entropy
Article
Entropy Generation on MHD Casson Nanofluid Flow
over a Porous Stretching/Shrinking Surface
Jia Qing
1
, Muhammad Mubashir Bhatti
2
, Munawwar Ali Abbas
3
,
Mohammad Mehdi Rashidi
1,4
and Mohamed El-Sayed Ali
5,
*
1
Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University,
Shanghai 201804, China; qing.jia@sawtc.com (J.Q.); mm_rashidi@sawtc.com (M.M.R.)
2
Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China;
muhammad09@shu.edu.cn
3
Department of Mathematics, Shanghai University, Shanghai 201804, China; munawwar@shu.edu.cn
4
ENN-Tongji Clean Energy Institute of Advanced Studies, Tongji University, Shanghai 200072, China
5
Mechanical Engineering Department, College of Engineering, King Saud University, P.O. Box 800,
Riyadh 11421, Saudi Arabia
* Correspondence: mali@ksu.edu.sa; Tel.: +966-11-467-6672; Fax: +966-11-467-6652
Academic Editors: Giulio Lorenzini and Omid Mahian
Received: 24 February 2016; Accepted: 30 March 2016; Published: 6 April 2016
Abstract:
In this article, entropy generation on MHD Casson nanofluid over a porous Stretching/
Shrinking surface has been investigated. The influences of nonlinear thermal radiation and chemical
reaction have also taken into account. The governing Casson nanofluid flow problem consists of
momentum equation, energy equation and nanoparticle concentration. Similarity transformation
variables have been used to transform the governing coupled partial differential equations into
ordinary differential equations. The resulting highly nonlinear coupled ordinary differential equations
have been solved numerically with the help of Successive linearization method (SLM) and Chebyshev
spectral collocation method. The impacts of various pertinent parameters of interest are discussed for
velocity profile, temperature profile, concentration profile and entropy profile. The expression for
local Nusselt number and local Sherwood number are also analyzed and discussed with the help of
tables. Furthermore, comparison with the existing is also made as a special case of our study.
Keywords:
nanofluid; entropy generation; successive linearization method; Chebyshev spectral
collocation method; Casson fluid
1. Introduction
Nanofluid is a fluid that is generated by a suspension of solid particles with the dimensions less
than 100 nm in fluids. Basically, Nanofluid is a nano-scale colloidal suspension containing condensed
nanomaterials. Choi [
1
] was the first who describe the combination of nanoparticles and base fluid
and subsequently termed as nanofluid. In fact, it has two phase system with one phase (liquid phase)
and another (solid phase). It can be found to exhibit enlarged thermophysical effects like thermal
diffusivity, viscosity, and thermal conductivity compared to those of base liquids such as water, oil
and ethylene glycol mixture, etc. It also has many diverse assets in an industrial application, for
instance, fuel cell, biomedicine, nuclear reactors and transportation. The performance of heat transfer
flow problems in nanofluid has been discussed by Xuan and Li [
2
] for the assumption of turbulent
flow conditions. The declaration of their experimental results emphasized that Nusselt number of
nanofluid and convective heat transfer coefficient is enhanced by increasing the Reynolds number and
volume fraction of nanoparticles. Khalili [
3
] solved the model of unsteady convective heat transfer
of nanofluid over a stretching wall numerically. In addition, Xiao et al. [
4
,
5
] developed a novel form
of thermal conductivity of nanofluid with Brownian motion effect and heat transfer of nanofluid by
Entropy 2016, 18, 123; doi:10.3390/e18040123 www.mdpi.com/journal/entropy
Entropy 2016, 18, 123 2 of 14
the convection in a pool. He solved both models by a novel technique named as Fractal method
and concluded some important results. Natural convection heat transfer inside circular enclosures
filled with Alumina nanofluid and heated from below or above has been reported by Ali et al. [
6
,
7
].
Nanofluid impingement jet heat transfer and its effect on cooling of a circular dick were studied by
Zeitoun and Ali [
8
,
9
]. Consequently, a number of review articles with the association of nano fluid
performance have been investigated by various researchers [10–12].
The fluid flow over stretching surfaces has many important applications in engineering systems,
such as metal spinning, drawing on plastic films, the continuous casting of metals, glass blowing
and spinning of fibers. All the above-mentioned applications have involved some aspects of flow
over stretching sheet, like stagnation point flow over stretching sheet [
13
,
14
], Magnetohydrodynamics
(MHD) free convection flow with heat transfer [
15
–
17
] and so on. Moreover, stretching sheet with a
porous medium also has received great attention from researchers in the last few years. Hamad and
Ferdous [
18
] examined nanofluid with internal heat generation/absorption and suction/blowing for
boundary layer stagnation-point flow over a stretching sheet in a porous medium. Copper–water and
silver–water Nanofluid flow over a stretching sheet through a porous medium has been analyzed by
Kameshwaran et al. [19].
