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Influence of inclined magnetic field on peristaltic flow of a Jeffrey fluid with heat and mass transfer in an inclined symmetric or asymmetric channel.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2012; 7: 33–44
Published online 22 July 2010 in Wiley Online Library
(wileyonlinelibrary.com) DOI:10.1002/apj.488
Research article
Influence of inclined magnetic field on peristaltic flow
of a Jeffrey fluid with heat and mass transfer in an inclined
symmetric or asymmetric channel
S. Nadeem1 * and Safia Akram2
1
2
Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan
Department of Electrical (Telecom) Engineering, Military College of Signals, National University of Sciences and Technology, Pakistan
Received 3 February 2010; Revised 31 May 2010; Accepted 11 June 2010
ABSTRACT: In this article, we have investigated the peristaltic flow of a Jeffrey fluid with heat and mass transfer
under the influence of inclined magnetic field in an inclined symmetric or asymmetric channel. The dissipation term in
the energy equation is also taken into account. The governing equations are transformed from moving to fixed frame of
reference and the resulting equations have then been simplified using the assumptions of long wavelength and low but
finite Reynolds number approximation. The reduced equations have been solved numerically by Adomian decomposition
method and the exact solutions have also been computed for stream function, temperature and concentration. The
expression for pressure rise is computed numerically. Lastly, the physical features of various parameters have been
discussed through graphs.  2010 Curtin University of Technology and John Wiley & Sons, Ltd.
KEYWORDS: peristaltic flow; Jeffrey fluid; inclined magnetic field; Adomian; exact solution
INTRODUCTION
Peristaltic flows in symmetric and asymmetric channels have renewed the interest among the researchers
due to its applications in physiology and especially in
myometrial contractions.[1 – 7] The study of heat transfer analysis in connection with peristalsis has recently
acquired a special status. Some recent applications in
this area are hypothermia, laser therapy, cryosurgery,
hemodialysis, transport of water from the ground to
upper branches of trees, sanitary fluid transport, blood
pumps in heart–lung machines and transport of corrosive fluids where contact of the fluid with the machinery
parts is prohibited. Vajravelu et al .[8] have analyzed the
peristaltic flow and heat transfer in a vertical porous
annulus with long wavelength approximation. The influence of heat transfer and magnetic field on the peristaltic flow of a Newtonian fluid in a vertical annulus
under a zero Reynolds number and long wavelength
approximation has been studied by Mekheimer and
Abd elmaboud.[9] Nadeem and Akram[10] have examined the heat transfer in a peristaltic flow with partial
*Correspondence to: S. Nadeem, Department of Mathematics,
Quaid-i-Azam University 45320, Islamabad 44000, Pakistan.
E-mail: snqau@hotmail.com
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Curtin University is a trademark of Curtin University of Technology
slip. They[11] also studied the slip effects on the peristaltic flow of a Jeffrey fluid in an asymmetric channel under the effect of an induced magnetic field. The
effects of heat transfer on the peristaltic transport of
MHD Newtonian fluid with variable viscosity using the
Adomian decomposition method have been analyzed by
Nadeem and Akbar.[12] In another study, Nadeem and
Akbar[13] have discussed the influence of heat transfer
on a peristaltic transport of Herschel–Bulkley fluid in
a non-uniform inclined tube. The peristaltic transport
of a Newtonian fluid in a vertical asymmetric channel
with heat transfer and porous medium has been examined by Srinivas and Gayathri.[14] The combined effect
of heat and mass transfer is mostly useful in the chemical industry and in reservoir engineering in connection
with thermal recovery process, and may be found in
salty springs in the sea.[15] To our knowledge, only three
papers have discussed the combined effects of heat and
mass transfer in peristaltic flows. Eldabe et al .[16] have
analyzed the mixed convection heat and mass transfer in a non-Newtonian fluid at peristaltic surface with
temperature-dependent viscosity. Heat and mass transfer of blood in a single lymphatic blood vessel with
uniform magnetic field has been discussed by Ogulu.[17]
Srinivas and Kothandapani[18] have studied the influence of heat and mass transfer on MHD peristaltic flow
34
S. NADEEM AND S. AKRAM
Asia-Pacific Journal of Chemical Engineering
through a porous space with compliant walls. For some
relevant work of interest we refer the reader to.[19 – 25]
The study of flows on an inclined plane is also
very important. Elshehawey et al .[25] have examined the
effects of inclined magnetic field on magnetofluid flow
through porous medium between two inclined wavy
porous plates. Various authors have studied different
kinds of variable magnetic fields here.[25] They considered the magnetic field as a function of inclination
angle.
The aim of the present investigation is multidimensional: to highlight the heat and mass transfer in
non-Newtonian fluids and inclined magnetic field on
an inclined symmetric or asymmetric channel. To our
knowledge, no attempt has been reported yet to discuss the inclined magnetic field in peristaltic phenomena
even in the absence of heat and mass transfer. This study
is useful in filling the gap in this direction. The governing equations of motion, energy and concentration
are simplified using the assumptions of long wavelength approximation. The exact solution of the reduced
equation has been arrived at. The expressions for velocity, pressure gradient, pressure rise, stream function,
heat transfer and concentration are found and discussed
graphically for different physical parameters.
