MHD Heat Transfer Mixed Convection Flow Along a Vertical ... · Abstract-- The steady...

International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:10 No:02 71

109602-3737 IJBAS-IJENS © April 2010 IJENS I J E N S

MHD Heat Transfer Mixed Convection Flow Along a

Vertical Stretching Sheet in Presence of Magnetic

Field With Heat Generation

M. Mohebujjaman1, Tania S. Khaleque

2 and M.A. Samad

2, b

1Department of Textile Engineering, Southeast University, Banani, Dhaka-1213, Bangladesh

2Department of Mathematics, Dhaka University, Dhaka-1000, Bangladesh

E-mail: [email protected] b Corresponding author

Abstract-- The steady two-dimensional MHD heat transfer

mixed convection flow of a viscous incompressible fluid near a

stretching permeable sheet in presence of a uniform magnetic

field with heat generation is considered when the buoyancy force

assists or opposes the flow. The equations governing the flow and

temperature fields are reduced to a system of coupled non-linear

ordinary differential equations. These non-linear differential

equations are integrated numerically by using Nachtsheim-

Swigert [1] shooting iteration technique along with sixth order

Runge-Kutta integration scheme. Critical values of buoyancy

parameter are obtained for vanished shear stress. The numerical

results are benchmarked with the earlier study by Mohamed Ali

and Fahd Al-Yousef [2] and found to be in excellent agreement.

Finally the effects of the pertinent parameters which are of

physical and engineering interest are presented in tabular form.

Index Term-- boundary layer, buoyancy force, electric

conductivity, variable wall temperature.

NOMENCLATURE

0B Uniform magnetic field strength xNu Local Nusselt number

wf Suction parameter Pr Prandtl number

M Magnetic field parameter Q Dimensionless heat source parameter

g Acceleration due to gravity 0Q Heat source parameter

pC Specific heat at constant pressure wq Local heat flux

fC Skin friction coefficient wv Suction velocity

f Dimensionless stream function Buoyancy parameter

wT Temperature of fluid at the sheet

T Temperature of fluid within the boundary layer

T

Temperature of fluid far away from the plate

u

Component of velocity in the x -direction

v Component of velocity in the y -direction x Coordinate along the sheet

y Coordinate normal to the sheet

Greek Symbols

Electric conductivity

Similarity parameter Step size

Dimensionless temperature Coefficient of volume expansion

Thermal conductivity Density of the fluid

Coefficient of dynamic viscosity Coefficient of kinematic viscosity

I. INTRODUCTION

Both the hydromagnetic flow and heat transfer in a viscous

incompressible fluid over a moving continuous stretching

surface is a significant type of flow has considerable practical

applications in industries and engineering. For example,

materials manufactured by extrusion processes, E.G. Fisher

[3], and heat-treated materials traveling between a feed roll

and a wind-up roll or on a conveyor belt possess the

characteristics of a moving continuous surface. Many

metallurgical processes involve the cooling of continuous

strips or filaments by drawing them through a quiescent fluid

and that in the process of drawing, these strips are sometimes

stretched. Mention may be made of wire drawing, annealing,

tinning of copper wires, crystal growing, spinning of

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filaments, continuous casting, glass fiber production and paper

production, T. Atlan, S. Oh, H. Gegel [4], Z. Tadmor and I.

Klein [5]. In all the cases the properties of the final product

depend to a great extent on the rate of cooling and the

processes of stretching as explained by M.V. Karwe and Y.

Jaluria [6], [7]. By drawing such strips or filaments in an

electrically conducting fluid subjected to a magnetic field, the

rate of cooling can be controlled and a final product of desired

characteristics can be achieved. Another interesting

application of hydromagnetics to metallurgy lies in the

purification of molten metals from nonmetallic inclusions by

the application of a magnetic field. The study of heat transfer

has become important industrially for determining the quality

of the final product. The dynamics of the boundary layer flow

over a moving continuous solid surface originated from the

pioneer work of B.C. Sakiadis [8] who developed a numerical

solution for the boundary layer flow field of a stretched

surface, many authors have attacked this problem to study the

hydrodynamic and thermal boundary layers, M.E. Ali [9],

[10], E. Magyari and B. Keller [11], [12].Due to entrainment

of ambient fluid, this boundary layer flow situation is quite

different from the classical Blasius problem of boundary flow

over a semi-infinite flat plate. Suction or injection of a

stretched surface was studied by L.E. Erickson, L.T. Fan and

V.G. Fox [13] and V.G.Fox, L.E. Erickson and L.T. Fan [14]

for uniform surface velocity and temperature and investigated

its effects on the heat and mass transfer in the boundary layer.

