PERISTALTIC PUMPING OF COUPLE STRESS NON ERODIBLE … · Peristaltic pumping of couple stress fluid...

INTERNATIONAL JOURNAL OF ADVANCE RESEARCH, IJOAR .ORG ISSN 2320-9143 7

IJOAR© 2015 http://www.ijoar.org

International Journal of Advance Research, IJOAR .org

Volume 3, Issue 6, June 2015, Online: ISSN 2320-9143

PERISTALTIC PUMPING OF COUPLE STRESS

FLUID THROUGH NON - ERODIBLE POROUS

LINING TUBE WALL WITH THICKNESS OF

POROUS MATERIAL USING MAGNETIC FIELD N.G.Sridhar Government First Grade College, Sedam, District Gulbarga, Karnataka, India Email: [email protected]

Abstract

This paper is devoted to study the effect of thickness of porous material on the peristaltic

pumping of couple stress fluid when the tube wall is provided with non- erodible porous

lining with magnetic field. Long wavelength and low Reynolds number approximation is

used to linearize the governing equations. The expression for axial velocity, pressure

gradient and frictional force are obtained by using Beavers-Joseph Boundary conditions.

The effect of various parameters on pumping characteristics is discussed with the help of

graphs.

Keywords:

Peristaltic pumping, Magnetic field, volume flow rate, pressure rise, couple stress fluid,

pumping Characteristics, Darcy’s law.



1. INTRODUCTION:

Peristaltic pumping is now well known to physiologists to be one of the major

mechanisms for fluid transport in many biological systems. In particular, Peristalsis is an

important mechanism generated by the propagation of waves along the walls of a channel

or tube. It occurs in the gastrointestinal, urinary, reproductive tracts and many other

glandular ducts in a living body. Blood is a suspension of cells in plasma. It is a

biomagnetic fluid, due to the complex integration of the intercellular protein, cell

membrane and the hemoglobin, a form of iron oxide, which is present at a uniquely high

concentration in the mature red cells, while its magnetic property is influenced by factors

such as the state of oxygenation.

A number of studies containing couple stress have been investigated by Alemayehu and

Radhakrishnamacharya [1], Elshehawey and El-Sebaei [3], Raghunath Rao and Prasad

Rao [11], Sohail Nadeem and Safia Akram [19] and Srivastava [20]. Couple Stress in

peristaltic transport of fluids is studied by Elshehawey and Mekheimer [2]. The initial

mathematical model of peristalsis obtained by train of sinusoidal waves in an infinitely

long symmetric channel or tube has been investigated by Fung and Yahi [4] and Shapiro

et.al [18]. Peristaltic transport to a MHD third order fluid in a circular cylindrical tube

was investigated by Hayat and Ali [5]. Hayat et al. [6] have investigated peristaltic

transport of a third order fluid under the effect of a magnetic field. Effect of thickness of

the porous material on the peristaltic pumping when the tube wall is provided with non-

erodible porous lining has been studied by Hemadri et.al [7]. Jayarami Reddy et al. [8]

have studied the Peristaltic flow of a Williamson fluid in an inclined planar channel under

the effect of a magnetic field. Peristaltic transport of a Couple-stress fluid in a uniform

and Non- uniform Channels is studied by Mekheimer [9]. The consideration of blood as a

MHD fluid helps in controlling blood pressure and has potential for therapeutic use in the

diseases of heart and blood vessels by Mekheimer [10]. Peristaltic pumping of couple

stress fluid through non - erodible porous lining tube wall with thickness of porous

material Rathod, Sridhar and Mahadev [12]. Rathod and Sridhar [13] have studied the

peristaltic transport of couple stress fluid in uniform and non-uniform annulus through

porous medium. Effect of thickness of the porous material on the peristaltic pumping of a



Jeffry fluid with non - erodible porous lining wall is studied by Rathod and Mahadev

[14]. Rathod and Sridhar [15] have studied the effects of Couple Stress fluid and an

endoscope on peristaltic transport through a porous medium. Rathod and Sridhar [16]

have studied the peristaltic flow of a couple stress fluid through a porous medium in an

inclined channel. Ravi Kumar et.al [17] studied the unsteady peristaltic pumping in a

finite length tube with permeable wall. It is now well known that blood behaves like a

magneto hydrodynamic (MHD) fluid Stud et al. [21]. Subba Reddy et al. [22] have

studied the peristaltic transport of Williamson fluid in a channel under the effect of a

magnetic field. Vijayaraj et. al [23] studied the Peristaltic pumping of a fluid of variable

viscosity in a non-uniform tube with permeable wall.

