The Influence of Wall Thickness and Additional Mass Flux on Peristaltic Pumping of a Jeffrey Fluid

The Influence of Wall Thickness and Additional Mass Flux on Peristaltic Pumping of a Jeffrey Fluid

Mohd. Riyazuddin M. V. Ramana Murthy

Department of Mathematics, Stanley College of Department of Mathematics, Osmania university, Engineering and Technology for Women, Hyderabad, INDIA

Hyderabad, INDIA

P. Subhash Y. V. K. Ravi Kumar

Department of Mathematics, JNTU College of Practice School Division, Birla Institute of Technology and Engineering, Science (BITS), Pilani,

Nachupally (Kondagattu), Karimnagar, INDIA Rajasthan, INDIA.

Abstract – This paper discusses the influence of wall thickness and additional mass flux on peristaltic pumping of a Jeffrey fluid in a cylindrical tube. The flow is investigated in a wave frame of reference moving with velocity of the wave. Using the perturbation technique, for modified Reynolds number, solutions are obtained for the non-linear differential equations with suitable boundary conditions. The axial velocity, pressure rise, friction force are obtain. The effect of various parameters of interest on flow rate and frictional force are discussed with help of graphs.

Keywords: wall thickness, additional mass flux, Jeffrey fluid, Peristaltic pumping.

1. INTRODUCTION

   Peristaltic motion is a form of fluid transport that occurs when a progressive wave of area contraction or expansion propagates along the length of a distensible tube or channel containing the fluid. Peristalsis is a well known major mechanisms for fluid transport in many biological systems like swallowing food through the esophagus, urine transport from the kidney to the bladder through the ureter, movement of chyme in the gastro intestinal tract, movement of ovum in the female fallopian tubes, the transport of spermatozoa in the ducts efferent us of the male reproductive tract and in the cervical canal, the transport of lymph in the lymphatic vessels,

   and in the vasomotion in small blood vessels, such as arterioles and capillaries. Peristaltic pumping is one of the most important characteristics of fluid transport mechanism in many biological systems. It pumps the fluid against pressure rise since the first investigation of Latham reference [1], a number of analytical, numerical and experimental studies of peristaltic pumping of different fluids have been reported under different conditions with reference to physiological and mechanical situations. Srinivas et al. [2], discussed the influence of heat and mass transfer, wall properties and slip conditions for Newtonian fluid. Most of bio-fluids such as blood exhibit the behavior of non-Newtonian fluids may help to get better understanding of the working of biological systems.

   Ravi Kumar et al.[3], studied unsteady peristaltic pumping in a finite length tube with permeable wall. Vajravelu et al.[4], considered the peristaltic transport of Herschel-Bulkley fluid in an inclined tube. Alsaedi et al.[5], considered couple stress fluid in their investigations. Subba Reddy et at.[6], studied the peristaltic motion of a power-law fluid in an asymmetric channel. Hayat and Ali [7], Srinivas and Kothandapani [8], Nadeem and Akram [9], and Pandey and Tripathi [10], have considered Jeffrey fluid in their study.

   Ravi kumar [11], considered Jeffry fluid in their analysis related to peristaltic pumping. Vajravelu et al.[12],

   z = Z ct , r = R , w(r , z) W (R, Z ct ) c

   (2)

   Kothandapani [13] considered Jeffrey fluid, and Hina et

   u (r , z) U (R, Z ct )

   (3)

   al.[14] considered third grade fluid in their work.

2. FORMULATION OF THE PROBLEM

   We consider the problem of peristaltic transport of

   where u and w are the dimensional velocity components in the direction of r and w respectively.

   The governing equations in the wave frame are given as

   a Jeffrey fluid in a circular cylindrical tube to study the effects of wall thickness and constantly added flux per unit

   1 ru w

   q

   r

   (4)

   volume.

   r 1 1 z

   The geometry of the wall surface is described as

   u u (w c) u P

   r z r

   H (Z , t ) a bd / a sin 2 (Z ct )

   (1)

   2

   1

   ru

   u qu

   r r r 1 z 2

   1

   (5)

   u w (w c) w P

   r z r

   1 r

   w

   2 w

   q w c

   r r 1 r z 2

   1

   (6)

   Figure1. Physical Model

   where a is the mean radius of the tube, is the wave amplitude, is the tube wall thickness, is the wavelength of

   where P is the pressure, is the coefficient of viscosity, is the density and q is the constantly added mass flux per unit

   volume, 1 is the Jeffrey parameter.

