Results in Physics 7 (2017) 139–146
Contents lists available at ScienceDirect
Results in Physics
journal homepage: www.journals.elsevier.com/results-in-physics
Computational analysis of magnetohydrodynamic Sisko fluid flow
over a stretching cylinder in the presence of viscous dissipation
and temperature dependent thermal conductivity
Arif Hussain a,⇑, M.Y. Malik a, S. Bilal a, M. Awais a, T. Salahuddin a,b
a
b
Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan
Department of Mathematics, Mirpur University of Science & Technology, Mirpur 10250, AJK, Pakistan
a r t i c l e
i n f o
Article history:
Received 28 October 2016
Received in revised form 30 November 2016
Accepted 5 December 2016
Available online 21 December 2016
Keywords:
Nanoparticles
Sisko fluid model
Stretching cylinder
Magnetohydrodynamic
Variable thermal conductivity
Viscous dissipation
Shooting method
a b s t r a c t
Present communication presents numerical investigation of magnetohydrodynamic Sisko fluid flow over
linearly stretching cylinder along with combined effects of temperature depending thermal conductivity
and viscous dissipation. The arising set of flow govern equations are simplified under usual boundary
layer assumptions. A set of variable similarity transforms are employed to shift the governing partial differential equations into ordinary differential equations. The solution of attained highly nonlinear simultaneous equations is computed by an efficient technique (shooting method). Numerical computations are
accomplished and interesting aspects of flow velocity and temperature are visualized via graphs for different parametric conditions. A comprehensive discussion is presented to reveal the influence of flow
parameters on wall shear stress and local Nusselt number via figures and tables.Furthermore, it is
observed that magnetic field provides noticeable resistance to the fluid motion while both material
parameter and curvature accelerates it. The progressing values of both Eckert number and thermal conductivity parameter have qualitively same effects i.e. they rise the temperature. Additionally, material
parameter and curvature parameter increase the coefficient of skin friction absolutely and qualitively
similar effects are noticed for Nusselt number against variations in Prandtl number and curvature parameter. On the other hand local Nusselt diminishes for larger values of Eckert number and power law index.
The present results are compared with existing literature via tables, they have good covenant with previous results.
Ó 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://
creativecommons.org/licenses/by/4.0/).
Introduction
MHD is the field of science which addresses the dynamics of
electrically conducting fluid. The key fact behind the MHD is that
the applied magnetic field induces the current, the consequences
of this phenomenon produces Lorentz force which affects fluid
motion significantly. The dynamics of electrically conducting fluids
is mathematically formulated with well-known Navier–Stokes
equations along with Maxwell equations. The nature contains
variety of MHD fluids like plasmas, salt water, electrolysis etc.
Currently, MHD is a topic of intense research due to its use in many
industrial processes like magnetic materials processing, glass manufacturing, magnetohydrodynamics electrical power generation.
Furthermore, it has applications in astrophysics and geophysics
as well, e.g. it is used in solar structure, geothermal energy extrac⇑ Corresponding author.
E-mail address: alihassan721214@gmail.com (A. Hussain).
tion, radio propagation etc. Whereas, MHD pumps and MHD flow
meters are some products of engineering which utilized the magnetohydrodynamics phenomenon. So, magnetohydrodynamics is
recently experienced a period of great enlargement and differentiation of subject matter. Alfven [1] discovered electromagneto–hy
drodynamic waves in his work. Liao [2] discussed the magnetohydrodynamics power law fluid flow over continuously stretching
sheet. Solution was computed by utilizing HAM. In this analysis,
he described effects of magnetohydrodynamics on skin friction
coefficient by considering variations in power-law index. He suggested that effects of MHD are prominent for shear thinning than
shear thickening fluids. Power law fluid flow in the presence of
MHD over stretching sheet was studied by Cortell [3]. In this problem, he calculated the numerical solution by way of shooting
method and analyzed the model for variations in power law index
and Hartmann number. He concluded that Hartmann number is a
source of decrease in both Newtonian and non-Newtonian fluids
velocity. Ishak et al. [4] investigated the incompressible flow of
http://dx.doi.org/10.1016/j.rinp.2016.12.006
2211-3797/Ó 2016 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
140
A. Hussain et al. / Results in Physics 7 (2017) 139–146
Newtonian fluid over stretching cylinder under the influence of
magnetohydrodynamics. The solution was calculated numerically
by applying Keller-Box method. They illustrated that MHD causes
decline in velocity profile while enhances the skin friction coefficient. Nadeem et al. [5] examined the flow of MHD Casson fluid
over a porous stretching sheet in three dimensions. Numerical
solution of the problem was found by applying Runge–Kutta Fehlberg numerical scheme. They proved that Hartmann number
reduces both horizontal and vertical velocity profiles, but it
increases skin friction coefficients. Akbar et al. [6] investigated
the Jeffrey fluid in small intestine under the impact of magnetic
field and found exact solution. Ellahi et al. [7] discussed the blood
flow of two-dimensional Prandtl fluid through taperd arties. The
solution was calculated analytically and variations are found
against physical parameters. Mabood et al. [8] deliberated the heat
transfer analysis of boundary layer flow of nanofluid over a nonlinear stretching sheet with aid of Runge–Kutta fifth order integration
scheme. They deduced that the applied magnetic field enhances
skin friction coefficient while decreases both Nusselt and Sherwood numbers. Rashidi et al. [9] analyzed the Darcy–Brinkman–F
orchheime fluid under the influence of transverse magnetic field
and computed the numerical solution via finite element technique.