In view of these, we investigate the peristaltic transport of a porous material on the

peristaltic pumping when the tube wall is provided with non- erodible porous line using

couple stress fluid with magnetic field. The Navier stokes equations are governed by the

free flow past the porous material and the flow in the permeable wall is described by

Darcy’s law. The axial velocity distribution, the stream function, the volume flow rate,

the pressure rise and the frictional force are calculated. The effect of thickness of porous

lining on the pumping characteristics is discussed with the help of graphs.

2. MATHEMATICAL FORMULATION:

Consider the peristaltic transport of a couple stress fluids in a tube of radius ‘a’. The wall

of the tube is lined with porous material of permeability ‘k’. The thickness of the porous

lining is 1h . The axisymmetric flow in the porous lining is governed by Navier-Stokes

equation. The flow in the porous layer is according to Darcy’s law. In a cylindrical

coordinate system ( *r , *z ), the dimensional equation for the tube radius for an infinite

wave train is represented by

2( , ) ( )R H Z t a bSin Z ct

(2.1)

Where b is the wave amplitude, is the wavelength, c is the wave speed.



Fig1. Physical Model

The transformation from fixed frame to wave frame is given by 2

; ; ( ) ( , );2

Rr R z Z ct p z P Z t (2.2)

Where, and are stream functions in the wave and laboratory frames respectively.

We assume that the flow is inertia- free and the wavelength is infinite.

In the wave frame, the equations governing the flow are

* * * * 2 *

* * * 2 2 * 2 *

* * * * * * *2

1{ } { ( ) } ( ( )) ( )

u u p u uu w r u u

r z r r r r z

(2.3)

* * * * 2 ** * * 2 2 * 2 *

* * * * * * *2

1{ } { ( ) } ( ( )) ( )

w w p w wu w r w w

r z z r r r z

(2.4)

* * *

* * *0

u u w

r r z

(2.5)

Where, 2 *

* * *

1( )r

r r r

*u and *w are radial & axial velocities in the wave frame, is density, *p is pressure,

is coefficient of viscosity, is coefficient of couple stress parameter , is electric

conductivity and is applied magnetic field.



It is convenient to non-dimensionalize variables and introducing Reynolds number Re,

wave number ratio as follows:

2* * * * *

*

1* *

* ** * 2

2 * * * * 2

, , , ( ), Re , ,

,

, , ,

1 1, , , , ,B

B

a az r Q u ca hz r Q u p p z

a c ac c a

ww

c

hh

w kw Da u w l M

c a r z r r a

(2.6)

The equations of motion (2.3), (2.4) and (2.5) becomes (dropping the stars),

2

3 2 4

2

1Re { } ( )

u u p u uu w r

r z r r r r z

-

22 2 2 2

2( ( )) ( )u M u

(2.7)

22

2

1Re { } ( )

w w p w wu w r

r z z r r r z

- 2 2 2

2

1( ( )) ( )w M w

(2.8)

0u u w

r r z

(2.9)

Where, 2

Ma

is the Hartmann number and

2a

couple-stress

parameter.

Using long wavelength approximation ( <<1) and dropping terms of order it follows

from equation (2.7) and (2.8) becomes,

0p

r

(2.10)

2 2 2

2

1 1( ) ( ( )) ( )

p wr w M w

z r r r

(2.11)

The dimensionless boundary conditions are:

0,w

r

2

20

w w

r r r

at r = 0 (2.12a)

2

21 , 0B

w ww w

r r r

at r h (2.12b)

( )B

ww Q at r h

r Da

(2.13)

Where, Da p

QZ

, 1

2



1

= Viscosity in the free flow region, 2

= Viscosity in the porous flow region

= Slip parameter

3. SOLUTIONS:

Solving the equation (2.10) and (2.11) using boundary conditions (2.12)-(2.13), we obtain

the velocity as

2 2

2 2

2 2 2

1 1 1

2 ( ( ) ) 2 Da 2 Da( 1) 1

(4 ) 4 4

X pXp r h pr pw

X M r X X

(3.1)

