, N = M 2 + 1/K ,
M is the Hartman number, K is the porosity parameter,
Pr is the Prandtl number and E is the Eckert number.
Furthermore, Eqns (18)–(20) are the slip conditions
for the velocity and temperature, respectively, and
β = α/d, γ = ξ/d (α and ξ are the dimensional slip
parameters corresponding to velocity and temperature).
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
sinh Ny
−y
N (cosh N η + βN sinh N η)
Br L4 N 2
8(cosh N η + βN sinh N η)2
cosh 2N η − cosh 2Ny
+2(y 2 − η2 )
2
N
sin 2N η
+4γ
−η
2N
y +η+γ
+
2(η + γ )
8επ 3 E3 sin 2π(x − t)
2
L =
2π
N2
(21)
θ=
−(E1 + E2 ) cos 2π(x − t)
(22)
(23)
Asia-Pac. J. Chem. Eng. 2011; 6: 649–658
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
MAGNETOHYDRODYNAMIC PERISTALTIC MOTION OF A VISCOUS FLUID
(a)
(b)
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-0.2
0.6
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
(c)
1.5
1
0.5
0
-0.5
-1
-1.5
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Figure 5. Streamlines for (a) K = 0.05, (b) K = 0.2, (c) K = ∞. The other parameters chosen are (ε = 0.17, β = 0.1,
E1 = 0.3, E2 = 0.1, E3 = 0.1, M = 4 and t = 0).
where Br = Pr E is the Brinkman number and the
longitudinal velocity is
∂
∂y
2
=L
u=
cosh Ny
−1
(cosh N η + βN sinh N η)
(24)
The heat transfer coefficient at the upper wall is
Z = ηx θ y
=
ηx Br L4 N 2
8(cosh N η + βN sinh N η)2
ηx
−2
sinh N η + 4η +
N
2(η + γ )
(25)
GRAPHICAL RESULTS AND DISCUSSION
The behavior of the longitudinal velocity, stream function, temperature and heat transfer coefficient is exhibited in this section. For this purpose, Figs (1–7) are
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
displayed. Figure 1 is plotted to examine the effect of
various parameters on the longitudinal velocity u. One
observes from Fig. 1(b) that the longitudinal velocity
u increases when the porosity parameter K increases,
whereas it decreases when the Hartman number M
is increased Fig. 1(a). Figure 1(c) and (d) records the
behavior of velocity slip parameter β and the occlusion parameter ε on the velocity, respectively. These
figures depict an increase in the velocity when M and
ε are increased. The effect of the elastic parameters E1 ,
E2 and E3 are evident in Fig. 1(e). It may be of interest to note from this figure that with increasing elastic
parameters, the velocity increases. It is also interesting
to note that the velocity profile is parabolic for fixed
values of the parameters and its magnitude is maximum
near the center of the channel. Moreover, it is observed
that the elastic tension E1 has a significant effect on the
axial velocity as compared with the mass characterizing parameter E2 and the damping nature of the wall E3 .
In Fig. 2, the nature of the temperature profile is also
parabolic. Here, the temperature decreases by increasing the Hartman number M (Fig. 2(a)) and the velocity
Asia-Pac. J. Chem. Eng. 2011; 6: 649–658
DOI: 10.1002/apj
655
656
T. HAYAT AND M. JAVED
Asia-Pacific Journal of Chemical Engineering
(a)
(b)
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
-0.2
-0.1
0.3
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
(c)
1.5
1
0.5
0
-0.5
-1
-1.5
-0.2
-0.1
0
0.1
0.2
0.5
0.6
Figure 6. Streamlines for (a) β = 0, (b) β = 0.1, (c) β = 0.2. The other parameters chosen are (ε = 0.17, E1 = 0.3,
E2 = 0.1, E3 = 0.1, M = 4, K = 0.05 and t = 0).
slip parameter β (Fig. 2(c)). Note that the temperature
decreases in the downstream. However, Fig. 2(b) and
(d) illustrates that the temperature increases by increasing the permeability parameter K and the thermal slip
parameter γ . The variations of the Brinkman number
Br and the occlusion parameter ε on the temperature
are sketched in Fig. 2(e) and (f). It is noted from these
figures that temperature is an increasing function of Br
and ε. Figure 2(g) elucidates the effect of the elastic
parameters E1 , E2 and E3 on the temperature. This figure
reveals that the amplitude of temperature increases with
an increase in E1 , E2 and E3 . It is further observed
that the effect of E1 on temperature is quite significant.
The results presented in Fig. 3 indicate the behavior of
M , K , β, γ , Br , E1 , E2 and E3 on the heat transfer
coefficient Z . This figure shows the typical oscillatory
behavior of heat transfer which may be due to the peristaltic phenomena. Figure 3(b), (d)–(f) depict that the
absolute value of the heat transfer coefficient increases
by increasing K , γ , Br , E1 , E2 and E3 , respectively,
while the behavior is quite opposite in the case of M and
β (Fig. 3(a) and (c)). The damping nature of the wall
E3 has a very insignificant effect on the heat transfer.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
The formation of an internally circulating bolus of
fluid by closed streamlines is shown in Figs 4–7.
Figure 4 displays the effect of the Hartman number M on the streamlines for fixed value of the
other parameters. This figure shows that the size
of the trapping bolus decreases with an increase
in the Hartman number M , whereas the behavior
is quite opposite in the case of the permeability
parameter K (Fig. 5). Figure 6 reveals the behavior
of velocity for slip parameter β on the streamlines.
Here, we observed that the number of the streamlines increases by increasing β. The effect of the
elastic parameters on the streamlines is plotted in
Fig. 7 . The number of the trapped bolus increases
with an increase in E1 , E2 and E3 . We also note
that the damping seems less effective in the trapping
phenomenon.
CONCLUSIONS
We have presented theoretical study for the slip effects
on the MHD peristaltic flow of a viscous fluid through
Asia-Pac. J. Chem. Eng. 2011; 6: 649–658
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
MAGNETOHYDRODYNAMIC PERISTALTIC MOTION OF A VISCOUS FLUID
(a)
(b)
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-0.1
(c)
(d)
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 7. Streamlines for (ε = 0.2, β = 0.1, K = 0.04, M = 5 and t = 0.1). (a) E1 = 0.5, E2 = 0.3, E3 = 0.1. (b)
E1 = 0.8, E2 = 0.3, E3 = 0.1. (c) E1 = 0.5, E2 = 0.5, E3 = 0.1. (d) E1 = 0.5, E2 = 0.3, E3 = 0.7.
a porous medium. The influence of heat transfer is
also seen. Closed form solutions are derived for the
velocity, stream function, temperature and heat transfer
coefficient. The salient features of the presented attempt
are as follows.
• The longitudinal velocity increases in the neighborhood of the walls and increases near the center of the
channel when the velocity slip parameter is increased.
• An increase in velocity slip parameter yields a
decrease in the temperature.
• There is an increase in the temperature when the
thermal slip parameter increases.
• The effects of Brinkman number and velocity slip
parameter on the temperature are quite opposite.
However, the variations of Brinkman number and
thermal slip parameter on the temperature are similar
in a qualitative sense.
• There is a decrease in the heat transfer coefficient
at the upper wall when velocity slip parameter is
increased.
• The influence of Hartman number on the heat transfer
coefficient at the upper wall is quite opposite to that
of a Brinkman number.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
• The absolute value of the heat transfer coefficient
increases in the upper part of the channel when an
elastic parameters increase.
Acknowledgements
The authors are grateful to the Higher Education Commission (HEC) of Pakistan for the financial support.
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