J.B. McLeod and K.R. Rajagopal [15] have investigated the

uniqueness of the flow of a Navier Stokes fluid due to a linear

stretching boundary. A. Raptis and C. Perdikis [16] have

studied the viscous flow over a non-linearly stretching sheet in

the presence of a chemical reaction and magnetic field. C.K.

Chen and M.I. Char [17] have studied the suction and

injection on a linearly moving plate subject to uniform wall

temperature and heat flux and the more general case using a

power law velocity and temperature distribution at the surface

was studied by M.E. Ali [18]. E. Magyari, M.E. Ali and B.

Keller [19] have reported analytical and computational

solutions when the surface moves with rapidly decreasing

velocities using the self-similar method. The study of heat

generation or absorption in moving fluids is important in

problems dealing with chemical reactions and these concerned

with dissociating fluids. Possible heat generation effects may

alter the temperature distribution; consequently, the particle

deposition rate in nuclear reactors, electronic chips and semi

conductor wafers. K. Vajravelu and A. Hadjinicolaou [20]

studied the heat transfer characteristics in the laminar

boundary layer of a viscous fluid over a stretching sheet with

viscous dissipation or frictional heating and internal heat

generation. Laminar mixed convection boundary layers

induced by a linearly stretching permeable surface was studied

by Mohamed Ali and Fahh Al-Yousef [2].

In the present study we aim to extend the analysis

by Mohamed Ali and Fahh Al-Yousef [2] considering a

uniform magnetic field which is normal to the stretching

surface with heat generation, which have been of interest

to the engineering community and to the investigators

dealing with the problem in geophysics, astrophysics,

electrochemistry and polymer processing.

II. MATHEMATICAL ANALYSIS

A steady-state two-dimensional motion of mixed convection

boundary layer flow from a vertically upward moving

permeable stretching sheet through a quiescent incompressible

fluid with suction or injection at the surface is considered. The

stretching sheet coincides with the plane 0y as shown in

fig. 1.

Fig. 1. Coordinate system, boundary

conditions and steady

boundary layers adjacent to the stretching wall issuing from a

narrow linear slot and

moving in the positive x direction.

A uniform magnetic field of strength 0B is imposed normal to

the stretching sheet, where the flow is confined to 0y . For

incompressible viscous fluid milieu with constant properties

using Boussinesq approximation, the equations governing this

convective flow are

0

y

v

x

u (1)

uB

y

u)TT(g

y

uv

x

uu

2

0

2

2

(2)

)TT(c

Q

y

T

cy

Tv

x

Tu

pp

0

2

2

(3)

subject to the following boundary conditions:

yTT,u

yxCTTTT

y),x(vv,xUuu

nw

wm

w

at

at

at

0

0

00

(4)

It should be mentioned that, positive or negative m indicate

that the sheet is accelerated or decelerated from the extruded

slit respectively. The x coordinate is directed along the

continuous stretching sheet and points in the direction of

motion, measured from the point where the sheet originates,

and the y coordinate is measured perpendicular to x axis

and to the direction of the slot ( z axis) where the continuous

stretching plane issues. In order to allow for fluid suction or



injection through the sheet, the sheet is regarded to be porous

with a transpired velocity )x(vw . Positive or negative wv

imply suction or injection at the surface respectively. u and

v are the velocity components in the x and y directions

respectively. is the kinematic viscosity, g is the

acceleration due to gravity, is the volumetric coefficient of

thermal expansion, T and

T are the fluid temperature within

the boundary layer and in the free-stream respectively, is

the electric conductivity, 0

B is the uniform magnetic field

strength (magnetic induction), is the density of the fluid,

is the thermal conductivity of the fluid, 0

Q is the volumetric

rate of heat generation/absorption, w

T is the uniform wall

temperature. In order to obtain a solution of equations (1)-(4),

we introduce the following similarity variables

),(fxUu 'm 0 )(xCTT n (5)

x

m

Rem

x

y

x

xUmy

2

1

2

1 0

(6)

'

m

fm

fm

xm

Uv

2

1

2

1

1

22

1

0 (7)

where 'f and are the dimensionless velocity and

temperature respectively, and is the similarity variable.