Where, 2 2 2

1 2

Da Da, 2 Da , ( ) 2( ) 4

dpp X M r X r h h

dz

Integrating the equation (3.1) and subjected to the boundary conditions 0 0at r ,

we get the stream function as,

3/2 4 2 2 3/2 3 2 2 2 2

1

2 3 6 2

2

1 1 1

4 4 3/2 6 42 2 2

2

2p (Da M r- DaM r -2Da M ArcTanh(Mr/2)+DaM Log(-4+M r )-S )-

Da M - DaM

S 2 S S-r -r 2r 2p Da -r 2 ( - )+p Da ( - - )- ( - )

DaM DaM Da M DaMDaM 2 DaM DaM

2p (h- ) (-2 DaM ArcTanh(Mr/2) (Log( -

2 2 2 2 44

1 2 2

4 2 2 2

224 2

3 3 3 32

2 2 3 6 2 4 3/2 6 4

2 2

3

4 2 2

-4+M r )-S )) S r S rp r- ( + - )

4DaM - 4M 16 4 2 2

2p (DaM Log(-4)-2 S ) 4S p Da 4S 4pDaSS r1 r r + ( - )+ - - + - +

4 4 2 2 Da M -DaM DaM Da M DaM

p(h- ) ( (Log(-4)-2S ))

2(DaM - M )

(3.2)

Where,

2 2 2

1 2 3 S =2 Log[ DaM r-2 ] , ( ) 2( ) 4 , S Log[-2 ]Da Da

S h h

The volume flux q through each cross section in the wave frame is given by

0

2

h

q wrdr

(3.3)



22 2 3/2 2 2 2 3 2

2 2 2

2 2 2 2

4

2 2 2 2 2

6

( - )[- (32 -32 ( - ) ( - ) -4 ( - ) 8

8 4

32( - ) ( )) ( ( - )-2 Log(-2 )+2 Log(-2 + M (h - ))) +

8( (h- )(4 + ( - ))-8 Log(-2 )+8 Log(-2 + M

DaM

(h -

p hq Da Da h h Da h Da

h DaM h DaM

DaM DaM h Da

2

4 2 2

2 2 2 3/2 4

6 2 2

8 (h- ) M(h- ))))+ (2 M ArcTanh( )+ (i +Log4-2Log(-2 )+

DaM -M 2

32 2Log(-2 + M (h - )))-Log(-4+M (h- ) ))) + (-Da hM +

DaM (-DaM + )

Da

Da

2 2 2 3/2 4 2 2 3/2 3

2 3 3 2 2 2 2

M(h- )iDaM + M h +Da M - M +2Da M ArcTanh( ) +

2

DaM Log4-2 Log(-2 )+2 Log(-2 + M (h - ))-DaM Log(-4+M (h- ) ))]+

Da Da

Da

2 22 2

2 4

2

(h- ) (8Da + (h- ) +4 (-h+ ) ) 4 [(h- ) - ( ( - )-2 Log(-2 )

8 DaM

+2 Log(-2 + M (h - )))] (3.4)

DaDaM h

Da

Where, dp

pdz

From equation (3.4), we have

2 22 2

2 4

2

22 2 3/2 2 2 2 3 2 2

2 2 2

2

4

(h- ) (8Da + (h- ) +4 (-h+ ) ) 32[8 (8(h- ) - ( ( - )

DaM

-2 Log(-2 ) +2 Log(-2 + M (h - ))))]

( - )[- (32 -32 ( - ) ( - ) -4 ( - ) 8 ( - )

4

32( )) (

Daq DaM h

dp Da

hdzDa Da h h Da h Da h

DM

2 2

6

2 2 2 2 2

2

4 2 2

2

8( - )-2 Log(-2 )+2 Log(-2 + M (h - ))) +

DaM

( (h- )(4 + ( - ))-8 Log(-2 )+8 Log(-2 + M (h - )))+

8 (h- ) M(h- )(2 M ArcTanh( )+ (i +Log4-2Log(-2 )+2Log(-2 +

DaM -M 2

M (h - )))-Log(-4+

aM h Da

DaM DaM h Da

Da

Da

2 2 3/2 4 2

6 2 2

2 2 3/2 4 2 2 3/2 3 2

3 3 2 2 2 2

32M (h- ) ))) + (-Da hM +iDaM +

DaM (-DaM + )