   Now, we introduce the following non-dimensional variables

   the peristaltic wave, is the wave propagation velocity and t

   r r

   R z z Z U u

   is the time. Here the flow is completely symmetrical about the

   , R

   a

   a , , Z

   , U

   , u ,

   ac ac

   axial coordinate Z .

   W w a 2 P q

   We choose the cylindrical coordinate system

   W , w

   c

   , P

   c

   c , q c

   (R , Z ) , where the Z -axis lies along the centerline of the tube,

   and R is the distance measured radially. Let U and W be

   the velocity components in the radial and axial directions respectively.

   And defining the modified Reynolds number and wave number as

   Re* ca a , a

   In the fixed frame of reference (R , Z ) , the flow is

   unsteady. However, in a coordinate frame moving with the wave speed c , (r , z) is stationary.

   The transformation from fixed frame to wave frame is given by

   H H

   a

   1 e sin2z

   (7)

   where is the amplitude ratio when thickness ratio tends to

   b

   Q

   1 T

   2

   Q dt F ca

   2

   1

   2e2

   (14)

   zero,

   a

   1, is thickness ratio and q is the additional T 0

   parameter which is equal to the ratio between period time of

   Q

   F ca

   2

   1

   2e2

   (15)

   2

   relaxation and contraction to additional time of secretion to

   flow region in the small intestine.

   Equations of motion in the dimensionless form have been

   The dimensionless time-mean volume flow rates and F

   respectively in the fixed and wave frame as

   reduced to

   Q

   and

   F F

   1 ru w

   2ca 2

   2ca 2

   q

   r

   (8)

   r 1 1 z

   and from Eq. (15) we can have

   P 0

   (9)

   1 2 e2

   r

   * w

   w

   P 1 r

   w

   F 1

   2 2

   H

   (16)

   Re u r (w 1) z z r

   where F

   rwdr

   (17)

   w 1

   r 1 1

   (10)

   r

   0

   On the symmetry plane r 0

   the normal velocity

   where is the additional Reynolds number which is equal to the ratio between additional force and viscosity force,

   q Re* .

   The instantaneous volume flow rate in the laboratory frame is given by

   component and the slope ofthe velocity profile are zero. Also, the tangential component of the fluid velocity on the wall is usually determined from the mechanical property of the wall. Choosing the simple condition here that the wall has only transverse displacements at all times in the fixed frame, then the tangential component of fluid velocity in the wave frame on the wall is

   H

   W c

   at r H (z)

   (18)

   Q 2 WRdR

   0

   (11)

   Using the concept of differentiation under the integral sign of

   The rate of volume flow in the wave frame is given by

   H

   1. and make use of (18) and (4), We have

      F 2 wrdr

      (12)

      dH qH

      0

      If we substitute equations (2) and (3) into equation (11) and

      u c

      dz 2

      at r H (z)

      (19)

      make use of (12), we find that the two rates of volume flow

      The dimensionless boundary conditions in the wave frame are

      are related through

      Q F cH 2

      (13)

      w 0

      r

      u 0

      dH qH

      at r 0

      (20)

      The time-mean flow over a period T at a fixed Z –

      c

      w 1

      u

      dz 2

      at r H

      (21)

      position is defined as

3. NUMERICAL SOLUTION

   We use perturbation method to find the solution. For this, we

   dP0 1

   dz 1 1

   16F0

   H 4

   8

   H 2

   (30)

   expand

   u, w, P, F

   in terms of the modified Reynolds

   where

   numbers Re* of first order is considered as

   c z 4F0 1, c z 4F0 2

   (31)

   u u Re* u …..

   1 H 2 2

   H 4

   H 2

   0 1

   d

   w w0 P P

   Re* w …….

   1

   Re* P ……

   ci z dz ci z , i 1,2.

   0

   F F0

   1

   1

   Re* F …..