Substitution in the governing equations gives rise to the

following two-point boundary-value problem.

01

2

1

2

1

2 2

m

fm

M)f(

m

mfff '''''''

(8)

01

2

1

2

m

Qpr)f

m

nf(pr ''''

(9)

The last term in equation (8) is due to the buoyancy force and

2

0

12

U

xCg mn

which serves as the buoyancy parameter,

when 0 the governing equations reduce to those of forced

convection limit. On the other hand, if is of a significantly

greater order of magnitude than one, the buoyancy force

effects will predominate and the flow will essentially be free

convective and combined convective flow exists when

).(O 1 Third term in equation (8) is due to the magnetic

field and 0

12

0

U

xBM

m

serves as the magnetic field

parameter. A consideration of equation (8) shows that and

M are functions of .x Therefore, the necessary and sufficient

conditions for the similarity solutions to exist is

that 12 mn , 01 m for f and be expressed as

function of alone. Which implies 1m and 1n .

Furthermore, if these conditions are not satisfied, local

similarity solutions are obtained, W.M. Kays and M.E.

Crawford [21], A. Mucoglu and T.S. Chen [22]. In this paper

local similarity solutions are also found for 1m at 1n

and 0 . The transformed boundary conditions are:

as,,f

at,ff,f w

00

011 (10)

where1

2

0

1

mU

xvf

m

ww

is the suction ( or injection )

parameter,

pcpr is the Prandtl number and

0

0

Uc

QQ

p is the heat source/sink parameter. The

parameters of engineering interest for the present problem are

the skin friction coefficient and local Nusselt number which

indicate physically wall shear stress and local wall heat

transfer rate respectively. From equation (5) we write

x

''m Rem

x.)(fxU

y

u

y

u

2

110

)(fRem

xU ''

x

m 2

11

0

At the stretching sheet i.e. at 0y ,

)(fRem

xUy

u ''

x

m

wall

02

11

0

, hence the

expression for wall shear stress can be developed from the

similarity solution in the form

)(fRem

xUy

u ''x

m

wall

w 02

110

The wall shear stress can be expressed in a dimensionless

form which is known as the skin friction coefficient is given

by 2

2

1w

wf

u

C

which

implies

x

''

m

''x

m

fRe

)(fm

xU

)(fRem

xU

C

02

12

2

1

02

1

220

10

)(fm

ReC ''xf 0

2

12

(11)

and for 1m which is given by ).(fReC ''xf 02

The local wall heat flux )q( w may be written by Fourier’s

law as



x'n

y

w Rex

m)(xC

y

Tq

1

2

10

0

Hence

the local Nusselt number is obtained as

)(Rem

xC

qxNu '

xn

wx 0

2

1

which implies

)(m

Re

Nu '

x

x 02

1

(12)

and for 1m it is given by ).(Re

Nu '

x

x 0 The numerical

values proportional to fC and xNu , corresponding to 1m

are shown in Table III-Table IV.

III. NUMERICAL COMPUTATION The numerical solutions of the non-linear differential equation

(8)-(9) under the boundary conditions (10) have been

performed by applying a shooting method namely

Nachtsheim and Swigert [1] iteration technique (guessing the

missing values) along with sixth order Runge-Kutta

integration scheme. The boundary conditions, equation (10),

associated with the non-linear ordinary differential equations

(8)-(9) are the two-point asymptotic class, that is, values of

the dependent variable are specified at two different values of

the independent variable. Specification of an asymptotic

boundary condition implies that the first derivative (and

higher derivatives of the boundary layer

equation, if exists) of the dependent variable approaches zero

as the outer specified value of the independent variable is

approached. For the method of numerically integrating a two-

point asymptotic boundary-value problem of the boundary-

layer type, the initial-value method is similar to an initial

value problem. Thus, it is necessary to estimate as many

boundary conditions at the surface as were (previously) given

at infinity. The governing differential equations are then

integrated with these assumed surface boundary conditions. If

the required outer boundary condition is satisfied, a solution

has been achieved. However, this is not generally the case.

