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Journal of Applied Science and Technology

Vol.7, No.2, December 2010, pp 9-16

ISSN 2218-841X

*Corresponding author:

E-mail: ibrahim25si@yahoo.com (M.I. Abdullah)

The Effects of Radiation and Viscous Dissipation on MHD Natural Convection

Flow along a Vertical Flat Plate in Presence of Joule Heating

M.M. Ali1, R. Akhter2, N.H.M.A. Azim3, M.I. Abdullah4*

1 Faculty of life science, Mawlana Bhashani Science and Technology University,

Tangail-1902, Bangladesh

2 Department of Mathematics, Bangladesh University of Engineering and Technology,

Dhaka-1000, Bangladesh

3 School of Business Studies, Southeast University, Dhaka-1000, Bangladesh

4 Department of Computer Science and Engineering, Islamic University, Kushtia-Jhenidah,

Bangladesh

(Received 14 April 2010; Revised 26 July 2010)

ABSTRACT

In this analysis, the effects of radiation, joule heating and viscous dissipation on Magneto-Hydrodynamic

(MHD) natural convection flow along a vertical flat plate in presence of heat conduction are investigated. The

governing equations associated with boundary conditions for this analysis are transformed into dimensionless

form using appropriate similarity transformations and then solved numerically adopting implicit finite

difference method. The resulting numerical solutions are presented graphically in terms of velocity profile,

temperature distribution, local skin friction coefficient in terms of shear stress and local heat transfer rate in

terms of Nusselt number and the effects of magnetic parameter (M), radiation parameter (R), Prandtl number

(Pr), viscous dissipation parameter (N) and joule heating parameter (J) on the flow have been studied.

Keywards: Radiation, Viscous dissipation, Joule heating, MHD, Finite difference method, Vertical flat plate.

INTRODUCTION

The concept of natural convection flow of an

incompressible, viscous and electrically conducting

fluid in presence of transverse magnetic field is

important from technical point of view. Such

problems have received much attention due to its

potential application in nuclear power plants, cooling

of transmission lines and electric transformer etc. In

addition, the significant applications of MHD natural

convection flow are observed in the field of stellar

and planetary magnetospheres, aeronautics, chemical

engineering and electronics. As the engineering

processes closely related with temperature,

accordingly radiation heat transfer has significant

influence on engineering. Due to its wide

applications in space technology such as cosmical

flight, aerodynamics rocket, propulsion systems,

plasma physics, spacecraft re-entry aerodynamics and

at high operator’s temperature, many researchers

have studied the effect of radiation on MHD free

convection flow. Takhar and Soundalgekar1 have

studied the effect of radiation on MHD free

convection flow of a gas past a sami-infinite vertical

plate using the Cogley-vincenti-Giles equilibrium

model (Cogley et al.). The problem of natural

convection-radiation interaction on boundary layer

flow with Rosscland diffusion approximation along a

vertical thin cylinder has been investigated by

Hossain and Alim2. Radiation effect on free

convection flow of fluid from a porous vertical plate

was studied by Hossain et al.3

Abdel-naby et al.4 have studied the radiation effects

on MHD unsteady free convection flow over a

vertical plate with variable surface temperature.

Furthermore, the viscous dissipation heat in the

natural convection flow is important when the flow

field is of extreme size or in high gravitational field.

Israel Cookey et al.5 have investigated the influence

of viscous dissipation and radiation on unsteady

magnetohydrodynamic free convection flow past a

finite vertical heated plate in an optically thin

environment with time dependent-suction. Effect of

viscous dissipation and radiation on natural

convection flow in a porous medium embedded with

vertical annulus was studied by Budruddin et al.6.

Hossain7 have analyzed the effect of viscous and

joule heating effects on MHD free convection flow

with variable plate temperature. Combined effect of

viscous dissipation and joule heating on coupling of

conduction and free convection along a vertical flat

plate have been discussed by Alim et. al.8. Analytical

solutions for momentum and energy equations have

been arrived and the effect of radiation on mean

velocity and temperature discussed.