The induced magnetic field on carbon nanotubes through a permeable channel was examined by Akbar et al. [10]. The resulting
equations are solved exactly and intersted quantities are delibrated. Malik and Salahuddin [11] examined the stagnation point
flow of Williamson fluid under magnetohydrodynamics effects
over stretching cylinder and calculated solution with shooting
method. Kandelousi and Ellahi [12] inspected the Fe3O4-plasma
nanofluid flow in a vessel under the combined influence of ferrohydrodynamic and magnetohydrodynamic. They computed numerical solution via Boltman technique and interpreted that magnetic
field subtantially affected the fluid movement. Ellahi et al. [13]
delibrated the generalized Coutte flow of Powell-Eyring fluid with
magnetohydrodynamics and slip effects and found numerical solution via pseudo-spectral collocation. Also, Malik et al. [14] discussed the flow of Powell-Eyring nanofluid over linearly
stretching sheet with the effect of normally impinging magnetic
field. The nonlinear set of resulting equations was solved with
Runge–Kutta Fehlberg method. A theoratical study on peristaltic
flow of MHD Jefferey fluid through a rectangular duct with Hall
and ion slip effects has been performed by Ellahi et al. [15]. The
influnce of magnetic field on water based nanofluids in the presence of Brownian motion was studied by Akbar et al. [16]. Khan
et al. [17] found analytic solution of the problem addressing
MHD peristaltic flow of Psedoplastic fluid along with varying viscosity and bionic effects. The heat transfer of viscous fero magnetic
fluid along with thermal radiation effects over stretching surafce
was discussed by Zeeshan et al. [18]. A numerical investigation
on MHD slip flow of nanofluid in sollar collectors was designed
by Afzal and Aziz [19]. They considered heat transfer problem
along with thermal radiation and thermal conductivity effects.
Hayat et al. [20] and Latif et al. [21] also discussed the MHD flows
of non-Newtonian fluids along with different physical situations.
Flow of non-Newtonian fluids is one of the most influential
issue in few recent years, because they have wide range of applications in many fields of daily life such as food engineering, chemical
engineering, power engineering and petroleum production. Also,
the recent advancement in industry and improved engineering
expertise, motivated researchers to explore non-Newtonian fluids
characteristics in more systematic way. As non-Newtonian fluids
have lot of varieties in nature, so numerous constitutive equations
were proposed to examine physical properties of these fluids. Most
of the non-Newtonian fluids such as lubricating greases, waterborne coatings, multiphase mixers etc. which encountered in
chemical engineering obeys the Sisko fluid model. This model
was presented by Sisko [25], he studied the properties of lubricating greases. Nadeem et al. [26] also designed theoretical model of
peristaltic motion in the endoscope by using constitutive equations
of Sisko fluid model. They computed both analytical and numerical
solutions and analyzed the pressure gradient and peristaltic
motion of Sisko fluid. In addition, they presented comparison
between Sisko and Newtonian fluid and found that Newtonian fluids have best peristaltic transference. Khan and Shehzad [27]
explored the physical features of Sisko fluid flow over linearly
stretching sheet in radial direction. The analytic solution was calculated by homotopy analysis method. The interesting quantities
were discussed by varying power law index. And showed that fluid
parameter accelerates the non-Newtonian fluid motion as well as
wall shear stress. Also, contrast of Sisko fluid with both viscous
and power law fluids was presented. The results show that friction
of wall is greater for Sisko fluid as compared to both viscous and
power law fluids. Khan and Shehzad [28] also examined Sisko fluid
flow over stretching sheet. In this investigation, velocity profile
was described under the influences of power law index and Sisko
parameter. They remarked that the power law index diminishes
velocity profiles for non-Newtonian fluids. Recently, Malik et al.
[29] discussed effects of transverse magnetic field on Sisko fluid
over stretching cylinder. Shooting technique was implemented to
solve governing equations.
The theory of boundary layer flows over stretching surfaces
accompanied by heat transfer analysis has many potential applications in practice like extrusion, paper production, fiber-glass production etc. Also, many industrial processes such as hot rolling,
condensation process, crystal growing, polymer sheets, artificial
fibers and plastic films etc. are based on heat transfer in boundary
layer flows. Thus, due to numerous applications in industrial manufacturing and engineering, the boundary layer flows are still topic
of great interest for researchers. The work of Sakiadis [30] is providing base to boundary layer flows. He analyzed the boundary
layer viscous fluid flow over continues solid surfaces in twodimension. The analysis of heat transfer problem was initiated by
Gupta and Gupta [31]. They studied the heat transfer of viscous
fluid flow over a stretching cylinder. Naseer et al. [32] formulated
the problem on tangent hyperbolic fluid flow over a exponentially
varying vertical cylinder. The calculation of solution was performed with the aid of Runge–Kutta Fehlberg technique. Since
thermal conductivity of the fluid changes as temperature varies,
so it leads to consideration of variable thermal conductivity in heat
transfer analysis. Rangi and Ahmad [33] studied the flow of viscous
fluid over a stretching cylinder along with the effect of variable
thermal conductivity. They computed numerical solution via
Keller-Box method and recommended that the temperature profile
is significantly affected due to variable thermal conductivity. Abel
and Mahesha [34] analyzed the problem of MHD viscoelastic fluid
over the stretching sheet. They considered the impacts of nonuniform heating, thermal radiation and temperature dependent
thermal conductivity. It was recognized that variable thermal conductivity increases temperature profile i.e. fluid with less thermal
conductivity are better for cooling. Abel et al. [35] also examined
the power law fluid flow over stretching sheet under the effects
of non-uniform heating along with variable thermal conductivity
and found analytical expressions for temperature profile. Recently,
effects of linearly varying thermal conductivity on MHD Sisko fluid
flow over continuously stretching cylinder was investigated by
Malik et al. [36]. The interesting quantities were computed with
shooting method to analyze problem physically.
