Iran. J. Chem. Chem. Eng. Vol. 36, No. 2, 2017
107
Influence of Nanoparticles Phenomena
on the Peristaltic Flow of Pseudoplastic Fluid
in an Inclined Asymmetric Channel
with Different Wave Forms
Akram, Safia*+
Department of Basic Sciences, MCS, National University of Sciences and Technology,
Islamabad 44000, PAKISTAN
Nadeem, Sohail
Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, PAKISTAN
ABSTRACT: The influence of nanofluid with different wave forms in the presence of inclined
asymmetric channel on peristaltic transport of a pseudoplastic fluid is examined. The governing
equations for two dimensional and two directional flows of a pseudoplastic fluid along with
nanofluid are modeled and then simplified under the assumptions of long wavelength and
low Reynolds number approximation. The exact solutions for temperature and nano particle volume
fraction are calculated. Series solution of the stream function and pressure gradient are carried out
using perturbation technique. The flow quantities have been examined for various physical
parameters of interest. It was found that the magnitude value of the velocity profile decreases
with an increase in Q and and increases in sinusoidal, multisinsoidal, trapezoidal and triangular
waves. It was also observed that the size of the trapping bolus decreases with the decrease
in the width of the channel d and increases with an increase in .
KEYWORDS: Nano fluid particles; Peristaltic flow; Pseudoplastic fluid; Different wave forms;
Inclined asymmetric channel.
INTRODUCTION
The study of nanofluids has achieved considerable
importance among the researchers because of its applications
in sciences and industry. Nanofluids are the embryonic
mixtures consist of solid particles disseminated
in the conventional heat transfer base fluids. Base fluids
(like water, ethylene glycol etc.) have ability to increase
the effective thermal conductivity of nanofluids.
The theory of nano fluids was first given by Choi [1].
Selvakumar & Suresh [2] have examined the convective
performance of CuO/water nanofluid in an electronic heat sink.
* To whom correspondence should be addressed.
+ E-mail: [email protected] ; [email protected] 1021-9986/2017/2/107-124 18/$/6.80
Iran. J. Chem. Chem. Eng. Akram S. & Nadeem S. Vol. 36, No. 2, 2017
108
In their analysis they pointed out that nanoparticles
dispersed in the base fluids have overcome the limitations
of micron sized particles. Bachok et al. [3] have examined
the unsteady boundary layer flow and heat transfer
of a two dimensional nanofluid over a permeable stretching
or shrinking sheet. Natural convective boundary layer flow
of a nanofluid past a convectively heated vertical plate
has been examined by Aziz & Khan [4]. In another study,
Aziz et al. [5] have presented the free convection
boundary layer flow past a horizontal flat plate embedded
in porous medium filled by nanofluid containing
gyrotactic microorganisms. Some recent studies of nano
fluid on different flow problems are given in Refs. [6-12].
Peristalsis is a mechanism which is produced by
successive waves of contraction which pushes their fluid
(or fluid like contents) forward. Since the first
investigation done by Latham [13], many researchers
have discussed the peristaltic flows of Newtonian and
non-Newtonian fluids with different flow geometries.
Eytan & Elad [14] have highlighted the importance
of peristaltic flows in asymmetric channel. Tripath [15]
has examined the peristaltic transport of a viscoelastic fluid
in a channel. Nonlinear peristaltic transport of a Newtonian
fluid in an inclined asymmetric channel through a porous
medium has been investigated by Kothandapani and
Srinivas [16]. Srinivas et al. [17] have examined the mixed
convection heat and mass transfer in an asymmetric
channel with peristalsis. Effects of partial slip on the
peristaltic flow of a MHD Newtonian fluid in an asymmetric
channel have been done by Yilidirim & Sezer [18].
Nadeem & Akbar [19] have highlighted the study
of influence of heat and mass transfer on the peristaltic
transport of a Jeffrey-six constant fluid in an annulus.
Nonlinear peristaltic flow of a fourth grade fluid
in an inclined asymmetric channel have been discussed by
Haroun [20]. In another paper Haroun [21] has examined
the effect of Deborah number and phase difference
on peristaltic transport of third order fluid in an asymmetric
channel. For some relevant work of interest, the reader
is referred to [22-24].
Motivated from the above analysis, the aim of
the present paper is to examine the effects of nano-particles
on the peristaltic flow of a Pseudoplastic fluid in
an inclined asymmetric channel. The governing equations
of Pseudoplastic fluid for two dimensional flow in Cartesian
coordinate system are modelled along with heat transfer
analysis and nanoparticle volume fraction. The highly nonlinear
equations are simplified using some assumptions (like long
wave length and low Reynolds number). The reduced
equations are solved analytically with the help of regular
perturbation technique. The physical features of the
pertinent parameters are discussed by plotting the graphs of
velocity, pressure rise, pressure gradient and stream lines.
THEORITICAL SECTION
Mathematical formulation
Let us consider the peristaltic transport of an
incompressible nano non-Newtonian fluid (pseudoplastic
fluid) in a two dimensional channel of width d1+d2.
The channel is inclined at angle . The channel asymmetry
is produced due to different amplitudes and phases
of the peristaltic waves. Heat transfer along with nano particle
phenomena has been taken into description. The lower
wall of the channel is sustained at temperature T1 and
nano particle volume fraction C1 while the upper wall has
temperature T0 and nano particle volume fraction C0.
The geometry of the wall surface is defined as
1 1 1
2Y H d a cos X ct ,
(1)
2 2 1
2Y H d b cos X ct ,
Where a1 and b1 are the amplitudes of the waves, is
the wave length, 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, the phase difference
varies in the range 0 , = 0 corresponds to
symmetric channel with waves out of phase and =
the waves are in phase, and further a1, b1, d1, d2 and
satisfies the condition
22 2
1 1 1 1 1 2a b 2a b cos d d .
The equations governing the flow are given by the
continuity equation
0, V (2)
The equation of motion
fdiv ,
t
VV V f (3)
where
Iran. J. Chem. Chem. Eng. Influence of Nanoparticles Phenomena on the Peristaltic Flow ... Vol. 36, No. 2, 2017
109
P I S
in which the extra stress tensor S for pseudoplastic
fluid is defined as [25]
1 1 1 1 1 1
1, ( ) ,
2
S S A S SA A (4)
Td,
dt
S
S SL LS (5)
gradL V (6)
The energy equation
2 Tf p B
0
Dd( c) k ( c) (D )
dt T
TT C T T T (7)
The nanoparticle volume fraction equation
2 2TB
0
DdD
dt T
CC T (8)
In the above equations, V is the velocity vector, is
the dynamic viscosity of the fluid, S the upper-
convected derivative, 1 the relaxation times, f is
the body force, P is the pressure, f is density of fluid base,
is the kinematic viscosity, T is the temperature, DB is
the Brownian diffusion coefficient, DT is the thermophoretic
diffusion coefficient,
p
f
c
c
is the ratio of the effective
heat capacity of the nanoparticle material and heat
capacity of the fluid with being the density, C is
the volumetric volume expansion coefficient and p is
the density of the particles.
