close

Вход

Забыли?

вход по аккаунту

?

epjp%2Fi2017-11660-0

код для вставкиСкачать
Eur. Phys. J. Plus (2017) 132: 441
DOI 10.1140/epjp/i2017-11660-0
THE EUROPEAN
PHYSICAL JOURNAL PLUS
Regular Article
Numerical analysis of 3D micropolar nanofluid flow induced by an
exponentially stretching surface embedded in a porous medium
M. Subhania and S. Nadeem
Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan
Received: 3 July 2017 / Revised: 24 August 2017
c Società Italiana di Fisica / Springer-Verlag 2017
Published online: 27 October 2017 – Abstract. The present article is devoted to probe the behavior of a three-dimensional micropolar nanofluid
over an exponentially stretching surface in a porous medium. The mathematical model is constructed in
the form of partial differential equations using the boundary layer approach. Then by employing similarity transformations, the modelled partial differential equations are transformed to ordinary differential
equations. The solution of subsequent ODEs is derived by utilizing the BVP-4C technique alongside the
shooting scheme. The graphical illustrations are presented to interpret the salient features of pertinent
physical parameters on the concerned profiles for different kinds of nanoparticles, namely copper, titania
and alumina with water as the base fluid. For a better understanding of the fluid flow, the numerical
variation in the local skin friction coefficients, Cfx and Cfy , and local Nusselt number is analyzed through
tables. We can see, from the present study, that the omission of porous matrix enhances the flow of the
fluid. Microrotation has a decreasing impact on the skin friction whereas it increases the rate of the heat
transfer of the nanofluid.
1 Introduction
In view of the substantial practical applications in industrial and manufacturing fields, a lot of work has been performed on boundary layer flows for linear and non-linear stretching surfaces. The interminable list of its engineering
applications comprises extrusion of polymer sheet, paper production, tinning and annealing of copper wires, manufacturing of metal wires and plastic sheets, cooling of metallic plates inside a cooling reservoir and so forth. Sakiadis [1]
propounded the concept of the boundary layer flow past a continuous stretched sheet and devised the boundary layer
equations for flow in two dimensions. Tsou et al. [2] inspected the consequences of heat transfer on boundary layer
flows past a stretching surface. Gupta et al. [3] incorporated mass transfer analysis over stretching sheet and considered
suction or blowing effects. After that, numerous researchers have further delved into boundary layer flows involving
important conventional fluids [4,5].
In the contemporary period, flow due to an exponentially stretching surface is gaining much importance due to
its vast applications. For instance, in the process of drawing and tinning of copper wires, the rate of heat transfer
past a continuously stretching surface which has exponential modifications in its stretching velocity and temperature,
influences the form of the final product. In this regard, Magyari and Keller [6] forwarded the comparison of numerical
and analytical solutions. Liu et al. [7] considered the characteristics of flow and heat transfer of 3D flow over a surface
which was stretched exponentially. Recently Ahmad et al. [8] explored the influence of the Cattaneo-Christov heat
flux model induced by an exponentially stretching surface.
It was the need of the hour to upgrade the thermal conductivity of some important conventional fluids which
are useful in heat transfer, for example water, ethylene glycol and mineral oil which possess poor heat transfer
characteristics. A novel solution was found by introducing small metallic solid particles in the fluid; this revolutionized
the realm of technology and industry. These nanoparticles elevate the thermal conductivity of fluids improving the
heat transfer properties. The fluids so obtained were termed as nanofluids. Therefore, nanofluids are characterized as a
solid-fluid mixture with a base fluid of low conductivity and nano-meter sized particles with high thermal conductivity.
Due to their higher thermal conductivity, these fluids lower the pumping cost of heat exchangers to a greater extent.
This fluid was first introduced by Choi [9] in 1995. A thorough investigation of convective transport within nanofluids
a
e-mail: msubhani@math.qau.edu.pk (corresponding author)
Page 2 of 12
