International Journal of Mechanical and Materials Engineering (IJMME), Vol. 7 (2012), No. 2, 152-157.
PERISTALTIC PUMPING OF A JEFFREY FLUID BETWEEN POROUS WALLS WITH
SUCTION AND INJECTION
A. Kavitha 1*, R.H. Reddy1, A.N.S. Srinivas1 S. Sreenadh2 and R. Saravana3
1
School of Advanced Sciences, VIT University, Vellore, TamilNadu-632 014, India.
Department of Mathematics, Sri Venkateswara University, Tirupati-517 502, India.
3
Department of Mathematics, Srinivasa Institute of Technology and Management Studies, Chittoor-517127, A.P, India
*Corresponding author’s E-mail: [email protected]
2
Received 21 October 2011, Accepted 22 August 2012
ABSTRACT
Peristaltic pumping of a non-Newtonian Jeffrey fluid
between two permeable walls with suction and injection
is discussed in the wave frame moving with constant
velocity of the wave under the consideration of long
wavelength and low Reynolds number. The analytical
solution for the fluid velocity field, pressure gradient and
the frictional force are obtained. The effect of suction/
injection parameter k , amplitude ratio  and the
permeability parameter including slip  on the flow
quantities are discussed graphically. It is noticed for
pressure rise is positive (pumping region), the rate of
pumping decreases with increasing Jeffrey parameter 1 ,
whereas for pressure rise is negative (co-pumping region)
the behaviour is quite opposite. For pressure rise
P  0 (free pumping region) no variation is observed in
the pumping rate.
R


K
m


Q( X , t )
Q




c
q
NOMENCLATURE
(x, y)
Horizontal and vertical directions of
steady wave propagation in the channel
( X ,Y )
Horizontal and vertical directions of
unsteady wave propagation in the
channel
p
pressure in wave (steady) motion
pressure in laboratory (unsteady) motion
P
a
half distance between the channel
t
time
Velocity of the fluid along x and y
(u , v )
directions
Velocity of the fluid along X and Y
(U , V )
directions
b
Amplitude of the channel
Suction/injection velocity
V0
Stress tensor of Cauchy’s
Jeffrey fluid additional stress tensor
identity tensor
ratio of relaxation to retardation times
2
 ,
the retardation time
Da
Dimensional less wave number
density
Porosity
Wave length
Wave speed
volume flow rate in the steady state
1. INTRODUCTION
The investigation on the peristaltic pumping has been
attracting attention of biomechanical engineers because
of its importance in both physiological and mechanical
situations. Many contributions in the area of peristalsis
have followed by Jaffrin and Shapiro (1971). Several
investigations on the peristaltic pumping deal with
Newtonian fluids but some bio fluids like blood and
chyme etc. are either Newtonian (or) non-Newtonian.
The single fluid model was analysed by Shapiro et al. in
(1969) that was extended Brasseur et al. (1987) for a two
fluid model in a channel. Peristaltic mechanism is a
natural
phenomena
occurring
in
many
biological/physiological systems including the ureter and
oesophagus by treating them as symmetric/asymmetric
ducts. In consideration of above mentioned fact, Rao et
al.(1995), Misra et.al (1999), RadhaKrishnamacharya
(2007), Vajravelu et al. (2005, 2009), Hayat et al. (2008),
Srinivas et al. (2009) and Reddy et al. (2011), Azwadi,
C.S.N.et al(2012), Mobini, K.(2012) made detailed
investigations on the peristaltic pumping of Newtonian
(or) non-Newtonian fluid ducts of different cross
sections. All these investigations are made with the
assumptions that the wall of the duct is impermeable.
Keywords: Peristalsis, Jeffrey fluid, Suction and
Injection, Reynolds number.
T
s
I
1
Reynolds number
permeability parameter
amplitude
permeability
Slip parameter
Jeffrey fluid viscosity
permeability including slip parameter
Instantaneous volume flux in a fixed
frame
Time average flux
The study of flow of physiological fluids such as blood
between permeable layers became very important
especially in lungs. Fung and Tang (1975) and Gopalan
(1981) reported that the lung can be considered as a
channel surrounded between two skinny layers of
Shear rate, dots on the shear rate denotes
differentiation with time.
Darcy number
152
permeable media. A thesis on the peristaltic pumping in a
channel with flexible porous walls has been presented by
Reese (1988). Kumar et al. (2010) discussed the unsteady
peristaltic motion of a Newtonian fluid in a finite length
tube with permeable wall. Sreenadh and Arunachalam
(1986) studied the Couette flow between two permeable
beds with suction and injection. Hassan et al. (2003)
investigated the influence of suction/injection on the nonNewtonian micro polar fluid with heat transfer. Das
(2009) discussed the effects of blowing and suction on
MHD three dimensional couette flow.
For non-Newtonian incompressible Jeffrey fluid, the
Cauchy stress tensor is given by
T  p I  s
(2)
and the additional stress tensor for Jeffrey fluid s is
given by s 

