Advances in Mathematics and Statistical Sciences
Swallowing of Casson fluid in Oesophagus under the influence of peristaltic
waves of varying amplitude
SANJAY KUMAR PANDEY and SHAILENDRA KUMAR TIWARI
Department of Mathematical Sciences,
Indian Institute of Technology (BHU), Varanasi, INDIA
[email protected] [email protected]
Abstract: The experimentally verified fact that there is a high pressure zone in the
lower part of oesophagus has established that the earlier models fall short of
representing the realistic swallowing process in the oesophagus. Since the high pressure
is created by gradually increasing amplitudes of peristaltic waves, the swallowing is
remodeled for swallowing Casson fluid in oesophagus in light of this. The wave amplitude
is assumed to increase exponentially. It is revealed that unlike constant amplitude case, In
the case of exponentially increasing amplitude, pressure is not uniformly distributed for
all cycles. Pressure keeps increasing along the entire length of the oesophagus; and finally
towards the end of the oesophageal flow, pressure increases quite significantly probably
to ensure delivery into stomach through the cardiac sphincter. This observation is the
same for Newtonian as well as non-Newtonian fluids but Casson fluids need more
pressure and hence more efforts by the oesophagus to be transported forward. At lower
flow rates, reflux occurs for much higher flow rates of Casson fluids than Newtonian
fluids. This tendency gradually diminishes with increasing amplitude. For a particular
value of amplitude, there is no difference; and beyond that the trends are quite opposite.
Thus near the wall, Casson fluid is less prone to reflux. It is also concluded that for both
the Newtonian and non-Newtonian Casson fluids, reflux is more likely to occur with
increasing amplitude and further augmented by the addition of amplifying damping
parameter.
Key Words: Peristalsis, Casson fluid, oesophagus, variable amplitude.
flow in ureters, flows in vas deferens etc. are
1 Introduction
pumping processes based on peristalsis.
Peristalsis is a naturally occurring pumping
Though some creatures use this
mechanism for generating flows in the
bodies of living beings. It is observed to take
mechanism for locomotion, the more
place in the muscular ducts of the entire
interesting fact is that human beings have a
human body. This phenomenon is caused by
great tendency of emulating natural
phenomena and processes. They used the
continuously but alternately occurring
contractions and relaxations of the walls of
mechanism in mechanical roller pumps to
transport viscous fluids in the printing
those tubular vessels in full synchronization
so that the fluid flowing inside is propelled
industry and for pushing blood in heart lung
forward with total purity of the fluid
machine. This is also used for sanitary fluid
preserved. This is because no external piston
transport. Thus, this is a widely used
is supplemented for that purpose, which
pumping mechanism.
saves the fluid from getting infected or
In an experimental investigation,
contaminated. A self dependent nature is
Peter J. Kahrilas et al. (1995) located a high
another merit of this process. Swallowing in
pressure zone in the lower part of the
oesophagus, chyme flow in intestines, urine
oesophagus whose length varies from a
normal esophagus to that suffering from
hiatus hernia. The present models didn’t
Moreover, in the recent past, Xia F.
have an answer to it.
et al.(2009) measured esophageal wall
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Advances in Mathematics and Statistical Sciences
thickness in contracted as well as dilated
state through 110 consecutive CT images of
adult patients without esophageal diseases.
When the esophagus was dilating, the
average esophageal wall thickness was
between 1.87 and 2.70 mm. The thickest
part was cervical esophagus. Thickness of
esophageal wall was larger in males than
that of females (5.26 mm vs. 4.34 mm
p<0.001). Age and the thickness of
subcutaneous fat had no significant impact
on the thickness of esophageal wall (p-value
was 0.056 and 0.173, respectively).
The overall conclusion that Pandey
et al. (2014) drew was that in the dilated
state the upper oesophagus is thicker while
in the contracted state the lower oesophagus
is thicker. Thus, the change in the thickness
in the lower oesophageal wall is much more
than the upper part. Since the peristaltic
waves are created by the contraction of
circular muscles of oesophagus, a larger
thickness of wall is an indication of larger
degree of contraction.
This is in contrast to prevailing
knowledge that wave amplitude remains the
same in size when swallowing takes place in
human oesophagus. In all of the models for
oesophageal
swallowing,
the
wave
amplitude was assumed to be constant by
the present investigators including Li and
Brasseure (1993), Misra and Pandey (2002)
and Pandey and Tripathi (2010a-b, 2011ab), Tripathi et al. (2013), etc. As a
consequence of this supposition, they
observed uniform pressure variation along
the oesophageal length in the theoretical
investigations for the flows of various fluids.
Pandey et al. (2014) pointed out it on the
basis of reported experimental verifications
and presented a corrected model for wave
propagation with exponentially increasing
amplitude, and then reported modified
results for Newtonian fluids swallowing in
the oesophagus. The exploration culminated
into the final conclusion was that the
pressure is higher in the lower oesophageal
zone. Of course there were many other
observations distracting from earlier
knowledge about swallowing. This further
paved the way for a fresh investigation of
the several types of Non-Newtonian
oesophageal flows.
