Numerical Solutions of Radiation Effect on Magnetohydrodynamic Free Convection Boundary Layer Flow...

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


,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


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)


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.


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


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


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


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

Numerical Solutions of Radiation Effect on Magnetohydrodynamic Free Convection Boundary Layer Flow...

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