Conjugate Effect of Radiation and Thermal Conductivity Variation on MHD Free Convection Flow for a Vertical Plate

doi:10.4236/ajcm.2013.33035

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American Journal of Computational Mathematics, 2013, 3, 252-259

http://dx.doi.org/10.4236/ajcm.2013.33035 Published Online September 2013 (http://www.scirp.org/journal/ajcm)

Conjugate Effect of Radiation and Thermal Conductivity

Variation on MHD Free Convection Flow

for a Vertical Plate

Rowsanara Akhter1*, Mohammad Mokaddes Ali2, Babul Hossain2, M. Sharif Uddin1

1Department of Mathematics, Jahangirnagar University, Dhaka, Bangladesh

2Department of Mathematics, Mawlana Bhashani Science and Technology University, Tangail, Bangladesh

Email: *[email protected], mmali309@gmail.com, babulhossain@yahoo.com, [email protected]

Received August 20, 2013; revised September 5, 2013; accepted September 11, 2013

cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

A numerical investigation is performed to study the effect of thermal radiation on magnetohydrodynamic (MHD) free

convection flow along a vertical flat plate in presence of variable thermal conductivity in this paper. The governing

equations of the flow and the boundary conditions are transformed into dimensionless form using appropriate similar-

ity transformations and th en solved employin g the implicit fi nite d ifference metho d with Keller-box scheme. Results for

the details of the velocity profiles, temperature distributions as well as the skin friction, the rate of heat transfer and

surface temperature distributions are shown graphically. Results reveal that the thermal radiation is more significant in

MHD natural convection flow during thermal conductivity effect is considered. To illustrate the accuracy of the present

results, the results for the local skin fraction and surface temperature distribution excluding the extension effects are

compared with results of Merkin and Pop designed for the fixed value of Prandtl number and a good agreement were

found.

Keywords: Radiation; MHD; Thermal Conductivity; Finite Difference Method

1. Introduction

The physical phenomenon of free convection flow is

driven by temperature difference. Using these considera-

tions, the temperature variation generates a density gra-

dient which responsible for buoyancy forces. The buoy-

ancy effects are important in free convection flow of v is-

cous incompressible electrically conducting fluid. Many

practical applications of free convection flow exist, for

example in the heater and coolers of mechanical devices,

in chemical industries, in nuclear power plants, in the

formation of microstructures during the cooling of mol-

ten metal’s, in fluid flows around heat-dissipation fins,

and solar ponds etc. Moreover, MHD free convection

flow is used frequently in the field of stellar and plane-

tary magnetospheres, aeronautics, chemical engineering

and electronics. Furthermore, most of the engineering

processes are rela ted with a high temperature , accordingly ,

radiation heat transfer is significant to design the relevant

equipment of heat transfer process. In addition, radiation

effects on MHD free convection flow and heat transfer

are important in the context of space technology. Con-

sidering it’s important applications in engineering and

industrial fields, a number of theoretical and experimen-

tal work have been conducted extensively by many re-

searchers. Among them, Soundalgekar and Takhar [1]

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.

[2]). Hossain and Takhar [3] employed implicit finite dif-

ference methods to analyze the effect radiation on mixed

convection flow along a heated vertical flat plate with a

uniform free stream and a uniform surface temperature.

The effects of radiation and transverse magnetic field

near stretching sheet were investigated by Ghaly [4] in

the p r esence of a uniform free stream of constant v e lo c i t y,

temperature and concentration to show that radiation hav e

significant influences on the velocity and temperature

profiles. Abd El-Naby et al. [5] studied the radiation ef-

fects on MHD un steady free convection flow over a ver-

tical plate with variable surface temperature. Badruddin

et al. [6] explored the effect of radiation and viscous dis-

sipation on natural convection flow in a porous medium

*Corresponding a uthor.