Eur. Phys. J. Plus (2017) 132: 441
was done by Buongiorno [10]. Nanoparticles along with their little volume fraction, stability and remarkable useful
applications in optical, biomedical and electronic fields have opened new horizons of research. Recently many scholars
have discussed the nanoparticle phenomena in different geometries with pertinent physical properties of fluid, see
refs. [11–21]. Nadeem et al. [22] discussed the heat transfer phenomenon of a three-dimensional nanofluid over an
exponentially stretching surface.
Since nanoparticles are minute in size they can haul with them slip velocity through the base fluid molecules
(see Buongiorno [10]). Particle rotation is also an essential factor in the heat transfer enhancement which was observed by Ahuja [23]. In view of the fact that nanoparticle can schlep slip velocity with base fluid molecules, the
likelihood of translation and microrotation arises. The micropolar theory considers the effects of microrotation hence
the application of this hypothesis in case of nanoparticles gives a valuable comprehension to the unusual rise in
nanofluid’s thermal conductivity. The theory of micropolar fluids set forth by Eringen [24,25] was a breakthrough
in the study of rheologically complex fluids. This class of non-Newtonian fluids comprises randomly oriented particles which are rigid and spherical in shape suspended in a viscous medium. The gist of the theory undertakes the
extension of the classical Navier-Stokes equations so that more complex fluids containing certain additives such as
liquid crystals, particle suspensions, fluids with materials containing fibrous structures are also considered. The theory
of micropolar fluid manifests the intrinsic motion of fluid elements and the microrotational effects. Industrially the
examples of micropolar fluid include exotic lubricants, muddy fluids, colloidal solutions, polymer suspensions whereas
there are certain biological fluids as well which model micropolar fluid, for instance animal blood. Lukaszewicz [26]
and Eringen [27] gave ample details of the micropolar theory in their books. Chamkha et al. [28] analyzed the flow
behavior of three-dimensional micropolar fluids. In 2009, Mohamed abd el-aziz [29] deliberated over the effects of
viscous dissipation on micropolar fluid past an exponentially extending sheet. Mohanty [30] conducted a numerical
research on heat and mass transfer of micropolar fluids past a stretching sheet. Alongside a research conducted over
micropolar nanofluids gained momentum and [31,32] analyzed the impact of different attributes over the micropolar
nanofluid.
Since the past few decades, many researchers have inquired the characteristics of flow and heat transfer inside a
porous medium owing to its incessantly growing industrial and technological applications. Problems associated with
porous surfaces incorporate insulation engineering, geo-mechanics such as geo thermal reservoirs and enhanced oil
recovery. The main use of a porous medium is to insulate a heated body so that its temperature can be maintained.
Porous media are also considered to be useful in vanishing the natural free convection, which would otherwise affect
the stretching surface immensely. Kamel et al. [33] and Heruska et al. [34] have examined the micropolar fluid flow
across a porous medium. Very recently [35,36] have explored the behavior of micropolar fluid induced by a stretching
porous sheet. A comprehensive study of MHD flow of nanofluids through a porous medium was done by Zeeshan et
al. [37].
In the vast field of literature, there is no research conducted on the behavior of a three-dimensional micropolar
fluid with nanoparticles due to an exponentially stretching surface. In the present article, the influence of a porous
medium on the three-dimensional steady flow of a micropolar nanofluid induced by an exponentially stretching surface
has been critically examined. Three different nanoparticles, namely copper, titania and alumina are compared taking
water as the base fluid. Non-dimensional velocities, angular velocities and temperature profiles as well as the influence
of various physical parameters on them are exhibited by plotting graphs and are also presented in tabular form.
2 Momentum and temperature description
We have considered a steady 3D incompressible boundary layer flow of a micropolar nanofluid embedded in a porous
medium over an exponentially stretching surface. It is supposed that the surface is being stretched in two adjacent