To the best of our knowledge, no one has done the

combined effect of radiation, joule heating and

viscous dissipation under the process of steady flow

but it has importance in the field of engineering heat

transfer problem. So, the objective of the present

study is to study the effect of radiation, joule heating

and viscous dissipation on MHD natural convection

flow of an incompressible, viscous and electrically

conducting fluid along a vertical plate under the

influence of transverse magnetic field in presence of

heat conduction. The governing partial differential

equations are reduced to locally non-similar partial

M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16

10

differential forms by using appropriate

transformations. The transformed boundary layer

equations are solved numerically adopting implicit

finite difference method together with Keller Box

Scheme technique 9, 10. Here, the assumption is

focused on the evaluation of the surface shear stress

in terms of local skin friction and the rate of heat

transfer in terms of local Nussetl number, velocity

profiles as well as temperature distribution for

selected values of parameters consisting magnetic

parameter M , radiation parameter R, Prandtl number

Pr, viscous dissipation parameter N and joule heating

parameter J.

MATHEMATICAL ANALYSIS

Let us consider a steady natural convection flow of

an incompressible, viscous and electrically

conducting fluid along a vertical flat plate of length l

and thickness b. A constant temperature, Tb is

maintained at the outer surface, where ∞

>TTb. ∞

T is

the ambient temperature of the fluid and uniform

magnetic field of strength H0 is applied along the y-

axis. The flow configuration and the coordinates

system are shown in Fig.-1.

The governing equation of such flow under the

Bousinesq approximation can be expressed within the

usual boundary layer as 4, 8:

0=

∂

∂

+

∂

∂

y

v

x

u (1)

ρ

σ

βν

uH

TTg

yu

y

u

v

x

u

uf

0

2

2

2

)( −−+

∂

∂

=

∂

∂

+

∂

∂

∞ (2)

()

p

bf

p

f

p

fff

CuH

TT

y

u

c

y

T

C

k

y

T

v

x

T

u

ρ

σ

ν

ρ

22

0

2

2

2

4+−Γ−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

=

∂

∂

+

∂

∂

(3)

Where

λ

λ

λ

d

T

e

K

w

b

w⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

=Γ ∫∞

0

,

(

)

ww TKK

λλ

= is

the mean absorption coefficient,

λ

b

eis Plank’s

function and

T

is the temperature of the fluid in the

boundary layer. Where

ν

is the kinematic viscosity,

β is the Thermal expression co-efficient, σ is the

Electrical conductivity, Cp is the specific heat due to

constant pressure. The boundary conditions are:

()

()

⎪

⎪

⎭

⎪

⎪

⎬

⎫

>∞→→→

>=−=

∂

∂

===

∞0,,0

0,0

,0,,0

xyatTTu

xyatTT

kbk

y

T

xTTvu

f

bf

f

s

f

f

(4)

We observe that the equations (1) (2) and (3) together

with the boundary conditions (4) are nonlinear partial

differential equations. Equations (1) (2) and (3) may

now be non-dimensionalzed by using the following

dimensionless dependent and independent variables:

()

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎭

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎬

⎫

Γ

=

=

−

=

−

−

==

===

−

−

∞

∞

∞

−

−

2/1

2

2/1

2

0

2

3

4/1

2/1

4

,

,

,

,,,

Gr

l

R

Gr

lH

M

TTlg

Gr

TT TT

Gr

lv

v

Gr

lu

u

l

y

y

l

x

x

b

b

ν

μ

σ

ν

β

θ

ν

ν

(5)

Where ⎟

⎠

⎞

⎜

⎝

⎛

=

ρ

μ

v is the kinematics viscosity, Gr is

the Grashof number and θ is the non-dimensional

temperature. Using equation (5) in equations (1) (2)

and (3), we can obtain the governing equations in a

dimensionless form as:

0=

∂

∂

+

∂

∂

y

v

x

u (6)

θ

+

∂

∂

=+

∂

∂

+

∂

∂

2

2

yu

Mu

y

u

v

x

u

u (7)

2

2

2

2

)1(

Pr

1

uJ

y

u

N

R

y

y

v

x

u

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

−−

∂

∂

=

∂

∂

+

∂

∂

θ

θθθ

(8)

The corresponding boundary conditions are:

⎪

⎭

⎪

⎬

⎫

>∞→→→

>=

∂

∂

=−==

0,0,0

0,01,0

xyatu

xyat

y

pvu

θ

θ

θ

(9)

Here 2/1

2

2

0−

=Gr

lH

M

μ

σ

is the magnetic parameter,

2/1

2

4−

Γ

=Gr

l

R

ν

is the radiation parameter and the

In

su

lat

or

b

l

s

T

b

T

b

−

0

y

v

Interface

g

∞

T

Figure 1. Physical model and coordinate system.