An analysis of heat transfer with viscous dissipation effects was
firstly considered by Brinkman [41]. He investigated the effect of
viscous heating in Newtonian fluids. Lin et al. [42] discussed the
effect of viscous dissipation on thermal entrance heat transfer
region in the laminar pipe flow with convective boundary
141
A. Hussain et al. / Results in Physics 7 (2017) 139–146
conditions. The flow of incompressible viscous fluid over the porous stretching surfaces was analyzed by Anjali Devi and Ganga
[43]. The influence of MHD and viscous dissipation were discussed
in this problem. The solution was computed by hypergeometric
functions for special case when plate is stretched linearly and concluded that influence of Eckert number is insignificant on temperature. Singh [44] formulated the problem on heat and mass
transfer of MHD boundary layer flow of incompressible viscous
fluid past the moving vertical porous plate. He discussed heat
transfer with viscous dissipation and temperature dependent variable viscosity and solved the problem numerically. The viscous dissipation effects on time-dependent Newtonian flow over stretching
sheet was discussed by Reddy et al. [45]. They employed KellorBox method to find numerical solution of governing equations.
Mabood et al. [46] analyzed the impact of viscous dissipation on
MHD stagnation point flow of water based nanofluid over a flat
plate. The Runge–Kutta Fehlberg integration scheme was utilized
to compute solution of modeled differential equations.
In spite of all the aforementioned studies, the boundary layer
flow of MHD Sisko fluid with viscous dissipation and thermal conductivity has paid less attention. Motivated from above literature
and also due to various potential applications of problem in engineering and industry, authors formulated the mathematical model
for boundary layer flow of Sisko fluid in presence of magnetic field
and heat transfer problem with the effects of both variable thermal
conductivity and viscous dissipation. The modeled set of partial
differential equations are transfigured to coupled ordinary differential equations via similarity variables. For numerical solution of
similarity equations, shooting technique is employed and the problem is analyzed physically by varying governing flow parameters.
Mathematical formulation
Let assume the two dimensional, axisymmetric, steady state
flow of incompressible Sisko fluid along the continuously stretching cylinder. The flow is induced due to stretching of cylinder along
the axial direction with stretching velocity UðxÞ ¼ cx, here c > 0.
The magnetic field of strength B0 is applied on fluid along the radial
direction. The problem of heat transfer consists the effect of viscous dissipation. Additionally, thermal conductivity is considered
variable in this investigation. After implying boundary layer
approximations on governing equations, these equations are modified to
@ ðruÞ @ ðr v Þ
þ
¼ 0;
@x
@r
ð1Þ
n
n1 2
@u
@u a @
@u
b
@u
nb
@u
@ u rB20
u þv ¼
r
þ
u;
@x
@r r q @r @r
rq
@r
@r
@r 2
q
q
ð2Þ
2
nþ1
@T
@T 1 @
@T
a
@u
b
@u
þ
ar
þ
;
þv
¼
u
@x
@r
r @r
@r
@r
@r
qc p
qc p
along with presubcribed boundary conditions
u ¼UðxÞ ¼ cx;
T ¼T w
v ¼ 0 at r ¼ r0
and u ! 0 at r ! 1;
ð3Þ
at r ¼ r 0 and T ! T 1 atr ! 1:
In above set of equations, the components of fluid velocity along
x and r directions are denoted with u and v. The material constants
are n (power law index), a (high shear rate viscosity) and b(consistency index). The density of fluid particles is denoted with q and r
called the electrical conductivity of fluid. In Eq. (3), T is
temperature of the flow, T w represents temperature of fluid at wall,
T 1 is ambient temperature, cp is the specific heat and ashows the
thermal diffusivity.
The stream function W of flow velocity is defined as
u¼
1 @W
;
r @r
v¼
1 @W
:
r @x
ð4Þ
The suitable similarity transformations are defined as
g¼
r 2 r 20
2xr0
1
Renþ1
b ;
1
hðgÞ ¼ TTT
;
w T 1
1
W ¼ xr 0 URebnþ1 f ðgÞ;
ð5Þ
a ¼ a1 ð1 þ hÞ;
where is the thermal conductivity parameter, a1 is the extreme
thermal diffusivity, Reb denotes the Reynolds number which is
defined as Reb ¼ qx Ub .
After employing aforementioned similarity transformations in
the governing partial differential equations (1)–(3), the continuity
equation is satisfied identically, while the other two governing
equations are converted into the following ordinary differential
equations
n
2n
nþ1
000
00 n1 000
Að1 þ 2cgÞf þ nð1 þ 2cgÞ 2 ðf Þ f ðn þ 1Þcð1 þ 2cgÞ
2n 00
02
00
0
ff þ 2Acf Mf ¼ 0;
f þ
nþ1
n1
2
2n
ð1 þ 2cgÞðh00 þ ðhh00 þ h02 ÞÞ þ 2cð1 þ hÞh0 þ
Prf h0
nþ1
nþ1
00 2
00 nþ1
þ Að1 þ 2cgÞEcPr f
þ EcPrð1 þ 2cgÞ 2 f
¼ 0;
00 n
ðf Þ
ð6Þ
ð7Þ
boundary conditions of the problem reduce to
f ð0Þ ¼ 0;
0
f ð0Þ ¼ 1;
0
f ð1Þ ¼ 0;
hð0Þ ¼ 1;
hð1Þ ¼ 0:
ð8Þ
Curvature parameter c, magnetic field parameter M, Eckert
number Ec, material parameter A and Prandtl number Pr are
defined as
c¼
1
x nþ1
Re ;
r0 b
A¼
Renþ1
b
;
Rea
M¼
2
Ec ¼
rxB20
qUx
;
; Rea ¼
a
qU
U2
;
cp ðT w T 1 Þ
Pr ¼
xU
a
2
Re1þn
b :
ð9Þ
The physical objects which are practically important i.e. skin
friction coefficient and local Nusselt number of flow field distribution are defined in following equation
