oscillatory flow of mhd polar fluid with heat and mass … · oscillatory flow of mhd polar fluid...

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International Journal of Mechanical Engineering and Technology (IJMET)

Volume 7, Issue 2, March-April 2016, pp. 212–231, Article ID: IJMET_07_02_024

Available online at

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=7&IType=2

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ISSN Print: 0976-6340 and ISSN Online: 0976-6359

© IAEME Publication

OSCILLATORY FLOW OF MHD POLAR

FLUID WITH HEAT AND MASS TRANSFER

PAST A VERTICAL MOVING POROUS

PLATE WITH INTERNAL HEAT

GENERATION

P.H.VEENA

Dept. of Mathematics, Smt. V.G. College for Women,

Gulbarga, Karnataka, INDIA

N. RAVEENDRA

Dept. of Mathematics, RajivGandi Institute of Technology,

Cholanagar, Bangaluru, Karnataka, INDIA

V. K.PRAVIN

Dept. of Mechanical Engineering,

P.D.A. College of Engineering Gulbarga, Karnataka, INDIA

ABSTRACT

The study of unsteady two-dimensional laminar boundary layer flow of a

viscous incompressible fluid (polar fluid) through porous medium past a semi-

infinite vertical porous stretching plate in the presence of transverse magnetic

field is investigated.

The sheet makes with a constant velocity in the longitudinal direction and

the free stream velocity follows an exponentially increasing or decreasing

small perturbation law. A uniform magnetic field acts perpendicularly to the

porous sheet which absorbs the polar fluid with a suction velocity varying with

time component. The effects of all parameters encountering in the problem are

investigated for velocity and temperature fields across the boundary layer.

The present results of velocity distribution of polar fluids are compared

with the corresponding flow problems for a Newtonian fluid. For a constant

stretching velocity with prescribed magnetic and permeability parameters,

Prandtl and Grashof numbers, viscous dissipation parameter, the effects of

increasing values of suction velocity parameter increases the surface skin

friction. It is also situated that the surface skin friction decreases with

increasing values of sheet moving velocity.

Oscillatory Flow of MHD Polar Fluid with Heat and Mass Transfer Past a Vertical Moving

Porous Plate with Internal Heat Generation


Cite this Article P.H.Veena, N. Raveendra and V. K.Pravin, Study of Process

Parameters of Gravity Die Casting Defects. International Journal of

Mechanical Engineering and Technology, 7(2), 2016, pp. 212–231.

http://www.iaeme.com/currentissue.asp?JType=IJMET&VType=7&IType=2

INTRODUCTION

Unsteady flows are of importance from the practical point of view and those that are

concerned with the effects of free stream oscillation are of physical significance from

the point of view of Industrial applications. After the initiation by Sakiadis [1],

analytical solutions to such problems of flow have been presented by a number of

authors.

Polar fluids are fluids with microstructure belonging to a class of fluids with

non-symmetrical stress tensor. Physically, they represent fluids consisting of

randomly oriented particles suspended in a viscous medium. The study of heat

transfer with boundary layer flow for an electrically conducting polar fluid past a

stretching porous sheet has attracted the interest of many investigators in view of its

applications in many engineering problems such as magneto hydrodynamic

generators, plasma studies, nuclear reactors, oil exploration, geothermal energy

extractions, the boundary layer control in the field of aerodynamics and these fluids

have key importance in polymer devolatasation, bubble columns, composite

processing etc. Sakiadis [2] attempted the first problem regarding boundary layer

viscous flow over a moving surface having constant velocity. Later this problem has

been studied extensively through various aspects. Very recent investigations relevant

to this problem are dealt by [3-7]. Wang [8] was the best researcher who discussed the

viscous flow due to an oscillatory stretching surface. Although Oscillatory stretching

sheet induces the present flow but we also have a free stream velocity Oscillating

about a constant mean oscillatory flow [9-11].

A great number of Darcian porous MHD studies have been carried out examining

the effects of magnetic field on hydrodynamic flow without heat transfer in various

configuration, eg in channels and past stretching plates and wedges, etc [12, 13].

Gribban [14] considered the MHD boundary layer flow over a semi-infinite plate

with an aligned magnetic field in the presence of pressure gradient and he has

obtained the solutions for large and small magnetic Prandtl numbers using the method

of matched asymptotic expansion. Takhar and Ram[15] studied the effects of Hall

currents on hydro magnetic free convection boundary layer flow via porous medium

past a plate. Thakar and Ram [15] also studied the MHD free convective heat transfer

of water at 40 C through a porous medium.

Soundalgekar [17] obtained approximate solutions for viscous two-dimensional

flow past an infinite vertical porous plate with constant suction velocity normal to the

plate. The difference between the temperature of the plate and the free stream is

moderately large causing the free convection currents Raptis [18] studied

mathematically the case of time-varying two-dimensional natural convective heat

transfer of an incompressible MHD viscous fluid Via a highly porous medium

bounded by an infinite vertical plate. Despite recent advances in non-Newtonian and

polar fluids, it is still of interest to develop stretching flows involving polar fluids. For

example, no investigation.

