Variable-Properties-Of-Mhd-Third-Order-Fluid-With-Peristalsis.pdf

Results in Physics 6 (2016) 963–972

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Results in Physics
journal homepage: www.journals.elsevier.com/results-in-physics

Variable properties of MHD third order fluid with peristalsis
T. Latif a,⇑, N. Alvi a, Q. Hussain a, S. Asghar a,b
a
b

Department of Mathematics, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

a r t i c l e

i n f o

Article history:
Received 4 October 2016
Received in revised form 2 November 2016
Accepted 6 November 2016
Available online 12 November 2016
Keywords:
Peristaltic flow
Third order fluid
Temperature dependent properties
Viscous dissipation

a b s t r a c t

This article addresses the impact of temperature dependent variable properties on peristaltic flow of third
order fluid in a symmetric channel. The MHD fluid and viscous dissipation effects are taken into account.
Assumptions of long wavelength and low Reynolds number are employed to model the problem. The governing nonlinear coupled equations are solved using perturbation method. Approximate solutions are
obtained for the stream function, temperature and pressure gradient. The results are graphically analyzed
with respect to various pertinent parameters.
Ó 2016 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://
creativecommons.org/licenses/by-nc-nd/4.0/).

Introduction
Peristaltic flow is of great interest for engineers, mathematicians and biologists alike. The popularity lies in its wide range of
applications in both physiology and industry. Peristaltic transport
is the movement of biological fluids within the hollow vessels that
successively contract and expand. It is responsible for the movement of food through the digestive tract, urine from kidney to
bladder, blood through arteries, bile juice in bile duct, sperm
through vas deferens and eggs in the fallopian tube. Peristalsis
was first discussed by Bayliss and Starling [1] in medical terms
in 1899. They described it as the motility in which there is both
contraction and relaxation which propels the bolus of food through
esophagus and intestines. Mathematical analysis of peristalsis was
started long after it was done in physiology. Latham [2] provided
the engineering analysis of peristaltic action for the first time in
1966. Shapiro [3] carried on Lathams work and investigated on
renal peristaltic flows in infants. Subsequently significant developments in peristaltic transport investigations were made by Fung
[4], Barton and Raynor [5] and Weinberg et al. [6]. Peristaltic techniques are of practical significance in the industry, bioengineering
and medical devices. Using this mechanism, several industrial peristaltic pumps like roller, hose, finger and blood pumps, heart-lung
machines and dialysis machines have been designed. Peristalsis
has been the subject of many recent research works as mentioned
in the references [7–11].

⇑ Corresponding author.

Biological and industrial fluids are non-Newtonian in nature.
Newtonian fluids can be considered while studying peristalsis in
the ureter but this approach is not good for peristaltic flow analysis
in blood vessels, lymphatic vessels, intestine, reproductive tracts
and esophagus. Numerous researchers are now engaged in studying peristalsis in non-Newtonian fluids particularly viscoelastic fluids because of its applications in physiology, engineering, medical
science and industry[12–18]. Blood and almost all semi-solid edible things like bread, jam and yogurt possess both viscous and elastic properties [19,20]. Third order fluid is a sub-class of viscoelastic
fluids and it comprehensively represents the properties possessed
by non-Newtonian fluids. Continuum physicists and mathematicians have keen interest in third order fluid model. Study of third
order fluid flow problems provides a good deal of physical insight
about elastic effects and develops the better understanding of
mathematical procedures required for coping with nonlinear viscoelastic problems. The ordered fluids provide good insight into
some problems that cannot be studied by generalized Newtonian
fluids or by linear viscoelastic models. The third order fluid model
has been used to examine flows in a wide variety of systems such
as flow in eccentric annuli, flow in a channel, flow in a converging
tube, radial flow between parallel disks, boundary layer flows and
the motion of suspended, orientable particles. Few researchers discussed the peristaltic action on third order fluid and are mentioned
in [21–23].
Several investigators studied the effects of MHD (magnetohydrodynamics) on peristaltic flows due to its importance in industry
and medical sciences. Such considerations have played key role in
the design of MHD power generators, solidification processes of