The equations governing the flow with inclined
magnetic field and in the presence of gravity effects
are[25]
∂V
∂U
+
=0
(3)
∂X
∂Y
∂U
∂U
∂U
∂p
∂
ρ
+U
+V
=−
+
(SXX )
∂t
∂X
∂Y
∂X
∂X
∂
+
(SXY ) − σ B02 cos (U cos ∂Y
− V sin ) − ρg sin α
(4)
∂V
∂V
∂p
∂
∂V
ρ
+U
+V
=−
+
(SYX )
∂t
∂X
∂Y
∂Y
∂X
∂
+
(SYY ) + σ B02 sin (U cos ∂Y
− V sin ) − ρg cos α
(5)
The heat and mass transfer equations are
∂T
∂T
∂T
C
+U
+V
∂t
∂X
∂Y
=
where a1 and b1 are the amplitudes of the waves, λ is
the wavelength, d1 + d2 is the width of the channel, c
is the velocity of propagation, t is the time and X is the
direction of wave propagation, φ is the phase difference.
φ varies in the range 0 ≤ φ ≤ π; φ = 0 corresponds to
a symmetric channel with waves out of phase and for
φ = π the waves are in phase. Furthermore, a1 , b1 , d1 ,
d2 and φ satisfy the condition
a12 + b12 + 2a1 b1 cos φ ≤ (d1 + d2 )2
(2)
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
(6)
∂C
∂C
∂ 2C
∂ 2C
∂C
+U
+V
= Dm
+
∂t
∂X
∂Y
∂X 2
∂Y 2
Dm KT ∂ 2 T
∂ 2T
+
+
(7)
Tm
∂X 2 ∂Y 2
MATHEMATICAL FORMULATION
Let us consider the peristaltic flow of an incompressible non-Newtonian fluid (Jeffrey fluid) in a twodimensional channel having widths d1 and d2 , under the
effect of a constant magnetic field. Both the magnetic
field and channel are inclined at angles and α, respectively. The lower wall of the channel is maintained at
temperature T1 while the upper wall has temperature T0 .
The geometry of the wall surface is defined as
2π
(X − ct)
Y = H1 = d1 + a1 cos
λ
upper wall
2π
(X − ct) + φ
Y = H2 = −d2 − b1 cos
λ
lower wall
(1)
K 2
∇ T + υ
ρ
where
∇2 =
and
=
∂2
∂X 2
+
∂2
∂Y 2
∂
∂
∂
1
+U
+V
1 + λ2
1 + λ1
∂t
∂X
∂Y
2
2 ∂U
∂V 2
∂V
∂U
2
+
,
+2
+
∂X
∂Y
∂Y
∂X
in which U , V are the velocities in the X and Y
directions in a fixed frame, ρ is the constant density,
p is the pressure, ν is the kinematic viscosity, σ is
the electrical conductivity, g is the acceleration due
to gravity, K is the thermal conductivity, C is the
specific heat, T is the temperature, Dm is the coefficient
of mass diffusivity, Tm is the mean temperature, KT is
the thermal diffusion ratio and C is the concentration
of fluid.
The constitutive equation for the extra stress tensor S
is[7]
µ
(8)
S=
(γ̇ + λ2 γ̈ )
1 + λ1
Asia-Pac. J. Chem. Eng. 2012; 7: 33–44
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
PERISTALTIC FLOW OF A JEFFREY FLUID WITH HEAT AND MASS TRANSFER
Reδ(y θx − x θy )
In the above equation, λ1 is the ratio of relaxation
to retardation times, γ̇ the shear rate, λ2 the retardation
time and dots denote the differentiation with respect to
time.