We intend to investigate swallowing
of semi-fluids such as jelly, tomato puree,
honey, soup, and concentrated fruits juices
etc. through oesophagus. These popular
edible fluids were assumed as a Casson fluid
by Pandey and Tripathi (2010a, 2011a).
They studied how such fluids swallow in the
human oesophagus by considering the
oesophagis
a
finite
long
circular
channel/cylindrical tube. The peristaltic
waves were considered to be of constant
amplitude and accordingly the results were
reported. Since it is established that the
wave amplitude increases as the bolus
moves down the oesophagus, the said
motion is required to be reinvestigated and
modified results need to be reported. The
corresponding results will be compared with
the previous ones in the discussion section.
where 𝜏 is the shear, 𝜇 is the Casson
viscosity 𝛾 is the rate of shear strain. 𝜏0 Is
the yield stress.
The peristaltic wave model that has
amplitude increasing exponentially proposed
by Pandey et al. (2014) is being considered
the centre line, axial distance, time, semi
width of the channel, amplitude of the
𝜆
wave supposed to propagate along
the wall
to create motion as desired, wavelength,
wave velocity
2 Mathematical model
We consider food bolus of Casson fluid
whose constitutive equations are given by
𝜏 = 𝜇𝛾 + 𝜏0 𝑓𝑜𝑟 𝜏 ≥ 𝜏0
(1)
𝛾 = 0 𝑓𝑜𝑟 𝜏 ≤ 𝜏0
(2)
here to create peristaltic waves on
walls. It was given by
𝜋
ℎ 𝑥, 𝑡 = 𝑎 − 𝜙0 𝑒 𝑘 𝑥 𝑐𝑜𝑠 2 𝜆 𝑥 − 𝑐𝑡 , (3)
where
ℎ, 𝜉 , 𝑡, 𝑎, 𝜙 , 𝜆 𝑎𝑛𝑑 𝑐 respectively
denote transverse distance of the wall from
ISBN: 978-1-61804-275-0
455
Advances in Mathematics and Statistical Sciences
2.1 Analysis- We consider an axi-symmetric
𝜕𝑝
unsteady flow of a Casson fluid in a circular
cylindrical tube of length L (see Fig.1). The
governing equations of the motion of such a
flow are given by
𝜕𝑟
𝜕𝑢
𝑑𝑢
𝜕𝑝
𝑑𝑣
𝜕𝑝
𝜌 𝑑𝑡 = − 𝜕𝑥 + 𝛻. 𝜏
(4)
𝜌 𝑑𝑡 = − 𝜕𝑟 + 𝛻. 𝜏
(5)
𝜕𝑢
1 𝜕(𝑟 𝑣)
+ 𝑟 𝜕𝑟 = 0
(6)
where 𝜌, 𝑢, 𝑝, 𝜏 , 𝑣 𝑎𝑛𝑑 𝑟 are respectively
fluid density, axial velocity, pressure, shear
stress, transverse velocity and transverse
distance and
𝜕
𝜕
1 𝜕
𝜕
𝜕
≡ 𝜕𝑡 + 𝜕𝑥 + 𝑟 𝜕𝑟 ,
𝛻. ≡ 𝜕𝑥 + 𝜕𝑟
Introducing the following non-dimensional
quantities
𝑥
𝑟
ℎ
ℎ = 𝑎 ,𝝉 =
𝑝=
𝒑 𝒂𝟐
𝝁𝒄𝝀
𝝉𝒂
𝝁𝒄
𝑐𝑡
𝜆
𝑢
𝝉 𝒂
𝝁𝒄
, 𝜓 = 𝜋𝑎 2 𝑐 , 𝑄 =
𝒍
𝑸
, 𝑅𝑒 =
𝝓
𝝆𝒄𝜶
𝝁
, (7)
𝐻𝑝𝑙 =
where 𝛼 is a parameter that gives the ratio
of the semi breath of the channel and
wavelength, l is the length of the channel 𝜓
is stream function , Q is volume flow rate
and 𝑅𝑒 is Reynolds number.
The length of the human oesophagus from
pharynx to stomach is 30 cm in an adult and
its diameter is about 2 cm. Thus the ratio of
the length to radios is 15:1. So, even if 3-4
waves are accommodated at a time in
oesophagus,
long
wavelength
approximation may be applied. Eqs (1-6),
𝜏0 +
𝜕𝑢
𝜕𝑟
for 𝜏 ≥ 𝜏0
𝑟=ℎ
(15)
𝑟 = 𝐻𝑝𝑙
(16)
2𝜏 0
(18)
𝜕𝑝
𝜕𝑟
𝐴(𝑡)
𝑟 𝜕𝑝
𝜏 = 𝑟 + 2 𝜕𝑥
(19)
where 𝐴(𝑡) is a function of 𝑡.