E-mail address: (T. Latif).
/>2211-3797/Ó 2016 Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license ( />

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T. Latif et al. / Results in Physics 6 (2016) 963–972

metals, cancer therapy, magnetic resonance imaging (MRI), petroleum industry and nuclear industry. Recent work on MHD flows is
cited in [10,12,14,16,24]. Partial slip effect in peristaltic flow of
third grade fluid is studied by Hayat and Mehmood [25].
Heat transfer is the shifting of thermal energy from a body to
the relatively cold ambient objects. It is not the physiological manifestation of the phenomenon of peristaltic flow with heat transfer,
but it is the mathematical description that has harnessed its potential to be implemented in industry. Heat transfer occurs in many
engineering processes like chemical distillatory processes, channel
type solar energy collectors, heat exchangers, paper making and
food processing. The importance of studying heat exchange in biological systems is quite significant. It has obvious involvements in
hemodialysis, treatment of liver cancer and lung cancer. Various
researchers [26–29] have documented the effects of heat transfer
for both Newtonian and non-Newtonian fluids.
Why this paper: In the existing literature, much attention has
been given to the constant fluid viscosity which is not an efficient
approach. It is acceptable to consider viscosity to be constant for
the isothermal fluids but if there is viscous dissipation, one cannot
ignore the effects of variation of viscosity due to temperature difference [30]. Its role is highly significant in peristaltic movement
of physiological fluids such as polymer solutions, honey, blood
and syrups. The viscosity of these fluids cannot remain constant
when there is temperature variation. Few attempts have been
made in which viscosity is taken as function of space coordinates
[31,32] but it is more realistic to consider the temperature dependent viscosity as the temperature difference exists in biological
systems. In addition to that temperature dependent thermal conductivity is also another important fragment that must be

accounted for. Hence in the present paper, our emphasis is on
the significance of temperature dependent viscosity and thermal
conductivity.
This paper deals with the study of peristaltic flow of third order
fluid with temperature dependent viscosity and thermal

conductivity under the effect of magnetic field. In this work viscous
dissipation effects are also taken into account. The problem is
modeled in the Section ‘‘Basic equations” and the governing
equations are obtained using long wave length and low Reynolds
approximations. In Section ‘‘Problem description”, approximate
asymptotic solutions are achieved by applying perturbation
technique. Section ‘‘Perturbation solution” deals with the graphical
illustrations of various parameters. Concluding remarks are
presented in the Section ‘‘Graphical results and discussion”.
Basic equations
The basic equations governing the fluid flow and heat transfer
are expressed as follows:
Continuity equation:

  V ẳ 0;
r

1ị

Momentum equation:

dV

 s
 ỵ J  B0 ;
q  ẳr
dt

2ị

Energy equation:

dT

r
  k
 Tị;
qcp  ẳ s  L ỵ r
dt

Fig. 1. Pressure rise DP k versus flow rate Q for / = 0.4.

ð3Þ


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T. Latif et al. / Results in Physics 6 (2016) 963–972

in which V represent the fluid velocity, q is the fluid density, t is the
 VÞ is the gradient
time, d=dt is the material time derivative, Lð¼ r
 is the thermal
vector, cp is the specific heat, T is the temperature, k
conductivity of fluid, J is Joule current, and B0 is the magnetic field.
 is given by
Cauchy stress tensor s

s ¼ PI þ S;

ð4Þ

where P is the fluid pressure, I is the identity tensor, and S is the
extra stress tensor for third order fluid defined by [33]:




 ỵ b3 trA21 A1 ỵ a1 A2 ỵ a2 A21 ỵ b1 A3
Sẳ l



ỵ b2 A1 A2 ỵ A2 A1 ;

5ị

Anỵ1 ẳ

dAn
ỵ An L ỵ LT An ;
dt

@X

ỵ

@V
@Y

ẳ 0;

8ị

@U
@U
@U
q  ỵU ỵV
@t
@X
@Y
n ẳ 1; 2:

6ị

 is the fluid viscosity, A1  A3 are the first
In above equations, l
three Rivlin–Ericksen tensors whereas ai and bi ; i ẳ 1; 2; 3ị are the
material constants.

