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Sains Malaysiana 45(2)(2016): 289–296
Free Convection Boundary Layer Flow on a Horizontal Circular
Cylinder in a Nanouid with Viscous Dissipation
(Olakan Bebas Aliran Lapisan Sempadan pada Silinder Bulat Mengufuk
dalam Nanobendalir dengan Pelesapan Likat)
MUHAMMAD KHAIRUL ANUAR MOHAMED, NOR AIDA ZURAIMI MD NOAR, MOHD ZUKI SALLEH* & ANUAR ISHAK
ABSTRACT
In this paper, the problem of free convection boundary layer ow on a horizontal circular cylinder in a nanouid with
viscous dissipation and constant wall temperature is investigated. The transformed boundary layer equations are
solved numerically using nite difference scheme namely the Keller-box method. Numerical solutions were obtained
for the reduced skin friction coefcient, Nusselt number and Sherwood number as well as the velocity and temperature
proles. The features of the ow and heat transfer characteristics for various values of the Brownian motion parameter,
thermophoresis parameter, Lewis number and Eckert number were analyzed and discussed.
Keywords: Free convection; horizontal circular cylinder; nanouid; viscous dissipation
ABSTRAK
Dalam kajian ini, masalah olakan bebas aliran lapisan sempadan pada silinder bulat mengufuk dalam nanobendalir
dengan pelesapan likat dan suhu permukaan malar dikaji. Persamaan lapisan sempadan terjelma diselesaikan secara
berangka dengan menggunakan skim beza terhingga dikenali sebagai kaedah kotak Keller. Penyelesaian berangka
diperoleh bagi pekali geseran kulit diturunkan, nombor Nusselt dan nombor Sherwood diturunkan serta prol halaju
dan suhu. Ciri aliran dan pemindahan haba bagi pelbagai nilai parameter gerakan Brown, parameter termoforesis,
nombor Lewis dan nombor Eckert dianalisis dan dibincangkan.
Kata kunci: Nanobendalir; olakan bebas; pelesapan likat; silinder bulat mengufuk
INTRODUCTION
In recent years, many investigations have been made on
the ow of a nanouid in a convective boundary layer
past various types of surface such as stagnation point,
stretching sheet, horizontal circular cylinder as well as a
solid sphere. This type of uid is believed can enhanced
thermal conductivity, viscosity, thermal diffusivity and
convective heat transfer compared to those base uids like
water and oil. This has made nanouids employed in plenty
of important applications which involves uid as cooling
medium in industrial outputs for example act as smart uid
in battery devices, as nanouid coolant in car radiator,
brake uid, fuel catalyst to improve engine combustion and
also act to cooling microchip in electronic devices (Wong
& De Leon 2010). Based on the large contributions and
issues, this topic has attracted many researchers to study
and expand this knowledge for example from the works
by Arin et al. (2011), Bachok et al. (2010), Kakaç and
Pramuanjaroenkij (2009), Khan and Pop (2010), Nazar
et al. (2011), Tiwari and Das (2007), Yacob et al. (2011),
Tham and Nazar (2012) and recently by Anwar et al.
(2013), Roşca and Pop (2014), Tham et al. (2014) and
Yusoff et al. (2014).
The study of boundary layer ow on a horizontal
circular cylinder was rst studied by Blasius (1908).
Blasius (1908) successfully solved the momentum equation
of forced convection boundary layer ow. The energy
equation for this problem was then solved by Fr¨ossling
(1958) with considering the constant wall temperature
(CWT). Since then, this topic has attracted many researchers
to study the constant wall temperature and constant heat
ux. Merkin (1977, 1976) considered the free and mixed
convection boundary layer on an isothermal horizontal
cylinder with CWT and became the rst who obtained the
exact solution for this problem. Merkin and Pop (1988)
updated this topic with constant heat ux. Next, Nazar
et al. (2002) extended the work by Merkin (1976) and
Merkin and Pop (1988) to a micropolar uid. Molla et
al. (2006) investigated the heat generation effects on
free convection ow on an isothermal horizontal circular
cylinder before Tahavvor and Yaghoubi (2010) done the
experimental and numerical study of frost formation on a
free convection over a cold horizontal circular cylinder.
Salleh and Nazar (2010) extended Nazar et al. (2002) work
with Newtonian heating. Recently, Rosca et al. (2014)
studied the mixed convection boundary layer ow close to
the lower stagnation point of a horizontal circular cylinder.