   (22)

   3.2 FIRST-ORDER PROBLEM

   The governing equations

   Substituting Eq.(22) into Eqs. 8-10, and 20 -21 we get the

   1 ru1 w1

   0

   r

   (32)

   zeroth order and first order problems as mentioned below

   3.1 ZEROTH-ORDER PROBLEM The governing equations

   r 1 1 z

   P1 0

   r

   (33)

   1 ru w

   u w0 (w

   1) w0

   P1

   0 0 q

   (23)

   0 r 0 z z

   r r 1 1 z

   1 r w

   (34)

   P

   1 qw0 1

   0 0

   r

   (24)

   r r 1 1 r

   P0 1 r

   w0

   The boundary conditions are given by

   z r

   (25)

   w1 0,

   u 0

   at r 0

   (35)

   r 1

   1 r

   r 1

   The boundary conditions are given by

   w1 0,

   u1 0

   at r H

   (36)

   w0 0 ,

   r

   u0 0

   at r 0

   (26)

   On solving the Eqs (32) – (34) together with boundary conditions given by Eqs (35) and (36)

   w 1 , u

   dH qH

   at r H (27)

   we get the first order solution as

   0 0 dz 2

   A 2

   2 B 4

   4 C 6 6

   On solving Eqs. (23) – (25) subject to boundary conditions given by Eqs. (26) and (27), we get the solution of the zeroth order problem as

   w1 2 r H

   • r H

     2

   • r H

     9

     (37)

     4

     w0 c1 z c2

     zr 2

     (28)

     r

     r 3

     u0 1

     1 q c1

     2

     c2

     (29)

     where

     A r 2 H 2 B r 4 H 4

     4

     8F1

     2

     w c z c zr 2 Re* 2

     1 1 H

     1 2

     C r6 H 6

     8F 2 H 1

     0

     9

     H

     3H 4

     1 1 2

     1 2

     (42)

     A 1 2F0 H 1 1 4

     3

     dP 1 16F0 8

     1 2

     1 1

     dz 1

     H 4

     H 2

     H H 3

     H 1

     1

     2 8F 2 H

     6F H

     H

     H

     3

     1 1 4

     1 1 0

     3

     H 5

     0

     H 3 H

     q 2F0 H

     2F

     16F 2 H

     1 3H

     q

     0 1

     0

     2

     * 2

     1 H 2

     H 5 1

     (38)

     Re 3 8F H

     4H

     F 2 H

     F H

     0

     H 3

     1 3 5

     0

     7 8 41

     0 6 21

     5

     H

     2

     B 1 H

     H 2 16F0 H

   • 8F0 H

   2F0q q

   16F1

   1 H F

   1

   H 5 H 3 H 2

   H 4 (1 )

   q 0

   1

   H 3

   1 2H 4

   4H 2

   (39)

   (43)

   8F 2 H

   6F H

   H

   The above results can be expressed to first-order by defining

   1

   C 1

   2 0 0

   (40)

   0 1

   H 9 H 7 H 5

   F F Re* F

   then substituting

   F0 F Re* F1 into

   The pressure gradient, at this order, is given as

   * ,

   dP 2

   8F 2 H

   6F H

   H

   Eq. (42) and (43), neglecting the terms greater than Re

   we find

   1

   dz

   1 1 0 0

   3 H 5 H 3

   H

   2F 16F 2 H

   1q 0 1 0 1 2

   3 8F H

   2 H 2 H 5

   0 1 3

   4H

   H 3 H 5

   16F 2 H 8F H 2F q 16F

   1

   2 0 0 0 q 1

   H 5

   H 3 H 2

   H 4 (1 )

   (41)

   The expressions for the axial velocity component and the axial pressure gradient, up to first order, may be respectively written as

   4 F 4F

   2 2

   1 dP

   w 2 1 4

   2 r

   P dz

   (46)

   H H

   H

   0 dz

   1

   8F

   1 2

   dP

   1 H 4

   F H

   dz

   (47)

   1

   2

   0

   dz

   8F H 1 1 2

   2

   Substituting Eq (45) into Eq (46) and Eq (47) yields

   4 1

   3H H

   1

   8

   r 2 H 2 1 2 FH 1

   4

   P

   2FI4 I 2 2 2FI2 1 1 1 1 2e2 1

   1 2

   2 H

   1 1

   H

   1 3

   3

   1 1

   3

   3

   (48)