1  1
    
2
(3)
Here p is the pressure, I is the identity tensor, 1 is the
In examination of the some physiological applications it
is interesting to discuss the peristaltic pumping of a
biofluid with suction and injection. In this article, the
steady peristaltic pumping of a non-Newtonian fluid
namely Jeffrey fluid in a two dimensional channel with
blowing and suction is discussed under lubrication
approach. The investigative expressions for the velocity
of fluid flow, the pressure gradient and the frictional
force are obtained. The results are deduced and discussed
graphically.
ratio of relaxation to retardation times, 2 is the
retardation time,  is shear rate and dots on the shear
rate denotes differentiation with time.
We are considering the following assumptions:
1.
2. MATHEMATICAL MODEL
Consider the flow of a non-Newtonian Jeffrey fluid
between two porous walls. The half channel width is a .
The sinusoidal wave travelling occurs on the lower and
upper porous walls of the channel. The fluid is blowing
into the channel perpendicular to the lower porous wall
with constant velocity V0 and is sucked out into the
The flow is in horizontal direction and is
P
driven by a constant pressure gradient
.
X
2. The strengths of the blowing and suction are
same.
Under these assumptions, the governing equations for the
non-Newtonian Jeffrey fluid flow in the unsteady frame
are given by
upper porous wall with same V0 as shown in Figure 1.
U
Due to symmetric waves on the porous walls, it is
enough to discuss for half size of the channel.
The wall deformation of the peristaltic wave for an
infinite wave train is given by
2
(1)
H ( X , t )  a  b sin
( X  ct )
X
0
 U  V U     P  S XX  S XY

0
Y 
X
Y
 t
X


where b ,  and c indicates the amplitude, wave length
and the speed of the wave respectively.
P
Y
0
(4)
(5)
(6)
where  , U , P and  indicates density, velocity in X
direction, pressure and the coefficient of Jeffrey fluid
viscosity.
The change from the laboratory (unsteady) frame of
reference ( X , Y ) to wave (steady) frame of reference
( x , y ) is given by
x  X  ct , y  Y , u  U  c, v0  V0
and p ( x , y )  P ( X , Y , t )
(7)
where u , v0 and p represents velocity in x direction,
suction/injection velocity and pressure in the steady
frame respectively. Here we consider that the wavelength
is infinite. So the flow is of Poiseuille type at each local
cross-section.
Figure 1 Physical model
153
Now introducing the non-dimensional quantities
2 x
x

R
y
, y
, t
2 c

a
 ca

, p=
2 a
u
c
2
c
t, u 
P, S 
a
c
, v0 
S,  
b
a
v0
, h
c
, 
H
3. SOLUTION
Solving the equation (11) with equations (13) and (14),
we get the velocity field as
1
k (1 1 ) y
u  1  P (
(e
 e k (1 1 ) h )
2
(1  1 ) k
(15)

1
k (1 1 ) y
 (1  e
)  ( y  h))
k
k
The volume flow rate q in the steady state is given by
,
a
a
K
1

UD
2 a
 am 
, 

 ,UD 
c

 K
(8)
h

q  udy
0
where R ,  ,  ,  , K , m and  denotes Jeffrey fluid
Reynolds number, dimensionless wave number,
permeability parameter, amplitude ratio, permeability,
slip parameter and permeability including slip parameter.
Using the conditions (7) and (8) in equations (4), (5) and
(6), the dimensionless governing equations of the nonNewtonian Jeffrey fluid flow in the wave (steady) frame
as follows
u
x
0
1
 e k (1  ) h
k (1  ) h 
 h  P(

e
h
2 
(1  1 ) k  k (1  1 ) k (1  1 )

1


(h 
k
e
1
1
k (1 1 ) h
k (1  1 )

1
k (1  1 )
)
h
2
)
2k
(16)
The volume flux Q( X , t ) in the unsteady (laboratory
frame) state between the central of the channel and the
upper wall is
(9)
H
S xy
p S xx
 u 





x
x
y
 y 
R v0

Q( X , t ) = U ( X , Y , t ) dY
(10)
0


1
1
 e k (1  ) h
k (1  ) h 

e
h

2 

 (1  1 ) k  k (1  1 ) k (1  1 )
 P
k (1  ) h

 h2 

e
1
  h





k
k (1  1 ) k (1  1 )  2k


1
1
where
S xx 
S xy 
2  u

1  1  x
2 c   u 
2

u
2
1

a  x 
 u 2 c   2u 
  a  u xy  
1  1  y


(17)
1
From equation (16) we have
Under the long wavelength assumptions (   1 ) the
equation (10) becomes
1
u
2
(1  1 ) y
p
y
2
k
u
y
P
dp
dx
(11)
0
(12)
where k  R.v0 , P 
and u D   Da
p
p
x
y
 0 at y = 0
u  1  
u
y
( q  h)