sw
xqw
and Nux ¼
:
2
kðT w T 1 Þ
q
U
2
Cf ¼ 1
ð10Þ
In above formulas skin friction coefficient is expresses with C f
while Nux shows Nusselt number. Here sw is known as wall shear
stress on the other hand qw denotes wall heat flux which are
defined below
n
@u
@u
@T
b and qw ¼ k
:
@r r¼r0
@r r¼r0
@r r¼r0
sw ¼ a
ð11Þ
By inserting similarity transformations in Eqs.(11) and (12), the
interesting physical quantities i.e. coefficient of skin friction and
Nusselt number are converted to
1
1
1
n
00
00
0
nþ1
C f Renþ1
b ¼ Af ð0Þ ½f ð0Þ and Nux Reb ¼ h ð0Þ:
2
ð12Þ
Numerical solution
As flow govern equations i.e. Eqs. (7) and (8) are highly nonlinear simultaneous set of ordinary differential equations. In order to
find solution of these simultaneous equations along with boundary
conditions Eq. (9), shooting technique along with fifth order
Runge–Kutta integration scheme is applied. Both velocity and
142
A. Hussain et al. / Results in Physics 7 (2017) 139–146
temperature profiles are discussed for different values of dimensionless complexes curvature parameter c, fluid parameter A, Hartmann number M, variable thermal conductivity parameter ,
Prandtl number Pr and Eckert number Ec. Since Runge–Kutta Fehlberg method solves only initial value problems. So first Eqs. (7) and
(8) are converted into set of first order equations. For this aim, rewrite the above set of equations as given below
000
f ¼
h00 ¼
ðn þ 1Þcð1 þ 2cgÞ
n1
2
00 n
02
00
00
2n
ðf Þ 2Acf þ f nþ1
ff þ Mf
Að1 þ 2cgÞ þ nð1 þ 2cgÞ
nþ1
2
00 n1
ðf Þ
0
;
ð13Þ
1
2n
Prf h0 þ 2cð1 þ hÞh0 þ ð1 þ 2cgÞh02
ð1 þ 2cgÞð1 þ hÞ n þ 1
nþ1
00 2
00 nþ1
þ EcPr Að1 þ 2cgÞ f
þ ð1 þ 2cgÞ 2 f
:
ð14Þ
New variables are defined in Eq. (16), which are utilized to
reduce above higher order equations into system of first order differential equations
f ¼ y1 ;
0
00
f ¼ y2 ;
f ¼ y3 ;
000
f ¼ y03 ;
h ¼ y4 ;
h0 ¼ y5 and h00 ¼ y05 :
ð15Þ
After inserting Eq. (16), into Eqs. (14) and (15), new system of
ordinary differential equations Eqs. (17)–(21), is obtained
y01 ¼ y2 ;
ð16Þ
y02 ¼ y3 ;
ð17Þ
y03 ¼
ðn þ 1Þcð1 þ 2cgÞ
n1
2
2n
ðy3 Þn 2Acy3 þ y22 nþ1
y1 y3 þ My2
nþ1
2
Að1 þ 2cgÞ þ nð1 þ 2cgÞ
ðy3 Þn1
;
ð18Þ
y04 ¼ y5 ;
y05 ¼
ð19Þ
1
2n
Pry1 y5 þ 2cð1 þ y4 Þy5
½
ð1 þ 2cgÞð1 þ hÞ n þ 1
þ ð1 þ 2cgÞy25 þ 2cy5 þ EcPrðAð1 þ 2cgÞðy3 Þ2
þ ð1 þ 2cgÞ
nþ1
2
ðy3 Þnþ1 Þ;
ð20Þ
together with the boundary conditions
y1 ð0Þ ¼ 0;
y2 ð0Þ ¼ 1;
y2 ð1Þ ¼ 0;
y4 ð0Þ ¼ 1 and y4 ð1Þ ¼ 0:
ð21Þ
This system of equations is solved with shooting method, the
following procedure is adopted;
1. Firstly limit for g1 is chosen, the best suited limit for g1 is 5.
2. Now choose appropriate initial guesses for y3 ð0Þ and y5 ð0Þ.
Initially y3 ð0Þ ¼ 1 and y5 ð0Þ ¼ 1 are selected.
3. In next step, set of ordinary differential equations is solved with
the aid of Runge–Kutta Fehlberg scheme.
4. Finally, boundary residuals (absolute variations in given and
calculated values of y2 ð1Þ and y4 ð1Þ) is calculated. The solution
will converge if absolute values of boundary residuals are less
then tolerance error, which is considered 106 .
5. If values of boundary residuals are larger than tolerance error,
then values of y3 ð0Þ and y5 ð0Þ are modified by Newton’s
method.
The above procedure is continued unless the boundary residuals
are less than error tolerance.
Results and discussion
In current investigation, the non-Newtonian Sisko fluid flow
over stretching cylinder is examined under the impact of normally
applied magnetic field. In addition, effects of viscous dissipation
and variable thermal conductivity are also considered. As modeled
set of equations having non-linearity of higher order, so numerical
technique Runge–Kutta-Fehlberg method is employed to obtain
more realistic results. The accuracy of solution is verified by comparing results with existing literature. Presently calculated values
of skin friction coefficient for variations in power law index n are
compared with previously reported data i.e. Hassanien et al. [39],
Cortell [3], Liao [2], Andersson and Kumran [38] and Abel et al.
[35] presented in Table 1. It can be observed that present results
are same with previous up to desired significant digits. Table 2
illustrates the comparison of Nusselt number i.e. h0 ð0Þ against
alteration in Prandtl number Prwith previously published results
i.e. Grubka and Bobba [37], Ishak et al. [40], Yih [22], Machireddy
[23] and Ali et al. [24]. This table shows that present results have
excellent agreement with previous results. Also, for visualizing
computed results more clearly, the variations are displayed against
single parameter in each figure (while others keeping fixed).
Fig. 1a exhibits the behavior of magnetic field parameter M on
0
velocity f ðgÞ. It is worth mentioning that the transverse magnetic
field reduces the fluid velocity inside the momentum boundary
layer. Because consequences of electrically conducting fluid
dynamics produced Lorentz force. The resisting nature of this force
0
opposes the fluid motion and hence slows down the velocity f ðgÞ.
Also, it is noticed that increment in power law index n leads to
0
decrease in velocity profile f ðgÞ.