M(h- )M h +Da M - M +2Da M ArcTanh( ) +DaM Log4-

2

2 Log(-2 )+2 Log(-2 + M (h - ))-DaM Log(-4+M (h- ) ))] (3.5)

Da Da

Da



Following the analysis given by Shapiro et al. [18], the mean volume flow, Q over a

period is obtained as

2

2(1 )2

Q q

(3.6)

Which on using equation (3.5), yields

2 2 2

2 2

2 4

22 2 2 2

2 2 2

3/2 2 2 2 3 2

(h- ) (8Da + (h- ) +4 (-h+ ) ) 32[[8( (1 ) ) (8(h- ) -

2 DaM

( - )( ( - )-2 Log(-2 ) +2 Log(-2 + M (h - ))))] / [- (32 -

4

32 ( - ) ( - ) -4 ( - ) 8 ( - )

dp DaQ

dz

hDaM h Da Da

Da h h Da h Da h

2 2 2

4

2 2 2

6

32( )) (

8( - )-2 Log(-2 )+2 Log(-2 + M (h - ))) + ( (h- )(4 +

DaM

DaMM

h Da DaM DaM

2

2 2 2

4 2 2

8 (h- )( - ))-8 Log(-2 )+8 Log(-2 + M (h - )))+ (2 M

DaM -Mh Da Da

2 2

2 3/2 4 2 2 2 3/2 4 2

6 2 2

2 3/2 3 2 3 3

M(h- )ArcTanh( )+ (i +Log4-2Log(-2 )+2Log(-2 + M (h - )))-Log(-4+M

2

32(h- ) ))) + (-Da hM +iDaM + M h +Da M - M

DaM (-DaM + )

M(h- )+2Da M ArcTanh( ) +DaM Log4-2 Log(-2 )+2 Log(-2 +

2

Da

Da Da

D

2

2 2 2

M

(h - ))-DaM Log(-4+M (h- ) ))]] (3.7)

a

4. THE PUMPING CHARACTERISTIC:

Integrating the equation (3.7) with respect to z over one wavelength, we get the pressure

rise (drop) over one cycle of the wave as

1 2 2 2

2 2

2 4

0

22 2 2 2 3/ 2

2 2 2

2 2 2 3 2 2

(h- ) (8Da + (h- ) +4 (-h+ ) ) 32[[8( (1 ) ) (8(h- ) - (

2 DaM

( - )( - )-2 Log(-2 ) +2 Log(-2 + M (h - ))))] /[- (32 -32 ( - )

4

( - ) -4 ( - ) 8 ( - ) (

Dap Q Da

hM h Da Da Da h

h Da h Da h

2 2

4

32)) ( ( - )-2 Log(-2 )+2DaM h

M



2 2 2 2 2

6

22

4 2 2

2 2 2

6

8Log(-2 + M (h - ))) + ( (h- )(4 + ( - ))-8 Log(-2 )+8

DaM

8 (h- ) M(h- )Log(-2 + M (h - )))+ (2 M ArcTanh( )+ (i +Log4-2Log(-2 )+

DaM -M 2

322Log(-2 + M (h - )))-Log(-4+M (h- ) ))) +

DaM (-

Da DaM DaM h

Da Da

Da

3/2 4 2

2 2

2 2 3/2 4 2 2 3/2 3 2 3

3 2 2 2 2

(-Da hM +iDaM +DaM + )

M(h- )M h +Da M - M +2Da M ArcTanh( ) +DaM Log4-2 Log(-2 )+

2

2 Log(-2 + M (h - ))-DaM Log(-4+M (h- ) ))]]

Da Da

Da dz

(4.1)

