Applied Mathematics and Computation 244 (2014) 761–771
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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
Effect of coupled radial and axial variability of viscosity
on the peristaltic transport of Newtonian fluid
Mehdi Lachiheb ⇑
Faculty of Siences, Taibah University, Madinah Munawwarah, Saudi Arabia
a r t i c l e
i n f o
Keywords:
Peristaltic motion
Newtonian fluid
Trapping
Reflux
a b s t r a c t
Many authors assume viscosity to be constant or a radius exponential function in Stokes’
equations in order to study the peristaltic motion of a Newtonian fluid through an axisymmetric conduit. In this paper, viscosity is assumed to be a function of both the radius and
the axial coordinate. More precisely, it is dependent on the distance from the deformed
membrane given the fact that the change in viscosity is caused by the secretions released
from the membrane. The effect of this hypothesis on the peristaltic flow of a Newtonian
fluid in axisymmetric conduit is investigated under the assumptions of long wavelength
and low Reynolds number. The expressions for the pressure gradient and pressure rise
per wavelength are obtained and the pumping characteristics and the phenomena of reflux
and trapping are discussed. We present a detailed analysis of the effects of the variation of
viscosity on the fluid motion, trapping and reflux limits. The study also shows that, in addition to the mean flow parameter and the wave amplitude, the viscosity parameter also
affects the peristaltic flow. It has been noticed that the pressure rise, the friction force,
the pumping region and the trapping limit are affected by the variation of the viscosity
parameter but the reflux limit and free pumping are independent of it.
Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction
Several works have analyzed the mechanics of peristaltic pumping of a Newtonian fluid through an axisymmetric conduit. One of the remaining works is the treatment of the Stokes equation under an infinitely long wavelength approximation
and a negligible Reynolds number in order to evaluate the velocities and pressures at each point of the geometrical model for
a viscous fluid. Most works used the sine function for the geometry of the intestinal wall surface and a constant or an exponential function or its linear approximation for viscosity. For example, Latham [1] was the first to investigate the mechanism
of peristalsis in relation to mechanical pumping. Shapiro et al. [2] studied the peristaltic motion of Newtonian fluid with constant viscosity, under long-wavelength and low-Reynolds number assumptions, for plane and axisymmetric flow. They discussed the pumping characteristics and the phenomena of reflux and trapping. Shukla et al. [3] investigated the effects of
peripheral-layer viscosity on peristaltic transport of a bio-fluid in uniform tube, the shape of interface is specified by them
independently of the fluid viscosities. The invalidity of their analysis was proved by Brasseur et al. [4] with the limit of infinite peripheral layer viscosity, since the conservation of mass is not satisfied. They presented the effect of the peripheral
layer on the fluid motion and discussed the pumping characteristics and the phenomena of reflux and trapping. They concluded that, for a peristaltic motion, a peripheral layer more viscous than the core fluid improves the pumping performance,
⇑ Address: Faculté de siences de Gabes, Université de Sud, Gabes, Tunisie.
http://dx.doi.org/10.1016/j.amc.2014.07.062
0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
762
M. Lachiheb / Applied Mathematics and Computation 244 (2014) 761–771
while a less-viscous peripheral layer degrades performance. Elshehawey et al. [5] studied the peristaltic motion of an incompressible Newtonian fluid through a channel in the presence of three layers flow with different viscosities. Srivastava et al.
[6,7], discuss the pressure rise, peristaltic pumping, augmented pumping and friction force for Newtonian fluid on peristaltic
motion, with variable viscosity in uniform and non-uniform tubes and channel under zero Reynolds number and long wavelength approximations and comparison with other theories are given. Their computational results indicate that the pressure
rise decreases as the fluid viscosity decreases. The difference between two corresponding values (for constant and variable
viscosity) of the pumping region increases with increasing amplitude ratio. Further, the free pumping increases as viscosity
of fluid decreases. The pressure rise, in the case of non-uniform geometry is found to be much smaller than the corresponding value in the case of uniform geometry. Abd El Hakim et al. [8,9] have investigated the effect of endoscope and fluid with
variable viscosity on peristaltic motion. Abd El Hakim et al. [10] choose the viscosity parameter in the exponential function
as a perturbation parameter in order to study the hydromagnetics flow of fluid with variable viscosity in a uniform tube with
peristalsis under a negligible Reynolds number, a small magnetic Reynolds number, sine harmonic traveling wave and infinitely long wavelength approximation. The effect of variable viscosity on the peristaltic transport of a Newtonian fluid in an
asymmetric channel has been studied by Hayat and Ali [11]. They showed that, in addition to the effect of mean flow parameter, the wave amplitude also affects the peristaltic flow. Recently a number of analytical, numerical and experimental studies of peristaltic flows of different fluids have been reported. Rathod and Asha [12] studied the effects of magnetic field and
an endoscope on peristaltic motion and Rathod and Devindrappa [13] studied the slip effect on peristaltic transport of a porous medium in an asymmetric vertical channel by Adomian decomposition method. Hina et al. [14] studied the Slip Effects
on magnetohydrodynamic Peristaltic Motion with Heat and Mass Transfer. A mathematical model has been developed by
Misra and Maiti [15] with an aim to study the peristaltic transport of a rheological fluid for arbitrary wave shapes and tube
lengths.
