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Applied Mathematical Sciences, Vol. 8, 2014, no. 140, 6989 - 7000
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.48649
Numerical Solutions of Radiation Effect on
Magnetohydrodynamic Free Convection Boundary
Layer Flow about a Solid Sphere
with Newtonian Heating
Hamzeh Taha Alkasasbeh* , Mohd Zuki Salleh 1 Futures and Trends Research Group, Faculty of Industrial Science and Technology
University Malaysia Pahang, 26300 UMP Kuantan, Pahang, Malaysia * Corresponding author
Roslinda Nazar
2 School of Mathematical Sciences, Faculty of Science and Technology
Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia
Ioan Pop
3 Department of Mathematics, Babeş-Bolyai University
400084 Cluj-Napoca, Romania
Copyright © 2014 Hamzeh Taha Alkasasbeh et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract In this paper, the effect of radiation on magnetohydrodynamic free convection boundary of a
solid sphere with Newtonian heating has been investigated. The basic equations of boundary
layer are transformed into a non-dimensional form and reduced to nonlinear systems of
partial differential equations are solved numerically using an implicit finite difference scheme
known as the Keller-box method. Numerical solutions are obtained for the wall temperature,
the local skin friction coefficient and the local Nusselt number, as well as the velocity and
6990 Hamzeh Taha Alkasasbeh et al.
temperature profiles. The features of the flow and heat transfer characteristics for various
values of magnetic parameter M, radiation parameter RN and the coordinate running along
the surface of the sphere, x are analyzed and discussed.
Keyword: Magnetohydrodynamic (MHD); Newtonian Heating; Radiation Effects;
Solid Sphere
1. Introduction
The effect of radiation on magnetohydrodynamic (MHD) flow, heat and mass transfer
problems has become industrially more important. Many engineering processes occur
at high temperatures, the knowledge of radiation heat transfer leads significant role in
the design of equipment. Nuclear power plants, gas turbines and various propulsion
devices for aircraft, missiles, satellites and space vehicles are examples of such
engineering processes. At high operating temperature, the radiation effect can be
quite significant, see Sivaiah et al. [1]. Nazar et al. [2,3] considered the free
convection boundary layer flows on a sphere in a micropolar fluid without effect of
radiation and magnetohydrodynamic with constant heat flux (CHF) and constant wall
temperature (CWT), respectively. Molla et al. [4], Akhter and Alim [5] and Miraj et
al. [6] studied the radiation effect on free convection flow from an isothermal sphere
with constant wall temperature, constant heat flux and in presence of heat generation,
respectively. The viscous dissipation and magnetohydrodynamic effect on a natural
convection flow over a sphere in the presence of heat generation have been presented
byGanesan and Palani [7], Alam et al. [8] and Molla et al. [9].
For the condition of Newtonian heating, many research papers were published up
to now. It seems that Merkin [10] was the first to use the term Newtonian heating for
the problem of free convection over a vertical flat plate. Pop et al. [11] presented
results for the problem of free convection boundary layer flow along a vertical
surface in a porous medium with newtonian heating. Recently, Salleh et al. [12-15]
studied the free and mixed convection boundary layer flows past a sphere with
Newtonian heating in a viscous and micropolar fluid using the Keller-box method.
The situation of Newtonian heating arises in what are usually termed conjugate
convective flows, where the heat is supplied to the convective fluid through a
bounding surface with a finite heat capacity (Merkin [10]). This configuration occurs
in many important engineering devices, for example in heat exchanger, where the
conduction in solid tube wall is greatly influenced by the convection in the fluid
flowing over it. Further, for conjugate heat transfer around fins where the conduction
within the fin and the convection in the fluid surrounding it must be simultaneously
analyzed in order to obtain the vital design information and also in convection flows
Numerical solutions of radiation effect 6991
set-up when the bounding surfaces absorb heat by solar radiation see Chaudhary and
Jain [16,17]
Therefore, the aim of the present paper is to study the effect of radiation on MHD
free convection boundary layer flow on a solid sphere with Newtonian heating. On
the other hand, this paper extends the work of Salleh et al. [12], who studied the same
problem but without the effects of radiation and magnetohydrodynamic. The
governing boundary layer equations are first transformed into a system of non-
dimensional equations via the non-dimensional variables, and then into non-similar
equations before they are solved numerically by the Keller-box method, as described
in the book by Cebeci and Bradshaw [18].