Magnetic nanofluid is a colloidal suspension of carrier liquid and magnetic nanoparticles. MHD
was initially tested in geophysical and astrophysical problems. In recent years, MHD has received
significant attention due to its various applications in engineering and petroleum industries. Magnetic
nanofluid is also one of them and the main objective of this research area is that fluid flow and heat
transfer can be controlled by external magnetic field. Effect of magnetic field on nanofluid with
different geometries has been investigated by several researchers. For instance, Rashidi et al. [
20
]
studied buoyancy effect on MHD flow of nanofluid over stretching sheet in the presence of thermal
radiation while Sheikholeslami et al. [
21
] discussed flow and heat transfer in a semi-annulus enclosure
in the existence of magnetic and thermal radiation. Moreover, Chamkha [
22
] reported that MHD
flow of uniformly stretching vertical permeable surface in the presence of a chemical reaction and
heat generation.
The entropy generation is relaxed in all the studies mentioned above which motivates the current
research. Although several models have been proposed to describe MHD effect on stretching surface
with various types of fluids but their full potential has not been exploited yet and much work needs to
be done. For instance, industrial sectors heating and cooling is important in many aspects including
transportation, energy, and electronic devices. Moreover, heat transfer and magnetic effect on biofluids
are also great interest, particularly in a physiological system. The main concern in the present analysis is
to better understanding about the minimization or entropy generation on heat transfer process. Entropy
generation can be expressed as various thermal systems are subjected to irreversibility phenomena and
are connected to viscous dissipation, magnetic field and heat and mass transfer. Entropy generation
clarifies energy losses in a system evidently in many energy-related applications such as cooling of
modern electronic devices or system, geothermal energy systems, etc. Consequently, entropy is a
measure of the number of specific ways in which a measure of progressing towards thermodynamics
equilibrium. Bejan [
23
] originally formulated the analysis of entropy generation. Abolbashari et al. [
24
]
have investigated analytically the fluid flow with heat and mass transfer and entropy generation for
the steady laminar non-Newtonian nanofluid induced by stretching sheet in the presence of velocity
slip and convective surface boundary condition. Few more attempts are taken into account of entropy
generation on nanofluid with stretching surface with different geometry [25–29].
In view of the above literature and by realizing the growing need of entropy minimization, the
present study brings to its fold many previous studies as particular cases. One of more significance of
the current study is the solution of MHD Casson Nanofluid model using a numerical technique, such
as Successive Linearization method (SLM) and Chebyshev spectral collocation method. The solution
methods for the coupled nonlinear ordinary differential equations are quite interesting. The solution
is based upon a choice of a function satisfying the boundary condition and the unknown functions
Entropy 2016, 18, 123 3 of 14
are obtained by iteratively solving the linearized version of the governing equation. Numerical
solutions of a temperature profile, concentration profile, entropy generation and velocity profile
are computed and demonstrated graphically and mathematically. The obtained results reveal the
characteristics of Casson nanofluid and entropy generation. This paper is summarized as follows: after
the introduction in Section 1, Section 2 characterizes the mathematical formulation of the governing
flow problem, Section 3 shows some important formula of local Nusselt number and local Sherwood
number, Section 4 interprets the solution methodology of the problem and Section 5 illustrates the
mathematical modeling of entropy generation, while Sections 6 and 7 are devoted to numerical results
and conclusions, respectively.
2. Mathematical Formulation
Consider the MHD boundary layer flow over a porous stretching surface near a stagnation point
at
y “
0. The MHD flow occurs in the domain at
y ą
0. The fluid is electrically conducting by an
external magnetic field while the induced magnetic is assumed to be negligible (or zero). Cartesian
coordinate is chosen in a way such that x-axis is considered along the direction of the sheet whereas
is y-axis considered along normal to it (see Figure 1). Suppose that
C
w
be the nano particle fraction
at the sheet while the temperature and nano particle fraction at infinity is
T
8
and
C
8
, respectively.
The velocity of the sheet is considered along x-axis,
r
u
w
“ ax.
=0 0.
=.
=
+
√
,
,
+
,
,
Π
Π
∂
∂
+
∂
∂
=0,
∂
∂
+
∂
∂
=1+
1
β
∂
∂
+
+
ν
(
)
+
σ
(
)
,
∂
∂
+
∂
∂
=
∂
∂
1
ρ
∂
∂
+
∂
∂
∂
∂
+
∂
∂
,
Figure 1. Geometry of the problem.
The rheological equation of state for an isotropic and incompressible Casson fluid is
ø
ij
“
$
&
%
2e
ij
´
µ
b
`
P
y
?
2Π
¯
, Π ą Π
C
,
2e
ij
´
µ
b
`
P
y
?