We seek the velocity field for the two dimensional
and two directional flow of the form
U X,Y, t ,V X,Y, t ,0 V (9)
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 (10)
Using Eqs. (9) and (10) in Eqs. (2) to (8) the equations
in wave frame becomes
u v0,
x y
(11)
f xx xy
u u pu v S S
x y x x y
(12)
0 0gsin g T T g C C ,
f yx
v v pu v S
x y y x
(13)
yyS g cos ,
y
2 2
2 2
T T T Tu v
x y x y
(14)
22
TB
0
DC T C T T TD
x x y y T x y
2 2
B 2 2
C C C Cu v D
x y x y
(15)
2 2T
2 20
D T T
T x y
Where the stresses appearing in the above equations
are defined through these equations.
xx
u2 S
x
(16)
xx xx1 xx xy
S S u uu v 2 S 2 S
x y x y
1 1 xx xy
1 u u v4S 2S
2 x y x
xy
u vS
y x
(17)
xy xy
1 xx yy
S S v uu v S S
x y x y
1 1 xx xy
1 u v(S S )
2 y x
yy
v2 S
y
(18)
yy yy
1 yy xy
S S v vu v 2 S 2 S
x y y x
1 1 yy xy
1 v u v4S 2S
2 y y x
Iran. J. Chem. Chem. Eng. Akram S. & Nadeem S. Vol. 36, No. 2, 2017
110
Defining the following non-dimensional quantities
1
1
dx y u vx , y , u , v , ,
d c c
(19)
2
2 1 11
1 1
d d p Hctd , p , t , h ,
d c d
2 1 1 12
2 1 1 1
H a b cdh , a , b , Re , ,
d d d v cd
0 1 1xx xx xy xy
1 0
T T d d, S S ,S S ,
T T c c
T 1 01yy yy T
0
D T TdS S , Pr , N ,
c T
2
B 1 0 1 1 0
b
D C C g d T TN ,Gr ,
c
2
1 1 0
B
g d C CBr , Le ,
c D
With the help of Eq. (19), Eqs. (11) to (18) along with
velocity stream function relation (u , v )y x
after dropping the bars take the form
y xy x yy xx
pRe S
x x
(20)
xy r
ReS Gr B sin ,
y Fr
3 2
x xy y xx xy
pRe S
y x
(21)
yy
ReS cos
y Fr
2
y x x y yy xx
1Re
Pr (22)
222 2
b x x y y T x yN N
2
y x x y yy xxRe Le (23)
2 T Txx yy
b b
N N
N N
Where
xy xx2 S (24)
1 y x xx xy xx yy xyS 2 S 2 S
x y
2
1 1 xy xx yy xx xy
14 S 2 S
2
2
yy xx xyS (25)
2
1 y x xy xx xx yy yyS S S
x y
2
1 1 yy xx xx yy
1S S
2
xy yy2 S (26)
2
1 y x yy xx xy xy yyS 2 S 2 S
x y
2
1 1 yy xx xy xy yy
12 S 4 S
2
The corresponding boundary conditions in terms of
stream function are defined as
1
q at y h 1 a cos 2 x,
2 (27)
2
q at y h d bcos(2 x ),
2
1 21 at y h and y h ,
y
10 at y h , (28)
21 at y h ,
10 at y h , (29)
21 at y h
Where q is the flux in the wave frame, a, b, and d
satisfy the relation
22 2a b 2abcos 1 d .
Under the assumption of long wave length ‹‹ 1 and
low Reynolds number, Eqs. (20) to (26) become
xy
r
Sp Resin Gr B ,
x y Fr
(30)
Iran. J. Chem. Chem. Eng. Influence of Nanoparticles Phenomena on the Peristaltic Flow ... Vol. 36, No. 2, 2017
111
p0,
y
(31)
22
b t2
1N N 0,
Pr y y yy
(32)
2 2t
2 2b
N0
Ny y
(33)
xx 1 1 xy yyS S , (34)
yy
xy 2
yy
S ,1
(35)
yy 1 1 xy yyS S , (36)
Where 2 2
1 12
1 2.
Elimination of pressure from Eqs. (30) and (31),
yields
2yy
r2 2
yy
Gr B 0y yy 1
(37)
The above equation can also be written as
34 2 2
r4 2 2Gr B 0
y yy y y
(38)
Solution of the problem
In order to calculate the solutions for the given system
of linear and non-linear differential equations,
the treasured solution for Eq. (33) is defined as
1 2
t
b
N(x, y) a (x)y a (x)
N (39)
Where a1(x) and a2(x) are unknown functions. Now
substitute Eq. (39) into (32) we get
1
2
b2Pr N a (x) 0
yy
(40)
The exact solution of Eq. (40) give the temperature
distribution as
b3 1
4
1
a (x)Pr N y
b
a (x)
(x, y) a (x)ePr N a (x)
(41)
Where a3(x) and a4 (x) are unknown functions. Now
with the help of temperature distribution (Eq. (41)),
the nano-particle concentration is given from Eq. (39) as:
b3 1
4
1
a (x)Pr N yt
b b
a (x)N(x, y) a (x)e
N Pr N a (x)
(42)
1 2
a (x)y a (x)
By applying the boundary conditions, values
of unknown functions a1(x), a2(x), a3(x) and a4(x)
are defined as
t
b
1
2 1
N
N1
a (x) , h h
(43)
t
b
12
2 1
N
N1
a (x) h ,h h
b 11
b b3 1 2 11 1
a (x)Pr N h
b a (x)Pr N h a (x)Pr N h
ea (x) Pr N a (x)
e e
b b4 2 11 1
a (x)Pr N h a (x)Pr N h
1a (x)
e e
Thus the exact expressions for the temperature
distribution and nano-particle concentration
are given by
b b 11 1
b b2 11 1
a (x)Pr N y a (x)Pr N h
a (x)Pr N h a (x)Pr N h
e e(x, y)
e e
(44)
1
2 1
t
b
y hN(x, y) 1
N h h
(45)
b b 11 1
b b2 11 1
a (x)Pr N y a (x)Pr N h
t
a (x)Pr N h a (x)Pr N hb
N e e
Ne e
Eqs. (30) and (38) are highly non-linear equations,
so the exact solutions are looking difficult, Therefore,
Iran. J. Chem. Chem. Eng. Akram S. & Nadeem S. Vol. 36, No. 2, 2017
112
we apply regular perturbation technique. Now we expand
, p and q as:
0 1 0 1 0 1( ), p p (p ), q q q (46)
Substituting Eq. (41) into Eqs. (29) and (37) and then
solving the resulting zeroth and first order systems and
setting 0 1,
q q q we arrive at
4 3
0 1 2
1
24k (h h )
(47)
4
0 1 2 1 2k (h h )(h y)(h y)
2
2 1 2 1 2 1 2(Brk (h h ) (h y)(h y) 24(h h 2y))
2 0h k
3 0 1 2 1 2 124k k (h h )(h y)(h y)((h y)e
1 0 0 1 0h k k y h k3
2 3 1 2e (h y)) 24k ((h h ) e e
2
2 1 2(h y) ( 3h h 2y)
2 0h k2
1 1 2(h y) e ( h 3h 2y))
1 23
1 2
1q(h h 2y)
2(h h )
2 2 2
1 1 2 1 2 2(h 2y(h h ) 4h h h 2y )
3
00 01 02 03
8 3
0 1 2
b y 3y b y b b(
45360k (h h )
0 0k y k y2 2
0 8 98
0
114175k e (384k y k e
45360k
0k y3 3