R. AKHTER ET AL. 253

by imposing finite element method (FEM). Furthermore,

it is also known that the physical property may change

significantly with temperature. To obtain better pred ictio n

of the flow behavior, it is necessary to take into account

this variation of thermal conductivity of fluid. Mishra et

al. [7] employed alternating direction implicit scheme

and collapsed dimension method to investigate the effect

of temperature dependent thermal conductivity and ra-

diation heat transfer on transient conduction for a 2-D

rectangular enclosure containing an absorbing, emitting

and scattering medium. Seddeek and Salama [8] applied

perturbation technique and shooting method to analyze

the effects of variable viscosity and thermal conductivity

on MHD unsteady two-dimensional laminar flow of a

viscous incompressible conducting fluid past a semi-

infinite vertical porous moving plate considering variable

suction. Sharma and Singh [9] obtained the analytical

and numerical solutions of the effects of thermal conduc-

tivity on MHD steady free convection flow of a viscous

incompressible electrically conducting liquid along an

inclined isothermal non-conducting porous plate in pres-

ence of viscous dissipation and Ohmic heating. The ef-

fects of thermal conductivity on unsteady MHD free

convective flow over an isothermal semi infinite vertical

plate were studied by Loganathan et al. [10] using im-

plicit finite-difference method of Crank-Nicholson type.

In this paper, the effects of radiation and variable ther-

mal conductivity on free convection flow for a vertical

flat plate in presence of transverse magnetic field are

studied. The detail derivation of the governing equations

for the flow and the parametric discussion depending on

the numerical results of the present simulations are pre-

sented in the following section.

2. Governing Equations of the Flow

We consider the conduction inside a vertical heated flat

plate and free convection flow of an incompressible, vis-

cous and electrically conducting fluid along that vertical

flat plate of length l and width b. The fluid properties are

assumed to be constant and the temperature b of the

outer surface of the plate is greater than ambient tem-

perature and a uniform magnetic field of strength H0

is imposed along the

T∞

-axis. Here the

-axis is taken

along the vertical flat plate in upward direction and also

the

-axis is normal to that plate. The effects of radia-

tion from the heated plate and thermal conductivity var ia-

tion within the two dimensional flow region are consid-

ered in this analysis. Moreover, thermal conductivity of

the fluid is assumed as

()

{

}

kk T

=+ T

∞∞

−. The

flow configuration and the coordinates system are shown

in following Figure 1.

The governing equations of the flow under the Bous-

sinesq approximations can be expressed within the usual

boundary layer as follows:

Figure 1. Physical model and coordinate system.

∂∂

∂∂ (1)

()

uu u

uv gTT

xyy

νβ

∞

∂∂∂

+= +−−

∂∂∂ (2)

(

ff f

)

TT T

uvk TT

xyCyy

∂∂ ∂



∂

+= −Γ−



∂∂ ∂∂

 (3)

where 0d

∞

∂

Γ= 



∂



,

()

λλ

= is the

mean absorption coefficient [5], b is the Plank’s func-

tion. The boundary conditions based on conduction are:

()

0, ,0,

at 0, 0

0, at , 0

uvT Tx

TkTTy x

ybk

uTTy x

∞



== =



∂



=− =>



∂



→→ →∞>



(4)

Equations (1) to (3) are nonlinear dimensional partial

differential equations and these equations can be made

non-dimensional by using the following dimensionless

variables:

()

14 1/2

1/4

,, ,

, ,

xy ul

xyGruGr

vGr TT

gl TT

−

∞

−

∞



== =



−



== 

−



−





(5)

R. AKHTER ET AL.

254

Therefore, the dimensionless governing equations are:

∂∂

∂∂ (6)

uu u

uvMu

xy y

∂∂ ∂

++=+

∂∂ ∂ (7)

()

θθθ θ

γθ γ





∂∂∂ ∂



+= ++





∂∂ ∂

∂







−−











(8)

The corresponding boundary conditions are:

0,1 at 0,0

0,0 at ,0

uvpy x

uyx

∂

== −==>



∂



→→ →∞>

(9)

Here 2212

−

= is the magnetic parameter,

212

Ra Gr

−

= is the radiation parameter,

(

γγ

∞

=−

)

is the thermal conductivity parameter,

= is the Prandtl number and

()

pkkblGr= is a conjugate conduction pa-

rameter. The value of the conjugate conduction parame-

ter p depends on

()

bl ,

()

kk and but each of

which depends on the types of considered fluid and the

solid. The steam function and similarity variable and the

dimensionless temperature are considered in the follow-

ing form to solve the governing equations:

() ()

()

()()

120

xxfx

yx x

xxhx

θη

−

(10)

Using the above transformations, we obtain the fol-

lowing dim e nsionl ess g overning equatio ns:

() ()

()