@V
@V
@V
q  ỵU þV
@t
@X
@Y

qcp

Problem description
Consider a two-dimensional channel ðH < Y < HÞ of half

width a filled with an electrically conducting third order fluid.
The channel walls are flexible and subjected to the constant temperature T w . The flow in the channel is generated when sinusoidal
waves of small amplitude b propagate on the channel walls with
constant speed c. The shape of the walls can be described mathematically as

ð7Þ

in which k is the wavelength whereas X and Y are axes along and
perpendicular to the channel walls respectively.
If U is the Xcomponent of velocity, V is the Ycomponent of
velocity, r is the fluid electrical conductivity and B0 ẳ 0; B0 ; 0ị is
signifying the applied magnetic field in the direction normal to
the flow then the Eqs. (1)–(3) in component form can be written as

@U

with

A1 ¼ L ỵ L T ;



2p
Y ẳ HX; tị ẳ a ỵ b cos
X  ctị ;
k

!
ẳ

!

@T
@T
@T
ỵU
ỵV
@t
@X
@Y

ẳ
!

@P
@X
@P
@Y
@

ỵ

ỵ

@SXX
@X
@SXY
@X

 @T

ẳ
k
@X
@X


ỵ

ỵ

!
ỵ

@SXY
@Y
@SYY
@Y
@
@Y

 rB20 U;

;

 @T
k
@Y

9ị

10ị
!
ỵ SXX

!
@U
@V
@V @U
ỵ SYY
ỵ SXY
ỵ
;
@X
@Y
@X @Y

11ị

Note the effects of induced magnetic field are not taken into
account under the assumption of negligibly small magnetic Reynolds number [15]. Also SXX ; SXY ; SYY are the components of the
extra stress tensor as already mentioned in [34].

Fig. 2. Pressure gradient dp=dx versus x for / = 0.4, Q = 0.5.


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T. Latif et al. / Results in Physics 6 (2016) 963–972

 appearing in the Eqs.

 and thermal conductivity k
The viscosity l
(8)–(11) are assumed to vary linearly with temperature and given
by the following equations [30,17]:





l ¼ l0 1  cðT  T w ị ; k ẳ k0 1 ỵ bT  T w Þ ;

ð12Þ

in which l0 and k0 are respectively the fluid dynamic viscosity and
thermal conductivity at constant temperature T ẳ T w , and c and b
are constants.
ị moving with velocity c with respect
Defining a wave frame ð
x; y


to the fixed frame X; Y; t by the transformation

x ¼ X  ct;

 ¼ Y; u
 ðx; y
Þ ¼ UX; Y; tị  c;
y


v x; yị ẳ VX; Y; tị; px; yị ẳ PX; Y; tị;

13ị

 ; v and p
 respectively represent the velocities and preswherein u
sure corresponding to the wave reference frame.
We introduce the non-dimensional variables and parameters:

x
x¼ ;
k

v


a2 p
v ¼ ; p¼
;
ckl0
dc

w
a
H
b
l
;
d¼ ; h¼ ; /¼ ; w¼ ; l¼
ca

k
a
a
l0
rffiffiffiffiffiffi
a 
r
qca
b c2
S¼
; c1 ¼ 1 2 ;
B a; Re ¼
S; M ¼
l0 0
l0 c
l0
l0 a

k
b c2
b c2
c2 ¼ 2 2 ; c3 ¼ 3 2 ; k ¼ ;
k0
l0 a
l0 a
h¼


y
y¼ ;

a

T  Tw
;
Tw

 ¼ bT w ;


u
u¼ ;
c

Pr ¼

@w
;
u¼
@y

l0 c p
k0

v

;