14 13 0 864 3k y k ) 5670k e (128k y
0 0k y k y2
14 9 13 17 1296k y 2k ye 64k y k e 32k )
8 4 3 2
0 6 5 7 16 49k y (y(3k y 6k y 14k y 42k ) 210k )
0 0k y k y4 4 3 2
0 8 14 935k e (1296k y 1296k y 81k y e
0 0k y 2k y2
13 12 17 101296k y 1296k y 81k ye 16k e
0 0k y k y
15 11 0 8 1481k e 1296k ) 5443200k e 4k y k
0k y
838102400k e ),
t
b
N
1 N
2 1
y h 1dp
Brdx h h
(48)
1 b 1 1 b
1 2 b 1 1 b
a (x)Pr N y a (x)h Pr N
t
05 2a (x)h Pr N a (x)h Pr N
b
N e e6b Brk y
N e e
0 1 b 1 1 b
1 2 b 1 1 b
k y a (x)Pr N y a (x)h Pr)N3
a (x)h Pr N a (x)h Pr)N0
k e e e ReGr sin
k Fre e
0k y
2 304 05 2
0
12k e(b ( 72b 12Brk y )
k
0k y
04 0 05 2 3
3
0
6b k 6b Brk y k e
k
0k y2
0 05 2 3
3 8
0 0
k y 12b Brk y 2k e 1
k 7560k
0k y3 3 2
0 05 2 3 0 0(126k ( 15e (72b Brk k k y k y 3
2 3 3 2 2 4
05 3 0 0 0 0 2 3432b k k y k y 2 k (k (3Br k k y
8 144(y(y(y k y k
0
1
y0
dpp | ,
dx
(49)
Where the constants appearing in Eqs. (47) and (48)
are defined in appendix.
Expressions for different wave shape
The non- dimensional expressions for five considered
wave form are given by [26]. The expressions for
the triangular, square and trapezoidal wave are derived
from the Fourier series.
1) Sinusoidal wave
1 2h (x) 1 asin 2 x, h (x) d bsin(2 x ).
2) Multisinusoidal wave
1 2h (x) 1 asin 2n x, h (x) d bsin(2n x ).
3) Triangular wave
m 1
1 3 2m 1
18h (x) 1 a sin 2 2m 1 x ,
2m 1
Iran. J. Chem. Chem. Eng. Influence of Nanoparticles Phenomena on the Peristaltic Flow ... Vol. 36, No. 2, 2017
113
Fig. 1: Velocity profile for different values of Q for fixed
values of a= 0.7, b= 0.7, d= 1, = /2, Gr= 0.8, Nb= 0.5, Pr= 2,
Nt= 0.9, x= 0, Br= 0.6, = 0.08.
Fig. 2: Velocity profile for different values of for fixed
values of a= 0.7, b= 0.7, d= 1, = /2, Gr= 0.8, Nb= 0.5, Pr= 2,
Nt= 0.9, x= 0, Br= 0.6, Q= 0.08.
m 1
2 3 2m 1
18h (x) d b sin 2 2m 1 x .
2m 1
4) Trapezoidal wave
8
1 2 2m 1
sin 2m 132h (x) 1 a sin 2 2m 1 x ,
2m 1
8
2 2 2m 1
sin 2m 132h (x) d b sin 2 2m 1 x .
2m 1
5) Square wave
m 1
1m 1
14h (x) 1 a cos 2 2m 1 x ,
2m 1
m 1
2m 1
14h (x) d b cos 2 2m 1 x .
2m 1
Numerical results and discussion
The main objective of this portion is to revise
the graphical significances of the present flow problem.
Mathematica software is used to carry out the expressions
for pressure rise and pressure gradient because pressure
rise definition involves integration of dp/dx which is not
solvable analytically. Figs. 1 to 3 show the velocity
profile for different values of volume flow rate Q,
pseudoplastic parameter and different wave forms.
It is observed from Fig. 1 that the magnitude value of the
velocity profile decreases with an increase in Q. Fig. 2
shows the velocity profile for different values of .
It is depicted from Fig. 2 that near the center of the channel
the magnitude of the velocity profile decreases with
an increase in Fig. 3 shows the velocity profile
for different wave forms. It is observed from Fig. 3 that
the magnitude value of the velocity profile increases
in sinusoidal, multisinsoidal, trapezoidal and triangular
wave. In order to see the behavior of pressure rise for
different values of , Gr, d and Figs. 4 to 7 are
prepared. It is observed from Figs. 4 and 5 that the
behavior of pressure rise in augmented pumping
p 0, Q 0 , peristaltic pumping p 0, Q 0
and retrograde pumping p 0, Q 0 regions is same.
In these regions the pressure rise increases with an
increase in and Gr. It is depicted from Figs. 6 and 7 that
in the augmented pumping region p 0, Q 0 the
pressure rise increases with an increase in the width of
the channel d, while in the peristaltic pumping
p 0, Q 0 region the pressure rise decreases.
Figs. 8 to 10 indicate the pressure gradient for different
value of Fr and . It is depicted that for x [0,0.2] and
x [0.8,1], the pressure gradient is small i.e., the flow
can easily pass with out imposition of a large pressure
gradient, while in the region x [0.2,0.8], pressure
gradient decreases with an increase in Fr and , large
-1 -0.5 0 0.5 1 1.5-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
y
u
Q = 0.1
Q = 0.5
Q = 0.7
Q = 0.9
Fig. 1.
-1 -0.5 0 0.5 1 1.5-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
y
u
= 0.0
= 0.04
= 0.06
= 0.08
Fig. 2.
-1 -0.5 0 0.5 1 1.5-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
y
u
Q = 0.1
Q = 0.5
Q = 0.7
Q = 0.9
Fig. 1.
-1 -0.5 0 0.5 1 1.5-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
y
u
= 0.0
= 0.04
= 0.06
= 0.08
Fig. 2.
-1 -0.5 0 0.5 1 1.5
y
-1 -0.5 0 0.5 1 1.5
y
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
u
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
u
Iran. J. Chem. Chem. Eng. Akram S. & Nadeem S. Vol. 36, No. 2, 2017
114
Fig. 3: Velocity profile for different wave forms for fixed values of a= 0.7, b= 0.7, d= 1, = /2, Gr= 0.8, Nb= 0.5, Pr= 2, Nt= 0.9,
x=0, Br= 0.6, = 0.08. (a) for sinusoidal wave, (b) for multisinusoidal wave, (c) for Trapezoidal wave, (d) for Triangular wave.
