ELECTROKINETIC FLOW OF PERISTALTIC TRANSPORT ...joics.org/gallery/ics-1285.pdfThe instantaneous...

ELECTROKINETIC FLOW OF PERISTALTIC TRANSPORT

OF JEFFREY FLUID IN A POROUS CHANNEL

Laxmi Devindrappa* and N. B. Naduvinamani.

Department of Mathematics, Gulbarga University,

Gulbarga-585106, Karnataka, India.

E-mail address: [email protected]

[email protected]

Abstract

Present paper concludes a mathematical model to study the electrokinetic flow of peristaltic

transport of Jeffrey fluid in a porous channel. The Poisson-Boltzmann equation for electrical

potential distribution is assumed to accommodate the electrical double layer. The closed form

analytical solutions are presented by using low Reynolds number and long wavelength

assumptions. The influence of various parameters like electro-osmotic, Jeffrey fluid parameter,

External electric field and Darcy number on the flow characteristics are discussed through

graphs.

Keywords: Jeffrey fluid, Peristalsis, External electric field and Electroosmosis.

Introduction

Peristaltic transport is a form of fluid transport generated by a progressive wave of area

contraction or expansion along the length of a distensible tube containing fluid. Latham [1] has

coined the idea of fluid transport. The peristaltic transport with long wave length at low

Reynolds number has been analyzed by Jaffrin and Shapiro [2]. Peristaltic flow of non-

Newtonian fluids was first studied by Raju and Devanathan [3].

Another non-Newtonian fluid model that has attracted the attention of researchers in fluid

dynamics is the Jeffrey fluid model which describes the effects of the ratio of relaxation to

retardation times and retardation time. Kothandapani and Srinivas [4] have studied the peristaltic

transport of a Jeffrey fluid. Qayyumet al. [5] have discussed the unsteady squeezing flow of

Jeffery fluid. Akbar and Nadeem [6] have analyzed Jeffrey fluid model for blood flow.

Electrokinetic flow processes in porous media is a broad subject of great interest to scientists and

engineers of different disciplines. The driving force for these processes is an electric field

imposed on a porous medium and the flows may include fluid, electricity, dissolved and

undissolved chemical species and colloid size particles. A great variety of applications of

Journal of Information and Computational Science

Volume 9 Issue 9 - 2019

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mailto:[email protected]

mailto:[email protected]

electrokinetic flow processes can be found in geotechnical engineering, environmental

engineering, biology and medicine, coating processes, ceramic technology, rubber processing

etc. Goswami et al. [7] studied the electro-kinetically modulated peristaltic transport. Tripathi et

al. [8] analyzed electrokinetically driven peristaltic transport. Mondal and Shit [9] have discussed

the electro-osmotic flow. Ranjit and Shit [10] have analyzed entropy generation on asymmetric

micro-channel.

In this study electrokinetic flow of peristaltic transport of Jeffrey fluid in a porous channel were

investigated. We have considered the non-Newtonian Jeffrey fluid model with the use of linear

momentum. Consider the approximation of long wavelength and low Reynolds number, the

governing flow problem is simplified. The reduced resulting ordinary differential equations are

solved analytically and exact solutions are presented. The impact of all the physical parameters

of interest is taken into consideration with the help of graphs.

Mathematical model

2( , ) ( ).H x t a bcos X ct

(1)

Where b is amplitude of the waves and is the wave length, c is the velocity of wave

propagation and X is the direction of wave propagation.

The constitutive equations for Jeffrey fluid are given by

. (2)

. (3)

where and are the Cauchy stress tensor and extra stress tensor, is the pressure, is the

identity tensor, is the dynamic viscosity, is the ratio of relaxation to retardation times,

is the retardation time, is the shear rate and dots over the quantities denote differentiation.

The transformation between these two frames is given by . (4)

Where U and V are velocity components within in the laboratory frame and u and v are the

velocity components within the wave frame.

The equations governing the electro osmotic flow are taken as

0.u v

x y

(5)

( ) .xyxx

e x

u u pu v u c E

x y x x y k

(6)

T PI S

2

1

( )1

S

T S p I

1 2

, , ,x X ct y Y u U v V



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.xy yyv v p

u v vx y y x y k

(7)

where xE denote electro kinetic body force. The Poisson’s equation is defined as

2 .e

(8)

in which e is the density of the total ionic charge and is the permittivity. The Boltzmann

equation is expressed as

0 .

B

ezn n Exp

K T

(9)

Where 0n

represents concentration of ions at the bulk, which is independent of surface

electrochemistry, e is the electronic charge, is the charge balance, BK

is the Boltzmann

constant, and T is the average temperature of the electrolytic solution.

Introducing the non-dimensional quantities

2

0

2 2

0

, , , , , ,

, , , , , .e

x y u v a a px y u v p

a c c c

ct a b ca kt R Da

c a a a

(10)

The equations governing the flow become

0.u u

x y

(11)

21( 1) .

xyxxe h s

u u pR u v u m U

x y x x y Da

(12)

23 2 .

xy yy

e

v v pR u v v

x y x x y Da

(13)

where

2

1

21 ,

1xx

c uu v

a x y x

2

1

21 ,

(1 )yy

c uu v

a x y y



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22

1

11 ,

1xy

c u vu v

a x y y x

Using long wavelength approximation and dropping terms of order and higher, Eqs. (11) - (13)

reduces to

2 2

1

1( 1) .