From Eqs. (8), (17) and (19) we get
𝜕𝑢
1 𝜕𝑝
= 2 𝜕𝑟 𝑟 + 𝐻𝑝𝑙 − 2 𝑟𝐻𝑝𝑙
(20)
𝜕𝑟
From Eqs. (20) and (14), 𝑢 may given by
𝑢=
𝑡
1 𝜕𝑝
4 𝜕𝑥
𝑟 + 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡 −
1 + 𝑟 + 2𝐻𝑝𝑙 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 −
1
−
3
2
8
3
3
𝐻𝑝𝑙 𝑟 2 − (𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 −
𝑡 )
(21)
Substituting 𝑟 = 𝐻𝑝𝑙 in Eq.(21), we get as
the plug flow velocity given by
(8)
𝜕𝑢
=𝑜
for
𝜏 ≤ 𝜏0
(9)
𝜕𝑟
Eqs. (4), (5) & (6) reduce to the following
dimensionless forms:
𝜕𝑝
1 𝜕(𝑟𝜏 )
=
(10)
𝜕𝑥
𝑟 𝜕𝑟
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,
2.2 Solution
Integrating Eq.(10) once with respect to 𝑟
we get
under the application of long wave length and
low Reynolds number approximations, reduce
to
𝜏=
𝜕𝑡
= 0, 𝑟 = 𝐻𝑝𝑙
(17)
𝜕𝑟
where 𝐻𝑝𝑙 is the width of plug flow region
and it is defined by
𝑎
,𝒍 = 𝝀, 𝜙 = 𝒂 ,
𝜋𝑎 2 𝑐
𝜕ℎ
𝑣 𝑥, 𝐻𝑝𝑙 , 𝑡 = 0
𝜕𝑢 𝑥,𝐻𝑝𝑙 ,𝑡
, 𝑢 = 𝑐 , 𝑣 = 𝑐𝛼 , 𝛼 = 𝜆 ,
, 𝝉𝟎 =
𝜓
𝑣
1 𝜕(𝑟𝑣)
𝑣 𝑥, ℎ, 𝑡 =
𝑑𝑡
𝑥 = 𝜆 ,𝑟 = 𝑎 ,𝑡 =
+
(11)
=0
(12)
𝜕𝑥
𝑟 𝜕𝑟
Wall Eq. (3) too reduces to
ℎ 𝑥, 𝑡 = 1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡
(13)
The boundary conditions for the modeled
problem on the axial and radial velocities are
imposed as usual, i.e., no-slip condition on the
inner wall boundary, regularity condition and
no radial velocity at the centre of the tube and
the fluid assumed to move radially with the
tubular wall. These are symbolically given as
follows:
𝑢 𝑥, ℎ, 𝑡 = 0 , 𝑟 = ℎ
(14)
𝜕𝑥
𝑑
=0
𝑢𝑝𝑙 =
1
8
3
456
1 𝜕𝑝
4 𝜕𝑥
𝐻𝑝𝑙 + 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡 −
1 + 3𝐻𝑝𝑙 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡
3
−
𝐻𝑝𝑙 𝐻𝑝𝑙 2 − (1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 −
Advances in Mathematics and Statistical Sciences
3
𝑡 )2
(22)
Eq. (21) together with Eq. (13) and the
boundary condition (16) yields the radial
velocity given by
𝑣=
1
1 𝜕𝑝
𝑟 2 − 𝐻𝑝𝑙
2 2𝑟 𝜕𝑥
2
𝑟
8
2
𝑡
𝑟𝐻𝑝𝑙
6
8
21
1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡 𝐻𝑝𝑙 −
𝐻𝑝𝑙 2 −
(23)
𝜕𝑥
2ℎ𝐻𝑝𝑙+12𝜕2𝑝𝜕𝑥2ℎ3ℎ8+
𝐻𝑝𝑙6−27ℎ𝐻𝑝𝑙−𝐻𝑝𝑙2ℎ24+ℎ𝐻𝑝𝑙2−2ℎ3ℎ𝐻
𝑝𝑙−13168𝐻𝑝𝑙2 ,
𝑟 2 − 2 1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 −
+
(24)
2𝑟 − 3 1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡
𝑟 𝐻𝑝𝑙
1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 −
𝑟
By employing the boundary condition (15),
the radial velocity at the wall yields
1 𝜕𝑝 𝜕ℎ
𝜕ℎ
ℎ 𝜕𝑡 = 4 𝜕𝑥 𝑟 2 − 𝐻𝑝𝑙 2 ℎ + 𝐻𝑝𝑙 −
2 1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡 𝐻𝑝𝑙 −
𝜕𝑥 2
𝐻𝑝𝑙 2
8
168
𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡 + 𝐻𝑝𝑙 −
𝜕2𝑝
+
1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡 + 2𝐻𝑝𝑙 −
13
1−
3
2
𝑡
3
𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 −
2𝜋𝑠𝑖𝑛𝜋 𝑥 − 𝑡 − 𝑘𝑐𝑜𝑠𝜋 𝑥 − 𝑡
𝑡
𝑡
3
−
7
𝑟 2 − 4 1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 −
Solving Eq. (24), we find The pressure
gradient is given by
𝜕𝑝
𝜕𝑥
𝑥
𝐺 𝑡 + 0 𝜋𝜙 0 𝑒 𝑘𝑠 𝑠𝑖𝑛 2𝜋 𝑠−𝑡 1−𝜙 0 𝑒 𝑘𝑠 𝑐𝑜𝑠 2 𝜋 𝑠−𝑡 𝑑𝑠
=
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 −𝑡
𝐻𝑝𝑙 2
1
8
3 1
16
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
1
12
1
7
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
where G(t) is the arbitrary function of t.