Introducing a wave frame (x , y) moving with velocity
c away from the fixed frame (X , Y ) by the transformation
=
x = X − ct, y = Y , u = U − c, v = V ,
p(x ) = p(X , t)
and
1 2
(δ xx
Sc
+ yy ) + Sr(δ 2 θxx + θyy )
Reδ(y x − x y ) =
x
y
u
v
, y= , u= , v= ,
λ
d1
c
c
Sxx
Sxy
Syy
(10)
Making use of Eqns (3)–(10), we obtain the following equations in terms of the stream function ∂
(dropping the bars, u = ∂
∂y , v = −δ ∂x )
Reδ(y xy − x yy )
Re
sin α
Fr
Reδ 3 (−y xx + x xy )
+ δx sin ) +
= 0 on y = h1 ,
a 2 + b 2 + 2ab cos φ ≤ (1 + d )2
(11)
(18)
(19)
Under the lubrication approach, Eqns (11)–(14) become
∂p
∂
1 ∂ 2
−
+
− M 2 cos2 (y + 1)
∂x
∂y 1 + λ1 ∂y 2
∂
∂p
∂
+ δ 2 (Syx ) + δ (Syy )
∂y
∂x
∂y
+ M 2 δ sin ((y + 1) cos Re
cos α
Fr
(17)
where F is the flux in the wave frame, a, b, φ and d
satisfy the relation
−M 2 cos ((y + 1) cos + δx sin ) − δ
θ = 1 on y = h2 ,
= 1 on y = h2 ,
∂
∂
∂p
+ δ (Sxx ) +
(Sxy )
=−
∂x
∂x
∂y
=−
F
at y = h1 = 1 + a cos 2π x ,
2
F
= − at y = h2 = −d − b cos(2πx + φ)(15)
2
∂
= −1 at y = h1 ,
∂y
∂
= −1 at y = h2
(16)
∂y
θ = 0 on y = h1 ,
=
ρνC c2
Sd1
,
Pr
=
,
, S =
C (T1 − T0 )
K
µc
C − C0
σ
B0 d1 , =
µ
C1 − C0
∂
∂
λ2 cδ
2δ
1+
y
=
− x
xy ,
1 + λ1
d1
∂x
∂y
1
λ2 cδ
∂
∂
− x
(yy − δ 2 xx ),
=
1+
y
1 + λ1
d1
∂x
∂y
2δ
λ2 cδ
∂
∂
− x
xy .
=−
1+
y
1 + λ1
d1
∂x
∂y
The corresponding boundary conditions are
c2
T − T0
Fr =
, θ=
,
gd1
T1 − T0
ρDm Kt (T1 − T0 )
µ
Sr =
, Sc =
,
Tm µ(C1 − C0 )
ρDm
Ec =
(14)
where
d1
d2
d 2p
, d= , p= 1 ,
λ
d1
µcλ
ct
H1
t = , h1 =
,
λ
d1
H2
a1
b1
h2 =
, a= , b= ,
d2
d1
d1
cd1
Re =
, =
,
v
cd1
δ=
M =
(13)
2
(4δ 2 xy
+ (yy − δ 2 xx )2 )
(9)
Defining
x=
1
(θyy + δ 2 θxx )
Pr
Ec
λ2 cδ
+
1+
(1 + λ1 )
d1
∂
∂
− x
y
∂x
∂y
(12)
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
+
Re
sin α = 0
Fr
(20)
Asia-Pac. J. Chem. Eng. 2012; 7: 33–44
DOI: 10.1002/apj
35
36
S. NADEEM AND S. AKRAM
∂p
=0
−
∂y
(21)
1 ∂ 2θ
Ec
+
2
Pr ∂y
(1 + λ1 )
∂ 2
2
=0
∂y 2
(22)
∂ 2θ
1 ∂ 2
+
Sr
=0
Sc ∂y 2
∂y 2
(23)
Elimination of pressure from Eqns (20) and (21)
gives
∂2
∂y
2
1 ∂ 2
1 + λ1 ∂y 2
− M 2 cos2 yy = 0
Ec
1 ∂ 2θ
+
2
Pr ∂y
(1 + λ1 )
∂ 2
(24)
2
=0
(25)
∂ θ
1 ∂ + Sr 2 = 0
2
Sc ∂y
∂y
(26)
∂y 2
2
2
It is observed that the results of Kothandapani et al .[7]
can be recovered as a special case of our problem in the
absence of heat and mass transfer and when = α = 0.
The results of Srinivas et al .[6] can be recovered when
= λ1 = 0 and in the absence of heat and transfer
analysis. Moreover, the results of Mishra and Rao[3]
can be recovered as a special case of our problem
in the absence of heat and mass transfer and when
= M = α = 0 = λ1 .
SOLUTION OF THE PROBLEM
Exact solution
The exact solution of Eqn (24) can be written as
= A + By + A1 cosh M cos 1 + λ1 y
+ B1 sinh M cos 1 + λ1 y
(27)
Asia-Pacific Journal of Chemical Engineering
√

−2
1 + λ1 +√2 1 + λ1  cosh (h1 − h2 )M 1 + λ1 cos 


λ1 ) cos +FM (1 + √
sinh (h√1 − h2 )M 1√+ λ1 cos ,
B= 
2 1 + λ1
2 1 + λ1 −√
 cosh (h1 − h2 )M 1 + λ1 cos 
 +(h − h )M (1 + λ ) cos 
2
1
√ 1
sinh (h1 − h2 )M 1 + λ1 cos 
√
√
√
(F + h1 − h2 ) 1 + λ1 cosh h1 M 1 + λ1 cos √
λ1 cos − cosh
h2 M 1 +
√
√


A1 =
,
2 1 + λ1
2 1 + λ1 −√
 cosh (h1 − h2 )M 1 + λ1 cos 
 +(h − h )M (1 + λ ) cos 
2
1
√ 1
sinh (h1 − h2 )M 1 + λ1 cos √
√
(F + h1 − h2 ) 1 + λ1 sinh h1 M 1 + λ1 cos √
λ1 cos − sinh√ h2 M 1 +√


B1 =
−2
1 + λ1 +√2 1 + λ1  cosh (h1 − h2 )M 1 + λ1 cos 
 −(h − h )M (1 + λ ) cos 
2
1
√ 1
sinh (h1 − h2 )M 1 + λ1 cos (28)
Solution by Adomian decomposition method
In this section, the Adomian solutions will be determined for the velocity field.