Fig. 1b elucidates the impact of curvature parameter c on velocity profile for n ¼ 1; 2. As curvature have opponent relation with
radius of cylinder, so the consequences of increment in curvature
led to shrink the radius and surface area. Hence, fluid motion experienced less resistance which consequently escalates fluid velocity.
The presently computed results validated the above mentioned
fact.
Table 1
n
00
00
Comparison table of skin friction coefficient i.e. ½Af ð0Þ ðf ð0ÞÞ for different values of power-law index n and considering c ¼ 0, A = 0, M = 0.
n
0.2
0.4
0.5
0.6
0.8
1.0
1.2
1.4
1.5
1.6
1.8
2.0
Hassanien et al. [39]
Cortell [3]
Liao [2]
1.2730
1.16524
1.02883
1.0000
0.98737
1.0000
0.98090
1.0280
1.0000
0.9820
0.97971
0.9797
0.9800
Andersson and Kumaran [38]
Abel et al. [35]
Present Results
1.9287
1.2715
1.1605
1.0951
1.0284
1.0000
0.9874
0.9819
0.9806
0.9798
0.9794
0.9800
1.943685
1.272119
1.167740
1.095166
1.028713
1.000000
0.987372
0.981884
0.980653
0.979827
0.989469
0.979991
1.9200
1.2716
1.1640
1.0976
1.0282
1.0000
0.9888
0.9812
0.9805
0.9797
0.9791
0.9801
143
A. Hussain et al. / Results in Physics 7 (2017) 139–146
Table 2
Comparison of h0 ð0Þ by varying Pr and considering c ¼ 0; Ec ¼ 0; n ¼ 1.
Pr
Grubka and Bobba [37]
Ishak et al. [40]
Yih [22]
Machireddy [23]
Ali et al. [24]
Present Results
0.71
1.0
3.0
10.0
1
1.9237
3.7207
1
1.9237
3.7207
1
1.9237
3.7207
1
1.9237
3.7208
1.0001
1.9230
3.72028
0.8686
1.0067
1.9260
3.7267
1
1
0.9
0.9
M
M
M
M
M
M
Pr = 1, γ = 0.5,
A = 1, ε = 0.1,
Ec = 0.2,
0.8
0.7
= 0.1, n = 1
= 0.2, n = 1
= 0.3, n = 1
= 0.1, n = 2
= 0.2, n = 2
= 0.3, n = 2
0.8
0.7
0.6
f’(η)
f’(η)
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
1
2
3
η
4
5
6
A = 1, n = 1
A = 2, n = 1
A = 3, n = 1
A = 1, n = 2
A = 2, n = 2
A = 3, n = 2
Pr = 1, M = 0.3,
ε = 0.1, γ = 0.5,
Ec = 0.2,
7
0
Fig. 1a. Velocity profile f ðgÞ by varying magnetic field parameter M and power law
index n.
0
1
2
3
η
4
5
6
7
0
Fig. 1c. Variations of f ðgÞ for altering values of material parameter A and n ¼ 1; 2.
1
1
γ=
γ=
γ=
γ=
γ=
γ=
Pr = 1, Ec = 0.2,
A = 1, ε = 0.1,
M = 0.3,
0.8
0.7
0.1, n =
0.5, n =
1, n = 1
0.1, n =
0.5, n =
1, n = 2
0.8
1
1
Ec = 0.1, n = 1
Ec = 0.5, n = 1
Ec = 1, n = 1
Ec = 0.1, n = 2
Ec = 0.5, n = 2
Ec = 1, n = 2
0.7
2
2
0.6
θ(η)
0.9
0.6
f’(η)
Pr = 1, ε = 0.1,
γ = 0.5, A = 1,
M = 0.3,
0.9
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
1
2
3
η
4
5
6
7
Fig. 1b. Influence of curvature parameter c and power law index n on velocity
0
profile f ðgÞ.
Fig. 1c depicts the behavior of the fluid parameter A on velocity
0
0
profile f ðgÞ by varying power law index n. The fluid velocity f ðgÞ
enhances for large values of material parameter A, it holds physically because when A has large values viscous forces diminishes,
so it offers less resistance to fluid motion. Additionally, it increases
the boundary layer thickness. Also, this figure reflects that the
velocity profile decreases by varying power-law index n.
0
1
2
3
η
4
5
6
7
Fig. 2a. Influence of Eckert number Ec on temperature profile hðgÞ for power-law
index n ¼ 1; 2.
Fig. 2a reflects the influence of Eckert number Ec on temperature profile hðgÞ for power law index n ¼ 1; 2. Since enhancement
in Eckert number Ec rapids the advective transportation i.e. kinetic
energy. As a result fluid particles collide more frequently with each
other, the consequences of these collisions transformed kinetic
energy into thermal energy. And hence it increases temperature
profile hðgÞ which can be observed from the figure. Finally, it can
be seen that temperature profile hðgÞ decreases by increasing
power-law index n.
144
A. Hussain et al. / Results in Physics 7 (2017) 139–146
1
1
Pr = 1, A = 1, Ec = 0.2,
γ = 0.5, M = 0.3,
0.9
0.9
ε = 0.1, n = 1
ε = 0.3, n = 1
ε = 0.5, n = 1
ε = 0.1, n = 2
ε = 0.3, n = 2
ε = 0.5, n = 2
0.8
0.7
0.8
θ(η)
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1
2
3
η
4
5
6
Pr = 3, n = 1
Pr = 3, n = 1
Pr = 3, n = 1
Pr = 1, n = 2
Pr = 2, n = 2
Pr = 3, n = 2
0.7
0
7
Fig. 2b. Effects of thermal conductivity parameter and power-law index n on hðgÞ.
Fig.2b demonstrates the influence of thermal conductivity
parameter on temperature profile hðgÞ. Since enhancement in
thermal conductivity boosted the capability of heat transference,
this causes increase in the fluid temperature which can be shown
from the graph.
