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INTERNATIONAL JOURNAL OF ADVANCE RESEARCH, IJOAR .ORG ISSN 2320-9143 1

IJOAR© 2015 http://www.ijoar.org

International Journal of Advance Research, IJOAR .org

Volume 3, Issue 7, July 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]

KeyWords Peristaltic pumping, Magnetic field, volume flow rate, pressure rise, couple stress fluid, pumping Characteristics, Darcy’s law.

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.



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),



23 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



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

22(1 )

2Q 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

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



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 22 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

2 2 2 2 2

6

22

4 2 2

2

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

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

2Log(-2 + M (h

DaM hM

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

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

DaM (-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 22 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

2 2 2

6

22 2 2

4 2 2

2

32)) (

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 -

2

DaMM

h Da DaM DaM

h Da Da

Da

2)))-Log(-4+M

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(h- ) )))+ (-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 2

2 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.



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