The previously mentioned works are based on Newtonian fluids with a viscosity that is either constant or function of the
radius only. This is inconsistent with the situation in natural peristaltic motions. For instance, chyme motion within the
small intestine is subjected to water intake at a first stage, then to water absorption accompanied by some acids released
from the intestinal membrane to facilitate digestion and enable the absorption of necessary nutrients such as proteins, fat
and vitamins [16–18]. The movement of blood in the blood vessels is characterized by a concentration of red blood cells near
the axis and of white blood cells and plasma near the membrane, which makes the viscosity decrease closer to the wall [19].
This shows that viscosity depends on the distance from the membrane in its deformed state, which again contradicts the
assumptions of constant or radially varying viscosity since these imply a constant viscosity along lines parallel to the axis.
This led us in this work to consider the viscosity to be a function of two variables. We assume that the viscosity remains
constant on any surface such that the radius is proportional to that of the membrane with a factor less than unity. This choice
is consistent with the observation mentioned above concerning the distribution of viscosity in the blood vessels and the
small intestine. It is expected to provide more accurate and realistic explanations of the mechanisms of peristaltic movement
and blood circulation and the associated reflux and trapping phenomena. Moreover, in order to make this work even more
general an additional parameter has been introduced into the viscosity function. This parameter indicates the gradient of the
viscosity. We further studied the effect of this parameter on pumping, reflux and trapping and compared it with published
results based on the assumptions of viscosity being either constant or function of a single variable.
2. Formulation and analysis
We consider the creeping flow of an incompressible Newtonian fluid with variable viscosity through an axisymmetric
form in a uniform tube wall thickness with a wave traveling down its wall. We use the cylindrical coordinate system
Fig. 1. Peristaltic transport through a tube.
M. Lachiheb / Applied Mathematics and Computation 244 (2014) 761–771
763
ðZ; RÞ, where the z axis lies along the centerline of the tube, and r transverse to it. The geometry of the wall surface is
described in Fig. 1 and it is given by
2p
^
HðZÞ
¼ a þ b sin
ðZ ctÞ
k
ð1Þ
where a is the radius of the tube at inlet, k is the wavelength, c is the wave speed and t is the time. Let U and W be the velocity components in the radial and axial directions and W the stream function. In these coordinates the flow is unsteady but we
can eliminate the time variable if we choose moving coordinates ðz; rÞ which travel in the Z direction with the same speed as
the wave. Then the flow can be treated as steady. The coordinate frames are related through:
z ¼ Z ct r ¼ R
¼W c
w
¼U
u
HðzÞ ¼ a þ b sin
ð2Þ
w ¼ W r 2 =2
ð3Þ
2p
z
k
ð4Þ
and w
are the velocity components in the moving coordinates. The following equations are the continuity and
where u
Navier–Stokes equations and the boundary conditions in the moving coordinates:
Þ @ w
1 @ðr u
þ
¼0
r @r
@z
ð5Þ
ðr Þ @ u
u
@w
@u
@u
@P @
@u
2l
@
@u
ðr Þ
¼
þ
þ
q u þ w
þ
2l
l ðrÞ þ r
@r @r
@z
@r
@z
@r
@r r
@z @r
ð6Þ
@w
@w
@w
@P @
@w
1 @
@u
ðr Þ
r l
ðr Þ
¼
þ
q u þ w
þ
2l
þ
r @r
@z @ z
@r
@ z
@ z
@z @r
ð7Þ
@w
¼ 0 if r ¼ 0
¼0 u
@r
ð8Þ
¼ c
w
¼ c
u
@HðzÞ
@z
if r ¼ H
ð9Þ
is the viscosity function. We define the Reynolds number
where P is the pressure and l
Re ¼
where
caq
ð10Þ
l0
l0 is the viscosity on the axis, and the wave number
d¼
a
k
ð11Þ
and the non-dimensional variables:
r¼
r
a
z¼
z
k
u¼
ku
ac
w¼
w
c
P¼
a2 P
ckl0
l¼
l
ct
H
t¼
H ¼ ¼ 1 þ / sinð2pzÞ
k
a
l0
ð12Þ
where / ¼ ba is the amplitude ratio.
