The Influence Of Slip Conditions, Wall Properties And
Heat Transfer On Peristaltic Pumping Of A
Bingham Fluid
S. Dhananjaya
Dept. of Mathematics
Sri Venkateswara
University
Tirupati, (A.P)
India.
P.V. Arunachalam
Dept. of Mathematics
Sri Venkateswara
University
Tirupati, (A.P)
India.
S. Sreenadh
Dept. of Mathematics
Sri Venkateswara
University
Tirupati, (A.P)
India.
P. Lakshminarayana
Dept. of Mathematics
Sree Vidyanikethan
Engineering College
Tirupati, (A.P)
India.
Abstract
The effects of both wall slip conditions and heat transfer on the peristaltic flow of a Bingham
fluid in a porous channel with elastic wall properties have been investigated under the
assumptions of long wavelength and low Reynolds number. The expressions for velocity,
temperature, stream function and heat transfer coefficient are obtained. The effects of various
physical parameters on velocity and temperature are analyzed through graphs and the results
are discussed in detail.
Keywords: Peristaltic flow, Slip conditions, Wall properties, Heat transfer, Bingham fluid
and Non- uniform channel.
1. Introduction: Peristaltic transport is a
mechanism of fluid transport induced by a
progressive wave of area contraction or
expansion along the length of a distensible
tube, mixing and transporting the fluid in
the direction of the wave propagation. It
occurs widely in many biological and
biomedical systems. It plays an important
role in transporting many physiological
fluids in the body in various situations
such as swallowing food through the
oesophagus, urine transport from the
kidneys to the urinary bladder through the
ureter, movement of chyme in the gastrointestinal
tract,
the
transport
of
spermatozoa in the ductus efferentes of the
male reproductive tract and in the cervical
canal etc. It has many applications in
medicine and industry. Also the literature
on peristalsis is extensive. Some
investigations [1-7] are presented in the
references.
Most of the biofluids like blood are
observed to behave like non-Newtonian
fluids. In fact there is no unique model to
describe all non-Newtonian fluids in
physiological systems, several models are
proposed [8-18] to explain the behavior of
these biofluids. Several researchers studied
the peristaltic flow with heat transfer due
to its wide applications in fluid mechanics.
Vajravelu et al. [19] analyzed the
peristaltic flow and heat transfer in a
vertical porous annulus with long
wavelength approximation. Srinivas and
Kothandapani [20] studied the influence of
heat and mass transfer on MHD peristaltic
flow through porous space with compliant
walls. Mekheimer and Abd Elmaboud [21]
investigated the influence of heat transfer
and magnetic field on the peristaltic
transport of a Newtonian fluid in a vertical
annulus. Srinivas et al. [22] studied the
influence of slip conditions, wall
properties and heat transfer on MHD
peristaltic transport. The influence of heat
transfer on the peristaltic transport of a
Jeffrey fluid in a vertical porous stratum is
discussed by Vajravelu et al. [23]. Rathod
and Laxmi [24] examined the effects of
heat transfer on the peristaltic MHD flow
of a Bingham fluid through a porous
medium in a channel.
1
In several applications the flow pattern
corresponds to a slip flow, the fluid
presents a loss of adhesion at the wetted
wall making the fluid slide along the wall.
When the molecular mean free path length
of the fluid is comparable to the distance
between the plates as in nanochannels or
microchannels, the fluid exhibits noncontinuum effects such as slip-flow as
demonstrated experimentally by Derek et
al. [25]. Hron et al. [26] have presented
analytical solutions for the flows of a
generalized fluid of complexity two in
special geometries under the assumption
that the flows meet Navier slip conditions
at the boundary. Ali et al. [27] have
studied the slip effects on the peristaltic
transport of MHD fluid with variable
viscosity. Ebaid [28] studied the effects of
magnetic field and wall slip conditions on
the peristaltic transport of a Newtonian
fluid in an asymmetric channel. Arun
Kumar et al. [29] have investigated the
influence of partial slip on the peristaltic
transport of a micropolar fluid in an
inclined asymmetric channel.
Hayat et al. [30] have investigated the
MHD peristaltic channel flow of a Jeffrey
fluid with complaint walls and porous
medium. The interaction of peristalsis with
heat transfer for the motion of a viscous
incompressible Newtonian fluid in a
channel with wall effects has been studied
by Radhakrishnamacharya and Srinivasulu
[31].
In the present study the effects of both wall
slip conditions and heat transfer on
peristaltic flow of a Bingham fluid in a
porous channel with elastic wall properties
have been analyzed under the assumptions
of long wavelength and low-Reynolds
number. The expressions for velocity,
temperature, stream function and heat
transfer coefficient are obtained. The
effects of various physical parameters on
velocity and temperature are analyzed
through graphs and the results are
discussed in detail.
2. Mathematical formulation
Consider the flow of a Bingham fluid
through a two-dimensional channel of
uniform thickness. The walls of the
channel are assumed to be flexible and are
taken as stretched membranes, on which
travelling sinusoidal waves of moderate
amplitude are imposed.
The geometry of the wall surface is given
by
2
(1)
 ( x, t )  d ( x)  a Sin
x  ct