110

16 1565

20 110 1

ffff

Mxxfhxff

′′′′′ ′

+−

′

∂∂



′′′′

−+ +=−



∂∂









)11(

()

() () ()

()

110

310

PrPr1Pr 1

16 1511

20 151

hhh h

hfhRaxx

Raxxx fh

γγ

 

′′′′ ′

 

 

+′′

+−−+

∂∂



′′

++= −



∂∂



The boundary condition (9) become

() ()

() ()()()

() ()

14 120

,0,0 0,

,011,0 at 0

,0,,0 at .

fxfx

hxxxx hxy

fx hxy

′

== 



′=−+++= 



′∞→ ∞→→∞





)13(

In practical point of view, it is important to calculate

the values of the skin friction co-efficient in term of sur-

face shear stress and the rate of heat transfer in term of

the Nussetl number. These can be written in the dimen-

sionless form as:

()

34 214

and

GrlGr l

CNu

kT T

μν

−−

∞

−q

(14)

where

τμ



∂

=

∂



is the shearing stress and

∂



=− 

∂



is the heat flux. Thus the local skin

friction co-efficient and the local Nussetl number is ob-

tained using the new variable systems that is describes in

Equation (20) as follows:

() ()

()()

320

1,0

Cxx fx

Nuxh x

−

′′

=− +

and

(15)

The numerical value of the surface temperature distri-

bution are obtained from the following relation

() ()(

,0 1,0

)

xxhx

−

=+ (16)

We have discussed the velocity profiles and tempera-

ture distributions for various values of magnetic parame-

ter, radiation parameter, thermal conductivity variation

parameter and Prandtl number in the present investiga-

tion.

3. Method of Solution

The numerical solutions of this analysis are found by

using implicit finite difference method with Keller-box

[11] Scheme which is well documented by Cebeci and

Bradshaw [12].

4. Comparison of the Results

The comparison of the skin friction coefficients

()

and the surface temperature distribution be-

tween the present work and the work of Merin and Pop

[13] is presented in following Table 1. We observed in

this table, that the present analysis is an excellent agree-

ment with the published work.

()

(

,0x

)

)12(

5. Results and Discussion

The main objective of the present work is to analyze

R. AKHTER ET AL. 255

Table 1. Comparison of the present numerical results of the

skin friction (Cfx) and surface temperature (θ(x,0)) with

Prandtl number Pr = 1.00, and p = 1.00.

Merin and Pop [1 3] Present work (2013)

= fx

()

0.7

0.8

0.9

1.0

1.1

1.2

0.430

0.530

0.635

0.745

0.859

0.972

0.651

0.686

0.715

0.741

0.762

0.781

0.424

0.529

0.635

0.744

0.860

0.975

0.651

0.687

0.716

0.741

0.763

0.781

MHD free convection flow in presence of thermal con-

ductivity variation with radiation effects. In this analysis,

the numerical solutions are calculated from the trans-

formed momentum and energy equations. The value of

conjugate conduction parameter is considered

for the simulation of the present problem and the values

of Prandtl number are considered 0.733, 0.930, 1.241 and

1.630 which corresponds to air, ammonia water and glyc-

erin, respectively. The detailed numerical solutions have

been obtained in terms of velocity, temperature, local s kin

friction, heat transfer rate and surface temperature for a

wide range of values of the parameters as M = 0.10, 0.50,

0.70 and 1.00, Ra = 0.01, 0.03, 0.06 and 0.08, γ = 0.01,

0.10, 0.15 and 0.20 and then presented graphically in

Figures 2-11, respectively.

1.0p=

The numerical values of velocity and temperature are

obtained from the solution of the Equations (11) and (12)

with the boundary condition (13) for different values of

magnetic parameter M when Pr = 0.733, Ra = 0.01 and γ

= 0.01 and are illustrated in Figures 2(a) and (b), respec-

tively. Here we observed that the velocity decreases for

the increasing values of M. This is to be expected be-

cause, the magnetic field acting along the horizontal di-

rection that introduces a retard force due to the interac-

tion between applied magnetic field and fluid flow which

acts against the fluid motion, as a result, the velocity of

the fluid decreases. But near the surface of the plate the

velocity increases and become maximum and then de-

crease and finally approaches to zero. Moreover, the ve-

locity profiles meet together after certain position of η

and cross the side. This is because, the gradient of de-

creasing of velocity decrease with the increasing value of

mag n e ti c pa r ame t e r . I n Figure 2(b), the temperature within

the boundary layer increases with the increasing values

of magnetic parameter M due to the interaction of applied

magnetic field and fluid motion that tends to heat the

fluid. Furthermore, the temperature decreases monotoni-

cally with increasing of η for each value of M. Thus the

magnetic field works to retard the fluid motion but in-

crease the temperature within the thermal boundary layer

region.