   H 1

   1

   1 4

   1

   1

   2 2 1

   3 H

   F 16FI2 8

   2 2F 1 1

   e 1

   2 F

   1 1

   3

   3

   1q H

   (49)

   3H

   2

   where I

   1

   & I

   2 3

   2e2

   F

   H 8 4 FH 6 2

   2 2 2 3 / 2

   4 2 2 7 / 2

   * r 4 H 4

   1

   H 7

   1 H 5

   1

   1 e 2 1 e

   Re 1

   2

   H F 1

   (50)

   H

   r 6 H 6 1 2

   3 1q 4

   2H

   2 6FH

   2

   4H

   H

   substituting Eq. (16) and Eq. (50) into Eq.(48) and Eq. (49) gives

   1

   8F H

   2 2

   9 H 9 H 7 H 5

   1

   2e 2

   2 3 e

   1

   7 / 2

   P

   8

   2 2

   1 2e 2

   1

   1

   1

   1 2e 2 3 / 2

   1 2e 2 1

   2

   1

   3 / 2 1 1

   2

   2 2

   1 2e 2

   3

   (44)

   1 1 2e 2 1

   3

   (51)

   2 q 2F

   1

   2e 2

   1

   2 1

   2

   1

   dP 1 16F 8

   * 3 H

   1 2e 2 2 2

   4

   2 Re

   2 1

   dz 1 1 H H

   2F0q

   8

   2 2

   1

   2 2

   q

   F

   2 2

   3/ 2

   1 2

   1

   H

   1 1

   1 e

   3

   (45)

   2 2 1

   The pressure rise P

   and friction force F

   (at the wall) in

   1 e

   1

   3

   the tube of length in their non-dimensional forms, are given by

   (52)

   The trapping limits are determined by calculating the ratio From Fig.2 and Fig.3, it is observed that pressure

   between minimum time-mean flow

   (1)

   min

   and maximum

   rise at 0.2 is much smaller than the corresponding value

   at 0.6 for certain values of thickness ratio with flow

   time-mean flow

   min as a function of amplitude ratio

   as tends to zero. The minimum time-mean flow is obtained by Eq.(44) when we put w 0 at r 0, H 1 , and

   rate as a parameter.

   Fig.4, shows that the pressure rise decreases with increasing additional Reyn

   H 0 solving it for then we get

   olds number in the pumping P 0, free pumping

   min L1

   1

   1

   2e2

   (53)

   P 0 and co-pumping P 0 regions. The effect

   2

   2

   2 of Jeffrey parameter is observed from the Fig.4(a) and 4(b),

   H 4H 1 1 1

   which are drawn for

   0.5 and 1.5 respectively . It can be

   where L 8 1 1 1

   1 4 1

   3 4H

   seen that increase in Jeffrey parameter decreases the pressure

   H 2 1

   1 12

   rise for the fixed values of remaining parameters.

   The maximum time-mean flow (1)min is obtained by putting P 0 in equation (51) and solving it for (1)min The absolute values of min and (1)min must be real and satisfy the relation 0 min (1)min. Therefore trapping occurs such that 0 min (1)min 1 for all given

   The effect of various parameters on friction force can be observed from Figs 5 – 9. Fig.5 and Fig.6 are drawn to study the effects of flow rate F on friction force at

   0.6 and 0.2 respectively. From Fig.5(a), it is

   noticed that reflux occurs at 0.2,0.4 and the peristaltic pumping occurs at 0 for all values of but the reflux occurs at all the values of 0,0.2,0.4 can be observed

   values of ,

   and .

   from Fig.5(b). From these two, we can study the effect of

4. RESULTS AND DISCUSSIONS

   The effects of flow rate parameter on pressure rise at 0.6,0.2 for different values of Jeffrey parameter

   1 is discussed through Fig.2 and Fig.3. From Fig.2(a) and

   Fig.2(b), it is observed that the pressure rise decreases with increasing thickness ratio in the peristaltic pumping case and its absolute value increases with increasing thickness ratio in augmented pumping case for 0.6 . From Fig.2(a) and Fig.2(b), it can be seen that the increases in the

   value of Jeffrey parameter 1 decreases the value of P

   for a fixed values of thickness ratio. Similar phenomenon is observed from Fig.3 which is drawn for 0.2

   Jeffrey parameter i.e. increase in the Jeffrey parameter effects the pumping phenomenon. From Fig.6, it is noticed that increase in flow rate increases the friction force for any

   fixed value of thickness ratio .

