influence of nanoparticles phenomena on the peristaltic ...€¦ · iran. j. chem. chem. eng. akram...

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


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)


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,


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


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


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


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


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


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


2, d 1.1


2, d 1.0


2, d 1.1


2, d 1.0


2, d 1.1


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


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


2, 0.01


2, 0.07


2, 0.01


2, 0.07


2, 0.01


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


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







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


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 ),


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 ,


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 ) ,


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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influence of nanoparticles phenomena on the peristaltic ...€¦ · iran. j. chem. chem. eng. akram...

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