peristaltic flow of sutterby fluid in a vertical channel with radiative heat transfer and compliant walls a numerical study

Results in Physics 6 (2016) 805–810

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

Peristaltic flow of Sutterby fluid in a vertical channel with radiative heat
transfer and compliant walls: A numerical study
T. Hayat a,b, Hina Zahir a,⇑, M. Mustafa c, A. Alsaedi b
a

Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan
Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
c
School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan
b

a r t i c l e

i n f o

Article history:
Received 16 September 2016
Accepted 19 October 2016
Available online 24 October 2016
Keywords:
Mixed convection
Radiation effect
Convective conditions
Compliant walls

a b s t r a c t
Current study aims to investigate the impact of compliant walls on peristaltically induced flow of
Sutterby fluid in a vertical channel. The flow is subjected to uniform magnetic field in the transverse
direction. In addition heat transfer effects characterized by convective boundary conditions are considered. Energy equation contains heat dissipation and radiative heat transfer effects. Problem formulation
is developed by negligible inertial effects and long wavelength approximation. Shooting method based
NDSolve of the software Mathematica has been applied to evaluate numerical results of stream function,
temperature and heat transfer coefficient. We found that velocity and temperature distributions in
Sutterby fluid are greater than of viscous fluid. Radiation parameter tends to reduce the fluid temperature
inside the channel.
Ó 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND
license ( />
Introduction
Peristaltic motion has wide coverage in engineering and medical industry. Peristalsis is a process in which physiological fluid (or
biofluid) is propelled by means of sinusoidal waves advancing axially along the length of tube/channel. In physiological world, peristalsis works as vasomotion of small blood vessels, bile in bile duct,
transport of food from the mouth through oesophagus and chyme
movement in the entire gastrointestinal tract. Further, blood pump
in heart lung machine, design of several modern medical devices,
locomotion of worms and translocation of water in tall trees are
also due to peristaltic principle. In urinary track, peristalsis occurs
due to involuntary contractions of the ureter walls which transport
urine from kidneys to bladder. In esophagus, peristalsis is due to
the wave-like motions of smooth muscles which carry the food
to stomach. In view of such biomedical applications, reasonable
attention has been devoted to explore the peristaltic mechanism
of non-Newtonian fluids through planar channel. Peristalsis
through magnetohydrodynamics (MHD) has been widely
addressed research topic for several years because of its occurrence
in applications such as magnetic drug targeting, treatment of
cancerous tissues, treatment of nuclear fuel debris, blood pumps,

bleeding reduction during surgeries etc. Magnet force also serves
⇑ Corresponding author.
E-mail address: (H. Zahir).

as pump for performing cardiac operations for arterial flow. Some
recent attempts involving peristaltic motion of Newtonian and
non-Newtonian fluids under the influence of magnetic field can
be found in [1–10]. Buoyancy forces stemming from cooling/heating of surfaces alter the flow and thermal field and thereby heat
transfer characteristics of the process. The study of mixed convective flow with MHD in peristalsis has gain enormous attention due
to its extensive range of applications in industry and technological
fields. In drying technologies, hemodialysis, chemical processing
equipment, cooling of electronic devices, solar energy collectors,
heat exchangers, nuclear reactors and oxygenation, the analysis
of convective heat transfer in peristalsis cannot be undervalued.
Excellent articles related to mixed convection MHD flow have been
presented by many authors [11–21]. Further, thermal radiation
effects in the flow are vital in applications involving plasma at high
temperature, heat conduction in tissues, MHD generators, gas turbines and nuclear plants. Heat transfer by means of radiation is
also effective in laser surgery, cryosurgery, destruction of cancer
tumors, processing of fusing of metals in an electrical furnace, in
the polymer processing industry controlling heat transfer and prediction of blood flow rate.
Inspired by the promising applications mentioned above, our
main objective here is to analyze the mixed convection flow of Sutterby fluid in a planar channel having compliant wall properties.
Heat transfer due to viscous dissipation and radiation is analyzed.
The relevant equations are first modeled under adequate

/>2211-3797/Ó 2016 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license ( />

