Jan14

INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND CREATIVE ENGINEERING (ISSN:2045-8711) VOL.4 NO.1 JANUARY 2014

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January 2014

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From Editor's Desk

Dear Researcher, Greetings! Research article in this issue discusses about Pulsatile flow of a Jeffrey fluid. Let us review research around the world this month; Green Drive, is showing off the free app this week at the Consumer Electronics Show in Las Vegas. EnLighten feeds off real-time traffic data supplied by cities, then uses a phone's GPS and accelerometer to determine its user's location and velocity. Ginsberg aims to sell his patented technology to car-makers, so that it can be built into cars. That would avoid draining smartphone batteries and allow for integration into other car systems. It's hardly Chopin. But a soft robot can bend and change the shape of its four rubbery fingers fast enough to hit keys and play a simple tune on a piano. The fingers, which are secured to the edges of the keys, change shape in just 50 milliseconds when air is pumped into them. A computer-controlled valve regulates the pressure of each finger, or actuator, with the appropriate timing. The new actuators have already been used in a pneumatic glove developed for the rehabilitation of stroke sufferers. The system could also be used in robotic surgery, to create soft tools for handling fragile internal organs.The team is now looking at increasing the force that can be applied with the fingers. The next generation of soft robots may not only move faster – they may be more powerful as well. The plant racks in a vertical farm can be fed nutrients by water-conserving, soil-free hydroponic systems and lit by LEDs that mimic sunlight. And they need not be difficult to manage: control software can choreograph rotating racks of plants so each gets the same amount of light, and direct water pumps to ensure nutrients are evenly distributed. Advances in vertical farms could trickle through from other sources, too. The US Defense Advanced Research Projects Agency is using an 18-storey vertical farm in College Station, Texas, to produce genetically modified plants that make proteins useful in vaccines. Adversity also plays its part: the tsunami-sparked nuclear accident in Fukushima, Japan, in 2011 is leading to innovation in vertical farming because much of the region's irradiated farmland can no longer be used. It has been an absolute pleasure to present you articles that you wish to read. We look forward to many more new technologies related research articles from you and your friends. We are anxiously awaiting the rich and thorough research papers that have been prepared by our authors for the next issue. Thanks, Editorial Team IJITCE


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Contents

Pulsatile flow of a Jeffrey fluid in a channel bounded by porous lined plates with suction and injection by S.Sreenadh, P.Govardhan and Y.V.K.Ravi Kumar.............................................................................................[172]


172 www.ijitce.co.uk

Pulsatile flow of a Jeffrey fluid in a channel bounded by porous lined plates with suction

and injection S.Sreenadh

1 P.Govardhan

1 Y.V.K.Ravi Kumar

2

1 Department of Mathematics, S. V. University, Tirupati – 517502, INDIA

2 Practice School Division, Birla Institute of Technology and Science (BITS) – Pilani, INDIA ([email protected] ,

corresponding author)

Abstract— Pulsatile flow of a Jeffrey fluid in a channel bounded by porous lined plates with suction and injection is studied in this paper. The steady and unsteady velocities are obtained. The effect of various parameters on the flow phenomenon is discussed through graphs..

Key words — pulsatile flow, Jeffrey fluid, suction, injection.

I. INTRODUCTION

Viscous fluid flow through and past porous media is

attracting the attention of scientists and engineers

because of its wider applications in various branches of

science and technology. The movement of ground water

in soil, the seepage of water through earth fills and

concrete dams, the movement of oil fields can described

using the knowledge of flow through porous media. The

petroleum industry has been showing a lot of interest in

these problems in connection with the crude oil

production from the under ground reservoirs. These

reservoirs consists of porous materials like lime stone

and dolomite where oil is preserved. Oil can be obtained

by drilling wells down into the reservoir. In order to have

a better oil production, it is necessary to use the

knowledge of flow through porous media.

The oil available in the porous reservoir is a complex

fluid. The properties of such fluid have impact on the oil

production. The behavior of the oil may be Newtonian or

non-Newtonian. In view of this, it is interesting to study

non- Newtonian fluid flow through and past porous

media. Further there are many important applications in

biomechanics also (vide Fung and Tang, [1,2].

Muakat [3] made theoretical and experimental studies

on porous flow using Darcy law. Darcy law is observed

to be valid for low speed flows and agrees with several

experiments modeled in one dimensional motion. Yih [4]

suggested the modified Darcy law for describing

unsteady flow through porous media. Following this law,

Rudraiah et al. [5], analyzed several time dependent

flows through and past permeable beds.