0

H

0

H

(

)

0,xT

u

x

M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16

11

f

p

K

C

μ

=Pr is the Prandtl number,

()

∞

−

=TTlC

Gr

N

bp 2

2

ν

is

the viscous dissipation parameter,

)(

2/12

0

∞

−

=TTC GrH

Jbp

ρ

νσ

is

the joule heating parameter and 4/1

Gr

l

b

k

k

ps

f⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

= is

a conjugate conduction parameter. The value of the

conjugate conduction parameter p depends on

s

f

k

k

l

b,

and Gr but each of which depends on the types of

considered fluid and the solid. Therefore in different

cases p is different but not always a small number. In

the present analysis we have taken p=1.The steam

function, similarity variable and the dimensionless

temperature are considered in the following form to

solve the equations (7) and (8) and for the boundary

conditions described in equation (9):

),()1(

,)1(

),,()1(

5/15/1

20/15/1

20/15/4

ηθ

η

ηψ

xhxx

xyx

xfxx

−

−−

−

+=

+=

+=

(10)

Where ψ is the dimensionless stream function which

is related to the velocity components such as y

u∂

∂

=

ψ

and

x

v∂

∂

−=

ψ

, ),(

η

xh is a dimensionless

temperature. By substituting (10) in equations (7)

and (8) and transforming these equations in the new

co-ordinate system, we obtain the transformed

equations:

)()1(

)1(10

56

)1(20

1516

10/1

5/22

x

f

f

x

f

fxhfx

Mxf

x

x

ff

x

x

f

∂

∂

′′

−

∂

′

∂

′

=+

′

+

−

′

+

+

−

′′

+

+

+

′′′

(11)

)(

)1()1(

)1(

)1(5

1

)1(20

1516

Pr

1

2

210/15/710/35/1

10/15/2

x

f

h

x

h

fxfxN

fxxJxxR

hxxRhf

x

hf

x

x

h

∂

∂

′

−

∂

∂

′

=

′′

+

′

++++

+−

′

+

−

′

+

+

+

′′

(12)

The boundary condition (9) become

⎪

⎭

⎪

⎬

⎫

∞→→∞→∞

′

=++

+−=

′

=

′

=

yatxhxf

yatxhxx

xxhxfxf

0),(,0),(

0)0,()1(

)1()0,(,0)0,()0,(

20/15/1

4/1

(13)

The equations (11) and (12) together with the

boundary condition (13) are solved numerically by

applying Implicit finite difference method with

Keller-Box (1978) Scheme. Its application to the

boundary layer flow problems have been analyzed

Cebeci, Brad and Shaw (1984) from the process of

numerical computation in practical point of view10. It

is important to calculate the values of the rate of heat

transfer and the skin friction co-efficient surface

shear stress. This can be written in the dimensionless

form as:

wf lGr

C

τ

μν

24/3−

=&

()

w

bf

uq

TTk Grl

N

∞

−

−

=

4/1

(14)

where

0=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

=

y

wy

u

μτ

and w

q=

0=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

−

y

f

fy

T

kare the

shearing stress and the heat flux

Thus the local skin friction co-efficient and rate of

heat transfer as

)0,()1( 20/35/2 xfxxCfx ′′

+= − (15)

)0,()1( 4/1 xhxNux ′

+−= − (16)

We have discussed the velocity profiles and

temperature distributions for various values of

Prandtl number, magnetic parameter, radiation

parameter, viscous dissipation parameter and joule

heating parameter in this paper.

RESULTS AND DISCUSSION

The resulting numerical solutions for the velocity and

the temperature field are shown in Fig. 2 to fig. 11.