2. Mathematical Analyses
Consider a heated sphere of radius a, which is immersed in a viscous and
incompressible fluid of ambient temperature T . It is assumed that the surface of the
sphere is subjected to a Newtonian heating (NH). Under the Boussinesq and
boundary layer approximations, the basic equations are
0)()(
vr
yur
x
(1)
ua
xTTg
y
u
y
uv
x
uu
2
2
2
sin)(
(2)
y
q
cy
T
y
Tv
x
Tu r
12
2
(3)
along with the boundary conditions of Salleh et al. [12]
0u v s
Th T
y
at 0y
0u T T as y , (4)
Here )/sin()( axaxr , u and v are the velocity components along the x and
y directions, respectively. T is the local temperature, rq is the radiative heat flux, g
is the gravity acceleration, is the thermal expansion coefficient, is the kinematic
viscosity, is the fluid density, is the electric conductivity, c the specific heat,
is the thermal diffusivity, fT is the temperature of the hot fluid, k c is the
thermal conductivity, and sh is the heat transfer coefficient for Newtonian heating
condition.
We introduce now the following non-dimensional variables
6992 Hamzeh Taha Alkasasbeh et al.
,a
xx ,4/1
a
yGry ,
a
rr
,2/1 uGra
u
,4/1 vGr
av
T T
T
(5)
where 3 2Gr g T a is the Grashof number for Newtonian heating. Using the
Rosseland approximation for radiation (Bataller [19] or Magyari and Pantokratoras
[20]), the radiative heat flux is simplified as
y
T
kqr
4
*
*
3
4 (6)
where * and
*k are the Stefan-Boltzmann constant and the mean absorption
coefficient, respectively. We assume that the temperature differences within the flow
through the porous medium such as that the term 4T may be expressed as a linear
function of temperature. Hence, expanding 4T in a Taylor series about T and
neglecting higher-order terms, we get 4 3 44 3T T T T (7)
Substituting variables (5)–(7) into (1)–(3) then become
,0)()(
rv
yru
x (8)
,sin2
2
Muxy
u
y
uv
x
uu
(9)
2
2
1 41 ,
Pr 3Ru v N
x y y
(10)
where Pr / is the Prandtl number, 2 2 1\2/M a Gr is the magnetic
parameter, and 3(4 * ) ( * )R pN T k c
is the radiation parameter. The
boundary conditions (4) become
0,u v (1 )y
on 0y
0,u 0 as y (11)
where 1/4
sah Gr is the conjugate parameter for the Newtonian heating.
To solve (8) to (10), subjected to the boundary conditions (11), we assume the
following variables:
( ) ( , ), ( , ),xr x f x y x y (12)
where is the stream function defined as
Numerical solutions of radiation effect 6993
yru
1
and xr
v
1, (13)
which satisfies the continuity equation (8). Thus, (9) and (10) become
2
222
2
2
3
3 sincot1
y
f
x
f
yx
f
y
fx
y
fM
x
x
y
f
y
ffxx
y
f , (14)
2
2
1 41 1 cot
Pr 3R
f fN x x f x
y y y x x y
, (15)
subject to the boundary conditions
,0
y
ff (1 )
y
at 0y
0, 0f
y
as y (16)
It can be seen that at the lower stagnation point of the sphere, ,0x equations (14)
and (15) reduce to the following ordinary differential equations: 22 0f ff f Mf (17)
1 41 2 0
Pr 3RN f
(18)
and the boundary conditions (16) become
(0) (0) 0,f f (0) (1 (0))
0,f 0 as y (19)
where primes do denote differentiation with respect to .y
The physical quantities of interest in this problem are the local skin friction
coefficient, fC and the local Nusselt number, uN which are given by
2
2( ,0)f
fC x x
y
41 1 ( ,0)
3u RN N x
y
(20)
where 2/ ( )f wC U is the skin friction coefficient and 0( / )w yu y is the
wall shear stress.