Radhakrishnamacharya [6] investigated the pulsatile

flow of a dusty fluid containing small solid particles

uniformly, through a two-dimensional constricted

channel. The effect of non-Newtonion nature of blood

and pulsatality on flow through a stenosed tube is

analyzed by Chaturani and Samy [7].

The pulsatile flow in a porous channel is important in

understanding the process of dialysis of blood in an

artificial kidney. Recently, Chandra and Prasad [8]

discussed the pulsatile flow problems with periodic

acceleration and varying cross section of tubes.

Wang [9] studied the interesting problem of pulsatile flow

in a porous channel bounded by rigid walls. The

pulsatile flow between permeable walls is important in

understanding the blood flow in the circulatory system

where the nutrients are supplied to tissues of various

organs and waste products are removed. Vajravelu et

al.,[10]made a detailed study on pulsatile flow between

permeable beds. Avinash et.al [11] studied the pulsatile

flow of a viscous stratified fluid of variable viscosity

between permeable beds is studied. The interaction of

peristaltic flow with pulsatile flow through a porous

medium is discussed by Afifi and Gad [12].

In this paper an exact solution for the pulsatile flow of

Jeffrey fluid in porous lined plates is obtained. The flow

between the permeable layers is governed by Jeffrey

model where as the flow in the lower and upper beds

are governed by Darcy law. The velocity distributions in

the porous and non-porous regions are determined.

II. MATHEMATICAL FORMULATION OF THE PROBLEM

Consider the pulsatile flow of a Jeffrey fluid in a

channel bounded by porous lined plates (Fig.1).

The thickness of the porous lining on each of the

plates is . The fluid is injected into the channel

from the lower porous layer with a velocity V and is

sucked out into the upper porous layer with the

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same velocity. The permeabilities of lower and

upper beds are k1 and k2 . The flow between the

permeable beds is governed by the Jeffrey model

whereas the flow in the porous medium is

described by modified Darcy’s law

Fig.1. Physical Model

The following assumptions are made in the analysis of the problem

The flow is laminar and fully developed

The permeable beds are homogeneous

The flow is driven by unsteady pressure gradient. We assume that tiBeA

x

p

1

where A and B are constants and ‘ ’ is the frequency. In view of the above assumptions, the basic equations and boundary conditions of the flow take the following form.

Basic equations

0

y

v

x

u (1)

2

2

11

1

y

u

x

p

y

uV

t

u

(2)

0

y

p (3)

Boundary conditions

1Buu at y = (4)

)( 11

1

Quky

uB

at y = (5)

2Buu at y = h - (6)

)( 22

2

Quky

uB

at y = h - (7)

u, v are velocities of the fluid, is the coefficient of viscosity

p is the pressure, is the slip parameter,k1,k2a are permeabilities of the lower and upper permeable

beds, 21, BB uu are slip velocities in lower and upper

permeable beds and 1 is the Jeffrey parameter.

Separating equations (1) – (7) into steady part denoted by a bar (-) and unsteady part denoted by a tilde ( we get

Steady part:

0

x

u (8)

2

2

11 y

uA

y

uV

(9)

1Buu at y =

(10)

)( 11

1

Qukdy

udB

at y =

(11)

2Buu at y = h - (12)

)( 22

2

Qukdy

udB

at y = h - (13)

where )1( 1

1

1

AkQ , )1( 1

2

2

AkQ

Unsteady part:

0~

x

u (14)

2

2

1

~

1

~~

y

uBe

y

uV

t

u ti

(15)

1~~

Buu at y = ε (16)

)~~(

~

11

1

Qukdy

udB

at y = ε (17)

2~~

Buu at y = h - ε (18)



)~~(

~

22

2

Qukdy

udB

at y = h – ε (19)

where tieBk

Q

)1(~ 11

1

, tie

BkQ

)1(~ 12

2

Non – Dimensionalization of flow quantities

The following non – dimensional quantities are introduced to make the basic equations and the boundary conditions dimensionless

Steady part:

)( 1

*

v

hA

uu ,

)( 1

1*1

v

hA

uu

BB ,

)( 1

2*2

v

hA

uu

BB

h

xx * ,

h

yy * ,

)( 1

1*

1

v

hA

QQ

,

)( 1

2*

2

v

hA

QQ

,

h

* where AA 1

.

Unsteady part:

)(

~~

1

2

*

Bh

uu

)(

~~

1

2

1*

1

Bh

uu B

B

)(

~~

1

2

2*

2

Bh

uu B

B ,

)(

~2

*

h

tt ,

)(

~~

2

*

h

,

)(

~~

1

2

1*

1

Bh

QQ

)(

~~

1

2

2*

2

Bh

QQ ,

h

yy *

,h

*

where BB 1 .