Br ¼

@w

;
¼
@x

l0 c2
T w k0

;

a ¼ cT w ;

where d; M, Re, ci ði ¼ 1  3Þ, Pr, Br, a;  and w are used to denote the
wave number, Hartman number, Reynolds number, material
parameters, Prandtl number, Brinkman number, viscosity parameter, thermal conductivity parameter and stream function
respectively.
After using Eqs. (13) and (14) into Eqs. (8)–(11) and then adopting the long wavelength and low Reynolds approximation (see for
detail [35]), we arrive at

2
!3 3


@p @ 4
@2w
@2w 5
@w
 M2
ð1  ahị 2 ỵ 2C
ỵ1 ;
0ẳ ỵ

2
@x @y
@y
@y
@y
0ẳ

@p
;
@y

16ị

2
!3 3
2
@2 4
@2w
@2w 5
2@ w

M
;
0 ẳ 2 1  ahị 2 ỵ 2C
@y
@y2
@y2
@y
2
!2

!4 3


2
2
@
@h
@
w
@
w
5;
ỵ 2C
1 ỵ hị
ỵ Br41  ahị
0ẳ
@y
@y
@y2
@y2

14ị

15ị

17ị

18ị

in which Cẳ c2 þ c3 Þ is the Deborah number. Note that continuity

Eq. (8) is vanished automatically and the compatibility Eq. (17) is
obtained by taking the cross differentiation of Eqs. (15) and (16).
The physical boundary conditions with respect to wave frame
are

w ¼ 0;

@2w
¼ 0;
@y2

Fig. 3. velocity u versus y for / = 0.4, Q = 1.8, x = 0.1.

@h
ẳ 0;
@y

at

y ẳ 0;

19ị


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T. Latif et al. / Results in Physics 6 (2016) 963972

w ẳ F;

@w
ẳ 1;
@y

h ẳ 0;

at

y ẳ hxị ẳ 1 þ / cosð2pxÞ:

ð20Þ

It may be noted that the present analysis in the absence of heat
transfer and constant viscosity of the fluid reduces to the problem
in Reference [34]. The dimensionless time mean flow rates Q and F
in respective fixed and wave reference frames can be related by the
relation

Q ẳ F ỵ 1;

21ị

with

Z
Fẳ
0

hxị

@w
dy:
@y

22ị

In view of perturbation technique, we write




f ẳ f 00 þ Cf 01 þ C2 f 02 þ a f 10 ỵ Cf 01 ỵ C2 f 02 ;

C < 1; a < 1;
ð24Þ

where f is any flow quantity. Using Eq. (24) in the governing Eqs.
(15), (17), (19), (20) and (23), collecting the coefficients of like powers of a and C and dropping the terms of OðC2 Þ; OðaCÞ and OðaC2 Þ,
we get a system of linear equations. Solving the resulting system for
w; dp=dx and h, we get

w ¼ a2 sinhMyị ỵ a1 y ỵ Cẵa6 sinhMyị ỵ a7 sinh3Myị
ỵa3 a4 coshMyị ỵ a5 ịy ỵ aẵa8 a14 sinhMyị ỵ a9 sinh3Myịị
ỵa8 a10 ỵ a11 coshMyịịy ỵ a13 sinhMyịịy2 ỵ a12 coshðMyÞy3

Perturbation solution



ð25Þ

The system of Eqs. (15)–(18) comprises nonlinear coupled differential equations whose closed form solution is difficult to find.
So, we construct the series solution by utilizing the asymptotic
analysis. In order to achieve this, the viscosity parameter a and
the thermal conductivity parameter  are taken asymptotically
small and of the same order of magnitude. It may also be recalled
that the viscosity parameter c and the thermal conductivity
parameter b are of the same dimension 1=T. Thus, the energy equation can be written as:

2
!2
!4 3


2
2
@
@h
@
w
@
w
5 ẳ 0:
ỵ Br41  ahị
ỵ 2C
1 ỵ ahị
@y
@y
@y2
@y2

23ị

"
3 3
dp
3 32h M  12hM ỵ 24 sinh2hMị
ẳ aBrM5 F þ hÞ
4
dx
384ðsinhðhMÞ  hM coshðhMÞÞ
#
3 sinhð4hMÞ  48hM coshð2hMÞ
þ
4
384ðsinhðhMÞ  hM coshhMịị
"
#
3
M 7 F ỵ hị 12hM  8 sinh2hMị þ sinhð4hMÞÞ
C
4
16ðsinhðhMÞ  hM coshðhMÞÞ
þ

M3 ðF þ hÞ coshðhMÞ
sinhðhMÞ  hM coshðhMÞ

Fig. 4. Temperature profile h versus y for / = 0.4, Q = 1.8, x = 0.1.

ð26Þ

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T. Latif et al. / Results in Physics 6 (2016) 963972




h ẳ b1 2M2 h  yịh ỵ yị  cosh2hMị ỵ cosh2Myị

Z

DP k ẳ


ỵC b2 b3 y2 ỵ b2 b4 ỵ hMb5 cosh2Myị ỵ 5coshMh ỵ 4yịị

0

1

 
dp
dx:
dx

28ị

The heat transfer coefficient Z at y ẳ hxị is shown below:

ỵ5coshMh  4yịịị ỵ 45sinhMh ỵ 2yịị ỵ sinhM3h ỵ 2yịị

@h @h
:
@x @y

ỵ5sinhMh  2yịị ỵ sinhM3h  2yịịị

Zẳ

5sinhMh ỵ 4yịị  5sinhMh  4yịịị


2
2
ỵb2 y 48hM sinhMh ỵ 2yịị ỵ 48hM sinhMh  2yịị

Graphical results and discussion

ỵ48M coshMh ỵ 2yịị  48M coshMh  2yịịị
ỵaẵb6 b10 ỵ hMb11 coshMh ỵ 2yịị ỵ coshMh  2yịịịb12 cosh2Myị
12coshM3h ỵ 2yịị ỵ 9coshMh þ 4yÞÞ  12coshðMð3h  2yÞÞ
þ9coshðMðh  4yÞÞÞ þ 36sinhðMðh þ 2yÞÞ þ 36sinhðMð3h þ 2yÞÞ
9sinhðMðh þ 4yÞÞ þ 36sinhðMðh  2yịị ỵ 36sinhM3h  2yịị
9sinhMh  4yịịị ỵ b6 b7 y4  b6 y2 b8 ỵ 16M2 18sinhMh ỵ 2yịị
ỵsinhMh  2yịịịị  b6 y2 18hMcoshMh ỵ 2yịị
ỵcoshMh  2yịịịịị  b6 b9 ysinh2Myị


4
b6 y3 64hM sinhMh ỵ 2yịị
4

64hM sinhMh  2yịị  64M 3 coshMh ỵ 2yịị
i
ỵ64M3 coshMh  2yÞÞ
ð27Þ

The constants ai and bi appearing in the solution expressions
(25)–(27) are given in the Appendix A.
The dimensionless pressure rise per wavelength is evaluated by
the definition

ð29Þ

The effect of various parameters on pressure rise per wavelength, pressure gradient, velocity, temperature, heat transfer coefficient and streamlines are discussed here. Pertinent parameters
adhered to are a (viscosity parameter and thermal conductivity
parameter), Br (Brinkman number), C (Deborah number) and M
(Hartman number).
The most important aspect of peristalsis is pumping against the
pressure rise. To discuss this phenomenon the graphs of pressure
rise per wavelength DP k against volume flow rate Q for different
values of involved parameters are plotted in Fig. 1. The DP k versus
Q plane is divided in to different regions depending upon the algebraic signs of DP k and Q. The region, where Q < 0 and DPk > 0, is
known as the retrograde pumping region. In this region, the flow
of fluid is because of the pressure gradient and its direction is
opposite to the wave propagation. The peristaltic pumping region
occurs when Q > 0 and DPk > 0. The resistance of the pressure gradient in this region is overcome by the peristalsis of the walls and
fluid moved in the forward direction. When DPk ¼ 0 then we have
free pumping zone and the corresponding volume flow rate Q is
known as free pumping flux. In copumping region, Q > 0 and
DPk < 0, the pressure difference assists the flow due to peristalsis