The variation of velocity and temperature for distinct

values of the radiation parameter Ra together with an

0.0 2.0 4.0 6.0 8.010.0

0.0

0.2

0.4

0.6

0.8

Velocity

M = 0.10

M = 0.50

M = 0.70

M = 1.00

(a)

0.0 2.0 4.0 6.0 8.010.0

0.0

0.4

0.8

1.2

1.6

2.0

Temperature

M = 0.10

M = 0.50

M = 0.70

M = 1.00

(b)

Figure 2. (a) Variation of velocity and (b) Variation of tem-

perature against η for varying of M wit h Ra = 0.01 γ = 0.01,

and Pr = 0.733.

individual certain value of Prandtl number, Pr, magnetic

parameter, M and thermal conductivity variation parame-

ter, γ are presented in Figures 3(a) and (b), respectively.

It can be seen that both the velocity and the temperature

of the fluid increase within the velocity boundary layer

and the temperature boundary layer, respectively for the

increasing Ra due to the absorbsion of emitted heat from

the heated plate that caused by the radiation effect. The

trend is observed to shift upward and the peak velocity

increases gradually with the increasing values of Ra. As

the velocity and temperature of the fluid increases with

the increasing value of radiation parameter as shown in

Figures 3(b) and (b), the thickness of the velocity and

thermal boundary layer increase.

Figures 4(a) and (b) illustrate the effects of thermal

conductivity variation parameter on velocity and tem-

perature profiles associated with the certain value of M,

Ra and Pr. Both figures reflect that the velocity and the

temperature of the fluid increases with the increasing

value of γ. The fact behind it’s that the increasing value of

thermal conductivity increases the energy transfer ability.

It is also seen that near the surface of the plate the veloc-

ity becomes maximum with increasing of γ then after the

peak position start to decrease and finally approaches to

zero. On the other hand, the maximum values of tem-

perature are occurred on the surface of the plate for each

R. AKHTER ET AL.

256

0.0 2.0 4.0 6.0 8.010.0

0.0

0.2

0.4

0.6

0.8

Velocity

Ra = 0.01

Ra = 0.03

Ra = 0.06

Ra = 0.08

(a)

0.0 2.0 4.0 6.0 8.010.0

0.0

0.3

0.6

0.9

1.2

1.5

1.8

Temperature

Ra = 0.01

Ra = 0.03

Ra = 0.06

Ra = 0.08

(b)

Figure 3. (a) Variation of velocity and (b) Variation of tem-

perature against η for varying of Ra with M = 0.50 γ = 0.01,

and Pr = 0.733.

0.02.0 4.06.08.010.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Velocity

γ = 0.01

γ = 0.10

γ = 0.15

γ = 0.20

(a)

0.0 2.0 4.0 6.0 8.010.0

0.0

0.6

1.2

1.8

2.4

3.0

Temperature

γ = 0.01

γ = 0.10

γ = 0.15

γ = 0.20

(b)

Figure 4. (a) Variation of velocity and (b) Variation of tem-

perature against η for varying of γ with M = 0.50, Ra = 0.01

value of γ and

and Pr = 0.733.

then turn to decrease asymptotically and

ature dis-

tri

ively reveal that the skin

finally approach to zero. These phenomenons are dem-

onstrated in Figures 4(a) and (b), respectively.

The effect of Pr on the velocity and temper

butions is displayed in Figures 5(a) and (b), respec-

tively. From Figure 5(a) observed that the velocity of the

fluid decrease as Pr increases. It is due to the fact that for

increasing Pr, density of the fluid increases which creates

a negative force to flow and then fluid does not move

freely. Furthermore, Figure 5(b) shows that the tem-

perature profiles for change in Pr from 0.733 to 1.630

and seen that the thermal boundary layer thickness de-

crease for increasing Pr, because of the increased Pr de-

crease the thermal d iffusivity, which leads to the decre ase

of the energy transfer ability.