The values of Prandtl number are taken to be 0.733,

0.900, 1.000 and 1.446 which correspond to air,

ammonia, steam and water at temperatures KT,

1300 ,

KT,

273 and Ct,

120 , respectively. The detailed

numerical solutions have been obtained a wide range

of values of the magnetic parameter M= (0.10, 0.50,

0.80, 1.00), radiation parameter R= (0.001, 0.010,

0.30, 0.050), viscous dissipation parameter N= (0.01,

0.20, 0.50, 1.00) and joule heating parameter J=

(0.05, 0.20, 0.50, 0.80).

0.0 2.0 4.0 6.0 8.0 10.0

η

0.0

0.1

0.2

0.3

0.4

0.5

Velocity

M=0.10

M=0.50

M=0.80

M=1.00

(a)

0.0 2.0 4.0 6.0 8.0 10.0

η

0.0

0.2

0.4

0.6

0.8

Temperature

M=0.10

M=0.50

M=0.80

M=1.00

(b)

M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16

12

Figure 2. (a) Variation of velocity and (b) variation

of temperature against

η

for varying of M with Pr =

0.733, R = 0.001, N=0.01and J = 0.05.

Fig. 2 (a) shows that the velocity profiles of fluid

decreases with the increase of magnetic parameter

due to the interaction of the applied magnetic field

and flowing fluid particles. But for each value of M,

the velocity is zero at the boundary wall and then

increase to the peak value as

η

increases and then

turn to decrease, finally approach to zero. Moreover,

we have seen that the velocity profiles meet together

after certain position of

η

and cross the side. This is

because, the gradient of decreasing velocity increases

with the increase of magnetic parameter. Reverse

situation is observed from Fig. 2(b) in the case of

temperature field. It shows that the magnetic field

works to increase the values of temperature within

the flow region and also increase the thickness of the

thermal boundary layer.

0.0 2.0 4.0 6.0 8.0 10.0

η

0.0

0.1

0.2

0.3

0.4

0.5

Velocity

R=0.001

R=0.010

R=0.030

R=0.050

(a)

0.0 2.0 4.0 6.0 8.0 10.0

η

0.0

0.2

0.4

0.6

0.8

Temperature

R= 0.001

R= 0.010

R= 0.030

R= 0.050

(b)

Figure 3. (a) Variation of velocity and (b) variation

of temperature against

η

for varying of R with Pr =

0.733, M= 0.50, N=0.01and J = 0.05.

The effect of radiation parameter R on velocity and

the temperature distributions for selected value of

Prandtl number Pr, magnetic parameter M,

dissipation parameter N and joule heating parameter J

are presented in Fig. 3 (a) and Fig. 3 (b), respectively.

The velocity and temperature of the fluid increase

with the increase of R due to the absorption of heat

while radiation imitates from the heated plate. It can

be noted that, near the surface of the flat plate,

velocity increases significantly and becomes

maximum. It then decreases along horizontal

direction. The maximum values of the velocity are

0.3909, 0.4070, 0.4380 and 0.4650 for R = 0.001,

0.010, 0.030 and 0.050 at η = 1.3827, 1.4279 1.4741

and 1.5214. From Fig. 3 (a) and also numerical

values we have seen that the velocity profiles shift

upward and the position of the velocity peak moves

toward the boundary layer for the increasing R. As

the temperature of the fluid increases with increasing

radiation as shown in Fig. 3(b), accordingly the

thickness of the boundary layer and also boundary

layer velocity increases observed in Fig. 3.

0.02.04.06.08.010.0

η

0.0

0.1

0.2

0.3

0.4

0.5

Velocity

Pr=0.733

Pr=0.900

Pr=1.000

Pr=1.446

(a)

0.0 2.0 4.0 6.0 8.0 10.0

η

0.0

0.2

0.4

0.6

0.8

Temperature

Pr=0.733

Pr=0.900

Pr=1.000

Pr=1.446

(b)

Figure 4. (a) Variation of velocity and (b) variation

of temperature against

η

for varying of Pr with M =

0.50, R = 0.001, N=0.01and J = 0.05.