Fig. 2c reveals the impact of curvature parameter c on temperature profile hðgÞ. Since curvature parameter cenlarges fluid velocity and kinetic energy, as a result temperature increases because it
is well-known fact that temperature is the average kinetic energy.
Fig. 2d delineates the fluctuations in temperature profile for
variations in Prandtl number Pr. When values of Prandtl number
Pr increases it causes decrease in temperature profile hðgÞ. This is
practically true because Prandtl number Pr is ratio of momentum
diffusivity to thermal diffusivity, thus for larger values of Prandtl
number Pr thermal diffusivity decreases.
Effects of material parameter A and magnetic field parameter M
are illustrated on skin friction coefficient via Fig. 3a. Current graph
describes that material parameter A increases the skin friction
0
1
2
3
η
4
5
6
7
Fig. 2d. Impact of Prandtl number Pr on temperature profile hðgÞ for n ¼ 1; 2.
7
6.5
6
-[A f’’ (0) - (-f’’(0))n]
θ(η)
0.6
M = 0.3, A = 1,
ε = 0.1, γ = 0.5,
Ec = 0.2,
5.5
A = 1, n = 1
A = 2, n = 1
A = 3, n = 1
A = 1, n = 2
A = 2, n = 2
A = 3, n = 2
5
4.5
4
3.5
3
2.5
2
0.1
0.2
0.3
0.4
0.5
M
1
Fig. 3a. Combined effects of Sisko parameter A as well as magnetic field parameter
M on coefficient of skin friction by considering power law-index n ¼ 1; 2.
0.9
γ = 0.1, n = 1
γ = 0.5, n = 1
γ = 1, n = 1
γ = 0.1, n = 2
γ = 0.2, n = 2
γ = 1, n = 2
Pr = 1, M = 0.3,
ε = 0.1, A = 1,
Ec = 0.2,
0.8
0.7
θ(η)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
η
4
5
6
7
Fig. 2c. Temperature profile hðgÞ by varying curvature parameter c and power law
index n.
coefficient in absolute sense, it holds physically because larger values of material parameter A enhances boundary layer thickness.
Also, variations in magnetic field parameter M increases coefficient
of skin friction. Finally, this figure also shows that wall friction
coefficient has less values for n ¼ 2 as compared with values calculated for n ¼ 1.
Fig. 3b is graphical representation of variations in skin friction
coefficient by altering magnetic field parameter M and curvature
parameter c. Since, Reynolds number and skin friction coefficient
are in inverse relation as shown in Eq. (13). Also, curvature parameter chas inverse relation with Reynolds number. Thus increase in
curvature parameter cReynolds number becomes low, which alternatively enhances the skin friction coefficient values in absolute
manner.
Influence of Eckert number Ec and thermal conductivity parameter is discussed on wall temperature gradient i.e. h0 ð0Þ through
Fig. 4a. Both parameters decreases Nusselt number i.e. h0 ð0Þ. It is
true because both parameters rise fluid temperature, which
145
A. Hussain et al. / Results in Physics 7 (2017) 139–146
4
1
3.8
3.4
3.2
0.8
0.7
0.6
3
-θ’ ( 0 )
-[A f’’ (0) - (-f’’(0))n]
0.9
γ = 0.1, n = 1
γ = 0.5, n = 1
γ = 1, n = 1
γ = 0.1, n = 2
γ = 0.5, n = 2
γ = 1, n = 2
3.6
2.8
2.6
2.4
0.5
0.4
0.3
0.2
2.2
G
G
G
G
G
G
0.1
2
0
1.8
-0.1
1.6
0.1
0.2
0.3
0.4
0.5
0.1
M
0.8
0.3
0.4
0.5
0.6
Fig. 4b. Variations in heat transfer from the surface i.e. h0 ð0Þ by varying Eckert
number Ec, curvature parameter c and power law-index n ¼ 1; 2.
Table 3
Values of skin friction coefficient for different values of parameters c; M and A for
n ¼ 1; 2.
0.7
0.6
-θ ’ (0)
0.2
Ec
Fig. 3b. Influence of magnetic field parameter M and curvature parameter c on wall
shear stress for power law-index n ¼ 1; 2.
0.5
c
M
A
ðA þ 1Þf ð0Þ
Af ð0Þ f
0.1
0.
1.0
0.5
0.3
0.3
0.2
1.6455
1.9163
2.1818
2.4416
1.8265
2.3528
2.8454
3.3201
1.7640
1.8221
2.0223
2.2284
2.4396
1.9549
2.3716
2.8531
3.3594
1.8704
0.4
ε = 0.1, n = 1
ε = 0.5, n = 1
ε = 1, n = 1
ε = 0.1, n = 2
ε = 0.5, n = 2
ε = 1, n = 2
0.3
0.2
0.1
0.1
= 0.1, n = 1
= 0.5, n = 1
= 1, n = 1
= 0.1, n = 2
= 0.5, n = 2
= 1, n = 2
0.2
0.1
0.3
0.5
0.3
1
2
3
00
00
002
ð0Þ
Table 4
Nusselt number table for different values of parameters c; Ec and Pr for n ¼ 1; 2.
0.3
0.4
0.5
0.6
Ec
0
Fig. 4a. Variations of Nusselt number i.e. h ð0Þfor altered values of Eckert number
Ec, variable thermal conductivity parameter and power law-index n ¼ 1; 2.
reduces difference between surface and fluid temperature. Hence,
this decelerates the rate of heat transfer from the surface. Also, values of Nusselt number increases for power law index n.
Deviations in temperature gradient at surface i.e. h0 ð0Þ for
altered values of Eckert number Ec and curvature parameter c
are shown by Fig. 4b. This graph illustrates that Nusselt number
declines for large values of Eckert number Ec. Also, this figure illustrates that curvature parameter c causes enhancement in h0 ð0Þ,
because large values of curvature parameter creduces thermal
boundary layer.