Hence, a method must be devised to estimate logically the

new surface boundary conditions for the next trial integration.

Asymptotic boundary value problems such as those governing

the boundary-layer equations are further complicated by the

fact that the outer boundary condition is specified at infinity.

In the trial integration, infinity is numerically approximated

by some large value of the independent variable. There is no a

priori general method of estimating these values. Selecting

too small a maximum value for the independent variable may

not allow the solution to asymptotically converge to the

required accuracy. Selecting a large value may result in

divergence of the trial integration or in slow convergence of

surface boundary conditions. Selecting too large a value of

the independent variable is expensive in terms of computer

time. Nachtsheim-Swigert [1] developed an iteration method

to overcome these difficulties. In equation (10) there are two

asymptotic boundary conditions and, hence, two unknown

surface conditions such as )(''f 0 and )(' 0 . Within the

context of the initial-value method and the Nachtsheim-

Swigert [1] iteration technique, the outer boundary conditions

may be functionally represented as

4100 j,))('),(''f(Y)(Y jjmaxj , (13)

Where ''fY,Y,'fY 321 and 'Y 4 . The last two of

these represent asymptotic convergence criteria.

Choosing 10 g)(f '' , 20 g)(' and expanding in a first-

order Taylor’s series after using equation (13), we obtain

41

2

1

1

j

,gg

Y)(Y)(Y j

i i

j

maxC,jmaxj (14)

where the subscript 'C' indicates the value of the function at

max determined from the trial integration. Solution of these

equations in a least-squares sense required determining the

minimum value of

Error

4

1

2

j

j (15)

Now differentiating (15) with respect to ig 21,i we obtain

4

1

0

j i

j

jg

. (16) Substituting equation

(14) into (16) after some algebra, we obtain

,bga

k

iiik

2

1

(17)

where

21

4

1

4

1

,k,i

,g

YYb,

g

Y

g

Ya

j i

j

C,ji

j k

j

i

j

ik

(18)

Now solving the system (17) using Cramer’s rule, we obtain

the missing (unspecified) values of ig as

iii ggg . (19)

Thus, adopting the numerical technique aforementioned along

with the sixth order Runge-Kutta-Butcher initial value solver,

the solutions of the equations (8)-(9) with boundary

conditions (10) are obtained as a function of the coordinate

for various values of the parameters. We have chosen a step

size 0010. to satisfy the convergence criterion of 610

in all cases. The value of

was found to each iteration loop

by

. The maximum value of

to each group of

parameters Pr,,M,f w and Q determined when the values

of the unknown boundary conditions at 0 not change to

successful loop with error less than 610 . In order to verify

the effects of the step size , we ran the code for our model

with three different step sizes as ,.0050 001.0 ,

00010. and in each case we found excellent agreement



among them. The figures 2(a) and 2(b) show the velocity and

temperature profiles for different step sizes.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

f'

Curves

---------

O

0.005

0.001

0.0001

Fw=-0.6, Pr=0.72, M=0.5, Q=0.5,

(a)

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Curves

----------

O 0.001

0.0001

Fw=-0.6, Pr=0.72, M=0.5, Q=0.5,

(b)

Fig. 2. Velocity and temperature profiles for different step size : (a)

Velocity and (b) Temperature.

To assess the accuracy of the present code, we compare the

critical values of the buoyancy parameter s.)crt(

corresponding to vanished shear stresses at the surface with

those of Mohamed Ali and Fahh Al-Yousef [2] by

setting 0M , 0Q .

Table I demonstrate the comparison of the data produced by the present code and those of M. Ali (2002). In fact, the

results show a close agreement, and hence justify the use of the present code.