806

T. Hayat et al. / Results in Physics 6 (2016) 805–810

assumptions and then solved for numerical solutions. Graphical
results for axial velocity, temperature and heat transfer coefficient
are prepared and analyzed for broad range of embedded
parameters.
Mathematical formulation
Here peristaltic flow of incompressible Sutterby fluid in a vertical symmetric channel of half thickness d is considered. The sinusoidal wave of amplitude a and wavelength k with constant speed c
propagates axially across the flexible walls of channel in time t. Let
uðx; y; tÞ be the velocity component in the axial direction and
v ðx; y; tÞ in transverse direction. The wave shape is represented
by the following equation:



2p
y ẳ ặgx; tị ẳ ặ d ỵ a sin
x À ctÞ ;
k

ð1Þ

The extra stress tensor S for Sutterby fluid is given by

"

S¼

#m

_
l sinhÀ1 fBng
Bn_

2

A1 ;

where A; B are material constants,

ð2Þ

l the fluid dynamic viscosity,

(   
2
 2 )#  
2
mB2
@u
@u @ v
@v
@u
Sxx ẳ
ỵ
ỵ2
1
2
2

ỵ
;
11ị
2
6
@y
@x
@y @x
@x
"
(   
) #




2
2
2
l
mB2
@u
@u @ v
@v
@u @ v
ỵ
ỵ
1
2
ỵ

ỵ2
; 12ị
Sxy ẳ
@x
@y @x
@y @x
2
6
@y
"
(   
2
 2 )#  
2
l
mB2
@u
@u @ v
@v
@v
ỵ
1
2
ỵ
ỵ2
2
:
13ị
Syy ẳ
@x

@y @x
2
6
@y
@y

l

"

No slip conditions for this flow are:

u ¼ 0 at y ¼ Æg;

Convective boundary condition for heat transport are as
follows:

@T
¼ À hðT T 0 ị at y ẳ g;
@y
@T
k
ẳ hT 0 Tị at y ẳ g;
@y

r V ẳ 0;

q

3ị

dV
ẳ rp ỵ div S ỵ J B ỵ qbT gT T 0 ị;
dt

qcp

dT
ẳ kr2 T r qr ỵ L S;
dt

4ị



where k and

5ị

6ị


r are the mean absorption coefficient and Stefan-

Boltzmann constant respectively. In Eq. (6), T 4 we expand in Taylor
series about T 0 and then neglect square and higher powers of
ðT À T 0 Þ by assuming that temperature differences within the flow
are sufficiently small.
Eqs. (3)–(5) can be expressed in the component form as:

@u @ v
ỵ
ẳ 0;
7ị
@x @y


@u
@u
@u
@p @Sxx @Sxy
ẳ ỵ
q ỵu ỵv
ỵ
rB20 u ỵ qbT gT T 0 ị; 8ị
@t
@x
@y
@x @x
@y


@v
@v
@v
@p @Sxx @Sxy
ẳ ỵ
q
9ị
ỵu ỵv

ỵ
rB20 v ;
@y @x
@t
@x
@y
@y
"
#


@T
@T
@T
@T @ 2 T
@
@
qr ịx qr ịy
ỵu ỵv
ẳk
qcp
ỵ
@t
@x
@y
@x2 @y2
@x
@y



@u
@u @ v
@v
ỵ
ỵ Sxx ỵ Sxy
ỵ Syy ;
10ị
@x
@y @x
@y
where

16ị

where h is the heat transfer coefficient. and boundary condition for
compliant wall is as under:

"

#
2
@3
@3
0 @

s 3 ỵ m1
ỵd
g
@x
@t@x

@x@t 2
ẳ q

@u @Sxy @Sxx
ỵ
ỵ
ỵ qbT gT T 0 ị rB20 u at y ẳ Æg;
@t
@y
@x

ð17Þ

where s is the elastic tension in the membrane, mÃ1 the mass per
0
unit area, d the viscous damping coefficient and r the electrical
conductivity.
In order to non-dimensionalize the problem, we introduce the
following variables

w
x
y
ct
u
; xà ¼ ; yà ¼ ; tà ¼ ; uà ¼ ;
cd
k
d
k

c
2
v
g
T
À
T
d
p
0
v à ¼ ; gà ¼ ; h ¼
; pà ¼
;
c
d
T0
lck
RÃ
lc Ã
S ;
k ¼ ; Sij ¼
d ij
d
wà ¼