directions with different velocities, Uw which is along the x-axis and Vw which is along the y-axis. Further, Tw is
considered to be the temperature near the wall while ambient temperature is assumed to be T∞ . The surface is
located at the plane z = 0 and the flow is confined in the region z > 0 (see fig. 1). The transport equations are
= 0,
∇·V
ρnf
ρnf j
dV
+ k(∇ × N
) − νnf V ,
= −∇p + (μnf + k)∇2 V
∗
dt
K
dN
− 2k N
+ k(∇ × V
),
= χnf ∇2 N
dt
dT
= αnf ∇2 T,
dt
(1)
(2)
(3)
(4)
Eur. Phys. J. Plus (2017) 132: 441
Page 3 of 12
Fig. 1. Physical regime of the problem.
where
knf
μf
, μnf =
, ρnf = (1 − φ)ρf + φρs ,
(ρcp )nf
(1 − φ)2.5
= (1 − φ)(ρCp )f + φ(ρCp )s ,
(ks + 2kf ) − 2φ(kf − ks )
μnf
, νnf =
=
,
(ks + 2kf ) + φ(kf − ks )
ρnf
= (μnf + k/2)j.
αnf =
(ρCp )nf
knf
χnf
(5)
χnf is the spin gradient viscosity, αnf is the nanofluid thermal diffusivity, μnf is the dynamic viscosity of the nanofluid,
k is the vortex viscosity, ρnf is the density of the nanofluid, knf is the thermal conductivity of the nanofluid, (ρCp )nf
is the heat capacity of the nanofluid, φ is the volume fraction of nanoparticles, K ∗ is the permeability of the porous
sheet, j = 2νL
Uw is the micro-inertia per unit mass.
After applying boundary layer approximation, we are left with the following equations:
Momentum equations
∂u
∂u
∂u
μnf + k ∂ 2 u
νnf u
∂N2
k
u
+v
+w
=
−
−
,
2
∂x
∂y
∂z
ρnf
∂z
ρnf
∂z
K∗
∂v
∂v
μnf + k ∂ 2 v
νnf v
∂N1
k
∂v
+v
+w
=
−
+
.
u
2
∂x
∂y
∂z
ρnf
∂z
ρnf
∂z
K∗
(6)
(7)
Angular momentum equations
∂N1
∂N1
∂N1
∂v
∂ 2 N1
+v
+w
= χnf
,
− 2kN1 − k
ρnf j u
2
∂x
∂y
∂z
∂z
∂z
∂N2
∂N2
∂N2
∂u
∂ 2 N2
ρnf j u
+v
+w
= χnf
.
− 2kN2 + k
2
∂x
∂y
∂z
∂z
∂z
Energy equation
∂T
∂T
∂T
u
+v
+w
= αnf
∂x
∂y
∂z
∂2T
∂z 2
(8)
(9)
.
(10)
Boundary conditions
u = Uw ,
u → 0,
v = Vw ,
v → 0,
T = Tw ,
T → T∞ ,
∂v
∂u
, N2 = −n
at z = 0,
∂z
∂z
N1 → 0, N2 → 0, as z → ∞.
N1 = n
(11)
Page 4 of 12
Eur. Phys. J. Plus (2017) 132: 441
The stretching velocities at the wall surface and temperature are described as follows:
Uw = U0 e
x+y
L
,
Vw = V 0 e
x+y
L
,
Tw = T ∞ + T 0 e
B(x+y)
2L
,
(12)
where L denotes the reference length, U0 , V0 and T0 are constants, T∞ indicates the ambient temperature, B symbolizes
the temperature exponent and n the boundary parameter, n ranges from 0 ≤ n ≤ 1. Here n = 0 symbolizes a strong
concentration of microelements at the wall and in this case the microelements at the stretching surface are incapable
of rotation. Diminishing of the antisymmetric fragment of the stress tensor is depicted in the case when n = 0.5 and
it also exhibits weak concentration of microelements and n = 1 symbolizes the turbulent boundary layer flows.
Using appropriate similarity transformations given below:
u = U0 e
x+y
L
f (η),
v = U0 e
x+y
L
g (η),
w=−
νU0
2L
12
e
x+y
2L
{f + ηf + g + ηg },
x+y
x+y
1
1
U0
U0
(U0 2νL) 2 e3( 2L ) h1 (η), N2 =
(U0 2νL) 2 e3( 2L ) h2 (η),
2νL
2νL
12
B(x+y)
x+y
U0
T = T∞ + T0 e 2L θ(η), η =
e 2L z.
2νL
N1 =
(13)
After applying these transformations to eqs. (6)–(10), the continuity equation is satisfied identically while linear
momentum, angular momentum and energy equations take the form
ρf
νnf
ρf
(A(φ) + R1 )f + f (f + g) − 2f (f + g ) −
R 1 h2 −
Kp f = 0,
ρnf
ρnf
νf
(14)
ρf
ρf
νnf
(A(φ) + R1 )g + g (f + g) − 2g (f + g ) +
R 1 h1 −
Kp g = 0,
ρnf
ρnf
νf
(15)
ρf
R2 h1 − R1 R3 (2h1 + g ) − 3h1 (f + g ) + h1 (f + g) = 0,
ρnf
(16)
ρf
R2 h2 − R1 R3 (2h2 − f ) − 3h2 (f + g ) + h2 (f + g) = 0,
ρnf
(17)
1
(knf /kf )θ
− B(f + g )θ + (f + g)θ = 0.
Pr (1 − φ) + (φ(ρcp )s /(ρcp )f )
(18)
The associated boundary conditions become
f (0) = 0,
f (0) = 1,
g(0) = 0,
g (0) = λ,
h1 (0) = ng (0),
f → 0,
h1 → 0,
θ(0) = 0,
h2 (0) = −nf (0),
as
η → 0,
g → 0, θ → 0,
h2 → 0,
as η → ∞,
(19)
where R1 , R2 , R3 , Kp , Pr, λ, φ and n denote vortex viscosity parameter, spin gradient viscosity parameter, microinertia density parameter, porosity parameter, Prandtl number, stretching ratio parameter, solid nanoparticle volume
fraction and boundary parameter, respectively. They are defined as
R1 =
k
,
μ
R2 =
χnf
,
μj
R3 =
2νL
,
jUw
Kp =
K ∗ Uw
,
2νL
Pr =
(μCp )f
,
kf
λ=
V0
U0
and
A(φ) =
μnf
1
=
.
μf
(1 − φ)2.5
(20)
Further, the skin friction coefficients and Nusselt number can be described as
Cf x =
τwx
1
2
ρ
2 nf Uw
,
Cf y =
τwy
1
2
ρ
2 nf Uw
,
N ux =
xqw
,
kf (Tw − T∞ )
(21)
Eur. Phys. J. Plus (2017) 132: 441
Page 5 of 12
1
0.5
0.9
Cu
TiO
0.8
Al2O3
2
Water
Water
0.35
0.6
= 0.0, 0.1, 0.2
0.5
g ’()
f ’()
2
Al2O3
0.4
0.7
0.4
0.3
0.2
0.15
0.2
0.1
0.1
0.05
0
1
2