According to Adomian decomposition method, we
write Eqn (24) in the operator form as
Lyyyy = M 2 (1 + λ1 ) cos2 yy .
Applying the inverse operator L−1
yyyy
dydydydy we can write Eqn (29) as
= C 0 + C1 y + C 2
1
y2
y3
+ C3 + L−1
(yy ) (30)
2!
3! A yyyy
1 = M 2 (1 + λ ) cos2 and C , C , C , C are
where A
1
0
1
2
3
functions of x . Now we decompose where A, B , A1 , B1 are functions of x .
√
√

(h1 + h2 ) −2 1 + λ1 + 2 1 + λ1
√
 cosh (h1 − h2 )M 1 + λ1 cos 


λ1 ) cos +FM (1 + √
sinh (h√
+ λ1 cos 1 − h2 )M 1√
 ,
A= 
−4
1
+
λ
+
4
1 + λ1 1 √
 cosh (h1 − h2 )M 1 + λ1 cos 
 −2(h − h )M (1 + λ ) cos 
2
1
1
√
sinh (h1 − h2 )M 1 + λ1 cos 
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
(29)
[.]
=
=
∞
n .
(31)
n=0
Substituting into Eqn (30), we obtain
y2
y3
= C0 + C1 y + C2 + C3 , ei θ
2!
3!
1
(n )yy dydydydy, n ≥ 0(32)
n+1 =
A
Asia-Pac. J. Chem. Eng. 2012; 7: 33–44
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
PERISTALTIC FLOW OF A JEFFREY FLUID WITH HEAT AND MASS TRANSFER
Therefore,
4
5
1 C y +C y ,
1 = A
2 4!
3 5!
6
y
y7
1
2 = 2 C2 6! + C3 7! ,
A
..
.
y 2n+2
y 2n+3
1
+ C3
,
n = n C2
(2n + 2)!
(2n + 3)!
A
Integrating Eqn (39) over one wavelength, we get
p =
0
n>0
(33)
According to Eqn (31), the closed form of can be
written as
y
= C0 + C1 y + AC2 cosh √ − 1
A
√
y
y
(34)
+ A AC3 sinh √ − √
A
A
Now the Adomian solution (36) and exact solution (27) are exactly same, in which A, B , A1 , B1 are
calculated using boundary conditions which are defined
in Eqn (28).
The flux at any axial station in the fixed frame is
Q=
(u + 1)
h2
+
dy =
c1 =
c2 =
udy
(37)
h2
The average volume flow rate over one period (T =
λ/c) of the peristaltic wave is defined as
1 T
1 T
Qdt =
(F + h1 + h2 )
Q=
T 0
T 0
dt = F + 1 + d
(38)
The pressure gradient is obtained from the dimensionless momentum equation for the axial velocity as
Re
dp
= −M 2 cos2 (B + 1) +
sin α
dx
Fr
(39)
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
1
2h1 (4 + Brh2 (A21
8(h1 − h2 )
− B12 )(−h1 + h2 ) cos4 M 4 (1 + λ1 )
+ BrM 2 cos2 − A21 + B12 h2
cosh[2h1 M 1 + λ1 cos ]
+ A21 + B12 h1 cosh[2h2 M 1 + λ1 cos ]
+ 2A1 B1 −h2 sinh[2h1 M 1 + λ1 cos ]
. (42)
+ h1 sinh[2h2 M 1 + λ1 cos ]
h1
dy = F + h1 + h2
1
−8 + 2(A21 − B12 )
8(h1 − h2 )
Br(h12 − h22 )M 4 (1 + λ1 ) cos4 + BrM 2 cos2 (A21 + B12 )
cosh 2h1 M 1 + λ1 cos − (A21 + B12 ) cosh 2h2 M 1 + λ1 cos + 2A1 B1 sinh 2h1 M 1 + λ1 cos − sinh 2h2 M 1 + λ1 cos ,
h2
h1
(40)
Using solution (27) in Eqn (25), the exact solution of the resulting equation satisfying the boundary
conditions can be written as
Br
θ = − M 3 1 + λ1 cos4 (A21 − B12 )M 1 + λ1 y 2
4
A1 B1 sec2 sinh[2My cos 1 + λ1 ]
+
M 1 + λ1
2
A1 + B12 cosh[2My cos 
√
1 + λ1 ] sec2 
+
 + c1 y + c2 , (41)
2M 1 + λ1
y
= A + By + A1 cosh √
A
y
(35)
+ B1 sinh √
A
= A + By + A1 cosh M cos 1 + λ1 y
+ B1 sinh M cos 1 + λ1 y
(36)
h1
dp
dx
dx
where