where d ( x)  d  m x, m  1


The governing equations which describe
the flow are
u  v
 0
x  y


 t
u
(2)

 
p
2 u 
u
v u  
 2 
( 0   )
x
y
x
y
x  y
(3)


 t
u
 2 v 2 v 

 
p
v
v




 2


2
 x

x
y
y

y


(4)


 t
u

 
k 2
2
v
T

(

)T

 x 2  y 2
x
y
  u  2   v  2    v  u  2
 2 

 
 

  x    y     x  y 
where
(5)
u , v ,  ,  , p , d , a ,  , c, m ,  ,  , k , T
and  0 are the axial velocity, transverse
velocity, fluid density, viscosity of the
fluid, pressure, mean width of the channel,
amplitude, wavelength, wave speed,
dimensional non-uniformity of the
channel, specific heat at constant volume,
kinematic viscosity, thermal conductivity
of the fluid, temperature and yield stress.
The equation of motion of the flexible wall
is expressed as
2
L* ( )  p  p0




m
d
a

m
,   ,   ,   1  mx   sin 2 ( x  t ), 

d

d
d

2
(T  T0 )
 cd

c

R
, 
, Pr 
, Ec 
,


(T1  T0 )
k
 (T1  T0 )

m1cd 3
 d 3
cd 3
h

E1  3 , E2  3 , E3  2 ,  

 c


d

(11)
where R is the Reynolds number,
 and  are the dimensionless geometric
x
y

d2 p
ct
k
x  , y  , 
, p
, t  ,K  2 ,

d
cd
 c

d
(6)
where L* is an operator, which is used to
represent the motion of stretched
membrane with viscosity damping forces
such that
2
L*  
x
2
 m1
2
t
2
C

t
(7)
where  is the elastic tension in the
membrane, m1 is the mass per unit area, C
is the coefficient of viscous damping
forces, p0 is the pressure on the out side
surface of the wall due to the tension in the
muscles and h is the dimensional slip
parameter. We assumed p0  0 .
parameters, Pr is the Prandtl number, Ec
is the Eckert number, E1 , E2 and E3 are
the dimensionless elasticity parameters m
is the non-uniform parameter and  is the
Knudsen number (slip parameter) .
Continuity of stress at y    and using
x- momentum equation, yield
 *
p
2 u 
u
L ( ) 
 2 
( 0   )
x
x
y
x  y
Using non-dimensional quantities the basic
equations (1) - (10) reduce to


 
  u
 v u
x
y
 t
  2   2   2 
p
R 



2 
x
 t y y xy x y 
3
2



 2 2 
( 0  2 )
x y y
y
(8)
u  h
u
at y  
y
T
 0 on y  y0
y
T  T1 on y  
(9)
(12)
  2   2   2 
p
R 3 