(a)
3
4
5
= 0.0, 0.1, 0.2
0.25
0.3
0
Cu
TiO
0.45
0
0
1
2

3
4
5
(b)
Fig. 2. (a) Impact of φ on velocity profile in x-direction f (η). (b) Impact of φ on g (η).
where τwx , τwy and qw are surface shear stress and surface heat flux, which can be defined as
∂u
τwx = (μnf + k)
+ k(N2 )z=0 ,
∂z
z=0
∂v
τwy = (μnf + k)
+ k(N1 )z=0 ,
∂z z=0
∂T
qw = −knf
.
∂z z=0
After simplification, the non-dimensional form of skin friction coefficients and local Nusselt number are
−1
1
A + (1 − n)R
1
√ Cf x (Rex ) 2 =
f (0),
(1 − φ) + φ(ρs /ρf )
2
−1
1
A + (1 + n)R
1
2
√ Cf y (Rey ) =
g (0),
(1 − φ) + φ(ρs /ρf )
2
√ L
1
knf 2 N ux Re− 2 = −
θ (0).
x
kf
(22)
(23)
3 Numerical scheme
The non-linear coupled differential equations (14)–(18) together with their boundary conditions given in eq. (19) are
solved numerically using the BVP-4C technique invoking the shooting methodology. In this process, firstly the system of
eqs. (14)–(18) escorted with boundary conditions are reduced to first order equations. Then appropriate initial guesses
are adopted which satisfy the boundary conditions. The results obtained depict the impact of various dimensionless
parameters such as material parameter R1 , stretching ratio parameter λ, temperature exponent B, porosity parameter
Kp and nanoparticle volume fraction radiation φ, on velocity, microrotation and temperature profiles. For attaining
the convergence criterion of 10−6 the shooting methodology is reiterated. Solutions to the given problem are given in
graphical and tabular form.
4 Results and discussion
In figs. 2, 3 and 4 analysis of velocity profiles f (η), g (η) in both x- and y-direction, respectively, microrotation profiles
h1 (η) and h2 (η) and temperature profile θ(η) has been performed for numerous values of the nanoparticle volume
fraction φ. It is learnt from figs. 2(a) and (b) that Cu-water gives a low velocity profile compared with Al2 O3 and TiO2 .
Page 6 of 12
Eur. Phys. J. Plus (2017) 132: 441
0.05
0.6
Cu
TiO
0
2
0.5
Al2O3
−0.05
Water
0.4
−0.15
= 0.0, 0.1, 0.2
2
= 0.0, 0.1, 0.2
−0.2
0.3
h ()
h1()
−0.1
0.2
−0.25
0.1
−0.3
Cu
TiO2
−0.35
Al O
0
2 3
Water
−0.4
0
1
2

3
4
−0.1
5
0
1
2
(a)

3
4
5
(b)
Fig. 3. (a) Variation of φ on microrotation h1 (η). (b) Effect of φ on microrotation h2 (η).
1
0.9
Cu
TiO
0.8
Al2O3
2
Water
0.7
()
0.6
0.5
= 0.0, 0.1, 0.2
0.4
0.3
0.2
0.1
0
0
1
2