where q is the density, p the pressure, B ¼ ð0; 0; B0 Þ the magnetic
field strength, J the current density, bT the coefficient of thermal
expansion, g the gravitational acceleration, T the fluid temperature,
T 0 denotes the convective temperature, cp the specific heat, k the
thermal conductivity, L the velocity gradient and qr denotes the

radiative heat flux given by

4rÃ
qr ¼ rT 4 ;
3k

15ị

k

t

A1 ẳ gradVị ỵ gradVị the first Rivlin Erickson tensor and
qffiffiffiffiffiffiffiffiffi2ffi
tracA1
.
n_ ¼
2
Thus, two-dimensional flow and heat transfer of Sutterby fluid
are governed by the following equations:

ð14Þ

ð18Þ

in which the stream function in terms of velocity components is
expressed below

u¼

@w
;
@y

v ¼ Àd

@w
:
@x

ð19Þ

Taking into account the assumptions of long wavelength of the
peristaltic wave compared to half width of channel and negligible
inertial effects, we arrive at the following equations



@p @Sxy
ỵ
ỵ Grh H2 wy ẳ 0;
@x
@y

20ị

@p
ẳ 0;
@y
1 ỵ RnPrị

21ị
@2h
ỵ Brwyy Sxy ẳ 0;
@y2

22ị

Sxy ẳ f1 bw2yy gwyy ;

23ị
B20

r

0
where Gr ẳ qblT gT
is the Grashof number, H2 ¼ l the Hartman
c
Ã
lcp
3
16r
number, Rn ¼ 3kk
Ã
lcp T 0 the radiation parameter, Pr ¼ k the Prandtl
2 2

c
the Sutterby

number, Br ¼ EPr the Brinkman number and b ¼ mB
6d2

fluid parameter. The boundary conditions (14)–(17) in view of
variables (18) become

wy ẳ 0 at y ẳ ặg;

24ị


T. Hayat et al. / Results in Physics 6 (2016) 805810

@h
ỵ Bih ẳ 0 at y ẳ g;
@y

25ị

@h
Bih ẳ 0 at y ẳ g;
@y

26ị

807

"

#

@3
@3
@2
@
E1 3 ỵ E2
ỵ E3
g ẳ Sxy
2
@y
@x
@x@t
@x@t
ỵ Grh H2 wy at y ẳ ặg
3

In above equations E1 ẳ ks3dlc ; E2 ẳ

m1 cd3

27ị

3 0

and E3 ¼ dlkd2 are the
non-dimensional elasticity and damping parameters, Bi ¼ hd
the
k
Biot number and  ¼ da the amplitude ratio parameter.
Through Eqs. (20) and (21) one obtains the following:



@ @Sxr
ỵ Grh À H2 wy ¼ 0:
@y @y

k3 l

Fig. 2. Variation in velocity u for wall parameters E1 ; E2 ; E3 when  ¼ 0:2; t ¼ 0:01;
x ¼ 0:2; H ¼ 0:1; Br ¼ 0:5; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5, Gr ¼ 4, and Rn ¼ 3.

ð28Þ

Numerical results and discussion
The governing nonlinear Eqs. (22) and (28) subject to the conditions (24)–(27) have been solved for the numerical solutions by
built in function NDSolve of Mathematica. This routine is based
on the standard shooting method with fourth order Runge–Kutta
integration technique.
Velocity distribution
The results presented in Figs. 1–4 reveal variation in velocity
profile uðyÞ with the change in Grashof number Gr, compliant wall
parameters E1 ; E2 , and E3 , Hartman number H and Sutterby fluid
parameter b. As the Grashof number Gr enlarges, the velocity distribution increases inside the channel (see Fig. 1). Physically, an
increase in Gr implies larger buoyancy forces which accelerates
the flow in axial direction. Fig. 2 shows that axial velocity uðyÞ is
an increasing function of tension and mass characterizing parameters E1 and E2 whereas velocity decays when damping parameter
E3 is increased. It is understandable since fluid experiences larger
resistance to flow in case of stronger wall damping. Moreover, larger values of E1 implies a reduction in wall tension due to which
velocity increases. In Fig. 3, the behavior of magnetic field effect
on axial velocity uðyÞ is portrayed. Hartman number H is defined

as the ratio of magnetic force to viscous force. As H enlarges, the
magnetic force normal to the flow direction dominates the viscous
effect. Consequently, velocity in the axial direction decreases upon
increasing the Hartman number H. Fig. 4 shows the influence of

Fig. 3. Variation in velocity u for Hartman number H when E1 ¼ 0:04; E2 ¼ 0:01;
E3 ¼ 0:01;  ¼ 0:2; t ¼ 0:01; x ¼ 0:2; Br ¼ 0:9; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4, and
Rn ¼ 3.