Fig. 4: Variation of pressure rise p with Q for a= 0.7, b= 0.7,
d= 1.5, = /2, = 0.01, Nt= 0.9, Nb= 0.5, Pr= 1, Gr= 0.8, Br=
0.9, Re= 0.5, Fr= 0.8.
Fig. 5: Variation of pressure rise p with Q for a= 0.7, b= 0.7,
= 0.2, = /4, = 0.001, Nt= 0.9, Nb= 0.5, Pr= 1.8, d= 1.5,
Br= 0.9, Re= 0.5, Fr= 0.8.
-1.5 -1 -0.5 0 0.5 1-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
y
u
Q = 0.1
Q = 0.5
Q = 0.7
Q = 0.9
Sinusoidal Wave
Fig. 3a
-1.5 -1 -0.5 0 0.5 1-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
y
u
Q = 0.1
Q = 0.5
Q = 0.7
Q = 0.9
Multisinusoidal Wave
Fig. 3b
-1.5 -1 -0.5 0 0.5 1-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
y
u
Q = 0.1
Q = 0.5
Q = 0.7
Q = 0.9
Sinusoidal Wave
Fig. 3a
-1.5 -1 -0.5 0 0.5 1-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
y
u
Q = 0.1
Q = 0.5
Q = 0.7
Q = 0.9
Multisinusoidal Wave
Fig. 3b
-2 -1.5 -1 -0.5 0 0.5 1-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
y
u
Q = 0.1
Q = 0.5
Q = 0.7
Q = 0.9
Trapezoidal Wave
Fig. 3c
-1 -0.5 0 0.5 1-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
y
u
Q = 0.1
Q = 0.5
Q = 0.7
Q = 0.9
Triangular Wave
Fig. 3d
-2 -1.5 -1 -0.5 0 0.5 1-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
y
u
Q = 0.1
Q = 0.5
Q = 0.7
Q = 0.9
Trapezoidal Wave
Fig. 3c
-1 -0.5 0 0.5 1-1
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
y
u
Q = 0.1
Q = 0.5
Q = 0.7
Q = 0.9
Triangular Wave
Fig. 3d
-1 -0.5 0 0.5 1 1.5 2 2.5 3-3
-2
-1
0
1
2
3
4
Q
p
= 0.0
= 0.3
= 0.6
= 0.9
Fig. 4
-1 -0.5 0 0.5 1 1.5 2 2.5 3-4
-3
-2
-1
0
1
2
3
4
5
Q
p
Gr = 0.1
Gr = 0.3
Gr = 0.5
Gr = 0.7
Fig. 5
-1 -0.5 0 0.5 1 1.5 2 2.5 3-3
-2
-1
0
1
2
3
4
Q
p
= 0.0
= 0.3
= 0.6
= 0.9
Fig. 4
-1 -0.5 0 0.5 1 1.5 2 2.5 3-4
-3
-2
-1
0
1
2
3
4
5
Q
p
Gr = 0.1
Gr = 0.3
Gr = 0.5
Gr = 0.7
Fig. 5
-1.5 -1 -0.5 0 0.5 1
y
-1.5 -1 -0.5 0 0.5 1
y
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
u
u
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
-2 -1.5 -1 -0.5 0 0.5 1
y
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
u
-1 -0.5 0 0.5 1
y
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1
u
-1 -0.5 0 0.5 1 1.5 2 2.5 3
Q
-1 -0.5 0 0.5 1 1.5 2 2.5 3
Q
4
3
2
1
0
-1
-2
-3
p
5
4
3
2
1
0
-1
-2
-3
-4
p
Iran. J. Chem. Chem. Eng. Influence of Nanoparticles Phenomena on the Peristaltic Flow ... Vol. 36, No. 2, 2017
115
Fig. 6: Variation of pressure rise p with Q for a=0.7, b=0.7,
= 0.2, = /4, = 0.001, Nt= 0.9, Nb= 0.5, Pr= 1, Gr= 0.8, Br=
0.9, Re=0.5, Fr= 0.8.
Fig. 7: Variation of pressure rise p with Q for a=0.7, b= 0.7,
= 0.2, = /4, Gr= 0.8, Nt= 0.9, Nb= 0.5, Pr= 1, d= 1.5,
Br= 0.9, Re=0.5, Fr= 0.8.
Fig. 8: Variation of pressure gradient dp/dx with x for
a= 0.7, b= 0.7, = 0.6, = /6, = 0.03, Q= 0.5, Nt= 0.9,
Nb= 0.5, Pr= 1.5, d= 1.5, Br= 0.9, Re= 0.8, Gr= 0.8.
Fig. 9: Variation of pressure gradient dp/dx with x for
a= 0.7, b= 0.7, = 0.6, = /4, Fr= 0.8, Q= 0.5, Nt= 0.9,
Nb= 0.5, Pr= 1.5, d= 1.5, Br= 0.9, Re= 0.5, Gr= 0.8.
amount of pressure gradient is required to maintain
the flux to pass. Figs. 10 indicate the pressure gradient
for different wave forms. Figs. 11 and 12 are displayed
to analysis the influence of temperature profile on Nt and Pr
It is explored from Figs. 11 and 12 that the temperature
profile increases with an increase in Nt and Pr. This is
physically valid because these parameters show a direct
relationship with temperature. To examine the effects
of concentration profile on Nt and Pr, Figs. 13 and 14
are prepared. It is illustrated from figures that the concentration
profile decreases with an increase in Nt and Pr.
Stream lines for different values of d and are shown
in Figs. 15 to 16. It is depicted from Figs. 15 that the size
of the trapping bolus decreases with the decrease in the
width of the channel. It is observed form Fig. 16 that the
size of the trapping bolus increases with an increase in .
It is also observed from Figs. 15 and 16 that the trapping
bolus also shifted towards right side of the channel and
this happens due to increase of phase angle. Stream lines
for different wave forms are shown in Fig. 17.