TABLE I

wf

1n , 1m

720.pr 03.pr 010.pr

M. Ali et al. Present M. Ali et al. Present M. Ali et al. Present

0.6 3.011 3.0102 6.2075 6.2142 13.335 13.4967

0.4 2.7250 2.72220 5.2069 5.2211 10.339 10.4267

0.2 2.4743 2.4697 4.397 4.3881 7.9374 7.9577

0.0 2.264 2.2499 3.7238 3.6989 ------- ---------

-0.2 2.078 2.0599 3.1447 3.1366 ------- ---------

-0.4 1.912 1.8968 2.698 2.6842 ------- ---------

-0.6 1.7735 1.7575 -------- 2.3248 ------- ---------

IV. RESULT AND DISCUSSION

For the purpose of discussing the result, the numerical

calculations are presented in the form of non-dimensional

velocity and temperature profiles. Numerical computations

have been carried out for different values of the Prandtl

number )pr( , magnetic field parameter )M( , buoyancy

parameter )( , heat source/sink parameter )Q( and suction (

or injection ) parameter ).f( w These are chosen arbitrarily

where 710.pr corresponds physically to air at C20 ,

01.pr corresponds to electrolyte solution such as salt water

and 07.pr corresponds to water. Equations (8) and (9)

were solved numerically subject to the boundary conditions

given in equation (10) for ,m 1 60.f w to 60. with a

step of ,.20 ,.pr 720 ,3 10 and for temperature exponent

,n 1 0 and .1 In Fig.3 we have plotted the dimensionless

velocity and temperature profiles for ,m 1 ,n 1 ,.f w 60

60. and for ..pr 720 showing the effects of the buoyancy

parameter . It can be seen that the velocity increases but

temperature decreases near the stretching sheet as we increase

the buoyancy parameter . It is also clear that the velocity

gradient at the surface increases from negative value to a

positive value. The velocity gradient at the surface is positive,

which signify that the stretching sheet velocity is smaller than

that of the adjacent fluid velocity. On the other hand negative

velocity gradient at the surface indicates that stretching sheet

velocity is greater than the adjacent fluid velocity. The

specific critical value of s.)crt( ,.1600822 3780413. are

for zero velocity gradient when ,.f w 60 60. respectively

as shown in Fig.3(a), where the surface shear stress are

vanished. As we increase the velocity gradient at the

surface increases for both injection and suction, as well as



they are almost identical which indicates that the flow is

predominated by the buoyancy effects. Since the velocity

gradient at the surface increases as we increase , so the

shear stress at the surface increases. It is also observed from

the Fig. 3(a) that, the hydrodynamic boundary layer thickness

is greater for 60.f w than for 60.f w since the earlier

for injection while the later is for suction.

Fig.3 (b) shows the temperature profiles for the same

parameters used in Fig.3 (a). It is clear that as increases the

thermal boundary layer thickness decreases for both suction

and injection. In this Fig. we see that the temperature gradient

as well as thermal boundary layer thickness are decreasing as

we increase . The temperature gradient is always negative

which means that heat is transferred from the sheet to the

ambient medium. Hence, the heat transferred rate from the

sheet to the ambient medium is getting large as increases

for both suction and injection. The thermal boundary layer

thickness is smaller for 60.f w than for 60.f w where

the earlier one is for suction while the later one is for

injection.

0 0.5 10

2

4

6

8

10

12

14

16

18

20

f'fw= - 0.6

fw= 0.6

2500.0250.025.02.160082

2500.0250.025.03.37804

(a)

0 1 2 3 40

0.25

0.5

0.75

1

fw= -0.6

fw= 0.6

2500.0250.025.02.160082

2500.0250.025.03.378041

(b)

Fig. 3. Dimensionless downstream velocity and temperature profiles

corresponding to ,m 1 1n and ..pr 720 showing the buoyancy

effects; (a) velocity and (b) temperature.

Moreover, the temperature profiles for suction are squeezed

together with reduced thermal boundary layer thickness than

that for injection therefore, suction enhances the heat transfer

coefficient from the stretched surface than injection.