2
x xy 
y
 t y y x
(10)

 3
 3 
  2  2


x 3 yy 2 

Introducing  such that
(13)
u

y

and v  
x
and the following
parameters are given by
       1  2  2  2 
R  

 2
   
2
y 
 t y x x y  Pr  x
2
2
2
2


  2
2   
2   

 E  4 
  2 



x 2  
 xy   y


non-dimensional
(14)


  2 at y  
y
y
(15)
2
3
 3

 2


( 0 
)
x 2 y y
y 2

(24)
and corresponding plug flow velocity is
given by
1


u p  A ( y0 2   y0   y0 )  ( y02   2  2  ) 
2


(25)
where

y0  0
A
and
E


A  8   3 ( E1  E2 ) cos 2 ( x  t )  3 sin 2 ( x  t ) 
2


  2
  2
  2 
 R 


y xy
x y 2 
 t y
  3
 3
 2 
  E1
 E2
 E3
3
xt 2
xt 
 x
(16)
Further, it is assumed that the zero value of
the streamline at the line y = 0, i.e.
 p (0)  0,
 yy (0)   0 at y  0
(17)

 0 on y  y0
y
  1on
y 

1
 p  A ( y0 2   y0   y0 )  ( y02   2  2 )  y
2


2
and the stream function in the non-plug
(18)
flow region as
 ( y 2  y0 2 )

 0 
  ( y  y0 )   ( y  y0 ) 
2


3. Solution of the problem
 y 3  8 y03  2 y

 A

  y  y02 (    ) 
2
 6

(26)
The corresponding velocity in the nonplug flow region is given by
A
u   0 ( y     )  ( y 2   2  2 )
2
(27)
Applying the long wavelength and low
Reynolds number approximation, the basic
equations (12) - (18) reduce to
p  
 2 
(19)
0     0  2 
x y 
y 
p
(20)
y
Equation (20) shows that p is not a
function of y
1  2
 2
(21)
0
 E(
)
2
0
Using equation (26) in equation (21)
subject to the condition (18) we obtain
the temperature as
 y 3 A2 y 4  0 Ay 3 
   Br  0 2 

  C1 y  C2
6
12
3 

(28)
2
2
3
 y

A y0
where C1  Br  0 2 0 
  0 Ay0 2 
2
3


Pr y
y
By differentiating equation (19) with
respect y we obtain
2 
 2 
(22)
  0  2   0
y 2 
y 
From equation (16) we get
 3

 2
3
 2 
( 0  2 )   E1 3  E2
 E3
y
y
xt 2
xt 
 x
(23)
By solving equation (22) with boundary
conditions (15), (17) and (23) we obtain
 2  3 A2 4
3 
C2  1  Br  0