which can be put in the simplest form
1
Using solution (41) in Eqn (26), the exact solution of the resulting equation satisfying the boundary
conditions can be written as
=
ScSrBr 3 M 1 + λ1 cos4 A21 − B12 M 1 + λ1 y 2
4
A1 B1 sec2 sinh[2My cos 1 + λ1 ]
+
M 1 + λ1
Asia-Pac. J. Chem. Eng. 2012; 7: 33–44
DOI: 10.1002/apj
37
S. NADEEM AND S. AKRAM
+
Asia-Pacific Journal of Chemical Engineering
2
A21 + B1 cosh[2My cos 
√
1 + λ1 ] sec2 
 + c3 y + c4
2M 1 + λ1
5
(43)
λ1 = 0.5
λ1 = 1.0
λ1 = 1.5
∆p
1
(−8 − 2(A21 − B12 )
c3 =
8(h1 − h2 )
ScSrBr(h12 − h22 )M 4 (1 + λ1 ) cos4 + BrM 2 ScSr cos2 −(A21 + B12 )
cosh 2h1 M 1 + λ1 cos + (A21 + B12 ) cosh 2h2 M 1 + λ1 cos + 2A1 B1 − sinh 2h1 M 1 + λ1 cos + sinh 2h2 M 1 + λ1 cos ,
0
-5
-1
0
1
Q
2
3
Figure 2. Variation of p with Q for different values of
λ1 for fixed b = 0.5, d = 1.2, φ = π/3, = π/4, = π/4,
M = 0.5, a = 0.9, Re = 1, α = 0.2, Fr = 0.8.
1
(2h1 (4 + ScSrBrh2
8(h1 − h2 )
10
(A21 − B12 )(h1 − h2 ) cos4 M 4 (1 + λ1 ))
+ ScSrBrM 2 cos2 A21 + B12 h2
cosh[2h1 M 1 + λ1 cos ]
− A21 + B12 h1 cosh[2h2 M 1 + λ1 cos ]
+ 2A1 B1 h2 sinh[2h1 M 1 + λ1 cos ]
− h1 sinh[2h2 M 1 + λ1 cos ]
.
(44)
8
Fr = 0.5
6
Fr = 1.0
4
∆p
c4 =
Fr = 1.5
2
0
-2
-4
-6
-1
6
Θ = π/8
2
0
1
Q
2
3
Figure 3. Variation of p with Q for different values of
Fr for fixed b = 0.5, d = 1.2, φ = π/3, = π/4, M = 2,
λ1 = 1, Re = 2, α = 1, a = 0.9.
Θ=0
4
∆p
38
Θ = π/4
NUMERICAL RESULTS AND DISCUSSION
0
-2
-4
-6
-1
-0.5
0
0.5
1
1.5
2
Q
Figure 1. Variation of p with Q for different values of
for fixed b = 0.5, d = 1.2, φ = π/3, a = 0.5, M = 2,
λ1 = 1, Re = 1, α = 0.2 Fr = 0.8.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
In this section, numerical results of the problem under
discussion are discussed through graphs. The expression for pressure rise is calculated numerically using
mathematics software Mathematica. Figs 1–4 are displayed to see the effects of pressure rise for various
values of the angle of inclination , Jeffrey parameter λ1 , Froude number Fr and Hartmann number M .
It is observed from Figs 1 and 2 that in the peristaltic pumping (p > 0, Q > 0), retrograde pumping
(p > 0, Q < 0) and free pumping (p = 0) regions
the pressure rise decreases with an increase in inclination angle and Jeffrey parameter λ1 , while in the
Asia-Pac. J. Chem. Eng. 2012; 7: 33–44
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
PERISTALTIC FLOW OF A JEFFREY FLUID WITH HEAT AND MASS TRANSFER
6
3
M = 0.5
4
Θ=0
2
Θ = π/8
M = 1.0
2
1
Θ = π/4
dp\dx
∆p
M = 1.5
0
-2
-1
-4
-2
-6
-1
0
1
Q
2
-3
3
0
0.2
0.4
0.6
0.8
1
x
Figure 4. Variation of p with Q for different values of
M for fixed b = 0.5, d = 1.2, φ = π/3, = π/4, a = 0.9,
λ1 = 1, λ1 = 1, Re = 1.0, Fr = 0.8.
Figure 6. Variation of dp/dx with x for different values
of for fixed b = 0.5, d = 1.2, a = 0.5, M = 2, φ = π/3,
λ1 = 1, α = 0.5, Q = 0.5, Fr = 0.8, Re = 0.5.