CONCLUSIONS
In the current research paper we have investigated
the influence of nanofluid on peristaltic transport of
a pseudoplastic fluid in the presence of inclined
asymmetric channel. With the help of long wavelength
-1 -0.5 0 0.5 1 1.5 2 2.5 3-4
-3
-2
-1
0
1
2
3
4
5
Q
p
d = 1.5
d = 1.6
d = 1.7
d = 1.8
Fig. 6
-1 -0.5 0 0.5 1 1.5 2 2.5 3-4
-3
-2
-1
0
1
2
3
4
5
Q
p
= 0.0
= 0.01
= 0.02
= 0.03
Fig. 7
-1 -0.5 0 0.5 1 1.5 2 2.5 3-4
-3
-2
-1
0
1
2
3
4
5
Q
p
d = 1.5
d = 1.6
d = 1.7
d = 1.8
Fig. 6
-1 -0.5 0 0.5 1 1.5 2 2.5 3-4
-3
-2
-1
0
1
2
3
4
5
Q
p
= 0.0
= 0.01
= 0.02
= 0.03
Fig. 7
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
x
dp
/dx
Fr = 0.2
Fr = 0.4
Fr = 0.6
Fr = 0.9
Fig. 8
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
x
dp
/dx
= 0.0
= 0.02
= 0.04
= 0.06
Fig. 9
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
x
dp
/dx
Fr = 0.2
Fr = 0.4
Fr = 0.6
Fr = 0.9
Fig. 8
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
x
dp
/dx
= 0.0
= 0.02
= 0.04
= 0.06
Fig. 9
5
4
3
2
1
0
-1
-2
-3
-4
p
5
4
3
2
1
0
-1
-2
-3
-4
p
-1 -0.5 0 0.5 1 1.5 2 2.5 3
Q
-1 -0.5 0 0.5 1 1.5 2 2.5 3
Q
8
7
6
5
4
3
2
1
0
dp
/dx
0 0.2 0.4 0.6 0.8 1
x
7
6
5
4
3
2
1
0
dp
/dx
0 0.2 0.4 0.6 0.8 1
x
Iran. J. Chem. Chem. Eng. Akram S. & Nadeem S. Vol. 36, No. 2, 2017
116
Fig. 10: Variation of pressure gradient dp/dx with x for a= 0.8, b= 0.1, = 0.6, = /4, Fr= 0.8, Nt = 0.9, Nb= 0.5,
Pr= 1.0, d= 1.8, Br= 0.9, Re= 0.5, Gr= 0.1, = 0.03. (a) For sinusoidal wave, (b) For multisinusiodal wave,
(c) For square wave, (d) For Triangular wave.
Fig. 11: Temperature profile for different values of Nt for
fixed values of a= 0.5, b= 1.0, Pr= 1, Nb= 0.3, d= 1.5, x=0,
= /6.
Fig. 12: Temperature profile for different values of Pr for
fixed values of a= 0.5, b= 1.0, Nt= 0.7, Nb= 0.3, d= 1.5, x=0,
= /6.
0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
1
1.5
2
x
dp
/dx
Q = 0.1
Q = 0.3
Q = 0.7
Q = 1.0
Fig. 10a
0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
1
1.5
2
x
dp
/dx
Q = 0.0
Q = 0.3
Q = 0.7
Q = 1.0
Fig. 10b
0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
1
1.5
2
x
dp
/dx
Q = 0.1
Q = 0.3
Q = 0.7
Q = 1.0
Fig. 10a
0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
1
1.5
2
x
dp
/dx
Q = 0.0
Q = 0.3
Q = 0.7
Q = 1.0
Fig. 10b
0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
1
1.5
2
2.5
x
dp
/dx
Q = 0.1
Q = 0.3
Q = 0.7
Q = 1.0
Fig. 10c
0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
1
1.5
2
2.5
x
dp
/dx
Q = 0.1
Q = 0.3
Q = 0.7
Q = 1.0
Fig. 10d
0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
1
1.5
2
2.5
x
dp
/dx
Q = 0.1
Q = 0.3
Q = 0.7
Q = 1.0
Fig. 10c
0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
1
1.5
2
2.5
x
dp
/dx
Q = 0.1
Q = 0.3
Q = 0.7
Q = 1.0
Fig. 10d
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
Nt = 0.1
Nt = 0.3
Nt = 0.5
Nt = 0.7
Fig. 11
-2 -1.5 -1 -0.5 0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
Pr = 0.2
Pr = 0.4
Pr = 0.6
Pr = 0.8
Fig. 12
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
Nt = 0.1
Nt = 0.3
Nt = 0.5
Nt = 0.7
Fig. 11
-2 -1.5 -1 -0.5 0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
Pr = 0.2
Pr = 0.4
Pr = 0.6
Pr = 0.8
Fig. 12
2
1.5
1
0.5
0
-0.5
dp
/dx
0 0.2 0.4 0.6 0.8 1
x
0 0.2 0.4 0.6 0.8 1
x
2
1.5
1
0.5
0
-0.5
dp
/dx
0 0.2 0.4 0.6 0.8 1
x
0 0.2 0.4 0.6 0.8 1
x
2.5
2
1.5
1
0.5
0
-0.5
dp
/dx
2.5
2
1.5
1
0.5
0
-0.5
dp
/dx
-2 -1.5 -1 -0.5 0 0.5 1 1.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
y y
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Iran. J. Chem. Chem. Eng. Influence of Nanoparticles Phenomena on the Peristaltic Flow ... Vol. 36, No. 2, 2017
117
Fig. 13: Concentration profile for different values of Nt for
fixed values of a= 0.5, b= 1.0, Pr= 1, Nb= 0.3, d= 1.5, x=0,
= /6.
Fig. 14: Concentration profile for different values of Pr for
fixed values of a= 0.5, b= 1.0, Nt= 0.7, Nb= 0.3, d= 1.5, x=0,
= /6.
Fig. 15: Stream lines for different values of d and for fixed values of
a= 0.7, b= 0.7, d= 1, Nt= 0.9, Nb= 0.5, Pr= 1, Q= 2, Br= 0.1, = 0.03, Gr= 0.1.
-2 -1.5 -1 -0.5 0 0.5 1 1.5-0.2
0
0.2
0.4
0.6
0.8
1
1.2
y
Nt = 0.1
Nt = 0.3
Nt = 0.5
Nt = 0.7
Fig. 13
-2 -1.5 -1 -0.5 0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
Pr = 0.2
Pr = 0.4
Pr = 0.6
Pr = 0.8
Fig. 14
-2 -1.5 -1 -0.5 0 0.5 1 1.5-0.2
0
0.2
0.4
0.6
0.8
1
1.2
y
Nt = 0.1
Nt = 0.3
Nt = 0.5
Nt = 0.7
Fig. 13
-2 -1.5 -1 -0.5 0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y
Pr = 0.2
Pr = 0.4
Pr = 0.6
Pr = 0.8
Fig. 14
Fig. 15afor0, d 1.1 Fig. 15bfor
2, d 1.1
Fig. 15cfor for0, d 1.0 Fig. 15dfor
2, d 1.0
Fig. 15afor0, d 1.1 Fig. 15bfor
2, d 1.1
Fig. 15cfor for0, d 1.0 Fig. 15dfor
2, d 1.0
Fig. 15afor0, d 1.1 Fig. 15bfor
2, d 1.1
Fig. 15cfor for0, d 1.0 Fig. 15dfor
2, d 1.0
Fig. 15afor0, d 1.1 Fig. 15bfor
2, d 1.1
Fig. 15cfor for0, d 1.0 Fig. 15dfor
2, d 1.0
-2 -1.5 -1 -0.5 0 0.5 1 1.5
y
-2 -1.5 -1 -0.5 0 0.5 1 1.5
y
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.5
1
0.5
0
-0.5
-1
-1.5
1.5
1
0.5
0
-0.5
-1
-1.5
-0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4
1.5
1
0.5
0
-0.5
-1
-1.5
1.5
1
0.5
0
-0.5
-1
-1.5
-0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4
Iran. J. Chem. Chem. Eng. Akram S. & Nadeem S. Vol. 36, No. 2, 2017
118
Fig. 16: Stream lines for different values of and for fixed values of
a= 0.7, b= 0.7, d= 1, Nt= 0.9, Nb= 0.5, Pr= 1.5, Q= 1.8, Br= 0.9, d= 1, Gr= 0.8.
and low Reynolds number approximation the governing
equations of a pseudoplastic fluid along with nanofluid
are modeled. The exact solutions for temperature and nano
particle volume fraction are calculated. Perturbation technique
is used to carry out the series solution of stream function and
pressure gradient. Graphical results were plotted and reported
for different involved physical parameters of interest. The main
results of the present study can be summarized as follows:
● The magnitude value of the velocity profile
decreases with an increase in Q and
● The magnitude value of the velocity profile
increases in sinusoidal, multisinsoidal, trapezoidal and
triangular waves.