  0 A   C1
6
12
3

and Br  Ec Pr is the Brinkman number
The coefficient of heat transfer at the wall
the stream function in the plug flow region
is given by
Nu    y 
at y 
as
(29)
4
number Br. non-uniform parameter m ,
amplitude ratio  and
yield stress
parameter  0 can be examined through the
figures Fig.9 - Fig.12. It is noticed that due
to peristalsis, the rate of heat transfer ( Nu )
shows oscillatory behaviour. From Fig.9
we observe that the Nusselt number
increases with increasing Brinkman
number Br . Figures Fig.10 to Fig.12
depict that the Nusselt number increases
with increasing non-uniform parameter m ,
amplitude ratio  and
yield stress
parameter  0 .
4. Results and Discussions
Equation (27) gives the expression for
velocity as a function of y. Velocity
profiles are plotted from Fig.1 to Fig.4 to
study the effects of different parameters
such as slip parameter  , non-uniform
parameter m , amplitude ratio  and yield
stress parameter  0
on the velocity
distribution. Fig.1 is plotted for different
values of slip parameter  . It is observed
that the velocity profiles are parabolic and
the velocity increases with increasing  .
Fig.2 depicts that the velocity for a
divergent channel (m > 0) is higher
compared with uniform channel (m = 0)
where as it is lower for a convergent
channel (m < 0). From Fig.3 and Fig.4 we
noticed that the velocity increases with
increasing amplitude ratio  and
decreasing yield stress parameter  0 .
Equation (28) gives the expression for
temperature as a function of y.
Temperature profiles are plotted from
Fig.5 to Fig.8 to study the effects of
different parameters such as non-uniform
parameter m , Brinkman number Br ,
amplitude ratio  and yield stress
parameter  0
on the temperature
distribution. It is observed that the
temperature profiles are almost parabolic.
Fig.5 reveals that the temperature is higher
diverging channel (m > 0) compared with
uniform (m = 0) and convergent (m < 0)
channels. Fig.6 and Fig.7 are plotted to
study the effect of Brinkman number Br
and amplitude ratio  . We notice that the
temperature increases with increasing
Brinkman number Br and amplitude ratio
 . Fig.8 shows that the temperature
decreases with increasing yield stress
parameter  0 .
The rate of heat transfer ( Nu ) is
calculated in equation (29). The variation
in Nusselt number for different values of
the interesting parameters Brinkman
Trapping phenomenon
Trapping is an interesting phenomenon
which refers to closed circulating
streamlines that exist at every high flow
rates and when occlusions are very large.
Streamlines are plotted to study the effect
of slip parameter  and non-uniform
parameter m on trapping through Fig.13
and Fig.14. From Fig.13 we observe that
the number of trapped boluses increases
with increasing slip parameter. Fig.14
reveals that the number of trapped boluses
increases with increasing non-uniform
parameter.
5
Fig.1 Velocity profiles for different '  'with fixed y0 = 0.2,
Fig. 4 Velocity profiles for different ' 0 'with fixed x = 0.2,t = 0.1,
x = 0.2,t = 0.1,m = 0.1,Br = 2,  0.1, E1 = 0.5,E 2 = 0.3,E3 = 0.2
 = 0.1, = 0.1,Br = 2, m  0.1, E1 = 0.5,E 2 = 0.3,E3 = 0.2
Fig.2 Velocity profiles for different 'm 'with fixed y0 = 0.2,x = 0.2,