4
2
3
2
M = 0.5
1.5
M = 1.0
1
M = 1.5
0.5
λ1 = 0.0
0
λ1 = 2.0
-0.5
λ1 = 4.0
1
y
dp/dx
0
0
-1
-1
-1.5
-2
0
0.2
0.4
0.6
0.8
1
x
-2
-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
u
Figure 5. Variation of dp/dx with x for different values of
M for fixed b = 0.5, d = 1.2, a = 0.9, φ = π/2, = π/8,
λ1 = 1, α = 0.5, α = 0.5, Q = 0.5, Fr = 0.8, Re = 0.5.
Figure 7. Velocity profile for different values of λ1 for fixed
a = 0.7, b = 1.2, d = 2, x = 0, Q = 1, φ = π/2, = π/8,
M = 1.0.
copumping region (p < 0, Q > 0), the pressure rise
increases with an increase in inclination angle and
Jeffrey parameter λ1 . From Fig. 3, it can be observed
that with an increase in Froude number Fr the pumping rate decreases in all the regions. It is also seen from
Fig. 4 that with an increase in Hartmann number M , the
pressure rise increases in peristaltic pumping, retrograde
pumping and free pumping regions, while in the copumping region the pressure rise decreases. The pressure
gradient for different values of M and against x is
plotted in Figs 5 and 6. It is shown that for x ∈ [0, 0.2]
and x ∈ [0.8, 1], the pressure gradient is small, i.e. the
flow can easily pass without imposition of a large pressure gradient, while in the region x ∈ [0.2, 0.8], the
pressure gradient increases with an increase in M and
decreases with an increase in , so a large pressure
gradient is required to enable the flux to pass. In order
to see the effects of the Jeffrey parameter λ1 , Hartmann
number M and volume flow rate Q on axial velocity
Figs 7–9 are plotted. It is observed from Figs 7 and
8 that increase of Jeffrey parameter λ1 and Hartmann
number M result in increasing resistance to the flow in
the central part of the channel. It is also observed from
Fig. 9 that the magnitude of velocity profile decreases
with an increase in volume flow rate Q. Figs 10–12
show the variation of temperature profile for different
values of Brinkman number Br, inclination angle and
Hartmann number M . It is seen from Figs 10 and 11
that the temperature profile increases with an increase in
Brinkman number Br and inclination angle . It is seen
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2012; 7: 33–44
DOI: 10.1002/apj
39
S. NADEEM AND S. AKRAM
Asia-Pacific Journal of Chemical Engineering
2
1.5
1.5
Br = 1.0
1
1
Br = 1.5
0.5
Br = 2.0
M = 0.5
y
y
0.5
0
M = 1.0
-0.5
M = 1.5
0
-0.5
-1
-1
-1.5
-2
-1
-1.5
-0.8
-0.6
-0.4
u
-0.2
0
0
0.2
0.4
0.6
0.2
Figure 8. Velocity profile for different values of M
for fixed b = 1.2, d = 2, a = 0.7, = π/8, φ = π/2,
λ1 = 1, Q = 2.
0.8
1
1.2
θ
Figure 10. Variation of temperature profile for
different values of Br for fixed b = 1.2, d = 1.5,
a = 0.5, = π/8, φ = π/2, λ1 = 1, x = 0, Q = 2,
Br = 2.
2
1.5
Q = 0.0
1
Q = 0.5
0.5
Q = 1.0
1.5
Θ=0
1
Θ = π/4
0.5
Θ = π/3
0
y
y
40
0
-0.5
-1
-0.5
-1.5
-1
-2
-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
u
Figure 9. Velocity profile for different values of Q for
fixed a = 0.5, b = 1.2, d = 2, x = 0, λ1 = 1, φ = π/2,
= π/8, M = 1.0.
from Fig. 12 that with an increase in Hartmann number M the temperature profile decreases. In order to see
the effects of concentration on Brinkman number Br,
Hartmann number M , Schmidt number Sc and Soret
number Sr, Figs 13–15 are prepared. It is observed
from Fig. 13 that the concentration decreases with an
increase in Brinkman number Br. It is also observed
from Fig. 14 that with an increase in Hartmann number M the concentration increases. It is also seen from
Fig. 15 that with an increase in Schmidt number Sc and
Soret number Sr, the concentration decreases.
Trapping phenomena
Another interesting phenomena in peristaltic motion is
trapping. It is basically the formation of an internally
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
-1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
θ
Figure 11. Variation of temperature profile for
different values of for fixed b = 1.2, d = 1.5,
a = 0.5, M = 2, φ = π/2, λ1 = 1, x = 0, Q = 2,
Br = 2.
circulating bolus of fluid by closed stream lines. This
trapped bolus pushed a head along peristaltic waves.