   It can be seen from Fig.6 that the friction force increases with an increase in the flow rate for a given thickness ratio . From Figs.5 and 6 it can be concluded that the decreases in for a given flow rate , decreases the

   friction force ( F ).

   The effects of additional Reynolds number on friction force for 0.6 and 0.2 is observed from

   Figs. 7 and 8. It is observed that the friction force increases with increasing at 0.6 and 0.2 For a given ,

   15

   0

   0.2

   0.4

   F increases with an increase in value is noticed from

   Figs.7 and 8. And also for a fixed Jeffrey parameter 10

   1 and

   additional Reynolds numbers , the frictional force increases with decreasing . Also, it is observed that the increase in the Jeffrey parameter 1 decreases the friction

   p

   5

   0

   0.00 0.05 0.10 0.15

   force for any given value of .

   From Fig.9, it is observed that the time-mean flow ratio decreases with the increasing additional Reynolds

   number for a given amplitude ratio . It is noticed

   from Fig.10, that the time mean flow ratio increases with increasing .

5. CONCLUSIONS

p

(a)

0

0.2

0.4

30

25

20

15

10

5

0

0.00 0.05 0.10 0.15

(b)

The influence of wall thickness and additional mass flux on the peristaltic pumping of non-Newtonian fluid is studied by considering Jeffrey fluid model.

The following are the conclusion drawn from this study.

1. The pressure rise decreases with increasing thickness ratio.

2. The pressure rise decreases with increasing Jeffrey

   parameter for any given thickness ratio and flow rate

   . 2

   with = 0.6

   1 = 0.5, ()1 = 1.5.

   0

   0.02

   0.04

   1.9

   2.0

   2.1

   p

   2.2

   2.3

   2.4

   parameter.

3. Increase in the additional Reynolds number decreases

   0.00 0.05 0.10 0.15

   (a)

   the pressure rise.

4. The friction force increases with increasing thickness ratio for any given flow rate parameter.

5. The friction force increases with increasing additional Reynolds number for a fixed thickness ratio.

   2.50

   2.55

   p 2.60

   2.65

   2.70

   0

   0.02

   0.04

   0.00 0.05 0.10 0.15

6. The trapping limit increases with increasing thickness ratio and the additional Reynolds number.

(b)

. 3

with 0.2

1 = 0.5 () 1 = 1.5.

3

2

1

p 0

1

2

3

2

1

0

0.2

0.4

0.00 0.05 0.10 0.15

(a)

3.60

3.55

3.50

F 3.45

3.40

3.35

3.30

3.25

3.52

3.50

0

0.02

0.04

0.00 0.05 0.10 0.15

0

0.02

0.04

(a)

0

0.2

0.4

p 0

1

2

0.00 0.05 0.10 0.15

(b)

F

3.48

3.46

3.44

3.42

0.00 0.05 0.10 0.15

b)

. 4

with = 0.6

1 = 0.5 ()1 = 1.5.

. 6

with 0.2

1 = 0.5 1 = 1.5.

2.8

5

4

3

2

F 1

0

1

2

0

0.2

0.4

0.00 0.05 0.10 0.15

F

2.6

2.4

2.2

2.0

1.8

1.6

1.4

0

0.05

0.09

0.00 0.05 0.10 0.15

(a)

0

0.05

0.09

(a)

F

3.5

0

0.2

0.4

3.0

2.5

2.0

1.5

1.0

0.5

0.0

F

1.6

1.4

1.2

1.0

0.00 0.05 0.10 0.15

0.00 0.05 0.10 0.15

(b)

. 5

with 0.6

1 = 0.5 () 1 = 1.5.