The variation of velocity profile and temperature

distribution for different values of Prandtl number Pr

with magnetic parameter M = 0.50, radiation

parameter R = 0.001, viscous dissipation parameter N

= 0.01 and joule heating parameter J=0.05 are shown

in Fig. 4(a) and Fig. 4 (b) respectively. The viscosity

of the fluid is measured with the increase of Pr. As a

results fluid does not move freely. Then it can be

concluded that, velocity of fluid decrease when the

value of Pr increases. Fig. 4 (b) shows temperature

distribution of boundary layer that decreases due to

the increase of Pr. The temperature at the solid/fluid

interface is reduced because the temperature at the

plate considered is constant. As a result, thermal

boundary layer thickness decreases with the increase

of Pr.

Fig. 5 (a) and Fig. 5(b) illustrates the effects of N on

velocity and temperature for Pr = 0.733, M = 0.50, R

=0.001 and J =0.05. From Fig. 5(a), it is seen that the

velocity profiles increases slightly due to the increase

of viscous dissipation parameter N along the η

direction and also we have seen that near the surface

of the plate the velocity is increased to a maximum

value with the increase of viscous dissipation

parameter N and then after the peak position start to

decrease and finally approaches to zero. The

maximum values of the velocities are 0.3909, 0.3954,

0.4026 and 0.4151 which occurs for N = 0.01, 0.20,

M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16

13

0.50 and 1.00 respectively. On the other hand, from

Fig. 5(b), we have seen that the same result is holds

for temperature distributions within the boundary

layer due to increasing of viscous dissipation

parameter N. The maximum value of temperature are

0.8800, 0.8941, 0.9178 and 0.9615 for viscous

dissipation parameter N =0.01, 0.20, 0.50 and 1.00,

respectively. Each of which occurs at the surface of

the plate. This is because; the increased value of

dissipation parameter N produces heat and also

dissipates viscosity of fluid.

0.0 2.0 4.0 6.0 8.0 10.0

x

0.0

0.1

0.2

0.3

0.4

0.5

Vlocity

N= 0.01

N= 0.20

N= 0.50

N= 1.00

(a)

0.0 2.0 4.0 6.0 8.0 10.0

η

0.0

0.2

0.4

0.6

0.8

1.0

Temperature

N=0.01

N=0.20

N=0.50

N=1.00

(b)

Figure 5. (a) Variation of velocity and (b) variation

of temperature against

η

for varying of N with Pr =

0.733, M=0.50, R = 0.001 and J = 0.05.

0.02.04.06.08.010.0

η

0.0

0.1

0.2

0.3

0.4

0.5

Velocity

J=0.05

J=0.20

J=0.50

J=0.80

(a)

0.0 2.0 4.0 6.0 8.0 10.0

η

0.0

0.2

0.4

0.6

0.8

1.0

Temperature

J= 0.05

J= 0.20

J= 0.50

J= 0.80

(b)

Figure 6. (a) Variation of velocity and (b) variation

of temperature against

η

for varying of J with Pr =

0.733, M=0.50, R = 0.001 and N=0.01.

The figures (6.a-b) shows that the effect of joule

heating parameter on velocity and temperature field

while the controlling parameters are Pr=0.733,

M=0.50, R=0.001 and N=0.01. As the effect of joule

heating parameter produce temperature in the

conductor, therefore the velocity and temperature of

the fluid increase associated with the increasing

values of joule heating parameter J. Moreover, the

maximum values of the velocity are 0.3909, 0.3987,

0.4146 and 0.4209 for J =0.05, 0.20, 0.50 and 0.80

and each of which occurs at

η

=1.3827. Furthermore,

the maximum value of temperature are 0.8800,

0.8889, 0.9078 and 0.9282 for joule heating

parameter J= 0.05, 0.20, 0.50 and 0.80, respectively

and each of which occurs at the surface of the plate.

Fig. 7(a) and Fig. 7(b), reveal that the skin friction

coefficient and the rate of heat transfer for some

values of M with Pr = 0.733, R = 0.001, N =0.01 and

J=0.05. The velocity decreases as shown in the Fig.