of the walls. The effect of a on DPk is shown in Fig. 1a. It is impor-

Fig. 5. Heat transfer coefficient Z versus x for / = 0.4, Q = 1.8.


T. Latif et al. / Results in Physics 6 (2016) 963–972

tant to note that increasing a reduces viscosity and raises thermal
conductivity and for a ¼ 0, we attain the fluid with constant
viscosity and thermal conductivity. Fig. 1a shows that in retrograde
region, the pressure rise per wavelength decreases with the
increase in a. However its behavior is opposite in the copumping
region. The effects of Br on DPk are shown in Fig. 1b and these
are similar to that of a. Figs. 1c and 1d present the effects of C
and M respectively and it is noted that DPk increases with an
increase in these parameters in retrograde region whereas it
behaves in opposite manner in copumping region. It can be seen
that under all sorts of variations, DP k shows no deviation in the
peristaltic pumping region.
Fig. 2 illustrates the influence of a, Br, C and M on the pressure
gradient dp=dx within one wavelength i.e., x 2 ½0:5; 0:5. Also, the
pressure gradient is small at x ¼ 0, which is the wider part of the
channel and this is physically justified as fluid can easily pass without the assistance of large pressure gradient. While much greater
pressure gradient is required at the narrow part of the channel in
order to maintain the same flux of the fluid to pass through it. From
Figs. 2a and 2b, it is evident that in the narrow part of the channel,
where x 2 ½0:5; 0:2 [ ½0:2; 0:5, the pressure gradient is decreasing for increasing values of a and Br. However, no variation can be
seen in the wider part of the channel. From Figs. 2c and 2d, one can
see that pressure gradient is small for Newtonian and hydrodynamic fluid when compared respectively with third order and
hydromagnetic fluid. There is an additional comment on Fig. 2d.

969

Unlike the other three parameters, the variations in M do cause
the curves to have a visible difference even in the wider part of
the channel.
The Fig. 3 shows the impact of a, Br, C and M on velocity. The
behavior of the velocity near the channel walls is opposite to that
of the center but variation is insignificant. So, it is for this reason,
we have only emphasized the peak velocities that occur at the center of the channel. It is evident from Fig. 3a that peak velocity has a
positive correlation with a. This is quite relevant to the physical
situation because increase in a causes decrease in viscosity and
consequently the resistance to deformation due to stresses
becomes low which triggers a rise in velocity. It is also manifested
in the figure that fluids with variable viscosity and thermal conductivity have more velocity than those with constant viscosity
and thermal conductivity. The effect of Br on velocity is illustrated
in Fig. 3b, which demonstrates that fluids with higher Br values
have relatively greater peak velocities. Like a and Br, C too has a
positive correlation with the peak velocity of the fluid as is shown
in Fig. 3c. Moreover the velocity of third order fluid is high as compared to that of Newtonian fluid. Of the four parameters, only Hartman M has a negative correlation with the peak velocity of the fluid
as is depicted in Fig. 3d. The decrease in velocity is caused by the
drag effects of the Lorentz force which itself increases with M.
The temperature profiles h are plotted against y for different
values of a, Br, C and M, and are displayed in Figs. 4. We observe
from Fig. 4a that a has a negative correlation with h in the flow

Fig. 6. Streamlines for variation of a for / = 0.2, Q = 0.97, Br = 0.8, C = 0.01, M = 0.8.

Fig. 7. Streamlines for variation of Br for / = 0.2, Q = 0.97, a = 0.4, C = 0.01, M = 0.8.