● The pressure gradient decreases with an increase
in Fr and .
● The temperature profile increases with an increase
in Nt and Pr.
● The concentration profile decreases with an increase
in Nt and Pr.
● The size of the trapping bolus decreases with the
decrease in the width of the channel d and increases with
an increase in .
Appendix
0 1 bk a (x)Pr N ,
2 0 1 0
01 h k h k
kk ,
e e
t
b
N
N
22 1
1k ,
h h
1 t3 1
b
Brk Nk Grk ,
N
Fig. 16afor0, 0.01 Fig. 16bfor
2, 0.01
Fig. 16cfor0, 0.07 Fig. 16dfor
2, 0.07
Fig. 16afor0, 0.01 Fig. 16bfor
2, 0.01
Fig. 16cfor0, 0.07 Fig. 16dfor
2, 0.07
Fig. 16afor0, 0.01 Fig. 16bfor
2, 0.01
Fig. 16cfor0, 0.07 Fig. 16dfor
2, 0.07
Fig. 16afor0, 0.01 Fig. 16bfor
2, 0.01
Fig. 16cfor0, 0.07 Fig. 16dfor
2, 0.07
1.5
1
0.5
0
-0.5
-1
-1.5
1.5
1
0.5
0
-0.5
-1
-1.5
-0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4
1.5
1
0.5
0
-0.5
-1
-1.5
1.5
1
0.5
0
-0.5
-1
-1.5
-0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4
Iran. J. Chem. Chem. Eng. Influence of Nanoparticles Phenomena on the Peristaltic Flow ... Vol. 36, No. 2, 2017
119
Fig. 17: Stream lines for different wave forms for fixed values of
a= 0.7, b= 0.7, d= 1, Nt= 0.9, Nb= 0.5, Pr= 1.5, Q= 1.8, Br= 0.9, d= 1, Gr= 0.8.
2 2
4 04 2 05 04k 12b Brk 432b b ,
2 2
5 05 2k 90b Br k ,
3 3
6 2
15k Br k ,
4
2 2 2
7 04 2 05 2k 18b Br k 648b Brk ,
2 2
8 2 3
3k Br k k ,
4
2
2 39 2
0
6Brk kk ,
k
3
310 4
0
9kk ,
k
204 2 3 05 04 311 04 32
00
12b Brk k 144b b kk 12b k
kk
2
05 3
2
0
216b k,
k
2
05 2 3 04 2 3 05 312 2
0 00
108b Brk k 24b Brk k 432b kk
k kk
04 05 372b b k ,
205 2 313 04 2 3 05 3
0
108b Brk kk 6b Brk k 108b k
k
2 2
2 3
2
0
9Br k k,
k
Fig 17afor sinusoidal wave Fig 17bfor multisinusoidal wave
Fig 17cfor Triangular wave Fig 17dfor Trapezoidal wave
Fig 17afor sinusoidal wave Fig 17bfor multisinusoidal wave
Fig 17cfor Triangular wave Fig 17dfor Trapezoidal wave
Fig 17afor sinusoidal wave Fig 17bfor multisinusoidal wave
Fig 17cfor Triangular wave Fig 17dfor Trapezoidal wave
Fig 17afor sinusoidal wave Fig 17bfor multisinusoidal wave
Fig 17cfor Triangular wave Fig 17dfor Trapezoidal wave
1.5
1
0.5
0
-0.5
-1
-1.5
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
1.5
1
0.5
0
-0.5
-1
-1.5
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
1.5
1
0.5
0
-0.5
-1
-1.5
1.5
1
0.5
0
-0.5
-1
-1.5
-0.2 0 0.2 0.4 0.6 -0.2 0 0.2 0.4 0.6
Iran. J. Chem. Chem. Eng. Akram S. & Nadeem S. Vol. 36, No. 2, 2017
120
2 2
3 214 05 3 2
0
6Br k kk 18b Brk k ,
k
2 2 2
04 3 05 3 2 315 2 3 4
0 0 0
24b k 72b k 3Brk kk ,
k k k
3
16 04 05 2 05k 216b b Brk 1296b ,
2 2
05 3 2 317 2 3
0 0
72b k 12Brk kk ,
k k
1 0 2 0h k h k
18 8k 76204800k (e e )
1 0h k
0 8 1 2 145443200k (e (k (11h 3h ) 2k )
2 0h k
8 1 2 14e (k (3h 11h ) 2k )),
1 0h k2
19 0 1 8 1 2k 28350k (32e (12h k (2h h )
2 0h k
14 1 2 13 2 8 1 23k (3h h ) 2k ) 32e ( 12h k (h 2h )
1 0 2 02h k 2h k
14 1 2 13 9 93k (h 3h ) 2k ) k e k e ),
8 3 2 2
20 0 1 2 16 1 1 2 2k 18k ((h h ) (21k (3h 4h h 3h )
4 3 2 3 4
5 1 1 2 1 2 1 2 23k (5h 8h h 9h h 8h h 5h )
2 2
1 2 7 1 1 2(h h )(14k (2h h h 2h2 )
4 3 3 4
6 1 1 2 1 2 2 4 1 23k (3h 2h h 2h h 3h ) 210k (h h ))),
1 0h k3 2
21 0 1 8 1 2k 5670k ( 32e (2h k (13h 9h )
1 14 1 2 13 1 2 123h k (5h 3h ) k (7h 3h ) 2k )
2 0h k 2
2 8 2 1 2 14 2 132e (2h k (13h 9h ) 3h k (5h 3h )
2 02h k
13 2 1 12 9 2 1 17k (7h 3h ) 2k ) e (k (7h 3h ) 2k )
1 02h k
9 2 1 17e (k (3h 7h ) 2k ))),
1 0h k 3
22 1 8 1 2k 1296e (2h k (7h 6h )
1 13 1 2 1 14 1 2h (k (8h 6h ) h k (11h 9h ))
1 0 2 03h k 3h k
12 1 2 11 10 10k (5h 3h ) 2k ) 32k e 32k e ,
2 0h k4 3
23 0 2 8 2 1k 35k ( 1296e (2h k (7h 6h )
2 13 2 1 2 14 2 1h (k (8h 6h ) h k (11h 9h ))
2 02h k
12 2 1 11 2 9 2 1k (5h 3h ) 2k ) 81e (2h k (4h 3h )