The pressure at any point along the
oesophageal length is determined by
−
5
5
2 1
− 𝐻𝑝𝑙 2
3
integrating Eq. (25) from the inlet to the
axial point, which gives
𝑥
𝐺 𝑡 + 0 𝜋𝜙 0 𝑒 𝑘𝑠 1 𝑠𝑖𝑛 2𝜋 𝑠1 −𝑡 1−𝜙 0 𝑒 𝑘𝑠 1 𝑐𝑜𝑠 2 𝜋 𝑠1 −𝑡 𝑑𝑠1
𝑥
0
3 1
16
1−𝜙 0 𝑒 𝑘𝑠 𝑐𝑜𝑠 2 𝜋 𝑠−𝑡
𝐻𝑝𝑙 2
1
8
1
12
1−𝜙 0 𝑒 𝑘𝑠 𝑐𝑜𝑠 2 𝜋 𝑠−𝑡 + 𝐻𝑝𝑙 −
13
1−𝜙 0 𝑒 𝑘𝑠 𝑐𝑜𝑠 2 𝜋 𝑠−𝑡
1−𝜙 0 𝑒 𝑘𝑠 𝑐𝑜𝑠 2 𝜋 𝑠−𝑡 +4𝐻𝑝𝑙 − 𝐻𝑝𝑙 2 −
336
1−𝜙 0 𝑒 𝑘𝑠 𝑐𝑜𝑠 2 𝜋 𝑠−𝑡
1
7
so that the pressure difference between the
two ends of the oesophagus is
ISBN: 978-1-61804-275-0
13
𝐻 2
336 𝑝𝑙
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 +4𝐻𝑝𝑙 −
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
𝑝 𝑥, 𝑡 − 𝑝 0, 𝑡 =
(25)
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 + 𝐻𝑝𝑙 −
457
5
5
1
1−𝜙 0 𝑒 𝑘𝑠 𝑐𝑜𝑠 2 𝜋 𝑠−𝑡 2 − 𝐻𝑝𝑙 2
3
𝑑𝑠
(26)
Advances in Mathematics and Statistical Sciences
𝑠
𝐺 𝑡 + 0 𝜋𝜙 0 𝑒 𝑘𝑠 𝑠𝑖𝑛 2𝜋 𝑠−𝑡 1−𝜙 0 𝑒 𝑘𝑠 𝑐𝑜𝑠 2 𝜋 𝑠−𝑡 𝑑𝑠
𝑙
0
𝑝 𝑙, 𝑡 − 𝑝 0, 𝑡 =
3 1
16
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
𝐻𝑝𝑙 2
1
8
𝑑𝑥 .
1
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 + 𝐻𝑝𝑙 −
12
(27)
13
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 +4𝐻𝑝𝑙 − 𝐻𝑝𝑙 2 −
336
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
1
7
5
5
1
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 2 − 𝐻𝑝𝑙 2
3
From which G(t) can be obtained as
𝑝 𝑙,𝑡 − 𝑝 0,𝑡 −
𝑙
0
𝑥
𝑘𝑠
0 𝜋 𝜙 0 𝑒 𝑠𝑖𝑛 2𝜋
3
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
1
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
𝐻 𝑝𝑙 2
8
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
𝐺 𝑡 =
𝑙
0
2
𝑄 + 1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
−
𝑟 2 1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 −𝑡
+2𝐻𝑝𝑙
2
16
21
1
7
−
3 1−𝜙 0
2
𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋
2 +84
−13𝐻𝑝𝑙
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 −𝑡 𝐻 𝑝𝑙
21
3𝜙 2
8
𝐻2
𝑝𝑙
𝑥−𝑡 +4𝐻𝑝𝑙 −
168
+ 𝑟2
(31)
×
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 𝐻𝑝𝑙 −
2
× 3(1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 )2 −7𝐻𝑝𝑙
𝑄𝜓 𝑥 = 𝜓 + 𝑟 2 (𝜓, 𝑥)
The stream function 𝜓𝑤 at the wall is
(33)
We average the above equation for one cycle
to obtain to obtain the average reflux rate
(32)
𝑄𝜓 = 𝜓 +
The reflux flow rate 𝑄𝜓 (𝑥) is defined as
ISBN: 978-1-61804-275-0
−
2
3
3
42 1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
𝜓𝑟=𝐻 = 𝑄 − 1 + 𝜙 −
×
𝑟 2 1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 −𝑡
7
4
24
2 1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 −𝑡
𝑟3
3
3𝜙 2
8
−1+𝜙−
𝐻𝑝𝑙 𝑟 2 − 𝑟 2 ((1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 ))2
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 −𝑡
(28)
𝑋 = 𝑥 − 𝑡, 𝑅 = 𝑟, 𝑈 = 𝑢 − 1, 𝑉 = 𝑣,
𝑞 = 𝑄 − 𝑟2 , 𝜓 = 𝜓 − 𝑟2
(30)
where the parameters on the left side are in the
wave frame and those on the right side are in the
laboratory frame. Using Eq. (29) and the
transformations (30), the stream function, in the
dimensionless form, is derived as
where 𝜓, 𝑈, 𝑉 , 𝑋 and 𝑅 are the stream function,
the axial velocity, the axial coordinate, the radial
2
𝑑𝑥
𝑑𝑥
velocity and the radial coordinate respectively.