We have delineated, In Fig4, the dimensionless velocity and

temperature profiles showing the effect of the Prandtl number

Pr when ,m 1 60.fw and 025. but 1n and

.0 1m indicates that the sheet is stretching linearly. We see

from Fig4 (a) and Fig4 (b) that the velocity and temperature

decrease rapidly as we increase Prandtl number Pr and the

temperature exponent n . The figures also indicate that

increasing Prandtl number Pr

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

n= -1

n = 0

n= -1

n = -1

n = 0

n = -1

n =0

pr = 0.72

pr = 3.0

pr = 10.0

f 'm =1, fw = 0.6,

(a)

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n=0

n=-1

n=-1

n=0

n=-1

n=0

pr = 0.72

pr = 3.0

pr = 10.0

m =1, fw = 0.6,

(b)


corresponding to ,m 1 ,n 0 1 and 025. showing the Prandtl

number effects; (a) velocity and (b) temperature.

and the exponent n reduce the momentum and thermal

boundary layer thickness that in turn reduce the shear stress

and increase the heat transfer coefficient at the surface

respectively. The velocity gradient at the surface is positive



for both 1n and 0 , when 720.Pr , 03. and 010. , which

means that the adjacent fluid velocity is higher than the

stretching sheet velocity. The velocity of the fluid within the

boundary layer, when ,n 1 is greater than that of the fluid

velocity when 0n (surface temperature is uniform), for

each value of the Prandtl number Pr . For example at

250. when 720.Pr and ,n 0 the velocity is %.9314

and temperature is %.718 less with in the boundary layer than

that of the velocity when 1n . On the other hand the

velocity reduces %.461 and temperature reduces %.1180 if

we increase Pr from 720. to 010. when 1 , 60.fw

and 025. .The momentum boundary layer thickness

remains same for both 1n and 0n for a fixed value

of Pr . But the difference of the velocities between these two

situations, that is 1n and 0n for each Prandtl number

decreases as we increase Pr . The specific critical (not shown

in figure) value of 5969520.PrPr )crt( is for zero

velocity gradient where the surface shear stress is vanished.

The dimensionless velocity and temperature profiles are

presented in Fig.5 for different values of the magnetic field

parameter when ,m 1 60.fw , 025. but 1n and

.0 It is clear from Fig.5 that velocity decreases but the

temperature increases as we increase the magnetic field

parameter M for both cases when 1n and .0 The

magnetic field lines act like strings and tend to retard the

motion of the fluid. The consequence of which is to increase

the temperature of the fluid. It is also noticeable that for a

fixed value of the magnetic field parameter, the velocity and

temperature corresponding to case of 0n are lower

compare to the case of 1n .

0 0.5 1 1.50

0.5

1

1.5

2

2.5

f '

M0.00.51.01.52.0

n=0

n= -1

m=1, fw= 0.6,

(a)

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

m=1, fw= 0.6,

n=0

n= -1

M2.01.51.00.50.0

M2.01.51.00.50.0

(b)


corresponding to ,m 1 ,n 0 1 and 025. showing the effects of

the magnetic field parameter; (a) velocity and (b) temperature.

That is, the thickness of the momentum and thermal boundary

layer are higher for the case 1n than for the case 0n .

The velocity gradients are positive but the temperature

gradients are negative for different situations of both cases.

Fig. 6 depicts the effects of the heat source parameter on

the velocity and temperature profiles. We see that velocity and

temperature increase rapidly as we increase the heat source

parameter Q . The figures also indicate that increasing heat

source parameter Q , increase the momentum and thermal

boundary layer thickness that in turn increase the shear stress

and reduce the heat transfer coefficient at the surface

respectively.

0 0.5 1 1.5 20

0.25

0.5

0.75

1

1.25

1.5

1.75

2

fw= 0.6, M=0.5, Pr= 0.72, m=1.0, n= -1,

Q=0.0, 0.5, 1.0, 1.5, 2.0

f'

(a)

0 1 2 30

1

2

3

4

5

fw= 0.6, M=0.5, Pr= 0.72, m=1.0, n= -1,

Q=0.0, 0.5, 1.0, 1.5, 2.0

(b)


corresponding to 01.m , 01.n showing the effects of the heat source

parameter Q ; (a) velocity and (b) temperature.



The critical (not shown in figure) value of the heat source

parameter is 6560.QQ )crt( which is for drag less

velocity profile. It is clear that the velocity and temperature

gradients at the surface increase from negative to positive

value. The specific critical value of 399760.QQ )crt( is

for zero temperature gradient at the surface. Critical values of

buoyancy parameter s.)crt( for which shear stress at the

surface become zero when

,m 1 1n , ,.pr 720 ,.03 ,.010 ,.M 50 50.Q and for

various values of wf are shown in the following table.