(b)

. 7

with 0.6

1 = 0.5 () 1 = 1.5.

4.15

4.14

4.13

F 4.12

4.11

4.10

4.09

0

0.005

0.009

0.00 0.05 0.10 0.15

(a)

min max 1

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0

0.005

0.009

0.6 0.7 0.8 0.9 1.0

(b)

0

0.005

0.009

. 9

2.49 with

0.2 , 0.4 0.08

F

2.48

2.47

2.46

0.00 0.05 0.10 0.15

1.6

1.4

min max 1 1.2

1

= 0.5 1

0

0.08

0.016

= 1.5.

(b)

. 8

with 0.2

1 = 0.5 1 = 1.5.

1.0

0.8

0.6 0.7 0.8 0.9 1.0

(a)

min max 1

0.15

0.20

0.25

0.30

0.35

0.40

0

0.005

0.009

0.6 0.7 0.8 0.9 1.0

(a)

min max 1

40

35 0

30 0.08

25 0.016

20

15

10

0.6 0.7 0.8 0.9 1.0

(b)

. 10

with

0.4 , 0.4

0.08

1 = 0.5 1 = 1.5.

REFERENCES

1. T.W. Latham, Fluid motion in a peristaltic pump, M.S.Thesis, MIT, Cambridge, MA, 1996.

2. S.Srinivas, K.Gayathri., M.Kothandapani, The influence of slip conditions, wall properties and heat transfer on MHD peristaltic transport, Comput Phy. Commn, 180, 2115-22, 2001.

3. Y.V.K.Ravi Kumar, S.V.H.N. Krishna Kumari, M.V.Ramana Murthy, S.Sreenadh, Unsteady peristaltic pumping in a finite length tube with permeable wall, ASME J.of fluids engg.,oct, , 132,101201-4 , 2010.

4. K.Vajravelu, S.Sreenadh, V.Ramesh Babu, Peristaltic transport of Herschel-Bulkley fluid in an inclined tube, Int.J.of non-linear. Mech., l40 (1), 83-90, 2005.

5. A.Alsaedi, N.Ali, D.Tripathi, T.Hayat, Peristaltic flow of couple stress fluid through uniform porous medium, Appl. Math. Mech. 35(4), 2014.

6. M.V.Subba Reddy, A.Ramachandra Rao, S.Sreenadh, Peristaltic motion of a power-Law fluid in an asymmetric channel, Int.J.of Non- Linear Mech., 42(10), 1153-1161, 2007.

7. T.Hayat, and N.Ali, Peristaltic Motion of Jeffrey fluid under the effect of a magnetic field in a tube, Commun. Nonlinear Sci Numer.Simul, 13, 1343-1352 , 2008.

8. S.Srinivas, M.Kothandapani, The influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls, Appl. Math. and Comput. 213(1), 197-208, 2009.

9. S.Nadeem, and S.Akram, Peristaltic flow of a Jeffrey fluid in a rectangular duct, Nonlinear Anal. Real world Appl. 11, 4238- 4247 , 2010.

10. S.K. Pandey, and D.Tripathi, Unsteady model of Transportation of Jeffrey fluid by peristalsis, Int.J. of Biomech, 3, 473-491, 2010.

11. S.V.H.N.Krishna Kumari, M.V Ramana Murthy,. Y.V.K.Ravi Kumar, S.Sreenadh, Peristaltic pumping of a conducting Jeffrey fluid in a vertical porous channel with heat transfer, Advances in Applied Science Research, 2(6), 439-453, 2011.

12. K.Vajravelu, S.Sreenadh, P.Lakshminarayana, The influence of heat transfer on peristaltic transport of a Jeffrey fluid in a Vertical porous stratum, Comm. Non linear sci. comm., 16(8), 3107-3125, 2011.

13. M.Kothandapani, and S.Sriniva, Peristaltic transport of a Jeffrey fluid under the effect of magnetic field in an asymmetric channel, Int.J. of Non Linear Mech., 43, 915-924, 2008.

14. S.Hina, M.Mustafa, T.Hayat, F.E.Alsaadi, Peristaltic motion of third grade fluid in curved channel, Applied Math. Mech, 35(1), 73-84 2014.