1hs

u pm U N u

y y x

(14)

.

Where 2 1.N

Da (15)

The non dimensional boundary conditions are

0 0 1 .u

at y and u at y hy

. (16)

where

02

B

nm a e z

K T is known as the electroosmotic parameter and x

h s

EU

c

is the

maximum electroosmotic velocity. Applying Debye-Hückel linearization approximation,

Poisson-Boltzmann equation reduces to 2

2

2m

y

. (17)

The boundary conditions for electrical potential are

0 0, 1 .at y at y hy

(18)

The solution of the Possion-Boltzmann equation (17) subjected to boundary conditions (18) give

rise to

[ ].

[ ]

Cosh m y

Cosh m h

(19)

Method of solution

Solving the Eq. (14) with the boundary conditions (16), we get

2 2 2

1

1.

(1 )u AB C

N m N

(20)

0p

y



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The volume flux q through each cross section of the micro-channel in the wave frame is given by

0

h

q u dy

2 2 2 2 2 2

1 1

1 1[ ]

(1 ) (1 )

dpD E F G Sech hm H I J

dxN m N N m N

(21)

Where 2 2 2 2 2

1 1(1 ) [ ] (1 ) [ ]hs

dpA N m N Cosh hm U m N Cosh my

dx

2 2 2 2

1 1 1

1

[ (1 )] (1 ) (1 )

[ ] [ (1 )]

hs

dpB Cosh hN U m N m N

dx

Cosh hm Cosh y N

2 2

1 1 1[ ] [ (1 )] [ (1 )] 1 [ (1 )C Sech hm Cosh h N Sinh h N Tanh h N

2 2 2

1 1(1 ) [ ] [ ] [ (1 )]D hN m N Sech hm Cosh hm Cosh h N

2

1 1(1 ) [ (1 )] [ ]hsE U m N Cosh h N Sinh hm

2

1 1(1 ) [ ] [ (1 )]hsF U m N Cosh hm Sinh hN

2 2

1 1 1[ (1 )] [ (1 )] 1 [[ (1 )]G Cosh h N Sinh h N Tanh h N

2 2

1 1(1 ) [ ] [ (1 )]H h m N Cosh hm Cosh h N

2 2

1 1

1

(1 ) [ ] [ (1 )]

(1 )

m N Cosh hm Sinh hNI

N

2 2

1 1 1[ (1 )] [ (1 )] 1 [[ (1 )]J Cosh hN Sinh h N Tanh h N

The expression for pressure gradient from Eq. (21) has the form

2 2 2

1

2 2 2

1

(1 ) 1.

[ ] (1 )

N m Ndpq D E F G

dx Sech hm H I J N m N

(22)



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The instantaneous volume flow rate ( , )Q x t in the laboratory frame between the central line and

the wall is

0

( , ) ( 1) .

h

Q x t u dy q h

(23)

0

11.

T

Q Q dt qT

(24)

The pressure rise per wave length is given by

1

0

.dp

p dxdx

(25)

Results and discussion

We have presented a set of Figures 1-2, which describe qualitatively the effects of various

parameters of interest on flow quantities such as pressure gradient and pressure rise per

wavelength.

Figures 1(a)-(d)show the variations of the axial pressure gradient dp

dxalong the length of the

channel, which has oscillatory behavior in the whole range of the x -axis for all other parameters.

From Figure 1(a) it is seen that for fixed values of all other parameters, the axial pressure

gradient decreases with increase in Jeffrey fluid parameter1 . The effect of electro-osmotic

parameter m is depicted in Figure 1 (b). It is noted that axial pressure gradient increases with

increasing Electro-osmotic parameter. From Figure 1(c) it is observed that, with an increase in

the Helmholtz- Smoluchowski velocity that is with an increase in the external electric field there

is enhancement in the pressure gradient. The effect of Darcy number Da is depicted in Figure 1

(d). It is noted that axial pressure gradient increases with increasing Da .

Figures 2(a)-(d) give the variations pressure rise p with time-averaged flux Q . It can be noticed

from Figure 2(a) that for an increase in Jeffrey fluid parameter1 causes the decrease in the

pumping region 0p , the free pumping region 0p and increases in the augmented

pumping region 0p . The effect of electro-osmotic parameter m on pumping characteristics

is depicted in Figure 2 (b). It is noted that Electro-osmotic parameter significantly elevates

pressure differences with increasing averaged volumetric flow rate in the pumping region 0p

, the free pumping region 0p and the augmented pumping region 0p . Figure 2(c) shows

the effect of Helmholtz-Smoluchowski velocity which is linearly proportional to external electric

on pumping characteristics (relation between pressure rise and averaged flow rate). It is revealed



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that with an increase in the external electric field there is enhancement in the pumping region

0p , the free pumping region 0p and the augmented pumping region 0p . From

Figure 2(d) it is revealed that with an increase in the Darcy number Da causes the decrease in

the pumping region 0p , the free pumping region 0p and increases in the augmented

pumping region 0p .