The behavior of the physical parameters n; c; A and M on the
skin friction coefficient is analyzed via Table 3. This table shows
that fluid parameters c; A and M enlarges the skin friction coefficient absolutely for both cases i.e. n ¼ 1 and 2.
Table 4 reveals the influence of curvature parameter c, Eckert
number Ec, thermal conductivity parameter and Prandtl number
Pr on wall temperature gradient i.e. h0 ð0Þ for power law index
n ¼ 1 and 2. The curvature parameter c and Pr increases rate of
c
Pr
Ec
h0 ð0Þ
h0 ð0Þ
0.1
0.5
1.0
0.5
0.1
1
0.2
0.4740
0.6493
0.8589
0.6493
0.5922
0.5497
0.6493
0.7762
0.8966
0.6493
0.4301
0.1011
0.6327
0.7914
0.9804
0.7914
0.7145
0.6569
0.7914
1.0428
1.2593
0.7914
0.6483
0.4336
0.1
0.3
0.5
0.1
1
2
3
1
0.1
0.3
0.6
heat transfer from the surface, while the Eckert number Ecand
thermal conductivity parameter reduces the rate of heat transfer
from wall.
Concluding remarks
In current investigation, the numerical scheme for simulations
of MHD Sisko fluid model is developed to explore the dynamics
over stretching cylinder. Effects of temperature dependent thermal
conductivity and viscous dissipation are taken into account. The
modeled non-linear coupled partial differential equations are
146
A. Hussain et al. / Results in Physics 7 (2017) 139–146
transfigured into ordinary differential equations by employing selfsimilar set of transformations, which then solved by shooting
method. The main outcomes of the current observation are:
An increment in material parameter A and curvature parameter
cleads to accelerate fluid velocity while the consequences of
magnetic field parameter M on fluid motion are opponent.
Prandtl number Pr is responsible of decrease in temperature of
the fluid hðgÞ while the flow field temperature hðgÞ rise up
against flow parameters Ec; c and .
The effects of all pertinent parameters i.e. M; A and c on skin
friction coefficient are qualitatively similar but Sisko parameter
A has prominent effects in quantitative sense.
Curvature parameter c and Eckert number Ec augments the
Nusselt number while Prandtl number Pr and thermal conductivity parameter are responsible to reduce it.
Acknowledgment
Authors are thankful to Higher Education Commission (HEC) of
Pakistan.
References
[1] Alfven H. Existence of electromagnetic-hydrodynamic waves. Nature
1942;405:150.
[2] Liao SJ. On the analytic solution of magnetohydrodynamic flows of nonNewtonian fluids over a stretching sheet. J. Fluid Mech. 2003;488:189–212.
[3] Cortell R. A note on magnetohydrodynamic flow of a power-law fluid over a
stretching sheet. Appl. Math. Comput. 2005;168:557–66.
[4] Ishak A, Nazar R, Pop I. Magnetohydrodynamic (MHD) flow and heat transfer
due to a stretching cylinder. Energy Convers. Manage. 2008;49:3265–9.
[5] Nadeem S, Haq R, Akbar NS, Khan ZH. MHD three-dimensional Casson fluid
flow past a porous linearly stretching sheet. Alexandria Eng. J.
2013;52:577–82.
[6] Akbar NS, Nadeem S, Lee C. Characteristics of Jeffrey fluid model for peristaltic
flow of chyme in small intestine with magnetic field. Results Phys.
2013;3:152–60.
[7] Ellahi R, Rahman SU, Nadeem S, Vafai K. The blood flow of Prandtl fluid
through a tapered stenosed arteries in permeable walls with magnetic field.
Commun. Theor. Phys. 2015;63:353–8.
[8] Mabood F, Khan WA, Ismail AIM. MHD boundary layer flow and heat transfer
of nanofluids over a nonlinear stretching sheet: a numerical study. J. Magn.
Magn. Mater. 2015;374:569–76.
[9] Rashidi S, Dehghan M, Ellahi R, Riaz M, Jamal-Abad MT. Study of stream wise
transverse magnetic fluid flow with heat transfer around a porous obstacle. J.
Magn. Magn. Mater. 2015;378:128–37.
[10] Akbar NS, Raza M, Ellahi R. Influence of induced magnetic field and heat flux
with the suspension of carbon nanotubes for the peristaltic flow in a
permeable channel. J. Magn. Magn. Mater. 2015;381:405–15.
[11] Malik MY, Salahuddin T. Numerical solution of MHD stagnation point flow of
williamson fluid model over a stretching cylinder. Int. J. Nonlinear Sci. Numer.
Simul. 2015;16:161–4.
[12] Kandelousi MS, Ellahi R. Simulation of ferrofluid flow for magnetic drug
targeting using Lattice Boltzmann method. J. Z. Naturforsch. A
2015;70:115–24.
[13] Ellahi R, Shivanian E, Abbasbandy S, Hayat T. Numerical study of
magnetohydrodynamics generalized Couette flow of Eyring-Powell fluid
with heat transfer and slip condition. Int. J. Numer. Methods Heat Fluid Flow
2016;26:1433–45.
[14] Malik MY, Khan Imad, Hussain Arif, Salahuddin T. Mixed convection flow of
MHD Eyring-Powell nanofluid over a stretching sheet: a numerical study. AIP
Adv. 2016;5:117118. doi: http://dx.doi.org/10.1063/1.4935639.
[15] Ellahi R, Bhatti MM, Pop I. Effects of hall and ion slip on MHD peristaltic flow of
jeffrey fluid in a non-uniform rectangular duct. Int. J. Numer. Methods Heat
Fluid Flow 2016;26:1802–20.
[16] Akbar NS, Raza M, Ellahi R. Impulsion of induced magnetic field for Brownian
motion of nanoparticles in peristalsis. Applied Nanosci. 2016;6:359–70.
[17] Khan AA, Muhammad S, Ellahi R, Zaigham Zia QM. Bionic study of variable
viscosity on MHD peristaltic flow of Pseudoplastic fluid in an asymmetric
channel. J. Magn. 2016;21:1–8.