Fig. 4. Variation in velocity u for Sutterby fluid parameter b when E1 ¼ 0:04;
E2 ¼ 0:01; E3 ¼ 0:01;  ¼ 0:2; t ¼ 0:01; x ¼ 0:2, H ¼ 0:1; Br ¼ 0:8; Bi ¼ 2; Pr ¼ 1:5;
Gr ¼ 4, and Rn ¼ 3.

Sutterby fluid parameter on the velocity profile uðyÞ. Flow appears
to accelerate along the axial direction when bigger values of b are
employed.
Temperature distribution

Fig. 1. Variation in velocity u for Grashof number Gr when E1 ¼ 0:04, E2 ¼ 0:01;
E3 ¼ 0:01;  ¼ 0:2; t ¼ 0:01; x ¼ 0:2; H ¼ 0:1; Br ¼ 0:5; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5, and
Rn ¼ 3.

Figs. 5–11 are sketched to observe the physical effects of
embedded parameters on temperature profile hðyÞ. From Fig. 5 it
is observed that temperature his inversely proportional to the radiation parameter Rn. Temperature is found to be maximum at the
centre of planar channel. Fig. 6 indicates that temperature


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T. Hayat et al. / Results in Physics 6 (2016) 805–810

Fig. 5. Variation in temperature h for Radiation Rn when E1 ¼ 0:03; E2 ¼ 0:02;
E3 ¼ 0:01;  ¼ 0:2; t ¼ 0:01; x ¼ 0:2, Br ¼ 0:9; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4, and
H ¼ 0:1.

Fig. 6. Variation in temperature h for wall parameters E1 ; E2 ; E3 when  ¼ 0:2;
t ¼ 0:01; x ¼ 0:2; Br ¼ 0:7; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4; H ¼ 0:1 and Rn ¼ 3.

Fig. 7. Variation in temperature h for Sutterby fluid parameter b when E1 ¼ 0:04;
E2 ¼ 0:01; E3 ¼ 0:01;  ¼ 0:2; t ¼ 0:01; x ¼ 0:2; Br ¼ 0:9; Rn ¼ 3; Bi ¼ 2; Pr ¼ 1:5;
Gr ¼ 4, and H ¼ 0:1.

Fig. 8. Variation in temperature h for Hartman number H when E1 ¼ 0:04;
E2 ¼ 0:01; E3 ¼ 0:01;  ¼ 0:2; t ¼ 0:01; x ¼ 0:2, Br ¼ 0:9; Rn ¼ 3; Bi ¼ 2; Pr ¼ 1:5;
Gr ¼ 4, and b ¼ 0:02.

Fig. 9. Variation in temperature h for Brinkman number Br when E1 ¼ 0:04;
E2 ¼ 0:01; E3 ¼ 0:01;  ¼ 0:2; t ¼ 0:01; x ¼ 0:2, H ¼ 0:1; Rn ¼ 3; Bi ¼ 2; Pr ¼ 1:5;
Gr ¼ 4, and b ¼ 0:02.

Fig. 10. Variation in temperature h for Prandtl number Pr when E1 ¼ 0:04;
E2 ¼ 0:01; E3 ¼ 0:01;  ¼ 0:2; t ¼ 0:01; x ¼ 0:2, H ¼ 0:1; Rn ¼ 3; Bi ¼ 2; Br ¼ 0:9;
Gr ¼ 4, and b ¼ 0:02.