Figures 6(a) and (b), respect

iction coefficient and the heat transfer rate for some

selected values of M with Pr = 0.733, Ra = 0.01, and

0.01

=. The increased value of M leads to decrease the

tion along the plate due to the fact the effect of

magnetic field parameter opposes the fluid flow. Increas-

ing fluid temperature for increasing M decrease the rate

of heat transfer from the plate to fluid. This is because,

the increased temperature reduces the temperature dif-

ference between the heated plate and fluid within the

boundary layer.

skin fric

0.02.04.06.08.010.0

0.0

0.2

0.4

0.6

0.8

Velocity

Pr =0.733

Pr =0.930

Pr =1.241

Pr =1.630

(a)

0.02.0 4.0 6.0 8.010.0

0.0

0.4

0.8

1.2

1.6

2.0

Temperature

Pr =0.733

Pr =0.930

Pr =1.241

Pr =1.630

(b)

Figure 5. (a) Variation of velocity and (b) Variation ofm- te

perature against η for varying of Pr with, M = 0.50, Ra =

0.01 and γ = 0.01.

R. AKHTER ET AL. 257

0.04.08.012.0 16.020.0

0.0

1.0

2.0

3.0

4.0

5.0

Skin

ricti o n

M = 0.10

M = 0.50

M = 0.70

M = 1.00

(a)

0.04.08.012.0 16.0 20.0

0.0

1.0

2.0

3.0

4.0

Heat transfer

M=0.10

M=0.50

M=0.70

M=1.00

(b)

Figure 6. (a) Variation of skin friction and (b) Variation of

Figures 7(a) and (b) plotted the numerical values of

on the skin friction (Cfx)

heat transfer against x for varying of M with Ra = 0.01, γ =

0.01 and Pr = 0.733.

e local sk in fr iction coefficie nt (Cfx) and heat transfer rate

(Nux) for different values of Ra associated with the dis-

tinct values of controlling parameter. As the radiation ef-

fect increases the fluid motion as well as temperature with-

in the boundary layer wh ich are mentioned earlier in Fig-

ures 3(a) and (b), respectively. Accordingly, th e correspon-

ding skin friction increases and heat transfer rate decreas-

es with the increasing value of Ra along the x direction.

The variation of skin friction and heat transfer rate fo

e effect of conductivity variation parameter are depi cted

in Figures 8(a) and (b), respectively. The incr easing value

of γ generates greater buoyancy force which therefore

increases the friction between the inner surface of the

vertical plate and moving fluid particles. Thus the skin

friction increases for the greater value of γ that is demon-

strate in Figure 8(a). Moreover, an increase in the value

of γ leads to increase the energy transfer ability within

the flow region, as a result, the heat transfers rate in-

creases with the increasing of γ.

The effects of Prandtl number

d heat transfer rate (Nux) against x for the fixed values

of M, Ra and γ are shown respectively in Figures 9(a)

and (b). The increased values of Pr decrease both the

velocity and temperature of the fluid within the boundary

layer, consequently, the related skin friction on the plate

decreases but the heat transfer rate from heated plate to

fluid increases that has been exposed in Figures 9(a) and

048 121620

0.8

1.6

2.4

3.2

4.0

Skin

icti o n

Ra = 0.01

Ra = 0.03

Ra = 0.06

Ra = 0.08

(a)

048 121620

0.0

0.8

1.6

2.4

3.2

4.0

Heat transfer

Ra =0.01

Ra =0.03

Ra =0.06

Ra =0.08

(b)

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

heat transfer against x for varying of Ra with M = 0.50, γ =

0.01 and Pr = 0.733.