Applying the following transformation between
the wave frame and the laboratory frame, given
by
Reflux is a phenomenon inherent to peristalsis
(Shapiro et al. [22]). It refers to the retrograde
motion near the wall. Analysis related to the
estimation of reflux limit is as follows:
Dimensional form of stream function in the
wave frame is defined as
(29)
𝑑𝜓 = 2𝝅𝑅 𝑈𝑑𝑅 − 𝑉 𝑑𝑋
𝜓 = −2 ×
1
1
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 + 𝐻 𝑝𝑙 −
16
12
13
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 +4𝐻 𝑝𝑙 −
𝐻 2 −
336 𝑝𝑙
5
5
1
1
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 2 − 𝐻 𝑝𝑙 2
7
3
1
3 1
1
𝑘𝑥
2
1−𝜙 0 𝑒 𝑐𝑜𝑠 𝜋 𝑥−𝑡
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 + 𝐻 𝑝𝑙 −
16
12
13
1
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 +4𝐻 𝑝𝑙 −
𝐻 2 −
𝐻 𝑝𝑙 2
336 𝑝𝑙
8
5
5
1
1
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 2 − 𝐻 𝑝𝑙 2
7
3
2.3 Reflux limit
𝑟4
4
𝑠−𝑡 1−𝜙 0 𝑒 𝑘𝑠 𝑐𝑜𝑠 2 𝜋 𝑠−𝑡 𝑑𝑠
458
1 2
𝑟 (𝜓, 𝑥)𝑑𝑥
0
(34)
Advances in Mathematics and Statistical Sciences
𝑟 2 𝜓, 𝑥 = 1 − 𝜙0 𝑒𝑘𝑥 𝑐𝑜𝑠2 𝜋 𝑥 − 𝑡 +
𝑎1 𝜀 + 𝑎1 𝜀 2 + ⋯
(35)
Solving Eqs. (31)and (35) and comparing the
coefficient of 𝜀, 𝜀 2 … on the sides we get
𝑎1 = −1
(36)
In order to evaluate above integration using
the perturbation method ,we expand 𝑟 2 (𝜓, 𝑥)
in power series in term of a small parameters
𝜀 about the wall as
2
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 − 𝐻𝑝𝑙
3𝜙 2
𝑄 + 1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 −1+𝜙 −
8
1−𝜙 0 𝑒 𝑘𝜉 𝑐𝑜𝑠 2 𝜋 𝜉−𝑡
1
𝑎2 = − ×
8
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
3
3 1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 +4𝐻𝑝𝑙 −
24
2
42 1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 𝐻 𝑝𝑙
21
3 1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
The reflux condition is
𝑄
> 1, as 𝜀 → 0,
ζnum =
0
×
2
−7𝐻𝑝𝑙
3𝜙 2
8
−
𝜁𝑛𝑢𝑚
𝜁𝑑𝑒𝑛
,
(38)
2
2
1 − 𝜙0 𝑒 𝑐𝑜𝑠 𝜋 𝑥 − 𝑡
1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡
24
𝑘𝑥
42 1 − 𝜙0 𝑒 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡
×
where 𝜻𝒏𝒖𝒎 and 𝜁𝑑𝑒𝑛 are integration as given
below:
which gives the reflux limit for the Casson
fluid as
1
2
𝑄 <1−𝜙+
𝑘𝑥
168
(37)
2 +84 1−𝜙 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡 𝐻
−13𝐻𝑝𝑙
0
𝑝𝑙 −
2 1−𝜙 0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥−𝑡
𝑄𝜓
𝐻2
𝑝𝑙
3
1 − 𝜙0
𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋
𝑥−𝑡
− Hpl
d𝑥
2
Hpl
3 1 − 𝜙0
𝑥 − 𝑡 + 4Hpl − 168 ×
2
2
− 13Hpl
+ 84 1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡 Hpl −
𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋
3 1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡
2
2
− 7Hpl
×
2(1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡 ) (1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡 )Hpl
21
and
2
1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡
1
𝜁𝑑𝑒𝑛 =
0
− 𝐻𝑝𝑙
𝑑𝑥
1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡
1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡
24
𝑘𝑥
42 1 − 𝜙0 𝑒 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡
3
2
𝐻𝑝𝑙
+ 4𝐻𝑝𝑙 − 168 ×
2
2
− 13𝐻𝑝𝑙
+ 84 1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡 𝐻𝑝𝑙 −
3 1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡
3 1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡
2
2
− 7𝐻𝑝𝑙
×
2(1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡 ) (1 − 𝜙0 𝑒 𝑘𝑥 𝑐𝑜𝑠 2 𝜋 𝑥 − 𝑡 )𝐻𝑝𝑙
21
position versus local pressure. The case
considered is free pumping, which excludes
impacts of additional pressure other than that
generated by propagating peristaltic waves,
3 Results and discussions
In order to analyse the physical impacts of
exponentially rising amplitude of the peristaltic
waves on Casson fluid, we plot graphs for axial
ISBN: 978-1-61804-275-0
459
Advances in Mathematics and Statistical Sciences
across the oesophageal length. This is achieved
simply by considering zero pressure at the two
ends of the oesophagus. Particularly, when the
fluid swallowed is not Newtonian, which is a
very ideal concept, only one bolus moves
practically at a time in the oesophagus. Fluids
such as water may be considered to be almost
Newtonian in nature. But Casson fluid which
possesses a plug flow region behaves closely to
many chewed food substances. For this reason,
one bolus swallowing has been considered for
pictorial demonstration required for discussion.