The pressure rise required to produce zero average flow rate is denoted by 0p and is

given by

1 2 2 2

2 2

2 4

0

22 2 2 2

2 2 2

3/2 2 2 2 3 2 2

(h- ) (8Da + (h- ) +4 (-h+ ) ) 32[[8( (1 ) ) (8(h- ) -

2 DaM

( - )( ( - )-2 Log(-2 ) +2 Log(-2 + M (h - ))))] / [- (32 -32

4

( - ) ( - ) -4 ( - ) 8 ( - ) (

Dap Q

hDaM h Da Da

Da h h Da h Da h

2 2

4

32)) ( DaM

M

2 2 2

6

22 2 2

4 2 2

2 2

8( - )-2 Log(-2 )+2 Log(-2 + M (h - )))+ ( (h- )(4 +

DaM

8 (h- )( - ))-8 Log(-2 )+8 Log(-2 + M (h - )))+ (2 M

DaM -M

M(h- )ArcTanh( )+ (i +Log4-2Log(-2 )+2Log(-2 + M (h - )))-Log(-4+M

2

(h

h Da DaM DaM

h Da Da

Da

2 3/2 4 2 2 2 3/2 4 2

6 2 2

2 3/2 3 2 3 3 2

2 2 2

32- ) )))+ (-Da hM +iDaM + M h +Da M - M

DaM (-DaM + )

M(h- )+2Da M ArcTanh( ) +DaM Log4-2 Log(-2 )+2 Log(-2 + M

2

(h - ))-DaM Log(-4+M (h- ) ))]]

Da Da

Da

dz

(4.2)



The dimensionless frictional force F at the wall across one wavelength in the tube is

given by

2 2 22 2 2

2

2 21

4

22 2 3/ 2 2 2 2 3 20

2 2 2

(h- ) (8Da + (h- ) +4 (-h+ ) )( - ) [-8( -(1- ) - ) (8(h- ) -

2

32( ( - )-2 Log(-2 ) +2 Log(-2 + M (h - ))))]

DaM[( - )

[- (32 -32 ( - ) ( - ) -4 ( - ) 84

Dah Q

DaM h Da

Fh

Da Da h h Da h Da

2 2 2 2

4

2 2 2 2

6

22

4 2 2

32( - ) ( )) ( ( - )-2 Log(-2 )+2 Log(-2 + M

8(h - ))) + ( (h- )(4 + ( - ))-8 Log(-2 )+8

DaM

8 (h- ) M(h- )Log(-2 + M (h - )))+ (2 M ArcTanh( )+

DaM -M 2

(i +Log4-2Log(-2 )+2Log(-2

h DaM h DaM

DaM DaM h

Da Da

2 2 2

3/2 4 2 2 2 3/2 4

6 2 2

2 2 3/2 3 2 3

3 2 2 2 2

+ M (h - )))-Log(-4+M (h- ) ))) +

32(-Da hM +iDaM + M h +Da M -

DaM (-DaM + )

M(h- )M +2Da M ArcTanh( ) +DaM Log4-2 Log(-2 )+

2

2 Log(-2 + M (h - ))-DaM Log(-4+M (h- ) ))]

Da

Da

Da

Da

]

(4.3)

dz

5. RESULTS AND DISCUSSIONS:

In order to see the effect of various pertinent parameters such as the thickness of porous

lining ( ), amplitude ratio ( ), ratios of viscosities in the free flow region and porous

region ( ), Darcy number (Da), slip parameter ( ), couple stress parameter ( ) and

Magnetic parameter (M) on pumping characteristics have plotted in Figs. 2-15.

The variation of pressure rise p with average flow rate Q for different values of Da

with = 0.7, = 0.01, = 0.2, = 0.6, = 0.5, M = 1 is presented in Fig. 2. It is

observed that, in pumping region the time averaged flow rate Q decreases with

increasing Darcy number. Further, it is noted that smaller Darcy number, larger pressure



rise. Fig.3. depicts the variation of on pumping characteristics with = 0.7, Da =

0.02, = 0.01, = 0.6, = 0.5, M = 1. It is seen that, an increase in increases the

pressure rise p against which the pumping works. In addition, it is noted that the flux

Q increases with increase of .The variation of pressure rise p with average flow rate