The trapping phenomena is discussed for different values of λ1 , and M for both symmetric and asymmetric
channels. It is observed from the figures that for a symmetric channel the trapping bolus is symmetric about
the center line of the channel (see panels (a) and (c)),
while in case of an asymmetric channel the bolus tends
to shift toward left side of the channel due to phase
angle (see panels (b) and (d)). Figure 16 shows the
stream lines for different values of Jeffrey parameter
λ1 . It is observed from the figure that the size of the
trapping bolus increases with an increase in λ1 . Further,
we also observed that the size of the trapping bolus is
small in the asymmetric channel as compared with the
Asia-Pac. J. Chem. Eng. 2012; 7: 33–44
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
PERISTALTIC FLOW OF A JEFFREY FLUID WITH HEAT AND MASS TRANSFER
1.5
1.5
M = 1.0
1
M = 1.0
1
M = 1.5
M = 1.5
0.5
M = 2.0
M = 2.0
y
y
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
0
0.2
0.4
0.6
0.8
1
-1.5
-1
1.2
-0.5
0
Φ
θ
Figure 12. Variation of temperature profile for
different values of M for fixed b = 1.2, d = 1.5,
a = 0.5, = π/8, φ = π/2, λ1 = 1, x = 0, Q = 2,
Br = 2.
0.5
1
Figure 14. Concentration profile for different values
of M for fixed b = 1.2, d = 1.5, a = 0.5, Br = 2,
φ = π/2, λ1 = 1, Sc = Sr = 1, Q = 2, = π/8.
1.5
1.5
1
Br = 1.0
1
0.5
Br = 1.5
y
0.5
0
y
Br = 2.0
0
-0.5
-0.5
-1
Sc = 1.0 , Sr = 0.5
Sc = 1.5 , Sr = 1.0
Sc = 2.0 , Sr = 1.5
-1
-1.5
-0.2
-1.5
-1
0
0.2
0.4
0.6
0.8
1
-0.5
0
Φ
0.5
1
1.2
Φ
Figure 13. Concentration profile for different values
of Br for fixed b = 1.2, d = 1.5, a = 0.5, M = 2,
φ = π/2, λ1 = 1, Sc = Sr = 1, Q = 2, = π/8.
Figure 15. Concentration profile for different values
of Sc and Sr for fixed b = 1.2, d = 1.5, a = 0.5,
M = 2, φ = π/2, λ1 = 1, Br = 2, Q = 2, = π/8.
CONCLUSIONS
symmetric channel. The stream lines for different values of and M are shown in Figs 17 and 18. It is
observed from the figures that the size of the trapping
bolus decreases with an increase in and M in both
symmetric and asymmetric channels.
The results of velocity field against different values of
flow rate Q, Jeffrey parameter λ1 and inclination angle
are tabulated in Tables 1–3. It is observed from these
tables that with the increase in all these three parameters
the magnitude of velocity decreases at different values
of y. The comparison of the present solution is compared with those available in the literature when some
of parameters are replaced by zero in our problem is
given in Table 4.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
We have presented the exact and new numerical solutions with the help of Adomian decomposition method
of the peristaltic flow of a Jeffrey fluid with heat and
mass transfer in an inclined symmetric and asymmetric channel under the influence of an inclined magnetic
field. We conclude with the following observations:
• It is observed that pressure rise decreases in all
the pumping regions with the increase in inclination
angle and Jeffrey parameter λ1 .
• The pumping rate decreases with the increase in
Froude number Fr.
• With the increase in Hartmann number M , pressure rise increases in peristaltic pumping, retrograde
Asia-Pac. J. Chem. Eng. 2012; 7: 33–44
DOI: 10.1002/apj
41
42
S. NADEEM AND S. AKRAM
1.5
Asia-Pacific Journal of Chemical Engineering
1.5
(a)
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
(b)
-1
φ=0
-1.5
-0.5
1.5
0
0.5
-0.5
1.5
(c)
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
φ=0
-1.5
-0.5
0
φ = π/2
-1.5
0
0.5
(d)
φ = π/2
-1.5
0.5
-0.5
0
0.5
Figure 16. Stream lines for different values of λ1 (a) and (b) for λ1 = 0.05, (c) and (d) for
λ1 = 0.07. The other parameters are Q = 1.5, = π , M = 2.6, b = 0.6, a = 0.6, d = 1.0.
8
1.5
1.5
(a)
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
φ=0
-1.5
-0.5
1.5
(b)
0
φ = π/2
-1.5
0.5
-0.5
1.5
(c)
1
1
0.5
0.5
0
0
-0.5
-0.5
0
0.5
(d)
f
-1
-1
φ=0
-1.5
-0.5
0
φ = π/2
-1.5
0.5
-0.5
0
0.5
Figure 17. Stream lines for different values of (a) and (b) for = 5π , (c) and (d) for = 5π .
4
6
The other parameters are Q = 1.5, λ1 = 0.05, M = 2.6, b = 0.6, a = 0.6, d = 1.0.
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2012; 7: 33–44
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
1.5
PERISTALTIC FLOW OF A JEFFREY FLUID WITH HEAT AND MASS TRANSFER
(a)
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
φ=0
-1.5
-0.5
1.5
(b)
0
φ = π/2
-1.5
0.5
-0.5
(c)
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
0
0.5
(d)
f
-1
-1
φ=0
-1.5
-0.5
0
φ = π/2
-1.5
0.5
-0.5
0
0.5
Figure 18. Stream lines for different values of M (a) and (b) for M = 2.0, (c) and (d) for M = 2.1. The
other parameters are Q = 1.5, λ1 = 0.1, = π , b = 0.6, a = 0.6, d = 1.0.