Fig. 11. Variation in temperature h for Biot number Bi when E1 ¼ 0:04;
E2 ¼ 0:01; E3 ¼ 0:01;  ¼ 0:2; t ¼ 0:01; x ¼ 0:2, H ¼ 0:1; Rn ¼ 3; Pr ¼ 1:5; Br ¼ 0:9;
Gr ¼ 4, and b ¼ 0:02.

henlarges by increasing tension and mass characterizing parameters while damping parameter cools down the fluid inside the

channel. Fig. 7 reveals that as Sutterby fluid parameter increases,
temperature distribution increases. It means that temperature in
Sutterby fluid is larger than that in viscous fluid. In Fig. 8, it is clear
that magnetic force has important role in reducing fluid temperature within the channel. Fig. 9 preserves the behavior of Brinkman
number Br on temperature. Brinkman number Br represents the
intensity of dissipation effects. Since temperature at the wall is
controlled convectively, therefore heat transfer is only considered
due to heat dissipation effects. Due to this reason, we expect temperature to be higher in case of larger Brinkman number, as also
observed in Fig. 9. Fig. 10 shows that temperature hhas direct


T. Hayat et al. / Results in Physics 6 (2016) 805–810

809

relationship with Prandtl number Pr. Fig. 11 indicates that temperature decreases when strength of convective heating is enhanced.
Heat transfer coefficient
In Figs. 12–16, we plot heat transfer coefficient ZðxÞ for different
values of embedded parameters. Interestingly, ZðxÞ has oscillatory
profile across channel walls. Fig. 12 indicates that buoyancy forces
reduce the heat transfer rate from the channel walls. The effect of
compliant wall parameters E1 ; E2 and E3 on ZðxÞ appears to be similar to that on the temperature profile (see Fig. 13). It can be concluded through Fig. 14 that radiative heat transfer opposes the
heat flow from the channel walls. Moreover this effect is preserved
when Hartman number is varied from H ¼ 0 to H ¼ 0:9.

Fig. 15. Variation in heat transfer coefficient Z for Hartman number H when
E1 ¼ 0:04; E2 ¼ 0:01; E3 ¼ 0:01;  ¼ 0:2; t ¼ 0:01; Br ¼ 0:9; Rn ¼ 3; Bi ¼ 2; Pr ¼ 1:5;
Gr ¼ 4, and b ¼ 0:02.

Fig. 12. Variation in heat transfer coefficient for Grashof number Gr when

E1 ¼ 0:04; E2 ¼ 0:01; E3 ¼ 0:01;  ¼ 0:02; t ¼ 0:01; H ¼ 0:1; Br ¼ 0:9; b ¼ 0:02;
Bi ¼ 2; Pr ¼ 1:5, and Rn ¼ 3.

Fig. 16. Variation in heat transfer coefficient Z for Sutterby fluid parameter when
E1 ¼ 0:04; E2 ¼ 0:01; E3 ¼ 0:01;  ¼ 0:2; t ¼ 0:01; H ¼ 0:1; Rn ¼ 3; Bi ¼ 2; Pr ¼ 1:5;
Gr ¼ 4, and b ¼ 0:02.

Conclusions
In this attempt, we presented numerical solutions for peristaltically driven flow of Sutterby fluid in planar channel. Radiative heat
transfer and wall properties effects are preserved in the mathematical model. The key observations of this study are listed below:

Fig. 13. Variation in heat transfer coefficient Z for wall parameters E1 ; E2 ; E3 when
 ¼ 0:02; t ¼ 0:01; Br ¼ 0:3; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4; H ¼ 0:1 and Rn ¼ 3.

 Axial velocity is directly proportional to both Sutterby fluid
parameter b and Grashof number Gr.
 Magnetic force acting normal to the flow resists the fluid
motion due to which velocity decreases in the axial direction.
 Magnetic field effect tends to reduce fluid temperature while
heat transfer coefficient enlarges when stronger magnetic force
is employed.
 Velocity and temperature distributions increase when either
wall tension reduces or wall mass per unit area enlarges.
 The influence of wall damping parameter E3 on the solutions is
qualitatively opposite to that of E1 and E2 .
 Temperature rises and heat flux from the wall diminishes when
convective heating at the walls is intensified.
 Role of radiation parameter Rn is to reduce temperature and
wall heat flux.

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Fig. 14. Variation in heat transfer coefficient Z for Radiation Rn when E1 ¼ 0:04;
E2 ¼ 0:02;
E3 ¼ 0:01;  ¼ 0:02; t ¼ 0:01; Br ¼ 0:3; b ¼ 0:02; Bi ¼ 2; Pr ¼ 1:5; Gr ¼ 4,
and H ¼ 0:1.

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