2(a), due to the increasing M .Accordingly, the skin

friction on the plate decreases as observed in Fig.

7(a). Moreover, the temperature within the boundary

layer increases (Fig. 2(b)) for the increasing M. As a

result, the heat transfer rate from the plate to fluid

decreases as shown in figure 7(b).

0.01.02.03.04.05.0

x

0.0

0.3

0.6

0.9

1.2

Skin friction

M=0.10

M=0.50

M=0.80

M=1.00

(a)

0.0 1.0 2.0 3.0 4.0 5.0

x

0.0

0.1

0.2

0.3

0.4

0.5

Heat transfer

M=0.10

M=0.50

M=0.80

M=1.00

(b)

Figure 7. (a) Variation of skin friction and (b)

variation of heat transfer against x for varying of M

with Pr = 0.733, R = 0.001, N = 0.01 and J=0.05.

The variation of the local skin friction coefficient Cfx

and local rate of heat transfer Nux for different values

of R associated with Pr = 0.733, M = 0.50, N = 0.01

and J=0.05 are illustrated in Fig. 8 (a) and Fig. 8 (b).

Radiation increases the fluid motion as mentioned in

Fig. 3(a) and increases the shear stress at the wall, for

which local skin friction increase with the increasing

M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16

14

R. This phenomenon demonstrated in Fig. 8(a).The

increased value of the radiation parameter increases

temperature (Fig. 3(b)) which after decrease the rate

of heat transfers along the x-direction. Moreover,

heat transfers depend on temperature variation. As

the temperature of fluid increase within the boundary

layer, the heat transfer rate also decreases.

0.0 1.0 2.0 3 .0 4.0 5.0

x

0.0

0.3

0.6

0.9

1.2

Skin friction

R=0.001

R=0.010

R=0.030

R=0.050

(a)

0.0 1.0 2.0 3.0 4.0 5.0

x

0.0

0.1

0.2

0.3

0.4

0.5

Heat transfer

R=0.001

R=0.010

R=0.030

R=0.050

(b)

Figure 8. (a) Variation of skin friction and (b)

variation of heat transfer against x for varying of R

with Pr = 0.733, M=0.50, N = 0.01 and J=0.05.

The effects of Prandtl number on the skin friction Cfx

and heat transfer rate Nux with the increasing of axial

distance x for the fixed values of magnetic parameter,

radiation parameter, viscous dissipation parameter

and joule heating parameter are shown in Fig. 9(a)

and Fig. 9(b), respectively. The values of Pr are

proportional to the viscosity of the fluid. So the

increased values of Pr decrease the velocity of the

fluid within the boundary layer, as a result,

corresponding skin friction coefficient also decreases

(Fig. 9(a)). For a particular value of Pr the local skin

friction coefficient increases monotonically due to

the increase of x. From Fig. 9(b), it is observed that

heat transfer rate increases due to increase of Pr.

Furthermore for a particular value of Pr the local heat

transfer rate decreases monotonically due to increase

of x.

Fig. 10(a) and Fig. 10(b) illustrates the local skin

friction coefficients and rate of heat transfer for

different values of viscous dissipation parameter N

with increasing of x and controlling parameter Pr =

0.733, M =0.50, R = 0.001 and J = 0.05. It can be

seen that an increase in the dissipation parameter N is

associate with the increase of skin friction

coefficients against x. Moreover, for a particular

value of N the shear stress coefficient increases

vertically. Furthermore, temperature of the fluid

increase with the increasing of N shown in Fig.5 (b),

in that case temperature difference between the

surface of the plate and the fluid reduces for which

the rate of heat transfer decreases. So, the heat

transfer rate decrease with the increase of viscous

dissipation parameter N (viewed in Fig. 10(b)).

0.0 1.0 2.0 3.0 4.0 5.0

x

0.0

0.3

0.6

0.9

1.2

Skin friction

R=0.001

R=0.010

R=0.030

R=0.050

(a)

0.0 1.0 2.0 3.0 4 .0 5.0

x

0.0

0.1

0.2

0.3

0.4

0.5

Heat transfer

Pr=0.733

Pr=0.900

Pr=1.000

Pr=1.446

(b)

Figure 9. (a) Variation of skin friction and (b)

variation of heat transfer against x for varying of Pr

with M=0.50, R = 0.001, N = 0.01 and J=0.05.