Fig.5 Temperature profiles for different 'm ' with fixed y0 = 0.2,
x = 0.2,t = 0.1,Br = 2,   0.1, E1 = 0.3,E 2 = 0.2,E3 = 0.1,
t = 0.1, = 0.1,Br = 2,   0.1, E1 = 0.5,E 2 = 0.3,E3 = 0.2
Fig. 3Velocity profiles for different '  ' with fixed y0 = 0.2,x = Fig.6
0.2, Temperature profiles for different 'Br ' with fixed y0 = 0.2,
t = 0.1, = 0.1,Br = 2, m  0.1, E1 = 0.5,E 2 = 0.3,E 3 = 0.2
6
x = 0.2,t = 0.1.m=0.1,   0.1, E1 = 0.3,E 2 = 0.2,E3 = 0.1,
Fig. 7 Temperature profiles for different '  ' with fixed y0 = 0.2,
x = 0.2,t = 0.1.m=0.1, Br  2, E1 = 0.3,E 2 = 0.2,E 3 = 0.1,
Fig. 8Temperature profiles for different ' 0 ' with fixed Br  2,
x = 0.2,t = 0.1.m=0.1,   0.1, E1 = 0.3,E 2 = 0.2,E3 = 0.1,
Fig. 10Variation of Nu for different 'm'with fixed x = 0.2,
t = 0.1.Br=2, 0  0.1,   0.05, E1 = 0.2,E 2 = 0.1,E3 = 0.1,
Fig. 11Variation of Nu for different '  ' with fixed x = 0.2,
t = 0.1.m=0.1, 0  0.1, Br  2, E1 = 0.2,E2 = 0.1,E3 = 0.1,
Fig.12 Variation of Nu for different ' 0 ' with fixed x = 0.2,
Fig. 9Variation of Nu for different 'Br 'with fixed x = 0.2,
t = 0.1.m=0.1, 0  0.1,   0.05, E1 = 0.2,E2 = 0.1,E3 = 0.1,
t = 0.1.m=0.1,Br  2,   0.05, E1 = 0.2,E 2 = 0.1,E 3 = 0.1,
7
8
(a)
(a)
(b)
(b)
(c)
(c)
Fig.13 Stream lines for (a)   0,(b)   0.4,(c)   0.6,
y0  0.1, m  0.1,t  0.5,   0.2, E1 = 0.6,E2 = 0.4,E3 = 0.2
8
Fig.14 Stream lines for (a)m  0.2,(b)m  0,(c)m  0.2,
y0  0.1,   0.2,t  0.5,   0.2, E1 = 0.6,E2 = 0.4,E3 = 0.2
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
Shapiro, A.H., Jaffrin, M.Y. and
Weinberg, S.L., 1969 Peristaltic
pumping with long wavelength
at low Reynolds number. J. Fluid
Mech., 37, 799-825.
Lew, G.S., Fung, Y.C. &
Lowensein, C.B. 1971 Peristaltic
carrying and mixing
of chime in small intestine.
J.Biomechanics., 4, 297-315.
Shukla, J.B. and Gupta, S.P. 1982
Peristaltic transport of a power-law
fluid with variable viscosity. Trans.
ASME. J. Biomech. Engg. 104,
182-186.
Radhakrishnamacharya, G., 1982
Long wavelength approximation
to peristaltic motion of a
power law fluid, Rheol. Acta., 21,
30-35.
Usha, S., Ramachandra Rao, A.,
1997 Peristaltic transport of twolayered power-law
fluids, J.
Biomech. Engrg., 119, 483-488.
Srinivasacharya, D., Mishra, M.,
Ramachandra Rao, A., 2003
Peristaltic
transport
of
a
micripolar fluid in a tube, Acta.
Mech., 161, 165-178.
Vajravelu, K., Sreenadh, S. and
Ramesh
Babu,
V.,
2005
Peristaltic
transport
of
a
Hershel-Bulkley fluid in an
inclined tube. Int. J. Nonlinear
Mech. 40, 83-90.
Raju, K.K., and Devanathan, R.,
1972 Peristaltic motion of a nonNewtonian fluid. Rheol. Acta.,
11, 170-178.
Srivastava, L.M. and Srivastava,
V.P. 1984 Peristaltic transport of
Blood:
Casson
model-II.
J
Biomech, 17, 821-829.
Srivastava,
L.M.
and
Srivastava,V.P. 1988 Peristaltic
transport of a power law fluid:
application to the ducts Efferentes
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
9
of
the
reproductive
tract.
Rheol.Acta, 27, 428-433.
Mishra, M., Ramachandra Rao, A.,
2004 Peristaltic transport of a
power law
fluid
in
a
porous tube, J. Non- Newtonian
Fluid Mech., 121, 163-74.
Subba Reddy, M.V., Ramachandra
Rao, A., 2007 Sreenadh, S.,
Peristaltic
motion
of
a
power- law fluid in an
asymmetric channel, Int. J. NonLinear Mech., 42 1153-61.
Hayat, T., Ali, N., and Asghar, S.,