Effect of exponentially increasing amplitude
By plotting graphs we intend to examine the
effects of various parameters on the flow
dynamics of food bolus in the oesophagus. The
primary concern is how pressure varies along
the axis of the oesophagus when a bolus travels
down the cardiac sphincter. In order to observe
this through computer simulation of the
mathematical model, we wrote a program to
determine pressure by setting ∅ = 0.7, 𝐻𝑝𝑙 =
0.01, 𝑘 = 0.0 − 0.03 . The values of all the
dimensionless parameters are merely suitable
assumptions
required
for
qualitative
examination.
Temporal variable 𝑡 has been
varied from 0-3 with an interval of 0.25 (0
indicating the initiation) during which the bolus
moves along the oesophageal length which is
four times the size of the bolus. The length of
the oesophagus is considered to be confined
within one peristaltic wavelength; and so
appears at different positions for different
temporal values. This was merely an
assumption that at the most 4 boluses are
accommodated in the oesophagus at a time. Of
course, this can vary. Since this is a qualitative
investigation, no physiological data have been
collected to substantiate the assumptions. The
observations depicted in Figures 2 are as
follows:
We consider that the bolus has already been
pushed into the oesophagus and it is confined
within a wave at 𝑡 = 0 as given in Figure 1(a).
The pressure falls sharp at its tail, which is
required for the bolus to travel down. But
finally, pressure rises to zero towards the end to
prevent any retrograde flow. It is observed that
pressure falls more for exponentially increasing
ISBN: 978-1-61804-275-0
wave amplitude than that for constant
amplitude wave. The depression increases with
the parameter 𝑘, i.e., higher the rate of increase
of amplitude, larger is the depression of
pressure.
But at 𝑡 = 0.25 , displayed in Figure
2(b), the bolus has travelled some distance.
Pressure increases from zero at the entrance
towards the tail of the bolus. This is required to
restrain the bolus from any possible backward
flow. But as soon as the tail is touched, it falls
sharp to facilitate flow of the bolus toward the
end of the oesophagus and finally rises to zero
as in the previous case. Pressure along the axis
of the bolus and within it is almost the same.
However, it is observed that here itself the
pressures corresponding to exponentially rising
wave amplitudes of different magnitudes
becomes more distinct.
Pressure rises further at the tail of the
bolus as it covers more distance (see Figure
2(c)), when 𝑡 = 0.50 . Pressure dips in the
similar manner to facilitate its motion towards
the hiatus, and then rises to zero at the cardiac
sphincter where the stomach starts waiting to
grab the bolus. The minimum pressures
corresponding to all values of 𝑘 are
comparatively high. A similar trend is observed
at 𝑡 = 0.75 as well as at 𝑡 = 1.00. However,
unlike the cases of peristaltic waves with
constant amplitudes, the difference between
the maximum and minimum pressures becomes
larger when amplitudes are increasing
exponentially. The quantitative differences are
more significant when the rate of increase of
the amplitude is more, i.e., 𝑘 is more.
With 𝑡 = 1.00, one cycle completes. If
the amplitude remains the same, a repetition of
the previously described event takes place. In a
sharp contrast to that the peristaltic waves
which gradually get mightier increase the
pressure immensely. With a meager
exponential growth 1%-3%, pressure increases
almost by 250% after the bolus undergoes 3
temporal cycles.
Pressure distributions
corresponding to 𝑡 = 1.25, 2.00 & 3.00 (see
Figures 1 (f-h)) have also been displayed to
show the rising trends of pressure, the
significance of exponential growth in amplitude
460
Advances in Mathematics and Statistical Sciences
when a bolus of Casson fluid is swallowed in the
oesophagus.
Effect of yield stress on flow dynamics
Another important aspect required to be
investigated is the effect of yield stress on the
flow dynamics of peristaltic transport when the
amplitude increases exponentially. The plug
flow region which is due to yield stress is the
only factor that contributes towards the nonNewtonian nature of the fluid named and
considered here as Casson fluid. That is the
thicker the plug flow region more is the nonNewtonian character present in the fluid.
For this too, we assume that the
oesophagus can accommodate four boluses at a
time. However, the presence of only one bolus
is at a time is the normal feature of an
oesophagus. The plug flow region is considered
to be thick of three different magnitudes 0.0,
0.1 and 0.2. Absence of plug flow region, i.e.,
plug flow thickness = 0, makes the fluid
Newtonian. This is all for qualitative analysis.
For comparisons, only four temporal values
𝑡 = 0.0, 1.0, 2.0, 3.0
are
taken
under
consideration. The other parameters that don’t
vary are considered to be ∅ = 0.7, 𝑘 = 0.01.
Graphs are plotted between pressure and axial
coordinates.
It is observed that for 𝑡 = 0.0, thicker
the plug flow region more is the fall in pressure.