[18] Zeeshan A, Majeed A, Ellahi R. Effect of magnetic dipole on viscous ferro-fluid
past a stretching surface with thermal radiation. J. Mol. Liq. 2016;215:549–54.
[19] Afzal K, Aziz A. Transport and heat transfer of time dependent MHD slip flow
of nanofluids in solar collectors with variable thermal conductivity and
thermal radiation. Results Phys. 2016;6:746–53.
[20] Hayat T, Bashir Gulnaz, Waqas M, Alsaedi A. MHD flow of Jeffrey liquid due to a
nonlinear radially stretched sheet in presence of Newtonian heating. Results
Phys. 2016;6:817–23.
[21] Latif T, Alvi N, Hussain Q, Asghar S. Variable properties of MHD third order
fluid with peristalsis. Results Phys. 2016;6:963–72.
[22] Yih KA. Free convection effect on MHD coupled heat and mass transfer of a
moving permeable vertical surface. Int. Commun. Heat Mass Transfer
1999;26:95–104.
[23] Machireddy GR. Inuence of thermal radiation, viscous dissipation and Hall
current on MHD convection flow over a stretched vertical at plate. Ain Shams
Eng. J. 2014;5:169–75.
[24] Ali FM, Nazar R, Arifin NM, Pop I. Effect of Hall current on MHD mixed
convection boundary layer flow over a stretched vertical at plate. Meccanica
2011;46:1103–12.
[25] Sisko AW. The flow of lubricating greases. Indus. Eng. Chem. Res.
1958;50:1789–92.
[26] Nadeem S, Akber NS, Vajravelu K. Peristaltic flow of a Sisko fluid in an
endoscope: analytical and Numerical solutions. Int. J. Comput. Math.
2011;88:1013–23.
[27] Khan M, Shahzad A. On axisymmetric flow of Sisko fluid over a radially
stretching sheet. Int. J. Non-Linear Mech. 2012;47:999–1007.
[28] Khan M, Shahzad A. On boundary layer flow of Sisko fluid over stretching
sheet. Quaestiones Mathematicae 2013;36:137–51.
[29] Malik MY, Hussain Arif, Salahuddin T, Malik M. Numerical Solution of MHD
Sisko fluid over a stretching cylinder and heat transfer analysis. Int. J. Numer.
Meth. Heat Fluid Flow 2016;26:1787–801.
[30] Sakiadis BC. Boundary-layer behavior on continuous solid surfaces: I
boundary-layer equations for two-dimensional and axisymmetric flow.
AIChE J. 1961;7:26–8.
[31] Gupta PS, Gupta AS. Heat and mass transfer on a stretching sheet with suction
or blowing. Can. J. Chem. Eng. 1977;55:744–6.
[32] Naseer M, Malik MY, Nadeem S, Rehman A. The boundary layer flow of
hyperbolic tangent fluid over a vertical exponentially stretching cylinder.
Alexandra Eng. J. 2014;53:747–50.
[33] Rangi RR, Ahmad N. Boundary layer flow past over the stretching cylinder with
variable thermal conductivity. Appl. Math. 2012;3:205–9.
[34] Abel MS, Mahesha N. Heat transfer in MHD viscoelastic fluid flow over a
stretching sheet with variable thermal conductivity, non-uniform heat source
and radiation. Appl. Math. Model. 2008;32:1965–83.
[35] Abel MS, Datti PS, Mahesha N. Flow and heat transfer in a power-law fluid over
a stretching sheet with variable thermal conductivity and non-uniform heat
source. Int. J. Heat Mass Transfer 2009;52:2902–13.
[36] Malik MY, Hussain Arif, Salahuddin T, Bilal S, Awais M. Magnetohydrodynamic
flow of Sisko fluid over a stretching cylinder with variable thermal
conductivity: a numerical study. AIP Adv. 2016;6:025316. doi: http://dx.doi.
org/10.1063/1.494247.
[37] Grubka LG, Bobba KM. Heat transfer characteristics of a continuous stretching
surface with variable temperature. ASME J. Heat Transfer 1985;107:248–50.
[38] Andersson HI, Kumaran V. On sheet-driven motion of power-law fluids. Int. J.
Non-Linear Mech. 2006;41:1228–34.
[39] Hassanien AI, Abdullah AA, Gorla RS. Flow and heat transfer in a power-law
fluid over a non-isothermal stretching sheet. Math. Comput. Model.
1998;28:105–16.
[40] Ishak A, Nazar R, Pop I. Hydromagnetic flow and heat transfer adjacent to a
stretching vertical sheet. Heat Mass Transfer. 2008;44:921–7.
[41] Brinkman HC. Heat effects in capillary flow I. Appl. Sci. Res. 1951;2:120–4.
[42] Lin TF, Hawks KH, Leidenfrost W. Analysis of viscous dissipation effect on the
thermal entrance heat transfer in laminar pipe flows with convective
boundary conditions. Warm Stoffubertrag 1983;19:97–105.
[43] Anjali Devi SP, Ganga B. Viscous dissipation effects on non linear MHD flow in
a porous medium over a stretching porous surface. Int. J. Math. Mech.
2009;5:45–59.
[44] Singh PK. Viscous dissipation and variable viscosity effects on MHD boundary
layer flow in porous medium past a moving vertical plate with suction. Int. J.
Eng. Sci. Technol. 2012;4:2647–56.
[45] Reddy MG, Padma P, Shankar B. Effects of viscous dissipation and heat source
on unsteady MHD flow over a stretching sheet. Ain Shams Eng. J.
2015;6:1195–201.
[46] Mabood F, Shateyi S, Rashidi MM, Momoniat E, Freidoonimehr N. MHD
stagnation point flow heat and mass transfer of nanofluids in porous medium
with radiation, viscous dissipation and chemical reaction. Adv. Powder
Technol. 2016;27:742–9.