The stability analysis for dual solution is discussed and
it is concluded that the upper branch solutions are stable
and physically realizable, while the lower branch solutions
290
are unstable. Singh and Makinde (2014) observed axis
symmetric slip ow on a vertical cylinder while Sarif et al.
(2014) updated Nazar et al. (2002) and Salleh and Nazar
(2010) works with convective boundary conditions. These
three problems were successfully solved numerically using
the Keller-box method.
It is known that the viscous dissipation or internal
friction is the rate of the work done againts viscous forces
which is irreversibly converted into internal energy. The
effect of viscous dissipation is signicant especially for
high velocity ow, highly viscous ow with moderate
velocity and for uid with moderate Prandtl number and
velocities. Hence, viscous dissipations effect is important
to study in order to understand the behavior of temperature
distributions when the internal friction cannot be neglected.
Gebhart (1962) is the rst person who studied viscous
dissipation in free convection ow. The viscous dissipation
effects on unsteady free convective ow over a vertical
porous plate was then investigated by Soundalgekar
(1972). Vajravelu and Hadjinicolaou (1993) then studied
the viscous dissipation effects on a stretching sheet. Chen
(2004) and Partha et al. (2005) observed the mixed and
MHD free convection heat transfer from a vertical surface
and exponentially stretching surface with Ohmic heating
and viscous dissipation, respectively. Recently, Yirga and
Shankar (2013) considered this topic with thermal radiation
and magnetohydrodynamic effects on the stagnation point
ow towards a stretching sheet.
Motivated by the above contributions, the purpose
of the present study is to investigate the free convection
boundary layer ow towards a horizontal circular cylinder
in a nanouid by including the viscous dissipation effect.
The governing partial differential equations were solved
numerically and the variation of pertinent physical
parameters were analyzed and discussed with the aid of
tables and proles.
MATHEMATICAL FORMULATION
Consider a horizontal circular cylinder of radius a, which
is heated to a constant temperature T
w
embedded in a
nanouid with ambient temperature T
∞
as shown in Figure
1. The orthogonal coordinates of and is measured along
the cylinder surface, starting with the lower stagnation
point = 0, and normal to it, respectively. Under the
assumptions that the boundary layer approximations is
valid, the dimensional governing equations of steady free
convection boundary layer ow are (Khan & Pop 2010;
Salleh & Nazar 2010):
(1)
(2)
(3)
(4)
subject to the boundary conditions
(5)
where and are the velocity components along the
and axes, respectively; μ is the dynamic viscosity; ν
is the kinematic viscosity; g is the gravity acceleration;
β is the thermal expansion; T is local temperature;
ρ
is
the uid density; and C
p
is the specic heat capacity at
a constant pressure. Furthermore, C is the nanoparticle
volume fraction, C
w
is the nanoparticle volume fraction C
at the surface and C
∞
is the ambient nanoparticle volume
fraction C.
FIGURE 1. Physical model of the coordinate system
Next, it introduced the governing non-dimensional
variables:
(6)
where θ and φ are the rescaled dimensionless temperature
and nanoparticle volume fraction of the fluid and
is the Grashof number. Using (6), (1)-
(4) becomes:
(7)
(8)
(9)
291
(10)
subject to the boundary conditions:
u(x, 0) = 0, v(x, 0) = 0, θ(x, 0) = 1, φ(x, 0) = 1,
u(x, ∞)→ 0, θ(x, ∞)→ 0, φ(x, ∞)→ 0. (11)
In order to solve (7) to (10), the following functions
were introduced:
ψ = xf(x, y) θ = θ(x, y), φ = φ(x, y), (12)
where ψ is the stream function dened as and
which identically satises (7). Substitute (12)
into (7)-(10), the following partial differential equations
were obtained:
(13)
(14)
(15)
where is the Prandtl number; is
the Brownian motion parameter, is the
thermophoresis motion parameter, is
an Eckert number; and is the Lewis number. The
boundary conditions (11) becomes:
(16)
The physical quantities of interest are the skin
friction coefcient C
f
, the local Nusselt number Nu
x
and
the Sherwood number Sh
x
which are given by Molla et al.