Q for different values of slip parameter with = 0.7, = 0.01, = 0.2, Da = 0.02,

= 0.5, M = 1 is presented in Fig. 4. It is observed that, an increase in the value of slip

parameter , decreases the pressure rise p .The effect of amplitude ratio on the

pumping performance is shown in Fig.5 with Da = 0.02, = 0.01, = 0.2, = 0.6,

= 0.5, M = 1, it is noted that, larger the amplitude ratio, greater the pressure rise against

which the pump works .i.e., p increases with increase in . Fig.6. shows the variation

of pressure rise p with average flow rate Q for different values of with = 0.7, Da

= 0.02, = 0.2, = 0.6, = 0.5, M = 2. It is found that, larger the thickness of porous

lining, greater the pressure rise against which the pump works. Fig.7. shows the relation

between the pressure rise p and averaged flux Q for different value of with = 0.7,

= 0.01, = 0.2, Da = 0.02, = 0.6, M = 1. It is found that in the entire pumping

region the volumetric flow rate increases with the increase in couple stress parameter.

Fig.8. shows the relation between the pressure rise p and averaged flux Q for different

value of M with = 0.7, = 0.01, = 0.2, Da = 0.02, = 0.6, = 0.5. It is

observed that, an increase in the value of Magnetic parameter M, decreases the pressure

rise p .

The variation of friction force F with averaged flow rate Q under the influence of all

emerging parameters such as Da, , , , , ,M. It is observed that the effect of all

the parameters on friction force are opposite to the effects on pressure with the averaged

flow rate is shown in figs. 9-15.



Fig.2. Effect of Da on p when =0.7, =0.2, =0.6, =0.01, =0.5,M=1

Fig.3. Effect of µ on p when =0.7,Da=0.02, =0.6, =0.01, =0.5,M=1

Fig.4. Effect of on p when =0.7, Da=0.02, =0.2, =0.01, =0.5,M=1



Fig.5. Effect of on p when Da=0.02, =0.2, =0.6, =0.01, =0.5,M=1

Fig.6. Effect of on p when =0.7,Da=0.02, =0.2, =0.6, =0.5,M=2

Fig.7. Effect of on p when =0.7,Da=0.02, =0.2, =0.6, =0.01,M=1



Fig.8. Effect of M on p when =0.7,Da=0.02, =0.2, =0.6, =0.01, =0.5

Fig.9. Effect of on F when Da=0.02, =0.2, =0.6, =0.01, =0.5,M=1

Fig.10. Effect of Da on F when =0.7, =0.2, =0.6, =0.01, =0.5,M=1



Fig.11. Effect of on F when =0.7,Da=0.02, =0.2, =0.6, =0.2,M=1

Fig.12. Effect of on F when =.7,Da=0.02, =0.2, =0.01, =0.5,M=1

Fig.13. Effect of µ on F when =0.7,Da=0.02, =0.6, =0.01, =0.5,M=1



Fig.14. Effect of on F when =.7,Da=0.02, =0.2, =0.6, =0.01, M=1

Fig.15. Effect of M on F when =0.7,Da=0.02, =0.2, =0.6, =0.01, =0.5

6. CONCLUSION:

In this analysis peristaltic transport of a porous material on the peristaltic pumping when

the tube wall is provided with non- erodible porous line using couple stress fluid with

magnetic field has been studied. We conclude the following observations:

Pressure with average flow rate

Q

Pressure decreases with increase in Da, & M and Pressure increases with

increasing in , , & .

Friction force with average flow rate

Q

It is observed that the effects of the parameters on friction force are opposite to the

effects on pressure with the averaged flow rate.



ACKNOWLEDGEMENT:

The author is highly grateful to the reviewers for careful reading of the manuscript and

for valuable suggestions. The present work is part of the Minor Research Project (plan)

[Grant No.: MRP(S)-0435/13-14/KAGU056/UGC-SWRO, Date: 28-Mar-14] sponsored

by the UNIVERSITY GRANTS COMMISSION (UGC), Bangalore, Karnataka,

India.

REFERENCES:

[1] Alemayehu and Radhakrishnamacharya.G, Dispersion of a Solute in Peristaltic

Motion of a Couple stress fluid in the presence of Magnetic Field, Word Academy

of Science, Engg. and Tech., 75(2011), 869-874.

[2] Elsehawey EF and El-Sebaei W, Couple-stress in Peristaltic transport of a

Magneto- Fluid, Physica Scripta, 64(2001), 401- 409.