0.04.08.012.0 16.0 20.0

0.0

1.0

2.0

3.0

4.0

5.0

Skin

rictio n

γ = 0.01

γ = 0.10

γ = 0.15

γ = 0.20

(a)

0.04.08.012.0 16.0 20.0

0.0

1.6

3.2

4.8

6.4

8.0

Heat trans

γ=0.01

γ=0.10

γ=0.15

γ=0.20

(b)

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

0.01 and Pr = 0.733.

heat transfer against x for varying of γ with M = 0.50, Ra =

R. AKHTER ET AL.

258

0.04.08.012.016.020.0

0.0

0.8

1.6

2.4

3.2

4.0

Skin

rictio n

Pr =0.733

Pr =0.930

Pr =1.241

Pr =1.630

(a)

0.04.08.012.0 16.0 20.0

0.0

0.8

1.6

2.4

3.2

4.0

Heat transfer

Pr =0.733

Pr =0.930

Pr =1.241

Pr =1.630

(b)

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

heat transfer against x for varying of Pr with M = 0.50, a =

r a particular value of Pr, the local skin

friction coefficient and heatfer rate increase mono-

n parameter and Prandtl

sis, we have studied numerically the

on MHD free convection flow under

0.01 and γ = 0.01.

(b). Moreover, fo trans

tonically along the x direction.

The influence of magnetic field parameter, radiation

parameter, conductivity variatio

mber on the interfacial temperature are depicted in Fig-

ures 10 and 11, respectively. The overall temperature

profiles shift upward as well as the thermal boundary

layer thickness increases with the increasing values of M,

Ra, and γ observed in Figures 2(b), 3( b) and 4(b), re-

spectively. Consequently, the surface temperature inc r ea se

for enlarging valu es of M, Ra and γ, respectively. On the

other hand, increasing Pr decreases the fluid temperature

which results interfacial temperature decreases that is

shown in Figure 11(b).

6. Conclusions

In the present analy

effects of radiation

the influence of thermal conductivity variation for a ver-

tical flat plate and the numerical solutions of the trans-

formed governing equations associated with the specified

boundary are obtained for different values of related

physical parameters including magnetic parameter, radia-

tion parameter, thermal conductivity variation parameter

and Prandtl number. The particular conclusions in this

0.04.08.012.0 16.0 20.0

0.0

0.4

0.8

1.2

1.6

2.0

Sur

ace temperature

M=0.10

M=0.50

M=0.70

M=1.00

(a)

048 121620

0.8

1.0

1.2

1.4

1.6

1.8

acetempe

atu

Ra = 0.01

Ra = 0.03

Ra = 0.06

Ra = 0.08

(b)

Figure 10. (a) Variation of surface temperature against x for

varying M and (b) Variation of surface temperature against

x for varying of Ra.

0.0 4.0 8.012.016.020.0

0.0

0.4

0.8

1.2

1.6

2.0

2.4

Sur

ace temperature

γ = 0.01

γ = 0.10

γ = 0.15

γ = 0.20

(b)

0.04.08.012.0 16.0 20.0

0.8

1.0

1.2

1.4

1.6

1.8

Sur

acetemperature

Pr =0.733

Pr =0.930

Pr =1.241

Pr =1.630

(b)

Figure 11. (a) Variation of surface temperature against x for

varying γ and (b) Variation of surface t emperature against x

for varying of Pr.

R. AKHTER ET AL.

259

e for decreasing

[1] V. M. Soundalgekar and H. S. Takhar, “Radiative Con-

vective Flow Pical Plate,” Model-

ling Measurem, 1992, pp. 31-40.

tural Convection in a Porous Medium Embedded within

Vertical Annulus,” International Journal of Thermal Sci-

ence, Vol. 46, No. 3, 2007, pp. 221-227.

http://dx.doi.org/10.1016/j.ijthermalsci.2006.05.005

[7] S. C. Mishra, P. Talukdar, D. Trimis a nd F. Durst, “T

Dimensional Transient Conduction and Radiation

study can be listed as follows:

The velocity of the fluid within the boundary layer and

the skin friction at the interface increas

values of magnetic parameter, M, Prandtl number, Pr and

increasing values of the radiation parameter, Ra and

thermal conductivity variation parameter, γ.

The increasing value of M, Ra and γ leads to increase

in the value of temperature within the thermal boundary

layer as well as the surface temperature on the plate a

wo-

Heat

Transfer with Temperature Dependent Thermal Conduc-

tivity,” International Communications in Heat and Mass

Transfer, Vol. 32, No. 3-4, 2005, pp. 305-314.

http://dx.doi.org/10.1016/j.icheat ma sst ran sfer.200 4.05.01 5

[8] M. A. Seddeek and F. A. Salama, “The Effects of Tem-

peratute Dependent Viscosity and Thermal Conductivity

the effect of M and Ra decrease heat transfer rate from

plate to fluid within the boundary layer but opposite re-

sults hold for increasing of γ and Pr.