In view of the above dimensionless quantities Eqs.(8) to (19) take the following form, neglecting the asterisks (*), we get,

Steady part

0dx

ud (20)

Rdy

udR

dy

ud)1()1( 112

2

(21)

1Buu at y = (22)

)1

(2

1

111 Ru

dy

udB

at y = (23)

2Buu at y = 1 - (24)

)1

(2

2

122 Ru

dy

udB

at y = 1- (25)

where

VhR is the Reynolds number,

1

1k

h ,

2

2k

h ( 21, are dimensionless parameters).

Unsteady Part:

0~

dx

ud (26)

tiedt

ud

dy

udR

dy

ud )1(~

)1(~

)1(~

1112

2

(27)

Letting tieyfu )(

~~

)1()(~

)1()(~

)1()(~

111 yfiyfRyf (28)

Using ti

B efu 11

~~ , ti

B efu 22

~~

The boundary conditions become

1

~~ff at y = (29)

)1~

(

~

2

1

111

f

dy

fd at y = (30)

2

~~ff at y = 1 - (31)

)1~

(

~

2

2

122

f

dy

fd at y = 1 - (32)

III. SOLUTION OF THE PROBLEM

Steady part:

Solving Eq.(21) subject to boundary conditions (22)-(25) we get the velocity field as

We get the velocity field as

yeCCuy

)1(

211 (33)

The slip velocities are 1Bu and 2Bu are given by

)(

)(

2341

14221

DDDD

EDEDuB

,

)(

)(

2341

21132

DDDD

EDEDuB

(34)

Unsteady part:

Solving equation (28) subject to boundary conditions (10) –(13), we get



iBeAeyf

ymym 21)(

~ (35)

The unsteady part of the velocity is given by

iwteyfu )(~~

Separating real and imaginary parts, we get

)()(~12121212 tSinAtCosBitSinBtCosAu

The slip velocities are given by

timm

ti

B eeBmeAm

efu

)

1(

~~2

1

1

2

2

1

111

21

ti

mm

ti

B eeBmeAm

efu

)

1(

~~

2

)1(

2

2

)1(

1

2

2

122

21

5. Deductions

(i) Taking k1 = k2 = k (i.e. in equation (33 )

and (35) we obtain the velocity field for pulsatile flow of Jeffrey fluid between porous lined plates with equal permeability as follows: Steady state velocity

= c1 + c2 ( + y

(36) Unsteady part:

(

)

(37)

(ii) When the permeabilities k1 and k2 tend to zero in equations (36) and (37) we obtain the velocity field for the pulsatile flow of the Jeffrey fluid in a channel bounded by rigid walls as follows

Steady part

(38)

The slip velocities and are zero. Unsteady part

(

) (39)

The slip velocities and are zero.

Further with , the results (38) and (39) reduce the

corresponding ones of Wang [9].

RESULTS AND DISCUSSION

From equation (33) we have calculated the steady part of the velocity as a function of y for different values of permeability parameter with fixed 0.5, 2, 0.5,R

0.02 and is shown in Fig.2. We observe that the

velocity decreases with the increase in h

k . This

may be due to the increase in the permeability ‘k’ of the upper and lower beds. Also, as the

increases, the gap between the velocity profiles becomes smaller i.e. there is not much change in the velocity due to variation for large values of .

The variation of steady part of the velocity with y is calculated from equation (33) for different values of permeability parameter with

fixed 0.1, 2, 0.5R , 0.02 and is shown

in Fig.3. It is noticed that the velocity decreases with the increase in . From Fig.2 and Fig.3 it is

noticed that the magnitude of the velocity increases with decrease in the slip parameter . Also, as the

increases, the gap between the velocity profiles

becomes smaller.

The variation of u with y is

calculated from equation (33) for different values of the Reynolds number R and for fixed

2, 25, 0.5, 0.02 and is shown in

Fig.4. We observe that the velocity increases with the increase in R.

From equation (33) we have calculated the steady part of the velocity as a function of y for different values of Jeffrey parameter with fixed

2, 25, 2R , 0.02 and is shown in

Fig.5. It is noticed that the velocity increases with the increase in the Jeffrey parameter . Similar

behavior due to is noticed by Srinivas et al.

(2008) and Hayat et al. (2008) for the peristaltic transport of viscous fluid in a flexible channel.

From equation (33) we have calculated the steady part of the velocity as a function of y for different values of the thickness of porous lined plate with fixed 2, 25, 2R , 0.5 and is

shown in Fig.6. We observe that the velocity decreases with the increase in .