(2006)
(17)
where
ρ
is the uid density. The surface shear stress τ
w
the surface heat ux q
w
and the surface mass ux j
w
are
given by:
(18)
with and k being the dynamic viscosity and the
thermal conductivity, respectively.
Using variables (6) and (12) give:
(19)
Furthermore, the velocity proles and temperature
distributions can be obtained from the following relations:
u = f´(x, y), θ = θ(x, y), (20)
NUMERICAL METHOD
The partial differential equations (13) to (15) subject
to boundary conditions (16) are solved numerically
using the Keller-box method, which is an implicit nite
difference method in conjunction with Newton’s method
for linearization. This made it suitable to solve parabolic
partial differential equations. The previous studies which
used Keller-box method in solving the boundary layer
problems including Ishak et al. (2007, 2006), Nazar et al.
(2004, 2003) and Salleh et al. (2011, 2009).
RESULTS AND DISCUSSION
Equations (13)-(15) subject to the boundary conditions (16)
were solved numerically using the Keller-box method with
ve parameters considered, namely the Prandtl number
Pr the Brownian motion parameter N
b
, thermophoresis
parameter N
t
, Lewis number Le and the Eckert number
Ec. The step size ∆y = 0.02, ∆x = 0.005 and boundary
layer thickness y
∞
= 8 and x
∞
= π are used in obtaining the
numerical results. Furthermore, the value of Pr is set to be
Pr = 7 which represents water that usually acts as the base
uid for the nanouid. Tables 1 and 2 show the comparison
values of Nu
x
Gr
–1/4
and CfGr
1/4
with previous results for
various values of x, respectively. It has been found that
they are in good agreement. It was concluded that this
method works efciently hence, the results presented here
are condently accurate.
Figure 2 illustrated the variations of reduced skin
friction coefcient C
f
Gr
1/4
against x for various values of N
b
and N
t
. From this gure, it was concluded that the effects
of N
b
and N
t
are more pronounced as x increases to the
middle of cylinder. It is clearly shown that the increase of
N
b
and N
t
results to the increase of C
f
Gr
1/4
.
292
Next, Figures 3 and 4 show the variations of the
reduced Nusselt number Nu
x
Gr
–1/4
and Sherwood number
Sh
x
Gr
–1/4
for various values of N
b
and N
t
, respectively. In
Figure 3, it was found that Nu
x
Gr
–1/4
decreases as N
b
and
N
t
increase. It was found that, the higher values of N
b
and
N
t
subsequently results to higher volume of nanoparticles
migrating away from the vicinity of the wall and thus
reduces the value of the Nusselt number. In Figure 4, the
trend is contrary with Figure 3 where an increase of N
b
and N
t
results to the increase of Sh
x
Gr
–1/4
. Furthermore,
from both gures, it was suggested that as the ow past
the cylinder, the inuence of N
b
and N
t
getting lesser than
the stagnation region (x ≈ 0).
Figures 5 and 6 display the velocity and temperature
proles for various values of N
b
and N
t
, respectively. It was
found that the increase of N
b
and N
t
results to the increase
in velocity and temperature distribution. This phenomenon
is realistic where the presence of nanoparticles in nanouid
will enhance the heat transfer characteristics as well as
thermal conductivity. Furthermore, it was worth to state
that N
b
and N
t
do not inuence much in the changes of
boundary layer thicknesses.
In order to understand the behaviour of Lewis number
Le and Eckert number Ec in this convective boundary layer
ow, Figures 7 to 11 were plotted. Figures 7 to 9 show the
variations of C
f
Gr
1/4
, Nu
x
Gr
–1/4
and Sh
x
Gr
–1/4
with various
TABLE 1. Comparison values of Nu
x
Gr
–1/4
with previous published results for various values
of x when Pr = 1 and N
b
= N
t
= Le = Ec = 0
x
Merkin
(1976)
Nazar et al.
(2002)
Molla et al.
(2006)
Salleh and
Nazar (2010)
Azim
(2014)
Present
0
π/6
π/3
π/2
2π/3
5π/6
π
0.4214
0.4161
0.4007
0.3745
0.3364
0.2825
0.1945
0.4214
0.4161
0.4005
0.3741
0.3355
0.2811
0.1916
0.4214
0.4161
0.4005
0.3740
0.3355
0.2812
0.1917
0.4214
0.4162
0.4006
0.3744
0.3360
0.2817
0.1939
0.4216
0.4163
0.4006
0.3742
0.3356
0.2811
0.1912
0.4214
0.4163
0.4008
0.3744
0.3364
0.2824
0.1939
TABLE 2. Comparison values of C
f
Gr
1/4
with previous published results for various values
of x when Pr = 1 and N
b
= N
t
= Le = Ec = 0
x Merkin
(1976)
Nazar et al.