970

T. Latif et al. / Results in Physics 6 (2016) 963–972

Fig. 8. Streamlines for variation of C for / = 0.2, Q = 0.97, a = 0.5, Br = 0.7, M = 0.7.

Fig. 9. Streamlines for variation of M for / = 0.2, Q = 0.97, a = 0.5, Br = 0.7, C = 0.01.

field. In fact a does not have the same correlation with viscosity as
it has with thermal conductivity. An increase in a causes a decrease
in viscosity and in turn decreases the inter-molecular forces and
viscous dissipation. Ultimately, there is decrease in temperature
h. On the other hand a has a positive correlation with the thermal
conductivity, which also explains the decrease in temperature. The
reason for such a behavior is that an increase in thermal conductivity increases the ability of the fluid to conduct heat and since the
temperature of the fluid is higher than the walls, consequently it
decreases. Hence, a decrease in viscosity and increase in thermal
conductivity decreases the temperature. The dependence of h on
Br is presented in Fig. 4b. It is a positive correlation. The reason
for such a manifestation is that the viscous dissipation causes the
conversion of kinetic energy to heat energy and obviously the temperature rises. Fig. 4c shows that C has a similar effect on h as Br
has. The dependence of h on M is shown in Fig. 4d. Rising values
of M have decremental effects on the amplitude of h because
according to Curie’s law, magnetization is inversely proportional
to temperature.
Variation of heat transfer coefficient Z at the wall y ¼ hðxÞ for
various values of sundry parameters is shown in Fig. 5. As Z is a
cyclic function, therefore, showing the plot for one wavelength is
sufficient. For the sake of demonstration, the wavelength over

the interval ½0; 1 is chosen. From these plots it is seen that Z is positive (negative) on the left (right) of the mean point x ẳ 0:5ị. The
absolute value, jZ j decreases for increasing values of a (see Fig. 5a)
and increases with Br, C and M (see Figs. 5d).

An intriguing phenomenon in transport of fluid is trapping and
is presented by sketching streamlines in the Figs. 6–9. A bolus is
enclosed by splitting of a streamline under certain conditions
and it is carried along with the wave in the wave frame. This process is called trapping. The trapped bolus is found to expand by
increasing a, Br, and C (see Figs. 6–8). However the size of bolus
decreases by the rising effects of M as shown in Fig. 9

Conclusion
In this paper, peristaltic flow of MHD third order fluid with temperature dependent viscosity and thermal conductivity along with
viscous dissipation effects are analyzed and series solutions are
obtained through perturbation method. The findings of the present
study are summarized as follows:
It is found that the effects of a (the viscosity and thermal conductivity parameter) are opposite to that of C (third order parameter) on pressure rise per wavelength and pressure gradient. a and
C cause a good variation in narrow part of channel in contrast to
wider part. Peak velocity is high for variable viscosity and thermal
conductivity as compare to constant viscosity and thermal conductivity. Similarly third order fluid has more velocity than Newtonian
fluid. It is found that the temperature has positive correlation with
thermal conductivity and negative correlation with viscosity. It is
found that the magnitude of heat transfer coefficient Z for a fluid
with constant viscosity and constant thermal conductivity is


971

T. Latif et al. / Results in Physics 6 (2016) 963972
4

Br2 M4 F ỵ hị

higher than that for a fluid with variable viscosity and variable
thermal conductivity. The Newtonian fluid has less heat transfer
coefficient as compared with the third order fluid. Besides that,
increasing values of a and C show increasing trend in the size of
the trapped bolus.