3. Results and Discussion
The nonlinear system of partial differential equations (14) and (15) subject to the
boundary conditions (16) were solved numerically using an efficient, implicit finite-
difference method known as the Keller-box scheme as described in the book by
Cebeci and Bradshaw [18]. The solution is obtained by the following four steps: (i)
6994 Hamzeh Taha Alkasasbeh et al.
reduce equations (14) and (15) to a first-order system, (ii) write the difference
equations using central differences, (iii) linearize the resulting algebraic equations by
Newton’s method, and write them in the matrix-vector form, (iv) solve the linear
system by the block tridiagonal elimination technique with several parameters
considered, namely, magnetic parameter M = 0, 5 , 10, radiation parameter
0,1,2,3RN , the Prandtl number Pr = 0.7 (air), the conjugate parameter 1 and
the coordinate running along the surface of the sphere, x. The numerical solutions
start at the lower stagnation point of the sphere, ),0( x and proceed round the sphere
up to the point x = 120o.
Table 1 present the values of the wall temperature distribution )(xw and the
local skin friction coefficient, fC for various values of x without effect of radiation
on magnetohydrodynamic free convection boundary layer flow of a solid sphere with
Newtonian heating (i.e. M = 0, 0RN ) when Pr = 0.7 and 1. In order to verify
the accuracy of the present method, the present results are compared with those
reported by Salleh et al. [12]. It is found that the agreement between the previously
published results with the present ones is excellent. We can conclude that this method
works efficiently for the present problem and we are also confident that the results
presented here are accurate.
Table 2 Values of the wall temperature and the skin friction coefficient 2 2( )f y with various values of RN when Pr = 0.7, M = 0, 5 and 1 . It is
observed that, when the magnetic parameter M is fixed an increasing of the radiation
parameter ,RN show that both values of and 2 2( )f y increases, and also when
RN is fixed, an increasing of M the values of and 2 2( )f y increases. It is clear
that the effect of magnetic parameter and radiation parameter are strongly on the
values of the wall temperature and skin friction coefficient
Figs 1 and 2 shows the temperature profiles (0, )y and velocity profiles
( )(0, ),f y y when Pr = 0.7, M= 5, 0,1,2,3RN and 1 , respectively. It is
found that as RN increases, the temperature and velocity profiles increases. The temperature and velocity profiles presented in Figs 3 and 4, respectively,
when Pr = 0.7, 1RN , M = 0, 5, 10 and 1 shows that when the value of M
increases, it is found that the temperature profiles also increases, but the velocity
profiles decreases.
Numerical solutions of radiation effect 6995
Table 1: Values of the wall temperature distribution )(xw and the local skin friction
coefficient fC for various values of x when Pr = 0.7, M = 0, 0RN and 1
x
)(xw
fC
Salleh et al.
[12]
Present Salleh et al.