2007 Hall effects on peristaltic
flow of a Maxwell fluid in a
porous medium, Phys. Lett. A 363,
397-403.
Sreenadh, S., Lakshminarayana, P.
and Sucharitha, G., 2011 Peristaltic
flow of micropolar fluid in an
asymmetric
channel
with
permeable walls. Int. J. of Appl.
Math and Mech., 7 (20), 18-37.
Noreen, S., Alsaedi, A., Hayat, T.,
2012 Peristaltic
flow
of
pseudoplasic fluid
in
an
asymmetric channel. Journal of
Applied
Mechanics.,
79,
054501-6.
Sucharitha
G,
Sreenadh
S,
Lakshminarayana P, 2012 Nonlinear peristaltic transport of a
conducting Prandtl fluid in a
porous asymmetric channel, Int. J.
of Eng. Research and Technology,
1 (10), 1-10.
Vajravelu, K., Sreenadh, S.,
Sucharitha,
G.,
and
Lakshminarayana,
P., 2014
Peristaltic
transport of a
conducting Jeffrey
fluid in an
inclined asymmetric channel,
International Journal
of
Biomathematics 7 (6) 1450064
(25 pages).
Sreenadh, S., Lakshminrayana, P.,
Arun kumar, M. and Sucharitha,
G.,
2014 Effect of MHD on
peristaltic
transport
of
a
[19]
[20]
[21]
[22]
[23]
[24]
Pseudoplastic fluid in an
asymmetric
channel
With
porous medium,
International
Journal of Mathematical Sciences
and Engineering Applications
(IJMSEA) 8 (V) 27- 40.
Vajravelu, K., Radhakrishna
macharya, G., and Radhakrishna
murty, V., 2007 Peristaltic flow
and heat transfer in a vertical
porous annulus, with long wave
length approximation, Int. J. NonLinear Mech. 42, 754-759.
Srinivas, S., and Kothandapani, M.,
2009 The influence of heat and
mass transfer
on
MHD
peristaltic
flow through a
porous space with
complaint
walls, Appl. Math.
Comput.
213, 197-208.
Mekheimer, Kh.S. and ABD
Elmaboud, Y., 2008 The influence
of heat transfer and magnetic
field on peristaltic transport of a
Newtonian fluid in a vertical
annulus: application of
an
endoscope, Phys.Lett. A 372 16571665.
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
Lakshminarayana, P., 2011 The
influence of heat transfer on
peristaltic
transport
of
a
Jeffrey fluid in a vertical
porous stratum, Commun.
Nonlinear Sci. Numer. Simulat. 16
3107-3125.
Rathod, V,P,. and Laxmi, D., 2014
Effects of heat transfer on the
peristaltic MHD flow of a
Bingham fluid through a porous
medium in a channel,
International
Journal
of
[25]
[26]
[27]
,
[28]
[29]
[30]
[31]
10
Biomathematics
7
(6),
1450060.
Derek, C., Tretheway, D.C. and
Meinhart, C.D, 2002 Phys. Fluids
14, L9.
Hron, J., Roux, C.L, Malik, J. and
Rajagopal, K.R, 2008 Comput.
Math. Appl. 56, 2128.
Ali, N., Hussain, Q., Hayat, T.,
Asghar, S., 2008 Phys. Lett. A 372
1477.
Ebaid, A., 2008 Effects of magetic
field and wall slip conditions on
the
peristaltic transport of a
Newtonian
fluid
in
an
asymmetric channel, Phys. Lett. A
372, 4493.
Arun Kumar, M., Venkataramana,
S.,
Sreenadh,
S.
and
Lakshminarayana,
P.,
2012
Influence of partial slip on the
peristaltic transport of a micropolar
fluid in an inclined asymmetric
channel, International Journal of
Mathematical Archive, 3, 11.
Hayat, T., Javad, M., and Ali, N.,
2008 MHD peristaltic transport
of Jeffrey fluid in a channel with
complaint walls and porous
space, Transp.
Porous Med.
74,
259.
Radhakrishnamacharya, G. and
Srinivasulu, Ch. 2007 Influence
of wall properties
on peristaltic
transport with heat transfer. C.R.
Mec., 335, 369-373.