e
1
1
k (1 1 ) h 


e
h




2

 (1  1 ) k  k (1  1 ) k (1  1 )
k (1 1 ) h
2
 

e
1
h
  (h 


)
k (1  1 ) k (1  1 ) 2k
 k

k (1 1 ) h
(18)
, 1 is the Jeffrey parameter
From equation (17), the time mean flow Q is given by
, (Darcy’s law)
Q
x
The non-dimensional boundary conditions becomes
u

1
T
T
 Qdt = q  1
(19)
0
4. CHARACTERISTICS OF PUMPING
Integrating the equation (18) with x per wavelength, we
get
(13)
1
at y  h (Saffman (1971) slip condition)
p  
0
(14)
154
dp
dx
dx
( q  h)
1

0
1
 e
 1
k (1  ) h  
h 
 (1   ) k 2  k (1   )  k (1   )  e


1
1
1

k (1  ) h
2
 

e
1
h

)
  (h 

k (1  1 ) k (1  1 )
2k
 k

k (1 1 ) h
at Q  0.37 . This is due to the suction/injection in the
dx
channel. For Q  0.37 we observe that the pressure rise
decreases with increasing the suction parameter k . The
1
behaviour is otherwise when Q  0.37 . For free pumping
the flux increases by increasing k .
1
(20)
The dimensionless friction force F at the wall per
wavelength is given by
1
 dp  dx
F  h