REFERENCES

on Unsteady MHD Convective Heat Transfer Past a Semi-

Infinite Vertical Porous Moving Plate with Variable Suc-

tion,” Computational Material Science, Vol. 40, No. 2,

2007, pp. 186-192.

http://dx.doi.org/10.1016/j.commatsci.2006.11.012

[9] P. R. Sharma and G. Singh, “Effects of Variable Therm

Conductivity, Viscous Dissipation on Steady MHD

Natu-

ast a Semi-Infinite Vert

ent and Control, Vol. 51ral Convection Flow of Low Prandtl Fluid on an Inclined

Porous Plate with Ohmic Heating,” Meccanica, Vol. 45,

No. 2, 2010, pp. 237-247.

http://dx.doi.org/10.1007/s11012-009-9240-0

[10] P. Loganathan, P. Ganesan and D. Iranian, “

Thermal Conductivity on Unsteady MHD Fr

[2] A. C. Cogley, W. G. Vincenti and S. E. Giles, “Differen-

tial Approximation for Radiative in a Non-Gray Gas Near

Equilibrium,” American Institute of Aeronautics and As-

tronautics Journal, Vol. 6, No. 3, 1968, pp. 551-553.

http://dx.doi.org/10.2514/3.4538

[3] M. A. Hossain and H. S. Takhar, “Radiation Effect on

Mixed Convection along a Vertical Plate with Unifo

Surface Temperature,” Heat and Mass Transfer

Effects of

ee Convec-

Fluid Mechanics, Vol. 10,

tive Flow over a Semi Infinite Vertical Plate,” Interna-

tional Journal of Engineering Science and Technology,

Vol. 2, 2010, pp. 6257-6268.

[11] H. B. Keller, “Numerical Methods in the Boundary Layer

Theory,” Annual Review of

, Vol. 31,

No. 4, 1996, pp. 243-248.

http://dx.doi.org/10.1007/BF02328616

[4] A. Y. Ghaly, “Radiation Effects on a Certain MHD Free

Convection Flow,” Chaos, Solitons &

No. 9, 2002, pp. 1843-1850.

1978, pp. 417-433.

http://dx.doi.org/10.1146/annurev.fl.10.010178.002221

[12] T. Cebeci and P. Bradshow, “Physical and Computation

Aspects of Convective Heat Transfer,” Springer, Ne

Fractals, Vol. 13,

http://dx.doi.org/10.1016/S0960-0779(01)00193-X

[5] M. A. Abd El-Naby, E. M. E. Elsayed and N. Y. Abde-

lazem, “Finite Difference Solution of Radiation Eff

MHD Unsteady Free Convection Flow over a V

York, 1984. http://dx.doi.org/10.1007/978-3-662-02411-9

[13] J. H. Merkin and I. Pop, “Conjugate Free Convection on a

Vertical Surface,” International Journal of Heat and

ect on

l ertica

Plate Variable Surface Temperature,” Journal of Applied

Mathematics, Vol. 2003, No. 2, 2003, pp. 65-86.

http://dx.doi.org/10.1155/S1110757X0320509X

[6] I. A. Badruddin, Z. A. Zainal, A. Zahid Khan and Z. Mal-

lick, “Effect of Viscous Dissipation and Radiation

Mass Transfer, Vol. 39, No. 7, 1996, pp. 1527-1534.

http://dx.doi.org/10.1016/0017-9310(95)00238-3

on Na-

Nomenclature

T: Temperature of the fluid

C: Local skin friction coefficient

C: Specific heat at constant pressure

imensionless stream function f: D

g: Acceleration due to gravity

Gr : Grashof number

h: Derature imensionless temp

k: Fluid and solid thermal conductivities

M: Magnetic parameter

Nux: Local Nusselt number

p: Conjugate conduction parameter

Pr: Prandtl number

u, v: Velocity components

v: Dimensionless velocity cou, mponents

, : Cartesian co-ordinates

x, y: Dimensionless Cartesian co-ordinate

β:oeion

γ: Thermal conductivity variation paramete

Cfficient of thermal expansr

η: Dimensionless similarity variable

θ: Dimensionless temperature

μ, v: Dynamic and kinematic viscositi

ρ: Density of the fluid

σ: Electrical conductivity