(2002)
Molla et al.
(2006)
Azim
(2014)
Present
0
π/6
π/3
π/2
2π/3
5π/6
π
0.0000
0.4151
0.7558
0.9579
0.9756
0.7822
0.3391
0.0000
0.4148
0.7542
0.9545
0.9698
0.7740
0.3265
0.0000
0.4145
0.7539
0.9541
0.9696
0.7739
0.3264
0.0000
0.4139
0.7528
0.9526
0.9678
0.7718
0.3239
0.0000
0.4121
0.7538
0.9563
0.9743
0.7813
0.3371
FIGURE 2. Variation of reduced skin friction coefcient C
f
Gr
1/4
against x for various values of N
b
and N
t
293
FIGURE 3. Variation of reduced Nusselt number Nu
x
Gr
–1/4
against x for various values of N
b
and N
t
FIGURE 4. Variation of reduced Sherwood number Sh
x
Gr
-1/4
against x for various values of N
b
and N
t
FIGURE 5. Velocity proles f́(y) against y for
various values of N
b
and N
t
FIGURE 6. Temperature proles θ(y) against y for various
values of N
b
and N
t
FIGURE 7. Variation of reduced skin friction coefcient C
f
Gr
1/4
against x for various values of Ec and Le
FIGURE 8. Variation of reduced Nusselt number Nu
x
Gr
–1/4
against x for various values of Ec and Le
294
values of Ec and Le. From Figure 7, it was found that the
value of C
f
Gr
1/4
is unique for all values of Ec and Le at
the early stage. From this gure, it was understand that
Ec and Le gave a small inuence on C
f
Gr
1/4
. The effects
of Ec and Le are signicance as x increase to the middle
of cylinder then converge back at the end of the cylinder
(x = π). In Figures 8 and 9, it is notice that the effect of
Ec is negligible at the lower stagnation region (x = 0). As
x increases, Nu
x
Gr
–1/4
decreases for all set of parameter
Ec and Le. Furthermore, the increase of both parameters
Ec and Le may results to the decrease of Nu
x
Gr
–1/4
while
Sh
x
Gr
–1/4
increase. This is due to the mass diffusivity is
more dominant than thermal diffusivity, where the transient
mass response is quicker than the transient thermal
response to the change of Le and temperature.
Lastly, Figures 10 and 11 display the temperature
and velocity proles for various values of Ec and Le,
respectively. It was found that the effect of Le is very small
while Ec gives no effects on the temperature and velocity
distributions as well as the boundary layer thicknesses. The
increase of Le results to the slightly increase in thermal
boundary layer thickness and velocity distribution.
CONCLUSION
In this paper, we have numerically studied the problem
of free convection boundary layer ow on a horizontal
circular cylinder in a nanouid with viscous dissipation
and constant wall temperature. It was concluded that the
increase of the Brownian motion parameter, thermophoresis
parameter, Lewis number and Eckert number results in the
increase of skin friction coefcient and Sherwood number
while Nusselt number decreases. This is due to the increase
of nanouid parameters which increase the volume of
nanoparticles migrating away from the vicinity of the
wall and thus reduce the value of the Nusselt number.
Furthermore, it was found that the Eckert number which
represents the viscous dissipation effect gives no effects
on the temperature and velocity proles.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge the financial
support received from the Universiti Malaysia Pahang
(RDU150101& RDU140111).
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Muhammad Khairul Anuar Mohamed, Nor Aida Zuraimi Md
Noar & Mohd Zuki Salleh*
Applied & Industrial Mathematics Research Group
Faculty of Industrial Sciences and Technology
Universiti Malaysia Pahang
26300 Kuantan, Pahang Darul Makmur
Malaysia
Anuar Ishak
School of Mathematical Sciences
Faculty of Science and Technology
Universiti Kebangsaan Malaysia
43600 Bangi, Selangor Darul Ehsan
Malaysia
*Corresponding author; email: zukikuj@yahoo.com
Received: 10 April 2015
Accepted: 2 July 2015