3
4
5
Fig. 4. Variation of various nanoparticle volume fractions on temperature distribution θ(η).
Furthermore, in the absence of nanoparticles, water has a higher velocity profile if compared with all three, i.e. copper,
alumina and titania. Therefore, it is discovered that increment in φ will hinder the motion of the fluid so velocity
as well as the boundary layer thickness decrease. This is because an increase in nanoparticles makes the thermal
conductivity increase resulting in a decrease in the boundary layer thickness. Figures 3(a) and (b) depict the behavior
of the microrotation profile with variation of nanoparticle volume fraction for fixed values of vortex viscosity parameter
R1 = 3. It is concluded that as we enhance the nanoparticle volume fraction φ, the behavior of h1 (η) is decreasing
while h2 (η) is fluctuating within the domain. In case of h2 (η) we learnt that for η < 1, the base fluid has a minimal
microrotation profile, whereas in case of η > 1, the base fluid exhibits a boost in the microrotation profile compared
with the rest of the mixture (see fig. 3(b)). Furthermore, it is noticed that near the wall, the effects of nanoparticles
for the microrotation profile are much sturdier if compared with the case away from the wall. Figure 4 elucidates the
impact of φ on temperature distribution θ(η). It is evident through this graph that the base fluid has a low temperature
profile in comparison with Cu-water, TiO2 -water and Al2 O3 -water. Increase in φ amplifies the thermal conductivity
of the nanofluid and hence the thermal boundary layer.
It is demonstrated through fig. 5 the impact of porosity parameter Kp on velocity distribution in the x- and
y-directions, microrotation and temperature profiles. Higher values of the porosity parameter correspond to lower
velocity in both directions as seen through figs. 5(a) and (b). The main dependence of the porosity parameter is
upon the permeability K ∗ of the medium. An increment in porosity results in lower permeability as they are inversely
proportional; this lower permeability instigates a reduction in velocity. Since the existence of a porous medium results
Eur. Phys. J. Plus (2017) 132: 441
Page 7 of 12
1
0.5
0.9
K =1
0.45
Kp= 1
K =2
0.4
K =2
Kp= 3
0.35
K =3
p
0.8
p
0.7
g’()
f’()
0.6
0.5
0.4
0.25
0.2
0.15
0.2
0.1
0.1
0.05
0
1
2

3
4
p
0.3
0.3
0
p
0
5
0
1
2
(a)

3
4
5
(b)
0.05
1.4
0
K =1
p
1.2
−0.05
Kp= 2
K =3
−0.1
p
1
h2()
h1()
−0.15
−0.2
−0.25
−0.3
Kp= 1
−0.35
0.6
0.4
K =2
p
−0.4
K =3
0.2
p
−0.45
−0.5
0.8
0
1
2

3
4
0
5
0
1
2
(c)

3
4
5
(d)
1
0.9
K =1
0.8
K =2
0.7
K =3
 ’()
p
p
p
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2

3
4
5
(e)
Fig. 5. (a) Influence of Kp on velocity profile in the x-direction. (b) Influence of Kp on velocity profile in the y-direction. (c) Influence of Kp on angular velocity h1 (η). (d) Influence of Kp on angular velocity h2 (η). (e) Influence of Kp on temperature profile.
in a greater hindrance to the fluid flow, velocity decelerates and so the angular velocity as witnessed through figs. 5(c)
and (d). It is observed in fig. 5(e) that an elevation in the porosity parameter rises the temperature noticeably. Thus,
it may be concluded that owing to the resistance offered by the porous medium temperature rises.
Page 8 of 12
Eur. Phys. J. Plus (2017) 132: 441
1
0.5
0.9
0.45
0.8
0.4
0.7
0.35
n= 0.0, 0.2, 0.4, 0.6, 0.8, 1
0.5
g’()
f’()
0.6
0.4
0.3
0.2
0.3
0.15
0.2
0.1
0.1
0.05
0
0
1
2

3
4
n= 0.0, 0.2, 0.4, 0.6, 0.8, 1
0.25
0
5
0
1
2
(a)

3
4
5
(b)
0.1
0.8
0
0.7
−0.2
0.6
−0.3
0.5
h ()
−0.4
n= 0.0, 0.2, 0.4, 0.6, 0.8, 1
−0.5
2
h1()
−0.1
0.4
n= 0.0, 0.2, 0.4, 0.6, 0.8, 1
0.3
−0.6
−0.7
0.2
−0.8
0.1
−0.9
−1
0
1
2