Replacing Eq. (12) in (5)–(9) and using the long wavelength approximation and the negligible wave number we obtain:
1 @ðruÞ @w
þ
¼0
r @r
@z
ð13Þ
@P
¼0
@r
ð14Þ
@P 1 @
@w
r lðrÞ
¼
@z r @r
@r
ð15Þ
@w
¼ 0 u ¼ 0 at r ¼ 0
@r
ð16Þ
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M. Lachiheb / Applied Mathematics and Computation 244 (2014) 761–771
@H
@z
w ¼ 1 u ¼ at r ¼ H
ð17Þ
The instantaneous volume flow rate in the fixed and moving coordinate system are defined by:
^ ¼ 2p
Q
Z
^
H
RWdR
ð18Þ
rwd
r
ð19Þ
0
¼ 2p
q
Z
H
0
From Eqs. (2)–(4) and (18) it follows that:
^ ¼q
^
þ pcH
Q
ð20Þ
Using the dimensionless variables, we find
Q¼
q
¼
2pa2 c
Z
H
rwdr
ð21Þ
0
The time-mean flow over a period T ¼ kc at a fixed Z-position is defined as
Q¼
1
T
Z
T
0
2
b
^ dt ¼ q
þ pc a2 þ
Q
2
!
ð22Þ
The dimensionless time-mean flow is
H¼
Q
1
/2
1þ
¼Qþ
2
2pca
2
2
!
ð23Þ
From Eq. (15) we obtain @r@ ðrlðr; zÞ @w
Þ ¼ r @P
. Since the pressure is independent to the variable r, then
@r
@z
1 2 @P
lðr; zÞr @w
¼
r
þ
c
ðzÞ,
where
c
ðzÞ
is
a
function
dependent
on z. Using the first condition of (16) we obtain c1 ðzÞ ¼ 0. Then
1
1
@r
2
@z
R
wðrÞ ¼ Mðr; zÞ @P
þ c2 ðzÞ, where Mðr; zÞ ¼ 12
@z
r
HðzÞ
t
lðt;zÞ dt
from the first condition of (17), c2 ðzÞ ¼ 1, then
@P
wðrÞ ¼ Mðr; zÞ
1
@z
ð24Þ
@w
@P
¼ rMðr; zÞ
r
@r
@z
ð25Þ
and
and the shear stress at wall is given by
@w
1 @P
¼ H
@r r¼H 2 @z
sðzÞ ¼ lðt; zÞ
ð26Þ
The dimensionless instantaneous volume flow rate in moving coordinate system is Q ¼ 2pqa2 c ¼
RH
ity component w by the right hand side of (24), we obtain Q ¼ 0 tMðt; zÞdt @P
12 H2 , then
@z
Q þ 12 H2
@P
¼ RH
@z
tMðt; zÞdt
0
RH
0
rwdr. Replacing the veloc-
ð27Þ
and since w ¼ 0 at r ¼ 0, it follows that
w¼
Rr
tMðt; zÞdt 1 2
1
Q þ H2 R H0
r
2
tMðt; zÞdt 2
ð28Þ
0
Since large quantities of water are secreted into the lumen of the small intestine during the digestive process, almost all of
this water is also reabsorbed in the small intestine [16–18] and the movement of blood in the blood vessels is characterized
by a concentration of red blood cells near the axis and of white blood cells and plasma near the membrane [19], which makes
the viscosity decrease closer to the vessels membrane. This shows that viscosity depends on the distance from the membrane in its deformed state, which again contradicts the assumptions of constant or radially varying viscosity since these
imply a constant viscosity along lines parallel to the axis. This led us in this work to consider the viscosity to be a function
of two variables in the following form:
lðr; zÞ ¼ f
where f ðrÞ ¼ ear .
r
HðzÞ
ð29Þ
M. Lachiheb / Applied Mathematics and Computation 244 (2014) 761–771
765
We assume that the viscosity remains constant on any surface such that the radius is proportional to that of the membrane with a factor less than unity. This choice is consistent with the observation mentioned above concerning the distribution of viscosity in the blood vessels and small intestine. Moreover, in order to make this work even more general the
parameter a has been chosen positive for decreasing viscosity and negative for increasing viscosity.