We have calculated the unsteady part of the velocity as the function of y from equation (35) for different values of wt with fixed 5, 5, 0R

, 0.02 , M=1, 1 and is shown in Fig.7.We

observe the velocity decreases with the increase in wt .

We have calculated the unsteady part of the velocity ( u ) as the function of y from equation (35)

for different values of wt with fixed



0.5, 25, 10R , 0.02 , M = 1, 1 and is

shown in Fig.8.We observe the velocity decreases with the increase in wt . From Fig.7. and Fig.8. for

fixed wt and , as R increases the maximum

velocity moves closure to the upper permeable bed.

We have calculated the unsteady part of the velocity ( u ) as the function of y from equation (35)

and are shown in Figs.9 and 10. For fixed wt , the

velocity decreases with the increment in . As R

increases the maximum velocity moves closure to the upper permeable bed.

From equation (35) we have calculated the unsteady part of the velocity as a function of y for different values of Reynolds number R and Jeffrey

parameter with fixed 5, M=1, 0.02,4

wt

, 10 and are shown in Fig.11. and Fig.12. It is

noticed that the velocity increases with the increase in . As R increases the maximum velocity moves

closure to the upper permeable bed.

The variation of u with y is calculated from

equation (35) for different values of the Reynolds number R and for fixed 2, 25 , M=1,

0.02,4

wt

, 0.5 and is shown in Fig.13. We

observe that the velocity decreases with the increase in R.

From equation (35) we have calculated the unsteady part of the velocity as a function of y for different values of the thickness of porous lined

plate with fixed 2, 10 , M = 1, 2,4

R wt

, 0.5 and is shown in Fig.14. We

observe that the velocity decreases with the increase in .

Fig. 2. Steady state velocity profiles for

0.5, 2, 0.5R and 0.02

Fig. 3. Steady state velocity profiles for with

fixed 0.1, 2, 0.5R and 0.02

Fig..4. Steady state velocity profiles for R with fixed

2, 25, 0.5 and 0.02



Fig..5. Steady state velocity profiles for with fixed

2, 25, 2R and 0.02

Fig. 6. Steady state velocity profiles for with

fixed 2, 25, 2R and 0.5

Fig.7. Unsteady state velocity profiles for t with

fixed 5, 5, 0R , 0.02 , M=1 and 1

Fig.8. Unsteady state velocity profiles for t with

fixed 0.5, 25, 10R , 0.02 , M=1 and

0.5

Fig.9. Unsteady state velocity profiles for with

0.5

Fig. 10. Unsteady state velocity profiles for with

1




fixed 5, 2R ,M=1, 0.02,4

wt

and

10 .


fixed 5, 10R , M=1, 0.02,4

wt

and

10

Fig. 13. Unsteady state velocity profiles for R with

fixed 2, 25 ,M=1, 0.02,4

wt

and

0.5 .


fixed 2, 10 , M=1, 2,4

R wt

and

0.5 .

Acknowledgements: Authors thank DST, Govt. of India for providing financial support under major research project to carry this work.



References

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[4].Yih,S.W.,”Dynamics of non-homogeneous fluids”, Macmillan Co. (1967).

[5].Rudraiah, N. et al., “Some flow problems in Porous media”PGSAM series, Bangalore University, Bangalore. (1979).

[6].Radhakrishnamacharya, G.,” Pulsatile flow of dusty fluid through a constricted channel” , J. of Applied. Math. Phys. (ZAMP), 29 (1981)

[7].Chaturani.P and Samy, R.P.,”Pulsatile flow of casson’s fluid through stenosed articles with applications to blood flow”, Biorheology, 23(5) (1986) 499-511.

[8]Chandra, P., Prasad, J.S.V.R.K,”Pulsatile flow in circular tubes of varying cross-section with suction/injection”, J.Aust. Math. Soc. Ser. B.35 (1994) 366-381.

[9].Wang, Y.C.,”Pulsatile flow in a porous channel”, Transaction of ASME, J. Appl. Mech. 38 (1971) 553-555.

[10].Vajravelu, K..,Ramesh, K.,Sreenadh, S.and Arunachalam, P.V.,” Pulsatile flow between permeable beds”, Int.Jr. of Non-Lin. Mech., 38 (2003) 999-1005.

[11].Avinash .K., Ananda Rao .J., Ravi Kumar. Y.V.K., Sreenadh.S.,” Pulsatile flow of a viscous stratified fluid of variable viscosity between permeable beds”, Jr.of Porous Media,14(12),2011,1115 – 1124,

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