(c)
3
4
5
0
0
0.5
1
1.5
2
2.5

3
3.5
4
4.5
5
(d)
Fig. 6. (a) Impact of n on velocity profile in the x-direction. (b) Effect of n on velocity profile in the x-direction. (c) Impact
of n on angular velocity h1 (η). (d) Influence of n on angular velocity h2 (η).
Now we analyze the impact of boundary parameter n on velocity and angular velocity profiles h1 (η) and h2 (η) in
x- and y-directions via fig. 6(a)–(d). n indicates the rotation of microelements close to the wall. As mentioned earlier
n = 0 signifies concentrated flow of particles i.e. strong concentration wherein the microelements near the surface of
the wall are not able to rotate, n = 0.5 is used to specify the diminishing of the antisymmetric fragment of the stress
tensor i.e. weak concentration and n = 1 denotes the turbulent flows of the boundary layer. It is illustrated through
figs. 6(a) and (b) that the gradient of velocity at the surface is greater for increasing values of n. And the couple stress
h(0) enlarges for higher values of n, as visible in fig. 6(c) and (d).
Through figs. 7(a) and (b) a comparison between 3 different types of nanoparticles is provided with a variation of
n against velocity f (η) and angular velocity h2 (η). It is observed that Cu has a low velocity profile in comparison
with Al2 O3 and TiO2 .
Figures 8(a) and (b) present the nature of the velocity profile in the x- and y-directions with a variation of
a parameter R1 = k/μ, the material parameter which gives the ratio between two viscosities of the fluid under
consideration, i.e. dynamic viscosity and vortex viscosity. For R1 these two viscosities possess the same order of
magnitude. It was learnt about the velocity that it enhances with an increase in material parameter R1 . This is
due to the vortex viscosity that makes fluid particles accelerate whereas the permeability and local inertia coefficient
decelerates the flow motion. It is apparent from this figure that the thickness of the boundary layer grows with R1 .
The absolute value of velocity gradient near the surface reduces as we increase R1 . Hence, there is a drag reduction in
micropolar fluids in comparison with viscous i.e. Newtonian fluids. The negative velocity gradient close to the surface,
f (0) < 0, as displayed in figs. 8(a) and (b) means that there is resistance to the fluid motion due to the stretching
Eur. Phys. J. Plus (2017) 132: 441
Page 9 of 12
1
0.8
Cu
TiO
0.9
Cu
TiO
0.7
2
Al O
2
Al O
2 3
0.8
2 3
0.6
0.5
0.6
h ()
n= 0, 1
0.5
2
f ’()
0.7
0.4
0.4
n= 0, 1
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
1
2

3
4
5
0
1
2
(a)

3
4
5
(b)
Fig. 7. (a) Behavior of different nanoparticles w.r.t. n on velocity profile in the x-direction. (b) Behavior of different nanoparticles
w.r.t. n on angular velocity profile h2 (η).
1
1
=0.5
=1.0
0.9
0.8
0.7
0.7
0.6
0.6
0.5
g’()
f’()
0.8
R1= 0, 1, 2
0.4
0.3
0.2
0.1
0.1
1
2

(a)
3
4
5
1
0.4
0.2
0
R = 0, 1, 2
0.5
0.3
0
=0.5
=1.0
0.9
0
0
1
2

3
4
5
(b)
Fig. 8. (a) Influence of R1 on velocity profile in the x-direction. (b) Impact of R1 on velocity profile in the y-direction.
surface. We can also notice from the figures that with a rise in stretching parameter λ there is a decrease in x-direction
velocity whereas the velocity profile in the y-direction amplifies. The latter behavior is due to the boundary condition
g (0) = λ.
Figures 9 and 10 are sketched to portray the behavior of R1 , R2 and R3 on microrotation profiles in x- and
y-directions. Figures 9(a) and (b) are plotted with a variation of n = 0 and n = 0.5 respectively, with h1 (η). In
both cases an increment in the profile behavior is reported for numerous values of micropolar parameters. Similarly,
figs. 10(a) and (b) are sketched to show variation of R1 , R2 and R3 on h2 (η) when n = 0 and n = 0.5. Four different
variations are evaluated and there is an enhancement witnessed in velocity profiles when n = 0 and n = 0.5. This
is because as there is an increment in micropolar parameters, rotation of the micropolar components is induced in
most of the boundary layer excluding the region near the surface since kinematic viscosity is dominating the flow over
there. Table 1 illustrates the thermo-physical properties of the base fluid and nanoparticles. In table 2 the effects of
various relevant parameters such as material parameter R1 , boundary parameter (n), porosity parameter (Kp ), volume
fraction of nanoparticles (φ) and stretching ratio parameter (λ) on skin frictions Cfx and Cfy and the Nusselt number
are displayed.
Page 10 of 12
Eur. Phys. J. Plus (2017) 132: 441
0.08
0.2
Case 1: R =1, R =0.05, R =0.1
1
0.07
2
3
0.1
Case 2: R1=2, R2=0.1, R3=0.2
Case 3: R =2.5, R =0.2, R =0.3
0.06
1
2
3
0
Case 4: R1=4, R2=0.3, R3=0.4
h ()
0.04
−0.1
1
h1()
0.05
−0.2
0.03
Case 1: R =1, R =0.05, R =0.1
1
−0.3
0.02
1
0
1
2
3

4
−0.5
5
3
1
2
3
Case 4: R =4, R =0.3, R =0.4
1
0
3
2
Case 3: R =2.5, R =0.2, R =0.3
−0.4
0.01
2
Case 2: R =2, R =0.1, R =0.2
0
1
2
(a)
2
3