[3] Elsehawey EF and Mekheime Kh S, Couple-stresses in Peristaltic transport of

fluids, J. of Phys. D: appl. Phys., 27(1994), 1163.

[4] Fung YC and Yih CS, Peristaltic transport. J. of Appl. Mech., Trans. ASME,

5(1968), 669-675.

[5] Hayat T and Ali N, Physica A: Statistical Mechanics and its Applications,

370(2006), 225-239.

[6] Hayat, T., Afsar, A., Khan, M. and Asghar, S. Computers and Mathematics with

Applications, 53(2007), 1074-1087.

[7] Hemadri Reddy R, Kavitha A, Sreenadh S, Hariprakashan P, Effect of

thickness of the porous material on the peristaltic pumping when the tube wall is

provide with Non-erodible porous lining, Adv. App. Sc. Research, 2(2011), 167-

178.

[8] Jayarami Reddy B, Subba Reddy M.V, Nadhamuni Reddy C and Yogeswar

Reddy P, Peristaltic flow of a Williamson fluid in an inclined planar channel

under the effect of a magnetic field , Adv. Appl. Sci. Res., 2012, 3 (1):452-461.

[9] Mekheimer Kh S, Peristaltic transport of a Couple-stress fluid in a uniform and

Non- uniform Channels, Biorheology, 39(2002), 755-765.

[10] Mekheimer, KH.S. Appl. Math. Comput, 153 (2004), 763-777.

[11] Raghunath Rao T & Prasad Rao D.R.V, Peristaltic flow of a couple stress fluids

through a porous medium in a channel at low Reynolds number, Int. J. of Appl.

Mech., 8(3)(2012), 97-116.

[12] Rathod V. P, Sridhar N. G and Mahadev M. Peristaltic pumping of couple

stress fluid through non - erodible porous lining tube wall with thickness of porous

material , Advances in Applied Science Research, 2012, 3 (4):2326-2336.

[13] Rathod V. P, Sridhar N. G, Peristaltic transport of couple stress fluid in

uniform and non-uniform annulus through porous medium, international journal of

mathematical archive, 3(4) (2012) 1561-1574.

[14] Rathod V.P and Mahadev M., Effect of thickness of the porous material on the

peristaltic pumping of a Jeffry fluid with non - erodible porous lining wall, Int. j.

of Mathematical Archive, 2(10), Oct.-2011.



[15] Rathod V.P and Sridhar N.G, Effects of Couple Stress fluid and an Endoscope

on Peristaltic Transport through a Porous Medium, International J. of Math. Sci. &

Engg. Appls., 9(1) (2015) 79-92.

[16] Rathod V.P and Sridhar N.G, Peristaltic flow of a couple stress fluid through a

porous medium in an inclined channel, Journal of Chemical, Biological and

Physical Sciences, 5(2) (2015) 1760-1770.

[17] Ravikumar S, Prabhakar Rao G, and Siva Prasad R, Peristaltic flow of

a Couple stress fluid in a flexible channel under an oscillatory flux, Int. J. of Appl.

Math and Mech., 6(13) (2010), 58-71.

[18] Shapiro A.H., Jaffrin M.Y, and Weinberg S.L, Peristaltic pumping Long Wave

Length at Low Reynolds Number J.Fluid Mech., 37 (1969) 799-825.

[19] Sohail Nadeem and Safia Akram, Peristaltic flow of a Couple stress fluids under

the effect of induced magnetic field in an asymmetric channel, Arch Appl. Mech.,

81(2011), 97-109.

[20] Srivastava LM, Peristaltic transport of a couple stress fluids, Rheol. Acta,

25(1986), 638–641.

[21] Stud, V.K., Sekhon, G.S. and Mishra, R.K. Bull. Math. Biol. 39(1977), 385-390.

[22] Subba Reddy, M. V., Jayarami Reddy, B. and Prasanth Reddy, D. International

Journal of Fluid Mechanics,3(1)(2011), 89-109.

[23] Vijayaraj K, Krishnaiah G, Ravikumar M.M., Peristaltic pumping of a fluid of

variable in a non-uniform tube with permeable, J. Theo. App. Inform. Science,

2009, 82-91.

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