Replacing Eq. (29) in Mðt; zÞ we obtain
1
Mðt; zÞ ¼ H2 ðzÞ
2
Z
t
HðzÞ
1
at
at 1 eHðzÞ
ð1 aÞea þ HðzÞ
s
2
ds ¼ H ðzÞ
f ðsÞ
2a2
and
Z
H
tMðt; zÞdt ¼ H4
0
ð6 þ ð6 6a þ 3a2 a3 Þea Þ
4a4
then
4
@P 4a
¼
@z
Q þ 12 H2
ð30Þ
bH4
and
sðzÞ ¼
2a4 Q þ 12 H2
ð31Þ
bH3
and
ar
ar ar Q þ 12 H2 6H2 1 þ e H þ 6aHre H þ a2 ð1 þ aÞea 2e H r2
1 2
w¼ r 2
bH2
ð32Þ
where b ¼ ð6 þ ð6 6a þ 3a2 a3 Þea Þ.
The pressure rise DP and friction force F in their non-dimensional forms are given by
Z
DP ¼
1
0
F¼
Z
1
0
@P
dz
@z
ð33Þ
@P
dz
H2 @z
ð34Þ
Replacing Eq. (30) in (33) and (34) we obtain
DP ¼
a4 ðð4 þ 6/2 ÞH ð8/2 12 /4 ÞÞ
7
bð1 /2 Þ2
0
F¼
a4 @/2 þ 2 4H
3
b
ð1 /2 Þ2
ð35Þ
1
2A
ð36Þ
If a approaches zero, we obtain the results given by [11]
w ¼ r2
"
#
1
ð2H2 r 2 Þ 1
Q þ H2
2
2
H4
DP ¼ 8
ð2 þ 3/2 ÞH 4/2 14 /4
7
ð1 /2 Þ2
1
2 þ /2 4HA
@
F ¼ 84
3
ð1 /2 Þ2
ð37Þ
ð38Þ
0
ð39Þ
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M. Lachiheb / Applied Mathematics and Computation 244 (2014) 761–771
3. Discussion of the results
3.1. Pumping characteristics
The maximum pressure rise DPmax against which the peristalsis works as a pump (i.e. DP corresponding to H ¼ 0). In the
same way the maximum volume flow rate Hmax is the value of H when the pressure rise DP ¼ 0 (free pumping). These are
given by
DPmax ¼
Hmax ¼
a4 12 /4 8/2
ð40Þ
7
bð1 /2 Þ2
8/2 12 /4
ð41Þ
4 þ 6/2
When DP > DP max one obtains negative flux and when DP < 0, one gets H > Hmax as the pressure assists the flow which is
known as copumping.
The relation between the pressure rise and the flow rate and the friction force and the flow rate given by (33) and (34),
respectively, are plotted in Figs. 2–7. Since the viscosity function, as reported in Srivastava and al.[7], is ear with a ¼ 0 (Shapiro et al. [2]) and a ¼ 0:1, Figs. 6 and 7 show the relation between the pressure rise and the friction force versus flow rate for
different viscosity functions at / ¼ 0:2; 0:4, and 0:6. It is clear that the pressure rise and the friction force are less under radial
and axial variability of viscosity than that for constant or under radial variability of viscosity as reported in [2]. The variation
of DP and F given by (33) and (34), respectively, with H for / ¼ 0:4 for different values of viscosity parameter a is shown in
Figs. 2–5. It is observed that the pressure rise and the friction force decrease with increasing a and the pumping region
ð0 6 DP 6 DPmax Þ is less for radial and axial variability of viscosity than that for constant or radial variability of viscosity
(Fig. 8) and it decreases as the viscosity parameter a increases (Fig. 9). But the free pumping is independent of alpha.
Fig. 10 shows the shear stress distribution along the wall for different values of the viscosity parameter for the three viscosity
functions ear and f ðr=HÞ ¼ ear=H and constant. Unlike in the viscosity function ear , in our choice the viscosity is constant
along the wall. Consequently, the shear stress function is independent of the gradient of the viscosity function f but it does
depend on the viscosity at the wall which is itself dependant on the viscosity parameter alpha.
3.2. Trapping limit
For certain combinations of H and / there is a region of closed streamlines. This region experiences a recirculation and
moves with a mean speed equal to that of the wave. This is termed as trapping. Peristalsis exhibits this phenomenon when
the tube is sufficiently occluded. The trapping limit is given by the value of H where w ¼ 0 at some r just greater then zero.
Having expressed w in a power series in terms of r, the flow-rate H for which trapping may occur lies between two extremes
as given below
1
H ¼
2
"
/2
1þ
2
!