3
4
5
(b)
Fig. 9. (a) Influence of R1 , R2 , R3 on angular velocity profile h1 (η). (b) Influence of R1 , R2 , R3 on angular velocity profile
h1 (η).
0.15
0.7
Case 1: R =1, R =0.05, R =0.1
Case 1: R =1, R =0.05, R =0.1
1
2
3
1
2
Case 4: R1=4, R2=0.3, R3=0.4
0.4
2
h ()
3
Case 3: R1=2.5, R2=0.2, R3=0.3
0.5
Case 4: R1=4, R2=0.3, R3=0.4
h2()
2
Case 2: R1=2, R2=0.1, R3=0.2
3
Case 3: R1=2.5, R2=0.2, R3=0.3
0.1
1
0.6
Case 2: R =2, R =0.1, R =0.2
0.3
0.05
0.2
0.1
0
0
1
2
3

4
0
5
0
1
(a)
2

3
4
5
(b)
Fig. 10. (a) Influence of microrotation parameters on velocity profile h2 (η). (b) Influence of microrotation parameters on
velocity profile h2 (η).
Table 1. Thermophysical properties of Base fluid and Nanoparticles.
Thermophysical properties
Cu
TiO2
Al2 O3
Fluid phase (water)
ρ (kg/m )
8933
4250
3970
997.1
Cp (j/kg)K
385
686.2
765
4179
k (W/mK)
400
8.9538
40
0.613
α × 107 (m /s)
1163.1
30.7
131.7
1.47
3
2
Eur. Phys. J. Plus (2017) 132: 441
Page 11 of 12
Table 2. Effects of vortex viscosity (R1 ), boundary parameter (n), porosity parameter (Kp ), nanoparticle volume fraction (φ)
and stretching parameter (λ) on skin friction and Nusselt number.
Cfx
Cfy
θ (0)
1.31474
0.98076
−1.3316
1.34378
1.17343
−1.4267
3
1.39027
1.25559
−1.5172
1
2.00873
1.00437
−1.3157
2.44874
1.22437
−1.4186
2.66306
1.33153
−1.5028
1.39027
1.25559
−1.3316
1.48147
2.59159
−1.3316
1.60774
4.22325
−1.3316
1.39027
1.25559
−1.3316
1.49663
1.29706
−1.3316
1.63583
1.40815
−1.3316
0
1.6855
1.25559
−1.0000
0.1
1.39027
1.17343
−1.3316
0.15
1.18793
1.09006
−1.5266
R1
N
λ
Kp
φ
1
2
2
0.5
0.0
0.5
0.5
1
1
0.1
0.1
3
0.5
3
0.5
1.0
1
0.1
1.5
1
3
0.5
0.5
2
0.1
3
3
0.5
0.5
1
5 Key results
A numerical survey is implemented to investigate the three-dimensional flow of a micropolar nanofluid induced by an
exponentially stretching surface within a porous medium. The influence of pertinent physical parameters on velocities
in the x-, y-directions, angular velocities as well as on the temperature profile is talked about in detail. The main
findings of the current analysis are itemized as follows.
– Increment in nanoparticle volume fraction φ has increasing influence on velocity profiles whereas it decreases the
temperature, while microrotation h2 (η) shows a fluctuating behavior at η = 1.
– The skin friction coefficient escalates for increasing values of φ, while the local Nusselt number shows a declining
behavior.
– The velocity profiles and momentum boundary layer thickness show an increasing behavior for vortex viscosity
parameter R1 .
– Microrotation parameter R1 exhibits decreasing effects on the skin friction coefficient while it shows an aggravating
impact on the rate of heat transfer of the nanofluid.
– The microrotation profiles h1 (η) and h2 (η) have a parabolic distribution when n = 0.
– Both the velocity and angular velocity profiles show a decelerating behavior with increasing values of porosity
parameter Kp , while there is a rise in the temperature distribution.
Page 12 of 12
Eur. Phys. J. Plus (2017) 132: 441
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
B.C. Sakiadis, AIChE J. 7, 26 (1961).
F.K. Tsou, E.M. Sparrow, R. Jh Goldstein, Int. J. Heat Mass Transfer 10, 219 (1967).
P.S. Gupta, A.S. Gupta, Can. J. Chem. 55, 744 (1977).