5
b7 ẳ  320hM coshhMị  320M 4 sinhhMị ;

Appendix A

b8 ẳ 16M2 34 sinhhMị þ sinhð3hMÞÞ  hMð4b15 þ 9 coshð3hMÞÞÞ;

FM coshðhMÞ þ sinhðhMÞ
;
hM coshhMị  sinhhMị
Fỵh
;
a2 ẳ 
hM coshhMị  sinhhMị
M 4 F þ hÞ





;



3
6hM  4h M 3 coshðhMÞ þ b20 þ hM cosh3hMị ;

b10 ẳ hM16b16 hM ỵ b22 ị ỵ 315 cosh3hMị ỵ 51 cosh5hMịị ỵ b17 ;



2
b11 ẳ 4 40h M2  33 ;



2
b12 ẳ 32hM sinhhMị 8h M2 þ 6 coshð2hMÞ þ 15 ;

3

16ðsinhðhMÞ  hM coshðhMÞÞ

5

12288ðhM coshðhMÞ  sinhhMịị

b9 ẳ 96M

a1 ẳ

a3 ẳ

b6 ẳ

;

4

a4 ẳ 12MsinhhMị  hM coshhMịị;




2
b13 ẳ sinhhMị 12h M 2 ỵ 2 cosh2hMị ỵ 13 ;

a5 ẳ M12hM  8 sinh2hMị ỵ sinh4hMịị;




2
b14 ẳ 16h M2  25 sinh3hMị ỵ sinh5hMị ỵ b18;

a6 ẳ

a7 ẳ

M 4 F ỵ hị

16sinhhMị  hM coshhMịị
M 4 F ỵ hị

4

a15 ;

4

a16 ;




3
2
b15 ẳ 8h M 3 sinhhMị þ 3  2h M2 coshðhMÞ  3hM sinhð3hMÞ;

3

16ðsinhðhMÞ  hM coshhMịị
2

a8 ẳ

3

3

BrM F ỵ hị




2
b16 ẳ 3 4h M2 þ 9 sinhð3hMÞ þ b21 þ 9hM coshð3hMÞ;

;

4

384ðsinhðhMÞ  hM coshhMịị

a9 ẳ 3hM coshhMị  sinhhMịị;

b17 ẳ 45 sinh3hMị  27 sinh5hMị;



4
2
b18 ẳ 16 12h M 4 ỵ 18h M 2 ỵ 1 sinhhMị;




3
a10 ẳ M 32h M 3 ỵ 12hM ỵ 48hM cosh2hMị ỵ a18 ;




2
2
b19 ẳ 60  72h M 2  8 6h M2 ỵ 5 cosh2hMị;




2

a11 ¼ 12M 4h M2  2 coshð2hMÞ  3 a16 ;



3
16BrM 2 M 3 F ỵ hị
a ;
a12 ẳ 
4 16
384sinhhMị  hM coshhMịị




2
b20 ẳ sinhhMị 4h M 2  2 cosh2hMị  5 ;

a13 ẳ

24M 4 BrF ỵ hị

3
4

384sinhhMị  hM coshhMịị

a16 ;
References

a14 ẳ hM4hMa17  9 cosh3hMịị ỵ 3 sinh3hMị;
a15 ẳ sinh3hMị ỵ 3hM4hM sinhhMị  cosh3hMịị;
a16 ẳ hM coshhMị  sinhhMị;

a17 ẳ 3 sinh3hMị ỵ 2hMcoshhMị  4hM sinhhMịị;
a18 ẳ 24 sinh2hMị  3 sinh4hMị;
b1 ẳ 
b2 ẳ

BrM 2 F ỵ hị

2
2

8sinhhMị  hM coshhMịị
BrM 6 F ỵ hị

;

4
5

512hM coshhMị  sinhhMịị




4
2
b21 ẳ 2 16h M 4 ỵ 18h M2  9 sinhhMị;



4
2
b22 ẳ 28h M4 ỵ 28h M 2  6 coshhMị:

;

b3 ẳ 16M2 3hM4 coshhMị ỵ cosh3hMịị  b13 ị;
b4 ẳ b14 ỵ 2hM coshhMị7 cosh4hMị ỵ b19 ị;
b5 ẳ 85 coshhMị ỵ 3cosh3hMị  4hM sinhðhMÞÞÞ;

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