When 𝑡 = 1.0, 2.0, the bolus is midway and
when 𝑡 = 3.0, it is at the end. Behind the bolus
the pressure is positive so that it does not move
in the backward direction. It is observed that in
all such cases, pressure corresponding to
thicker plug flow region, pressure required is
more. In previous investigations for constant
amplitude, similar was observation. But in sharp
contrast to that the wave amplitude increases
exponentially and as a consequence, pressure
too increases as the bolus progresses in the
oesophagus.
Reflux region
Flow rate generally enhances when wave
amplitude is increased. Shapiro et al ()
discovered retrograde motion for flow rates less
than a limit corresponding to a given wave
amplitude. This limit was termed as reflux limit
beyond which no reflux is possible. The analysis
ISBN: 978-1-61804-275-0
for large amplitude and high flow rates has
been carried out. Obviously, the wave
amplitude is then close to the tube radius. In
such a case, some fluid flows in the opposite
direction near the tubular wall. Consequently,
the amount of flow diminishes. For small and
large amplitudes, Shapiro et al () used different
perturbation approaches to estimate the limits.
Surprisingly, the two formulations were in a
good agreement for all amplitudes. Hence, the
same result has been applied for all amplitudes.
Diagrams are then drawn to show respectively
the impact of thickness of plug flow region
when the damping parameter, 𝑘, of the wave
amplitude is kept constant (Fig.(4)) and that of
damping parameter if the plug flow thickness is
kept unchanged (Fig.(5)).
It is observed that the curves representing
reflux limits for Casson fluids are higher
compared with that of Newtonian fluid for low
flow rates. The curves rise higher as plug flow
region widens. However, for particular
amplitude near the wall, all coincide and
beyond that the tendency is reversed, i.e., the
curves that rose are observed to descend. In
other words, reflux limit, i.e., the maximum
flow rate prone to reflux, is higher for Casson
fluid than that for Newtonian fluid; but for high
flow rates, it is lower for Casson fluid.
In order to examine the role of rising
amplitude, the plug flow thickness is kept
constant and then damping parameter is varied.
It is observed that reflux limit rises as wave
amplitude is augmented with the damping
parameter of higher magnitude. The trends
don’t alter with flow rates. This can be viewed
in Fig. 5. This is an indication that peristalsis
with exponentially increasing wave amplitude is
more prone to reflux.
4 Conclusion and physical
interpretation
Unlike constant amplitude case, In the case of
exponentially increasing amplitude, pressure is
not uniformly distributed for all cycles. Pressure
keeps increasing along the entire length of the
oesophagus; and finally towards the end of the
oesophageal flow, pressure increases quite
461
Advances in Mathematics and Statistical Sciences
significantly probably to ensure delivery into
the stomach through the cardiac sphincter. This
observation is the same for Newtonian as well
as non-Newtonian fluids. But the distinction
remarked is that Casson fluids need more
pressure and hence more efforts by the
oesophagus to be transported forward.
The plug flow region which represents
the non-Newtonian nature of Casson fluids
plays very significant role in reflux. At lower
flow rates which correspond to smaller wave
amplitude, reflux occurs for much higher flow
rates in comparison to Newtonian fluid. This
tendency gradually diminishes with increasing
amplitude. For a particular value of amplitude,
there is no difference; and beyond that the
trends are quite opposite. Thus near the wall,
i.e., when the flow rates are very high, Casson
fluid is less prone to reflux.
It is also concluded that for both the
Newtonian and non-Newtonian Casson fluids,
reflux is more likely to occur with increasing
amplitude and further augmented by the
addition of amplifying damping parameter.
References
[8] S. K. Pandey and Dharmendra Tripathi,
A mathematical model for swallowing of
concentrated fluids in oesophagus, Applied
Bionics and Biomechanics 8 (2011a) 309–
321.
[9] S. K. Pandey, D. Tripathi, Unsteady
peristaltic flow of micro-polar fluid in a
finite
channel,
Zeitschrift
für
Naturforschung 66a (2011b) 181-192.
[10] Dharmendra Tripathi, S. K. Pandey, O.
Anwar Bég c Mathematical modelling of
heat transfer effects on swallowing
dynamics of viscoelastic food bolus through
the human oesophagus, International
Journal of Thermal Sciences 70 (2013) 4153.
[1] Kahrilas P. J., Wu, S., Lin, S.,
Pouderoux, P. 1995, Attenuation of
esophageal shortening during peristalsis
with hiatus hernia. Gastroenterology 109,
1818.
[2] Xia F., Mao J. , Ding J., Yang H., 2009,
Observation of normal appearance and wall
thickness of esophagus on CT Images.
European Journal of Radiology 72, 406-11.
[3] S. K. Pandey, Gireesh Ranjan,
Shailendra Tiwari, 2014, A mathematical
model for swallowing in oesophagus
generating high pressure near distal end,
(communicated).
[4] Meijing, Li, Brasseure, J. G., 1993, Nonsteady peristaltic transport in finite-length
tubes. Journal of Fluid Mechanics 248, 129151.
[5] Misra, J.C., Pandey, S.K., 2001, A
mathematical model for oesophageal
swallowing of a food-bolus, Mathematical
and Computer Modeling 33, 997-1009.