1 02h k
17 2 1 15 1 9 1 2k (5h 3h ) 2k ) 81e (2h k (4h 3h )
17 1 2 15 22k (5h 3h ) 2k ) k ),
1 03h k5
24 0 1 2 10k 210k (h h )( 8k e
1 0h k
1 1 1 1 8 14 13 12 11216e (h (h (h (h k k ) k ) k ) k )
1 0 2 02h k 3h k
1 1 9 17 15 1027e h h k k k 8k e
2 0h k
2 2 2 2 8 14 13 12 11216e (h (h (h h k k k ) k ) k )
2 02h k
2 2 9 17 1527e h h k k k ),
1 0 2 0h k h k
25 8 1 2k 38102400k (h h ) e e ,
1 0h k 2 2
26 0 8 1 1 2 2k 5443200k (e k 5h 5h h 2h
2 0h k 2 2
14 1 2 8 1 1 2 2k (h h ) e k ( 2h 5h h 5h )
14 1 2k (h h ))),
1 0h k2 2 2
27 0 1 8 1 1 2 2k 14175k ( 64e (2h k (5h 5h h 4h )
2 2
14 1 1 2 2 13 1 22k (2h 2h h h ) k (h h ))
2 0h k 2 2
2 8 1 1 2 264e (2h k ( 4h 5h h 5h )
2 2
14 1 1 2 2 13 1 22k (h 2h h 2h ) k (h h ))
2 0 1 02h k 2h k
9 1 2 9 1 2k (h h )e k (h h )e ),
2 2 3
28 4 1 1 2 2 1 2k 70k h 4h h h (h h ) ,
3 2 2 2
29 1 2 1 1 2 2 7 1 2 2k (h h ) (14(h 3h h h )(k (h1 h h h )
16 1 2 5 1 22k (h h )) 4k (h h )
4 3 2 2 3 4
1 1 2 1 2 12 2 2(2h 6h h 5h h 6h h 2h )
6 5 4 2 3 3 6
6 1 1 2 1 2 1 2 2k (5h 20h h 29h h 32h h 5h
2 4 5
1 2 1 229h h 20h h ),
1 0h k 2 2 2
30 1 8 1 1 2 2k 32e (2h k (5h 5h h 6h )
2 2 2 2
1 14 1 1 2 2 13 1 1 2 26h k (h h h h ) k (3h 3h h 2h )
12 1 2k (h h )),
2 2 2
31 2 8 1 1 2 2k 2h k ( 6h 5h h 5h )
2 2 2 2
2 14 1 1 2 2 13 1 1 2 26h k ( h h h h ) k ( 2h 3h h 3h )
12 1 2k (h h ),
Iran. J. Chem. Chem. Eng. Influence of Nanoparticles Phenomena on the Peristaltic Flow ... Vol. 36, No. 2, 2017
121
1 02h k 2 2
32 9 1 1 2 2k e (k (3h 3h h 2h )
2 02h k 2 2
17 1 2 9 1 1 2 2k (h h )) e (k ( 2h 3h h 3h )
17 1 2k (h h )),
2 0h k3 2 2
33 8 2 1 1 2 2k 1296k (h ( 8h 5h h 5h )e
1 0h k3 2 2
1 1 1 2 2h (5h 5h h 8h )e ),
2 02h k2 2
34 9 2 1 1 2 2k 81k (h ( 4h 3h h 3h )e
1 02h k2 2
1 1 1 2 2h (3h 3h h 4h )e ),
1 0h k 2 2
35 12 1 1 2 2k 1296e (2k (h h h h )
2 2 2
1 13 1 1 2 2 1 14 1 1 2 2h (k (3h 3h h 4h ) 2h k (2h 2h h 3h ))
1 03h k
11 1 2 10 1 2k (h h )) 16k (h h )e
2 03h k
10 1 2 33 3416k (h h )e k k ,
2 0h k 2 2
36 12 1 1 2 2k 1296e (2k ( h h h h )
2 2
2 13 1 1 2 2h (k ( 4h 3h h 3h )
2 2
2 14 1 2 2 11 1 22h k ( 3h 2hh 2h ) k (h h )),
1 02h k4 2 2
37 0 17 1 1 2 2k 35k ( 81e (2k (h h h h )
2 02h k 2 2
15 1 2 17 1 1 2 2k (h h )) 81e (2k ( h h h h )
15 1 2 35 36k (h h )) k k ),
2 03h k5
38 0 1 2 10 1 2k 70k (h h )(8k (2h h )e
2 0h k
1 2 2 2 2 2 8 14 13 12216(2h h )e (h (h (h (h k k ) k ) k )
2 02h k
11 1 2 2 2 9 17 15k ) 27(2h h )e (h (h k k ) k )
1 0 1 0h k 2h k
1 2 10 1 1 1 1 8 14(h 2h )e (8k e 216(h (h (h (h k k )
1 0h k
13 12 11 1 1 9 17 15k ) k ) k ) 27e h h k k k )),
1 0h k
39 0 2k 5443200k (h e
2 2
8 1 1 2 2 1 14k 10h h h h 2h k
2 0h k 2 2
1 8 1 1 2 2 2 14h e (k (h h h 10h ) 2h k ))
1 0 2 0h k h k
1 2 876204800h h k e e ,
1 0h k2 2 2
40 0 2 1 8 1 1 2 2 13k 28350k (h e 64h 2k 5h h h h k
1 0h k2 2
14 1 1 2 2 1 932k 8h h h h h k e
2 0h k 2 2
1 2 8 1 1 2 232h e 4h k h h h 5h
2 02h k2 2
14 1 1 2 2 2 13 1 2 9k h h h 8h 2h k h h k e ),
8 3 2 2
41 1 2 0 1 2 16 1 1 2 2k 18h h k (h h ) (7k (4h 7h h 4h )
4 3 2 2 3 4
5 1 1 2 1 2 1 2 2k (8h 17h h 20h h 17h h 8h )
2 2
1 2 7 1 1 2 2(h h )(14k (h h h h )
4 3 2 2 3 4
6 1 1 2 1 2 1 2 2 4 1 2k (5h 6h h 8h h 6h h 5h )) 70k (h h )),
1 0h k 2 2 2
42 2 1 8 1 1 2 2k 32h e (2h k ( 10h 3h h 3h )
2 2 2 2
1 14 1 1 2 2 13 1 1 2 2 1 123h k ( 4h h h h ) k ( 6h h h h ) 2h k ),
2 0h k 2 2
43 1 2 2 8 1 1 2 2 12k h e (64h (h k ( 3h 3h h 10h ) k )
2 2
2 14 1 1 2 2 13 1 2 1 296h k (h h h 4h ) 32k (h 2h )(h 3h )
2 0h k
2 17 9 1 2 1 2e 2h k k (h 2h )(h 3h ) ),
1 02h k3 2 2
44 0 2 9 1 1 2 2k 5670k (h e (k ( 6h h h h )
1 17 42 432h k ) k k ),
2 0h k 2 2
45 1 2 13 1 1 2 2k 1296h e (h (2k (h h h 3h )
2 2
2 14 1 1 2 2 12 1 2 11h k (3h 3h h 8h ) k (h 4h ) 2k )
2 03h k3 2 2 2