0.01.02.03.04.05.0

x

0.0

0.3

0.6

0.9

1.2

1.5

Skin friction

N=0.01

N=0.20

N=0.50

N=1.00

(a)

0.0 1.0 2.0 3.0 4.0 5.0

x

-0.2

0.0

0.2

0.4

Heat transfer

N=0.01

N=0.20

N=0.50

N=1.00

(b)

Figure 10. (a) Variation of skin friction and (b)

variation of heat transfer against x for varying of N

with Pr = 0.733, M=0.50, R = 0.001 and J=0.05.

Fig. 11(a) and Fig. 11(b) illustrates the local skin

friction coefficients and heat transfer rate for

M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16

15

different values of J with increasing x and controlling

parameter Pr = 0.733, R = 0.001, M = 0.50 and

N=0.01. The velocity of the fluid increase with the

increase of J that has been shows in Fig.6 (a),

accordingly the corresponding skin friction

coefficient increase associated with the increase of J.

But reverse situation is observed in Fig.11 (b) for the

case of heat transfer rate within the boundary layer

for increasing of joule heating parameter J.

0.0 1.0 2.0 3.0 4.0 5.0

x

0.0

0.4

0.8

1.2

1.6

2.0

Skin friction

J=0.0 5

J=0.2 0

J=0.5 0

J=0.8 0

(a)

0.0 1.0 2.0 3.0 4.0 5.0

x

-0.4

-0.2

0.0

0.2

0.4

Heat transfer

J=0.0 5

J=0.2 0

J=0.5 0

J=0.8 0

(b)

Figure 11. (a) Variation of skin friction and (b)

variation of heat transfer against x for varying of J

with Pr = 0.733, M=0.50, R = 0.001 and N = 0.01.

CONCLUSION

In this analysis the effect of radiation and joule

heating parameter on Magneto Hydrodynamic

(MHD) natural convection flow along a vertical flat

plate in presence of heat conduction and viscous

dissipation have been investigated for some selected

values of pertinent parameters including magnetic

parameter, radiation parameter, Prandtl number,

viscous dissipation parameter and joule heating

parameter. From the present investigation, it may be

concluded that the velocity of the fluid and the skin

friction at the interface decrease with the increase of

magnetic field and Prandtl number while they

increase with the increase of radiation parameter,

viscous dissipation parameter and joule heating

parameter. The temperature of the fluid increases

with the increase of magnetic field, radiation

parameter, viscous dissipation parameter and joule

heating parameter but decrease for Prandtl number.

Moreover, the rate of heat transfer decreases with

increasing magnetic field, radiation parameter,

viscous dissipation parameter and joule heating

parameter but it increases for increasing Prandtl

number.

M.M. Ali et al. / J. Appl. Sci. Tech. Vol.7, No.2 (2010) 9-16

16

Nomenclature:

b Plate thickness

fx

C Local skin friction coefficient

p

C Specific heat at constant pressure

f Dimensionless stream function

g Acceleration due to gravity

r

G Grashof number

h Dimensionless temperature

0

H Strength of magnetic field

J Joule heating parameter

sf kk , Fluid and solid thermal conductivities

l Length of the plate

M Magnetic parameter

N Viscous dissipation parameter

ux

N Local Nusselt number

p Conjugate conduction parameter

P

r

Prandtl number

w

q Heat flux

R Radiation parameter

b

T Temperature at outside surface of the

plate

f

T Temperature of the fluid

w

T Average temperature of porous plate

∞

T Temperature of the ambient fluid

u,v Velocity components

u,v Dimensionless velocity components

x

,

y

Cartesian co-ordinates

x

,y Dimensionless Cartesian co-ordinate

β

Coefficient of thermal expansion

η

Dimensionless similarity variable

θ

Dimensionless temperature

μ

Viscosity of the fluid

ν

Kinematic viscosity

ρ

Density of the fluid

σ

Electrical conductivity

w

τ

Shearing stress

ψ

Shearing stress

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