Table II

,m 1 1n ,m 1 1n

wf 720.pr 03.pr 010.pr 720.pr 03.pr

0.6

3.378041

6.87491

15.18558

2.26354

4.96301

0.4

3.093619

5.8113

11.78121

1.93646

3.791265

0.2

2.845829

4.91055

8.98045

1.637226

2.74961

0.0

2.631609

4.16187

6.76494

1.3622

1.813537

-0.2

2.447956

3.55106

5.094422

1.10724

0.96093

-0.4

2.2918

3.06359

3.90799

0.86808

-----------

-0.6

2.16008

2.68366

3.11479

0.64065

-----------

TABLE III

NUMERICAL VALUES PROPORTIONAL TO fC

AND xNu FOR DIFFERENT VALUES OF

,Q

ANDn .

n fC xNu Q n

fC xNu

2500.0

1.0

-1.0

271.6381653

357.2735710

-3.9996076

-0.3222749

0.0

1.0

-1.0

-0.4577762

-0.1787497

-1.1735897

-0.4334700

250.0

1.0

-1.0

45.7724434

60.6641322

-2.3664587

-0.2413562

0.5

1.0

-1.0

-0.3425068

0.1279294

-0.9615057

0.1244760

25.0

1.0

-1.0

6.3019204

8.8952453

-1.4629240

-0.1085501

1.0

1.0

-1.0

-0.1609941

0.7442207

-0.6775980

1.3320847

2.5

1.0

-1.0

-0.3425068

0.1279294

-0.9615057

0.1244760

1.5

1.0

-1.0

0.1199152

1.7995660

-0.2610622

3.8806777

1.0

1.0

-1.0

-0.9892398

-0.7398144

-0.8191061

0.2837815

2.0

1.0

-1.0

0.4895942

3.2751343

0.3368945

8.3870186



TABLE IV

NUMERICAL VALUES PROPORTIONAL TOfC

AND xNu FOR DIFFERENT VALUES OF M , Pr AND

n .

M n fC xNu Pr n

fC xNu

0.0

1.0

-1.0

-0.0861588

0.4192974

-1.0091940

0.0946265

0.72

1.0

-1.0

-0.3425068

0.1279294

-0.9615057

0.1244760

0.5

1.0

-1.0

-0.3425068

0.1279294

-0.9615057

0.1244760

1.0

1.0

-1.0

-0.4855775

-0.0624840

-1.2034948

0.0049507

1.0

1.0

-1.0

-0.5701267

-0.1323773

-0.9168410

0.1560966

3.0

1.0

-1.0

-0.9567890

-0.7418091

-2.7618639

-1.0987332

1.5

1.0

-1.0

-0.7751737

-0.3667972

-0.8747137

0.1893687

7.0

1.0

-1.0

-1.2204772

-1.1310145

-5.5449711

-3.4881805

2.0

1.0

-1.0

-0.9619481

-0.5794918

-0.8345793

0.2243773

10.0

1.0

-1.0

-1.3000409

-1.2432359

-7.5168460

-5.2959597

V. CONCLUSION

A mathematical model has been derived for MHD heat

transfer mixed convection flow in the presence of

magnetic field with heat generation. A benchmarked

numerical solution has been obtained to the transformed

boundary layer equations using Nachtsheim-Swigert

(1965) shooting iteration technique along with sixth-order

Runge-Kutta integration scheme. The numerical code has

been verified by comparison with previous computation by

Mohamed Ali et al. (2002), for the absence of magnetic

field and heat generation. We see that velocity increases

but the temperature decreases as we increase the buoyancy

parameter both in the cases of suction or injection. But the

suction always stabilizes both the velocity and temperature

fields. If we increase Prandtl number, velocity and

temperature profiles decrease. The temperature index

n has great effect on both velocity and temperature

profiles and increasing it reduces the shear stress and but

increases the heat transfer coefficient at the surface.

Magnetic field parameter has significant effect on velocity

profiles. Velocity and temperature profiles increase rapidly

with the increase of the heat source parameter therefore

using heat source parameter the velocity and temperature

fields can be controlled.

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