2 8 1 1 2 2 1 12 1 2 102h k (2h 2h h 5h ) h k ) 32h h k e ,
2 02h k 2 2
46 1 2 9 1 1 2 2k 81h e 2h k h h h 3h
2 2
17 1 1 2 2 2 15 45k h h h 4h 2h k k ,
3 2 2 2
47 1 13 1 14 1 1 2 2k 6h k h k ( 8h 3h h 3h )
2 2 3 2 2
1 2 13 1 12 1 8 1 1 2 22h h k 4h k 2h k ( 5h 2h h 2h )
2 2
1 2 13 1 2 12 1 11 2 122h h k h h k 2h k h k ,
1 0 1 0h k h k 2 2
48 2 1 9 1 1 2 2k h e ( 81e (2h k ( 3h h h h )
2 2
17 1 1 2 2 1 15k ( 4h h h h ) 2h k )
1 02h k
1 10 47 4632h k e 1296k ) k ,
Iran. J. Chem. Chem. Eng. Akram S. & Nadeem S. Vol. 36, No. 2, 2017
122
2 03h k
49 1 10 1 2k 8h k (h 2h )e
2 0h k
1 1 2 2 2 2 2 8 14 13216h (h 2h )e (h (h (h (h k k ) k )
2 02h k
12 11 1 1 2 2 2 9 17 15k ) k ) 27h (h 2h )e (h (h k k ) k ),
1 0 1 0h k 2h k5
50 0 1 2 2 1 2 10k 70k (h h )(h (2h h )e (8k e
1 1 1 1 8 14 13 12 11216(h (h (h (h k k ) k ) k ) k )
1 0h k
1 1 9 17 15 4927e (h (h k k ) k )) k ),
2 0h k2
51 8 1 2 1k 38102400k (h (3h h )e
1 0h k2
2 2 1h (h 3h )e ),
2 0h k2
52 0 1 2 8 1 2k 5443200k (h e (h k (7h 15h )
1 0h k2
14 1 2 2 1 8 1 2 14 1 2k (h 3h )) h e (h k (15h 7h ) k (3h h ))),
1 0h k2 2
53 2 1 8 2 1 1 14 2 1k 64h e (6h k (3h 5h ) 6h k (h 2h )
1 02h k2
13 2 1 2 9 2 1k (h 3h )) h k (h 3h )e ,
2 0h k2 2 2
54 0 1 2 8 2 1k 14175k (h e (64(6h k (5h 3h )
2 0h k
2 14 1 2 13 1 2 9 2 1 536h k (h 2h ) k (h 3h )) k (3h h )e ) k ),
2 2
55 5 1 2 1 1 2 2k 12k (h h )(2h h h 2h )
2 2 4 3 2 2
7 1 1 2 2 6 1 1 2 1 214k (3h 4h h 3h ) 3k (5h 8h h 9h h
3 4
1 2 2 16 1 2 48h h 5h ) 84k (h h ) 210k ,
8 2 2 3
56 0 1 2 55 1 2k 9k (h h k (h h ) ),
2 0h k2 3
57 1 2 8 1 2k 32h e (2h k (11h 15h )
2 13 1 2 2 14 1 2 12 1 2h (k (5h 9h ) 6h k (2h 3h )) k (h 3h )),
1 0h k3 2 3
58 0 2 1 8 1 2k 5670k (h e (32(2h k (15h 11h )
1 13 1 2 1 14 1 2 12 1 2h (k (9h 5h ) 6h k (3h 2h )) k (3h h )) ,
1 0h k
1 9 1 2 17 1 2e (h k (9h 5h ) k (3h h )))
2 02h k2
1 2 9 1 2 17 1 2 57h e (h k (5h 9h ) k (h 3h )) k ),
2 0h k2 2 2
59 1 2 8 2 2 1k 1296h h k (h (15h 13h )e
1 0 1 0h k 3h k2 2
1 1 2 2 10 2 1h (15h 13h )e ) 16h k (h 3h )e
1 0h k2
60 2 11 2 1k 1296h e (k (h 3h )
1 12 2 1 1 13 2 1h (k (4h 6h ) h (k (7h 9h )
1 14 2 12h k (5h 6h )))),
1 02h k2 2
61 2 1 9 2 1 1 17 2 1k 81h e (h k (7h 9h ) 2h k (2h 3h )
15 2 1k (h 3h )),
2 0 2 0h k h k2 2
62 1 2 9 2 1k h e (81e (h k (9h 7h )
2 17 2 1 15 1 22h k (3h 2h ) k (h 3h ))
2 02h k
10 2 1 11 1 216k (3h h )e 1296(k (h 3h )
2 12 1 2 2 13 1 2h (k (4h 6h ) h (k (7h 9h )
2 14 1 22h k (5h 6h ))))),
2 03h k5
63 1 2 0 1 2 1 10k 210h h k (h h h )(8k e
2 0h k
1 2 2 2 2 8 14 13 12 11216h e (h (h (h (h k k ) k ) k ) k )
2 02h k
1 2 2 9 17 1527h e (h (h k k ) k )
1 0 1 0h k 2h k
2 10 1 1 1 1 8 14 13h e (8k e 216(h (h (h (h k k ) k )
1 0h k
12 11 1 1 9 17 15k ) k ) 27e (h h k k k ))),
00 18 19 20 21 23 24b k k k k k k ,
2 0h k3
01 0 31 30 32b 5670k ( 32k e k k )
8
28 29 0 25 26 27 37 389 k k k k k k k k ,
4
02 48 0 39 40 41 44 50b 35k k k k k k k ,
4
03 59 60 61 62 0 51b 35 k k k k k k
52 54 56 58 63k k k k k ,
2 0
h k4 4
04 1 2 2 0 2 3 03 4
1 2 0
1b h ( Brh k k 24h k k e
24 h h k
2 0 1 0h k h k 4
3 3 2 0 072k e 24k e h k 3 72k q)
2 0 1 0h k h k4 4
2 2 2 0 3h ( Brh k k 72(k e e
1 0 2 0h k h k4 3
0 2 0 3 0k q) 24h k k 2e e 3k )
5 4 4 4 2 3 4
1 0 2 2 1 0 2 2 1 0 2Brh k k Brh h k k 8Brh h k k
1 0 2 0h k h k2 3 3
1 0 0 2 2 38h k (k (Brh k 9) 3k e 2e ) ,
Iran. J. Chem. Chem. Eng. Influence of Nanoparticles Phenomena on the Peristaltic Flow ... Vol. 36, No. 2, 2017
123
1 0 2 0
h k h k
05 33 4
1 2 0
1b 24k e e
12 h h k
4 3 3 3
0 1 0 2 2 1 0 2k Brh k k 2Brh h k k
1 0 2 0h k h k3 3
1 0 2 2 32h k Brh k 12 6k e e
1 0 2 0h k h k4 3 3 3
2 0 2 0 0 2 3Brh k k 24k q 2k 12h k e e ,
Received : Dec. 8, 2014 ; Accepted : Aug. 1, 2016
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