Conclusion

In the present work, we have analyzed electrokinetic flow of peristaltic transport of Jeffrey fluid

in a porous channel. Closed form solutions are derived for the pressure gradient and pressure

rise. The main observations of the present analysis are as follows. It is observed that pressure

gradient decreases with the increase of Jeffrey fluid parameter1 and Darcy number Da , while it

increases by increasing electro-osmotic parameter m and external electric field. It is observed

that pressure rise decreases with the increase in Jeffrey fluid parameter1 and slip Darcy number

Da , However it increases with an increase in m and hsU .

.

0.0 0.2 0.4 0.6 0.8 1.0

-5

0

5

10

15

20

dp

dx

x

=0.1

=0.2

=0.3

(a)



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0.0 0.2 0.4 0.6 0.8 1.0

-1

0

1

2

3

4

5

6

dp

dx

x

m=1

m=2

m=3

(b)

0.0 0.2 0.4 0.6 0.8 1.0

-1

0

1

2

3

4

5

dp

dx

x

Uhs

=1

Uhs

=2

Uhs

=3

(c)



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0.0 0.2 0.4 0.6 0.8 1.0-1

0

1

2

3

4

5

dp

dx

x

Da=0.1

Da=0.2

Da=0.3

(d)

Fig. 1. Axial pressure gradient for (a) 0.6, 2, 0.1, 1.hsm Da and U

(b) 10.2, 0.3, 0.1 1.hsDa and U (c) 10.2, 0.3, 1 0.1.m and Da

(d)10.2, 0.3, 2 1.hsm and U

-1.0 -0.5 0.0 0.5 1.0

-10

0

10

20

30

40

p

Q

1=0.1

1=0.3

1=0.5

(a)



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-1.0 -0.5 0.0 0.5 1.0

-10

-5

0

5

10

p

Q

m=1

m=2

m=3

(b)

-1.0 -0.5 0.0 0.5 1.0

-10

-5

0

5

10

p

Q

Uhs

=1

Uhs

=2

Uhs

=3

(c)



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-1.0 -0.5 0.0 0.5 1.0-15

-10

-5

0

5

10

15

20

25

p

Q

Da=0.1

Da=0.3

Da=0.5

(d)

Fig. 2. Pressure rise with time-averaged flux for (a) 0.4, 2, 0.1 1.hsm Da and U (b)

10.2, 0.3, 0.1 1.hsDa and U (c) 10.3, 0.3, 2 0.1.m and Da (d)

10.2, 0.3, 2 1.hsm and U

Acknowledgments

This work is supported by UGC, Post Doctoral Fellowship for women (PDFWM). One of the

authors, Dr. Laxmi Devindrappa, acknowledges UGC for awarding the Post-Doctoral

Fellowship.

REFERENCES

1. Latham, T.W.: Fluid motion in a peristaltic pump, M. Sc. Thesis. MIT. Cambridge. M. A

(1966).

2. Jaffrin, M.Y., Shapiro, A. H.: Peristaltic pumping. Annual Review of Fluids Mechanics. 3,

13- 37 (1971).

3. Raju, K. K., Devanathan, R.: Peristaltic motion of a non-Newtonian fluid: Part I. Rheol.

Acta. 11, 170–178 (1972).

4. Kothandapani, M., Srinivas, S.: Peristaltic transport of a Jeffrey fluid under the effect of

magnetic field in an asymmetric channel. Int. J. Non-Linear Mech. 43, 915–924 (2008).

5. Qayyum, A., Awais, M., Alsaedi, A., Hayat, T.: Unsteady squeezing flow of Jeffery fluid

between two parallel disks. Chin. Phys. Lett. 29 (3), 034701 (2012).



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6. Akbar, N.S., Nadeem, S.: Simulation of variable viscosity and Jeffrey fluid model for

blood flow through a tapered artery with a stenosis. Commun. Theor. Phys. 57, 133–140

(2012).

7. Goswami, P., Chakraborty, J., Bandopadhyay, A., Chakraborty, S.: Electrokinetically

modulated peristaltic transport of power-law fluids. Microvasc. Res. 103, 41–54 (2016).

8. Dharmendra Tripathi., Ashu Yadav., Anwar Beg, A.: Electrokinetically driven peristaltic

transport of viscoelastic physiological fluids through a finite length capillary:

mathematica modeling. Mathematical Biosciences. 283, 155-168 (2017).

9. Mondal, A., Shit, G. C.: Electro-osmotic flow and heat transfer in a slowly varying

asymmetric micro-channel with Joule heating effects. Fluid Dynamics Research. 50,

065502 (2018).

10. Ranjit, N., Shit, G. C.: Entropy generation on electromagnetohydrodynamic flow through

a porous asymmetric micro-channel. European Journal of Mechanics / B Fluids.77, 135–

147 (2019).



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