2Π
c
¯
, Π ă Π
C
,
(1)
where
e
ij
is the component of the deformation rate,
Π
is the product of the deformation rate and
Π
C
is
the critical value of the product based. The governing equations of Casson nanofluid model can be
written as
B
r
u
Bx
`
B
r
v
By
“ 0, (2)
r
u
B
r
u
Bx
`
r
v
B
r
v
By
“ ν
ˆ
1 `
1
β
˙
B
2
r
u
By
2
`
r
u
e
d
r
u
e
dx
`
ν
p
r
u
e
´
r
u
q
r
k
`
σB
2
0
p
r
u
e
´
r
uq
ρ
, (3)
r
u
B
r
T
Bx
`
r
v
B
r
T
By
“
α
B
2
r
T
By
2
´
1
ρc
p
Bq
r
By
` τ
¨
˝
D
B
BC
By
B
r
T
By
`
D
T
T
8
˜
B
r
T
By
¸
2
˛
‚
, (4)
Entropy 2016, 18, 123 4 of 14
r
u
BC
Bx
`
r
v
BC
By
“ D
B
B
2
C
By
2
`
D
T
T
8
B
2
r
T
By
2
´ K pC
1
´ C
8
q . (5)
The nonlinear radiative heat flux can be written as
q
r
“ ´
4
σ
3k
B
r
T
4
By
“ ´
16
σ
r
T
3
3k
B
r
T
By
, (6)
and their respective boundary conditions are
r
u “ u
w
, v “ v
w
,
r
T “
r
T
w
, C “ C
w
at y “ 0, (7)
r
u “
r
u
e
,
r
v “ 0,
r
T Ñ
r
T
8
, C Ñ C
8
as y Ñ 8. (8)
The steam function satisfying Equation (1) are defined as
r
u “ Bϕ{B y
and
r
v “ ´Bϕ{Bx
. Defining
the following similarity transformation variables
ζ “
c
r
u
w
νx
y,
r
u “
r
u
w
f
1
pζq ,
r
v “ ´
c
ν
r
u
w
x
f
pζq , θ “
r
T ´
r
T
8
r
T
w
´
r
T
8
, φ “
C ´ C
8
C
w
´ C
8
, (9)
and using Equation (8) in Equations (3) and (7), we get
ˆ
1 `
1
β
˙
f
3
` 1 ´ f
12
` f f
2
` k
`
1 ´ f
1
˘
` M
`
1 ´ f
1
˘
“ 0, (10)
ˆ
1
P
r
` N
r
˙
θ
2
` f θ
1
` λθ ` N
b
θ
1
Φ
1
` N
t
θ
12
“ 0, (11)
Φ
2
` L
e
f Φ
1
`
N
t
N
b
θ
2
´ γΦ “ 0. (12)
Their corresponding boundary conditions are
f p0q “ S, f
1
p
0q “ α, f
1
p
8q “ 1, (13)
θ p0q “ 1, θ p8q “ 0, (14)
Φ p0q “ 1, Φ p8q “ 0, (15)
where
P
r
“ ν{α
,
k “ ν{
r
k
,
M “ B
2
0
σ{cρ
,
L
e
“
ν
D
B
,
N
b
“
τD
B
p
Φ
w
´Φ
8
q
ν
,
N
t
“
τD
T
´
r
T
w
´
r
T
8
¯
r
T
8
ν
,
α
is a stretching
parameter, i.e.,
α ą
0 corresponds to the stretching surface case,
α ă
0 correspond to shrinking surface
case and for
α “
0, planar stagnation flow towards a stationary surface occurs, and for
α “
1 the flow
having no boundary layer and S “ 0 corresponds to impermeable surface (See Tables 1 and 2).
Entropy 2016, 18, 123 5 of 14
Table 1. Nomenclature.
Symbol Names
r
u,
r
v Velocity components pm{sq
x, y Cartesian coordinate pmq
r
p
Pressure
`
N{m
2
˘
r
k
Porosity parameter
N
G
Dimensionless entropy number
Re Reynolds number
r
t
Time psq
k
Mean absorption coefficient
S Suction/injection parameter
N
b
Brownian motion parameter
N
t
Thermophoresis parameter
q
w
Heat flux
q
m
Mass flux
B
r
Brinkman number
T
8
Environmental temperature (K)
M Hartman number
B
0
Magnetic field
N
r
Radiation parameter
r
T, C
Temperature pKq and Concentration
g
Acceleration due to gravity
`
m{s
2
˘
D
B
Brownian diffusion coefficient
`
m
2
{s
˘
D
T
Thermophoretic diffusion coefficient
`
m
2
{s
˘
K Chemical reaction parameter
Table 2. Greek Symbol.
Symbol Names
α Thermal conductivity of the nano particles
β Casson fluid parameter
σ Stefan-Boltzmann constant
µ
Viscosity of the fluid
`
Ns{m
2
˘
χ,λ
1
Dimensionless constant parameter
Ω Dimensionless temperature difference
Φ Nano particle volume fraction
θ Temperature profile
σ Electrical conductivitypS{mq
ϕ Stream function
τ Effective heat capacity of nano particle pJ{Kq
ν
Nano fluid kinematic viscosity
`
m
2
{s
˘
γ Dimensionless chemical reaction parameter
P
y
Yield stress
µ
b
Plastic viscosity
3. Physical Quantities of Interest
The physical quantities of interest for the governing flow problem are local Nusselt number and
local Sherwood number, which can be written as [24]
Nu
x
“
xq
w
κ
´
r
T
w
´
r
T
8
¯
, Sh
x
“
xq
m
D
B
pC
w
´ C
8
q
, (16)