8
Table 1. Comparison of velocity for different values of Q for fixed a = 0.5, b = 1.2, d = 2, x = 0, λ1 = 1, φ = π/2,
= π/8, M = 1.0.
y
−2.0
−1.5
−1
−0.5
0.0
0.5
1.0
1.5
u(x , y) for Q = 0.0
u(x , y) for Q = 0.5
u(x , y) for Q = 1.0
−1
−1
−1
−0.883698
−0.767396
−0.651094
−0.826482
−0.652965
−0.479447
−0.803054
−0.606108
−0.409162
−0.803054
−0.606108
−0.409162
−0.826482
−0.652965
−0.479447
−0.883698
−0.767396
−0.651094
−1
−1
−1
Table 2. Comparison of velocity for different values of λ1 for fixed a = 0.5, b = 1.2, d = 2, x = 0, Q = 1, φ = π/2,
= π/8, M = 1.0.
y
−2.0
−1.5
−1
−0.5
0.0
0.5
1.0
1.5
u(x , y) for λ1 = 0.0
u(x , y) for λ1 = 2
u(x , y) for λ1 = 4
−1
−1
−1
−0.666221
−0.638719
−0.619705
−0.477239
−0.481724
−0.486237
−0.392004
−0.423035
−0.44407
−0.392004
−0.423035
−0.44407
−0.477239
−0.481724
−0.486237
−0.666221
−0.638719
−0.619705
−1
−1
−1
Table 3. Comparison of velocity for different values of for fixed a = 0.5, b = 1.2, d = 2, x = 0, Q = 1, φ = π/2,
λ1 = 1, M = 1.0.
y
−2.0
−1.5
−1
−0.5
0.0
0.5
1.0
1.5
u(x , y) for = 0.0
u(x , y) for = π /6
u(x , y) for = π /4
−1
−1
−1
−0.646577
−0.654479
−0.663391
−0.480225
−0.478901
−0.477608
−0.414243
−0.405341
−0.39523
−0.414243
−0.405341
−0.39523
−0.480225
−0.478901
−0.47608
−0.646577
−0.654479
−0.663391
−1
−1
−1
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2012; 7: 33–44
DOI: 10.1002/apj
43
44
S. NADEEM AND S. AKRAM
Asia-Pacific Journal of Chemical Engineering
Table 4. Comparison of velocity in case of Newtonian and non-Newtonian fluid.
y
−1.5
−1
−0.5
0.0
0.5
1.0
1.5
Ours
Kothandapani et al .[7]
Srinivas et al .[6]
Mishra[3]
−1
−1
−1
−1
−0.537189
−0.511971
−0.549315
−0.58333
−0.362456
−0.3843444
−0.354278
−0.333333
−0.317708
−0.354894
−0.299941
−0.25
−0.362456
−0.381344
−0.354278
−0.333333
−0.537189
−0.381344
−0.549315
−0.58333
−1
−1
−1
−1
pumping and free pumping regions while it decreases
in the copumping region.
• In the center of the channel, the pressure gradient
increases with an increase in M and decreases with
the increase in .
• The temperature profile increases with the increase
in Br and and decreases with an increase in M .
• The concentration field increases with an increase in
M and decreases with an increase in Sc and Sr.
• The size of the trapping bolus decreases with the
increase in and M .
• A nice comparison of our results and available results
available in the limiting case are also presented in the
table 4.
NOMENCLATURE
U , V Velocity components in X and Y directions in
fixed frame
u, v Velocity components in x and y directions in
wave frame
ρ
Constant density
p
Pressure
σ
Electrical conductivity
C
Thermal diffusion ratio
K
Thermal conductivity
M
Hartmann number
Br
Brinkmann number
Pr
Prandtl number
Re
Reynolds number
Sc
Schmidt number
Sr
Soret number
g
Acceleration due to gravity
Dm Coefficient of mass diffusivity
KT
Thermal diffusion
T
Temperature of fluid in dimension form
θ
Temperature of fluid in dimensionless form
C
Concentration of fluid in dimension form
Concentration of fluid in dimensionless form
φ
Amplitude ratio
a1, b1, Amplitude of waves
λ
Wavelength
c
Velocity of propagation
Q
Volume flow rate
δ
Long wavelength
 2010 Curtin University of Technology and John Wiley & Sons, Ltd.
Stream function
Fr
Froude number
α, Inclination angle of channel and magnetic field
respectively
Temperature at upper wall
T0
T1
Temperature at lower wall
S
Extra stress tensor
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Asia-Pac. J. Chem. Eng. 2012; 7: 33–44
DOI: 10.1002/apj
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