6 þ a2 þ ð6 6a þ 2a2 Þea
ð1 þ /Þ2
a2 ð1 þ ð1 þ aÞea Þ
#
ð42Þ
and
1
H ¼
2
þ
"
/2
1þ
2
!
6 þ a2 þ ð6 6a þ 2a2 Þea
ð1 /Þ2
a2 ð1 þ ð1 þ aÞea Þ
#
Fig. 2. The pressure rise DP versus the volume flow rate h with / ¼ 0:4 with decreasing viscosity.
ð43Þ
M. Lachiheb / Applied Mathematics and Computation 244 (2014) 761–771
767
Fig. 3. The friction force F versus the volume flow rate h with / ¼ 0:4 with decreasing viscosity.
Fig. 4. The pressure rise DP versus the volume flow rate h with / ¼ 0:4 with increasing viscosity.
Fig. 5. The friction force F versus the volume flow rate h with / ¼ 0:4 with increasing viscosity.
If a approaches zero, we have the range for Newtonian fluid with constant viscosity reported in [2]. Like in the case of
þ
þ
constant viscosity, HHmax is greater than 1 for all amplitude ratios h. Fig. 11 shows the lower limit of trapping region HHmax versus
the amplitude ratio with various values of viscosity parameter including zero (constant viscosity). The trapping region is
smaller for radial and axial variability of viscosity than that for constant viscosity and it decreases as the viscosity parameter
a. In Fig. 12, the trapping contours are plotted with various values of the viscosity parameter a and it can be seen that the size
of trapping increases as a decreases.
3.3. Reflux limit
Reflux refers to the presence of fluid particles that move, on the average, in a direction opposite to the net flow. Since the
laboratory frame has been found suitable for the study of reflux, W the flow-rate corresponding to Z and t, is transformed as
768
M. Lachiheb / Applied Mathematics and Computation 244 (2014) 761–771
Fig. 6. The pressure rise DP versus the volume flow rate h.
Fig. 7. The friction force F versus the volume flow rate h.
Fig. 8. The maximum pressure rise DP max versus / with a ¼ 0:1.
1
2
W ¼ w þ r2 ðw; zÞ
ð44Þ
Averaged over one cycle, this becomes
HW ¼ w þ
1
2
Z
0
1
r 2 ðw; zÞdz
ð45Þ
M. Lachiheb / Applied Mathematics and Computation 244 (2014) 761–771
Fig. 9. The maximum pressure rise DP max versus the parameter viscosity a with various values of amplitude ratio /.
Fig. 10. The shear stress at wall versus the variable z with various values of viscosity parameter a and with / ¼ 0:4 and H ¼ 0:1.
Fig. 11. The lower limit of trapping region with various values of viscosity parameter a.
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770
M. Lachiheb / Applied Mathematics and Computation 244 (2014) 761–771
Fig. 12. Streamlines in the wave frame showing the appearance of the trapping with various values of viscosity parameter a and with / ¼ 0:6 and H ¼ 0:1.
In order to evaluate the reflux limit, HW is expanded in power series in terms of a small parameter
and it is subjected to the reflux condition
about the wall, where
¼wF
HW
> 1 as
H
!0
ð46Þ
The coefficients of the first two terms in the expansion of r, i.e.,
r ¼ h þ a0 þ a1 2 þ ð47Þ
are found by using (30), as given below
1
a0 ¼ ;
h
a1 ¼
ð2F þ H2 Þa4 ea bH2
2H5 b
ð48Þ
Using (45)–(48) it follows that the reflux occurs whenever HH
< 1 and it is independent to the parameter a. That is, like in
max
the case of Newtonian fluids with constant viscosity, reflux does take place in the entire domain.
4. Conclusion
A coupled radial and axial variability of viscosity is considered to study the peristaltic transport of Newtonian fluid with
variable viscosity in axisymmetric conduit. The effects of the viscosity function’s slope and the amplitude ratio on the pressure rise, the friction force, trapping and reflux limits and pumping region are studied. It is observed that an increase in the
viscosity parameter decreases the pressure rise and the frictional forces. Moreover, the pressure rise, the friction force, the
pumping region and the trapping limit are affected by the variation of the viscosity parameter but the reflux limit and free
pumping are independent of it. It is found that the pressure rise, the friction force and the pumping region are lower under
radial and axial variability of viscosity than that for constant or under radial variability of viscosity. Moreover, the trapping
region and the size of trapping is smaller for radial and axial variability of viscosity than that for constant viscosity.
M. Lachiheb / Applied Mathematics and Computation 244 (2014) 761–771
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