[12]
Present
0 o 26.4590 26.4592 0.0000
0.0000
10o 56.8602 56.8605 2.8206
2.8208
20o 59.4033 59.4038 5.7090
5.7095
30o 61.1367 61.1369 8.7332
8.7334
40o 62.7065 62.7069 11.5864
11.5867
50o 64.3987 64.3988 14.3102
14.3105
60o 66.2689
66.2692
16.7934
16.7938
70o 68.5102
68.5107
19.1415
19.1419
80o 71.1541
71.1545
21.2356
21.2358
90o 74.2967
74.2970
23.0291
23.0293
100o 78.0623
78.0629
24.4695
24.4697
110o 82.6233
82.6237
25.4947
25.4949
120o 88.2340
88.2343
26.0269
26.0272
Table 2: Values of the wall temperature and the skin friction coefficient 2 2( )f y various values of RN when Pr = 0.7, M = 0, 5 and 1
M = 0 M = 5
RN 2 2( )f y 2 2( )f y
0 26.4592
8.9438 40.0103 10.2295
1 197.8462 24.2837 263.7182 26.5971
2 623.7598 43.4019 776.3894 46.3588
3 1403.9385 65.5576 1676.0197 68.9898
Figs 5 and 6 display the variation of the local Nusselt number uN and the local
skin friction coefficient, fC with various values of x when Pr = 0.7, 1RN M = 0, 5,
10 and 1 , respectively. It found that as x and M increases, the local Nusselt
number and the local skin friction coefficient increases
Again the variation of the local Nusselt number uN and the local friction
coefficient fC with various values of x when Pr = 0.7, M = 5, 0,1,2,3RN and 1
are plotted in Figs 7 and 8, respectively. It found that as x and RN increases, the local
Nusselt number and the local skin friction coefficient increases. It is noted that the
6996 Hamzeh Taha Alkasasbeh et al.
values of radiation parameter have very high effect on the values of local Nusselt
number and the local friction coefficient with increasing of the coordinate running
along the surface of the sphere, starting from the lower stagnation point of the sphere,
( 0),x and proceed round the sphere up to the point 120 .ox
Figure 1: Temperature profiles (0, )y
when Pr = 0.7, M= 5, 0,1,2,3RN
and 1
Figure 2: Velocity profiles ( )(0, ),f y y
when Pr = 0.7, M= 5, 0,1,2,3RN
and 1
Figure 3: Temperature profiles (0, )y ,
when Pr = 0.7, 1,RN M = 0, 5, 10
and 1
Figure 4: Velocity profiles ( )(0, ),f y y
when Pr = 0.7, 1,RN M= 0, 5, 10
and 1
Numerical solutions of radiation effect 6997
Figure 5: Variation of the local Nusselt number
uN with x when Pr = 0.7, 1RN M = 0, 5, 10 and 1
Figure 6: Variation of the local skin friction
coefficient, fC with x when Pr = 0.7,
1,RN M = 0, 5, 10 and 1
Figure 7: Variation of the local Nusselt number
uN with x when Pr = 0.7, M= 5, 0,1,2,3RN and 1
Figure 8: Variation of the local skin friction
coefficient, fC with x when Pr = 0.7, M = 5,
0,1,2,3RN and 1
4. Conclusions
In this paper, we have numerically studied the problem of the effect of radiation
on magnetohydrodynamic free convection boundary layer flow on a solid sphere with
Newtonian heating (NH). It is shown how the magnetic parameter, radiation
parameter, and the coordinate running along the surface of the sphere, x affects on the
6998 Hamzeh Taha Alkasasbeh et al.
values of the temperature and velocity profiles, the skin friction coefficient, the local
Nusselt number and the local friction coefficient.
We can conclude that
when the magnetic parameter M is fixed an increasing of the radiation parameter
,RN the values of wall temperature and the skin friction coefficient increases,
and also when RN is fixed and M increases, the values of wall temperature and
the skin friction coefficient increases
when the value of magnetic parameter M increases, it is found that the
temperature profiles also increases, but the velocity profiles decreases and also
when radiation parameter RN increases, the temperature and velocity profiles
increases
when the values of x, magnetic parameter M and radiation parameter
,RN increases the local Nusselt number and the local skin friction coefficient
increases significantly.
Acknowledgement
The authors gratefully acknowledge the financial supports received from Universiti
Malaysia Pahang (RDU 121302 and RDU 120390).
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Received: August 7, 2014