 dx 
0

( q  h)
1
h 
0
1
 e
k (1  ) h  
h 
 (1   ) k 2  k (1   )  k (1   )  e


1
1
1

k (1  ) h
2
 

e
1
h

)
  (h 

k (1  1 ) k (1  1 )
2k
 k

1
k (1 1 ) h
dx
1
1
(21)
Figure 3 The difference of pressure rise with flux for
various values of k with 1 =0.5,  = 0.6,  = 0.1.
5. RESULTS AND DISCUSSIONS
The pressure rise with flux is shown in Figure 2 for
various values of non-Newtonian Jeffrey fluid 1 with
We studied the pressure rise with flux for various values
of amplitude ratio  when 1 =0.5, k =0.1,  =0.1 is
shown in Figure 4. We observe that for fixed P , the
flux increases with increasing  . For a given flux, the
pressure rise increases with increasing  .
fixed k =0.1,  =0.6,  =0.1. As expected, we notice
that for pressure rise is positive (pumping region), the
rate of pumping decreases with increasing Jeffrey
parameter 1 , whereas for pressure rise is negative (copumping region) the behaviour is quite opposite. For
pressure rise P  0 (free pumping region) no variation
is observed in the pumping rate.
Figure 4 The difference of pressure rise with flux for
various values of  with 1 =0.5, k = 0.1,  = 0.1.
Figure 2 The difference of pressure rise with flux for
various values of 1 with k = 0.1,  = 0.6,  = 0.1.
We observed that the pressure rise with flux for various
values of  with fixed 1 =0.5,  =0.6, k =0.1 is shown
in Figure 5. We observe that for larger  the pressure
rise decreases in pumping region and free pumping
region where as the behaviour is opposite in co-pumping
region.
We studied the pressure rise with flux for emerging
values of k when 1 =0.5,  =0.6,  =0.1 is shown in
Figure 3. We have seen that for various values of k the
pumping curves coincide at a point in the first quadrant
155
Finally, from equation (21), we have calculated frictional
force F. The influence of 1 , k ,  and  on the friction
force are depicted in figures (6) to (9). From figures 6, 7
and 9, we observed that, the frictional force increases
first and then decreases with increasing 1 , k and  .
Figure 8. Shows frictional force initially decreases and
increases by increasing  . In general, we identified that
the F has opposite behaviour compared to P .
Figure 5 The difference of pressure rise with flux for
various values of  with 1 =0.5, k = 0.1,  = 0.6.
Figure 8 The difference of frictional force with flux for
various values of  with 1 =0.5, k = 0.1,  = 0.1.
Figure 6 The difference of frictional force with flux for
various values of 1 with k = 0.1,  = 0.6,  = 0.1.
Figure 9 The difference of frictional force with flux for
various values of  with 1 =0.5, k =0.1,  = 0.6
6. CONCLUSIONS
In this article, we discuss the peristaltic motion of an
incompressible non-Newtonian Jeffrey fluid between two
porous walls with suction and injection. The exact
solution is analysed for the velocity distribution. The
pressure rise (drop) and frictional force per wavelength
are derived. The results are analyzed by plotting graphs.
1. Increasing the Jeffrey parameter 1 , decreases the
pressure rise.
2. Increasing the suction/injection parameter k , decreases
the pressure rise.
Figure 7 The difference of frictional force with flux for
various values of k with 1 =0.5,  = 0.6,  = 0.1.
156
3. Increasing the amplitude ratio  , decreases the
pressure rise.
4. Frictional force shows opposite behaviour to that of
pressure rise.
Mobini, K. (2012). Drag reduction of submerged moving
bodies using stepped head shape, International
Journal of Mechanical and Materials Engineering 7
(1): 41-47.
Radhakrishnamacharya, G. and Srinivasulu, C. 2007.
Influence of wall properties on peristaltic transport
with heat transfer, Comptes Rendus Mécanique 335:
369-373.
Rao, A.R. and Usha, S. 1995. Peristaltic transport of two
immiscible viscous fluids in a circular tube, Journal
of Fluid Mechanics 298: 271-285.
Reddy, R.H., Kavitha, A., Sreenadh, S., and Saravana, R.
2011. Effect of induced magnetic field on peristaltic
transport of a Carreau fluid in an inclined channel
filled with porous material, International Journal of
Mechanical and Materials Engineering 6 (2): 240249.
Reese, G.1988. Modelle peristalticher Stromungen, Ph.D
Thesis, University of Bremen.
Saffman, P.G., 1971. On the Boundary Conditions at the
Surface of a Porous Medium, Studies in Applied
Mathematics 1 (2): 93–101.
Shapiro, A.H. 1967. Pumping and retrograde diffusion in
peristaltic waves, In: Proceedings of the workshop in
ureteral reflux in children, Washington, DC: 109126.
Sreenadh, S. and Arunachalam, P.V. 1986. Couette flow
between two permeable beds with suction and
injection, International Proceedings of National
Academy of Sciences 56 (A): 329-335.
Srinivas, S., Gayathri, R. and Kothandapani, M. 2009.
The influence of slip conditions, wall properties and
heat transfer on MHD peristaltic transport, Computer
Physics Communications 180: 2115-2122.
Vajravelu, K., Sreenadh, S. and Babu, V.R. 2005.
Peristaltic pumping of a Herschel-Bulkley fluid in a
channel, Applied Mathematics and Communications
169: 726-735.
Vajravelu, K., Sreenadh.S, Reddy, R.H. and
Murugeshan, K. 2009. Peristaltic Transport of a
Casson fluid in contact with a Newtonian Fluid in a
Circular Tube with permeable wall, International
Journal of Fluid Mechanics Research 36 (3): 244-254.
REFERENCES
Azwadi, C.S.N. and Salehi, M. 2012. Prediction of flow
characteristics in stenotic artery using CIP scheme,
International Journal of Mechanical and Materials
Engineering 7 (1): 101-106.
Brasseur, J.G., Corrsin, S. and LU, N.Q. 1987. The
influence of a peripheral layer of different viscosity
on peristaltic pumping with Newtonian fluids, Journal
of Fluid Mechanics 174: 495-519.
Das, S.S. 2009. Effects of Suction and injection on MHD
Three Dimensional Couette flow and Heat Transfer
through a Porous Medium, Journal of Naval
Architecture and Marine Engineering 6: 41-51.
Fung, Y.C. and Tang, H.T.1975. Longitudinal Dispersion
of particles in the blood flowing in a pulmonary
alveolar sheet, ASME-Journal of Applied Mechanics
42: 536.
Gopalan, N.P. 1981. Pulsatile blood flow in a rigid
pulmonary alveolar sheet with porous walls, Bulletin
of Mathematical Biology 43(5):563-577.
Hassan, A.M. and EL-Arabawy. 2003. Effect of
suction/injection on the flow of a micropolar fluid
past a continuously moving plate in the presence of
radiation, International Journal of Heat and Mass
Transfer 46: 1471-1477.
Hayat, T. and Ali, N. 2008. Peristaltic motion of a Jeffrey
fluid under the effect of a magnetic field in a tube,
Communications in Nonlinear Science and Numerical
Simulation 13: 1343-1352.
Jaffrin, M.Y. and Shapiro, A.H. 1971. Peristaltic
Pumping, Annual Review of Fluid Mechanics 3: 1336.
Kumar, Y.V.K.R, S.V.H.N. Kumari, P.K, Murthy,
M.V.R. and Sreenadh, S. 2010. Unsteady peristaltic
pumping in a finite length tube with permeable wall,
ASME Journal of Fluids Engineering 132: 1012011 –
1012014.
Misra, J.C. and Pandey, S.K. 1999. Peristaltic transport
of a non-Newtonian fluid with a peripheral layer,
International Journal of Engineering Science 37:
1841-1858.
157