Ramya, Dodda, R. Srinivasa Raju, J. Anand Rao, J. Nanofluids 6, 541 (2017).
R. Ellahi, R. Ellahi, A. Zeeshan, A. Zeeshan, Mohsan Hassan, Mohsan Hassan, Int. J. Numer. Methods Heat Fluid Flow
26, 2160 (2016).
E. Magyari, B. Keller, J. Phys. D: Appl. Phys. 32, 577 (1999).
I-Chung Liu, Hung-Hsun Wang, Yih-Ferng Peng, Chem. Eng. Commun. 200, 253 (2013).
B. Ahmad, Z. Iqbal, Front. Heat Mass Transf. 8, 22 (2017).
S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, in Proceedings of the 1995 ASME International
Mechanical Engineering Congress and Exposition, San Francisco, CA, USA, Vol. 231 (ASME, 1995) pp. 99-105.
Jacopo Buongiorno, J. Heat Transf. 128, 240 (2006).
J.A. Esfahani, M. Akbarzadeh, Saman Rashidi, M.A. Rosen, R. Ellahi, Int. J. Heat Mass Transfer 109, 1162 (2017).
N. Shehzad, A. Zeeshan, R. Ellahi, K. Vafai, J. Mol. Liq. 222, 446 (2016).
Kamel Milani Shirvan, Mojtaba Mamourian, Soroush Mirzakhanlari, Rahmat Ellahi, Powder Technol. 313, 99 (2017).
S.U. Rahman, R. Ellahi, S. Nadeem, QM Zaigham Zia, J. Mol. Liq. 218, 484 (2016).
M. Akbarzadeh, S. Rashidi, M. Bovand, R. Ellahi, J. Mol. Liq. 220, 1 (2016).
Saman Rashidi, Javad Aolfazli Esfahani, Rahmat Ellahi, J. Appl. Sci. 7, 431 (2017).
Kamel Milani Shirvan, Rahmat Ellahi, Mojtaba Mamourian, Mohammad Moghiman, Int. J. Heat Mass Transfer 107, 1110
(2017).
R. Ellahi, M.H. Tariq, M. Hassan, K. Vafai, J. Mol. Liq. 229, 339 (2017).
M.M. Bhatti, A. Zeeshan, R. Ellahi, Microvasc. Res. 110, 32 (2017).
Kamel Milani Shirvan, Mojtaba Mamourian, Soroush Mirzakhanlari, R. Ellahi, J. Mol. Liq. 220, 888 (2016).
Mohsan Hassan, Ahmad Zeeshan, Aaqib Majeed, Rahmat Ellahi, J. Magn. & Magn. Mater. 443, 36 (2017).
Sohail Nadeem, Rizwan Ul Haq, Zafar Hayat Khan, Alex. Eng. J. 53, 219 (2014).
Avtar Singh Ahuja, J. Appl. Phys. 46, 3408 (1975).
A. Cemal Eringen, Int. J. Eng. Sci. 2, 205 (1964).
A. Cemal Eringen, J. Appl. Math. Mech. 16, 1 (1966).
Grzegorz Lukaszewicz, Micropolar fluids: theory and applications, in Springer Science & Business Media (Springer, 1999).
A. Cemal Eringen, Microcontinuum field theories: II. Fluent media, Vol. 2, Springer Science & Business Media (Springer,
2001).
Ali J. Chamkha, M. Jaradat, I. Pop, Int. J. Fluid Mech. Res., https://doi.org/10.1615/InterJFluidMechRes.v30.i4.10
(2003).
Mohamed Abd El-Aziz, Can. J. Phys. 87, 359 (2009).
B. Mohanty, S.R. Mishra, H.B. Pattanayak, Alex. Eng. J. 54, 223 (2015).
Syed Tauseef Mohyud-Din, Saeed Ullah Jan, Umar Khan, Naveed Ahmed, Neural Comput. Appl.,
https://doi.org/10.1007/s00521-016-2493-3 (2016).
S.T. Hussain, Sohail Nadeem, Rizwan Ul Haq, Eur. Phys. J. Plus 129, 161 (2014).
M.T. Kamel, D. Roach, M.H. Hamdan, in Proceedings of the WSEAS International Conference on Mathematics and Cpmputers in Science and Engineering no. 11 (WSEAS, 2009).
Michael W. Heruska, Layne T. Watson, Kishore Kumar Sankara, Comput. Fluids 14, 117 (1986).
Heidar Hashemi, Zafar Namazian, S.A.M. Mehryan, J. Mol. Liq. 236, 48 (2017).
M. Turkyilmazoglu, Int. J. Nonlinear Mech. 83, 59 (2016).
A. Zeeshan, R. Ellahi, M. Hassan, Eur. Phys. J. Plus 129, 261 (2014).
Документ
Категория
Без категории
Просмотров
2
Размер файла
796 Кб
Теги
11660, epjp, 2fi2017
1/--страниц
Пожаловаться на содержимое документа