[6] S. K. Pandey and Dharmendra Tripathi,
Peristaltic transport of a Casson fluid in a
finite channel: application to flows of
concentrated
fluids
in
oesophagus,
International Journal of Biomathematics,
Vol. 3, No. 4 (December 2010a) 453–472.
[7] S.K. Pandey, D. Tripathi, Peristaltic flow
characteristics of Maxwell and Magnetohydrodynamic fluids in finite channels,
Journal of Biological Systems 18 (2010b)
621-647.
ISBN: 978-1-61804-275-0
c
H
a
𝛌
Figure1. This diagram shows propagation along
the wall of the oesophagus of peristaltic waves
whose amplitude is amplified exponentially by a
damping parameter. C, a, h and 𝛌 respectively
represents wave velocity, radius of the tube, wall
displacement due to wave propagation and wave
amplitude
462
Advances in Mathematics and Statistical Sciences
.
50
50
(a) t=0.00
0
𝛥p
-50
0
0
1
2
3
4
𝛥p
0
1
2
3
4
-50
-100
-150
-100
k = 0.0
k = 0.01
k = 0.02
k = 0.03
𝑥
-200
𝑥
-200
100
(c) t = 0.50
1
2
3
𝛥p 0
4
-50
-50 0
-100
-100
k = 0.0
k = 0.01
k = 0.02
k = 0.03
-150
-200
(e) t =1.00
t = 1.00
50
0
0
k = 0.0
k = 0.01
k = 0.02
k = 0.03
-150
50
𝛥p
(b) t=0.25
-150
-200
1
2
3
4
k = 0.0
k = 0.01
k = 0.02
k = 0.03
-250
50
(d) t = 0.75
100
0
𝛥p
0
1
2
3
50
4
-50
𝛥p 0
-50 0
-100
-150
-200
-100
k = 0.0
k = 0.01
k = 0.02
k = 0.03
-150
-200
-250
ISBN: 978-1-61804-275-0
1
463
k=0
k = 0.01
k = 0.02
k = 0.03
2
3
4
Advances in Mathematics and Statistical Sciences
50
200
3 (a) t=0
(g) t=2.00
150
0
100
𝛥p
𝛥p
50
0
-50 0
-100
-150
1
k = 0.0
k = 0.01
k = 0.02
k = 0.03
2
400
300
3
4
2
3
4
Hpl = 0.0
-150
Hpl = 0.01
Hpl = 0.02
-250
100
(h) t = 3.00
k = 0.0
k = 0.01
k = 0.02
k = 0.03
3(b) t = 1.0
50
𝛥p
200
0
-50
100
0
-100
0
-100 0
1
-200
600
𝛥p
0
-100
-200
500
-50
1
2
3
-150
4
1
2
3
4
Hpl = 0.0
Hpl = 0.01
Hpl = 0.02
-200
Figure 2. The diagrams show the pressure
distribution when peristaltic waves with
exponentially increasing amplitude propagate
along the oesophageal length. Pressure
distribution for four different damping
parameters ( 𝑘 = 0 − 0.03)are shown in the
diagrams. 𝑘 = 0 is a special case, which
represents constant wave amplitude. The eight
diagrams correspond to eight different temporal
values required for observation and discussion
about spatial and temporal effects on pressure.
ISBN: 978-1-61804-275-0
200
3(c) t=2.0
150
100
𝛥p 50
0
-50 0
-100
-150
464
1
2
Hpl = 0.0
Hpl = 0.01
Hpl = 0.02
3
4
Advances in Mathematics and Statistical Sciences
0.4
300
Hpl = 0.0
Hpl = 0.01
Hpl = 0.02
250
𝛥p
200
𝑄
3(d) t = 3.0
0.3
0.2
150
100
Reflux
region
No Reflux
region
0.0
RefluxK=Region
0.1
0.2
0.3
0.1
50
0
0.2
-50 0
1
2
3
Hpl=0.00
0.01
0.02
0.03
𝑄
0.2
No Reflux
Region
0.1
Reflux Region
0.2
0.4
0.6
∅
0.8
1.0
Figure 4. The diagram exhibits the relation
between time-averaged flow rate and wave
amplitude. Different lines represent reflux
limit corresponding to plug flow region. The
damp parameter is fixed as K=.01, whereas
the radius of plug flow region Hpl has been
varied from 0.0- 0.03.
ISBN: 978-1-61804-275-0
0.6
0.8
Figure 5. The diagram exhibits the relation
between time-averaged flow rate and wave
amplitude. Different lines represent reflux
limit corresponding plug flow region. The the
radius of plug flow region Hpl is fixed as 0.03,
whereas the damping parameter 𝑘 has been
varied from 0.0- 0.3.
Figure 3.The diagrams show the pressure
distribution when peristaltic waves with
exponentially increasing amplitude propagate
along the oesophageal length. Pressure
distribution for three different plug flowregion
(𝐻𝑝𝑙 = 0 − 0.03 are shown in the diagrams.
Hpl = 0 is a special case, which represents
Newtonian nature of fluid. The four diagrams
correspond to four different temporal values
required for observation and discussion about
spatial and temporal effects on pressure.
0.3
0.4
∅
4
465
1.0