In this regard Abbas et al [11] made an attempt to get analytical solution for a

second grade fluid flow due to MHD and unsteady stretching surface by HAM

technique. Further unsteady free convection MHD flow with heat transfer in a porous

http://www.iaeme.com/currentissue.asp?JType=IJMET&VType=7&IType=2

P.H.Veena, N. Raveendra And V. K.Pravin


medium has been studied extensively by several authors because of its wide spread

applications in a number of engineering and environmental applications [ref 19]- such

as fibred and granular insulations, and granular insulations, electronic system cooling,

Modern communication system, heat exchanger and geophysical studies etc.

Many works have already been done on transient free convection flow past a

vertical plate. Some of the pioneers Gephart [20]. Schetz et al [21], Gold stain and

Briggs [22], Soundalgekar [23] and Das et al [24].

However most of the previous works assume that the plate is at rest. Thus the

study reported presently here in considers the unsteady free convection flow of an

electrically conducting polar fluid in the presence of transverse magnetic field over a

semi-infinite moving porous plate with a constant velocity in the longitudinal

direction. we also consider the free stream to consist of a mean velocity and

temperature with a super imposed exponentially variation with time. Applying

perturbation technique, the solutions for velocity and temperature of the flow field are

obtained and the effects of the flow parameters are discussed with the help of graphs.

This problem is an extension of the work of Kim [16] to a porous medium with

viscous dissipation and internal heat generation/absorption in the energy transfer.

In general, the study of Darcian porous MHD is very complicated. It is necessary

to consider in detail the distribution of velocity and temperature across the boundary

layer in addition the surface skin friction. Thus the present work is an attempt made to

shed some light to these aspects.

MATHEMATICAL FORMULATION

We consider the unsteady flow of an two dimensional an incompressible laminar fluid

past a plate saturated in a porous medium and subjected to a transverse magnetic field

in the presence of a pressure gradient.

It is assumed that there is no applied voltage which implies the absence of an

electric field. The magnetic field and magnetic Reynolds number are of low

conduction and hence the induced magnetic field is negligible. Viscous and Darcy’s

resistance terms are consider with constant permeability of the porous medium.

Porous medium is considered as an assemblage of small spherical particles fixed in

space, following [31]. Due to semi infinite plane surface assumption, the flow

variables are further consider as the functions of y* and t

* only.

Under the above assumptions the governing equations of continuity, momentum,

angular momentum and heat conservation equations can be considered in a Cartesian

frame of reference as follows

0*

*

y

v (1)

y

wvuBu

kTTg

y

uvv

x

p

y

uv

t

urr

**2

0

*

*2*

*2

*

*

*

**

*

*

2)(1

(2)

2*

*2

*

**

*

**

y

w

y

wv

t

wj

(3)

TTQ

y

T

y

Tv

t

T2*

2

*

*

* (4)




2*

2

0*

*

* y

CD

y

Cv

t

C

(5)

Where u*

and v* are the components of longitudinal and transverse velocities

along the respective dimensional distances along x* and y

* directions.

[ρ is the density and υ is the kinematic viscosity, vr is the rotational viscosity, g is

the acceleration gravity, β is the coefficient of voluaicetric thermal expansion of the

fluid, k* is the co-efficient of permeability of the porous medium, σ is the electrical

conductivity]

The heat due to viscous dissipation term is neglected because it is of the same

order of magnitude as the viscous dissipation term but heat due to internal

generation/absorption is considered for the study. It is assumed that the porous plate

moves with constant velocity up*

and the temperature T, the concentration C and

suction velocity v* vary exponentially with time t

*. D is the mass diffusion co-

efficient

Under the above considerations the appropriate boundary conditions for the

velocity and temperature and concentration fields are as follows

u* = up

* ,

tn

ww eCCCC*

(6a)

2*

*2

*

*

y

u

y

w

at y

* = 0

tneUUu*

10

yaswCC 0; (6b)

Where n* is constant scalar and U0 is the scale of free stream velocity.

Solution for continuity equation yields

tneAVv*

10 (7)

Where V0 the non zero positive constant is the scale of suction velocity normal to

the plate and is a function of t and A is a real positive constant, ε and εA are very

small and are less than unity.

Solution of momentum, heat and mas transfer equations;

Out side the boundary layer equation (2) gives

*

0

*

*

*

*

*1

UBU

kdt

dU

x

p

(8)

We now set the following non-dimensional variables

2

0

**

000

*

0

**

0

00

,;,,,;t

twVU

wU

uU

U

UU

yy

VU

uU

p

pp

,

parametertyPermiabiliVk

kVCC

CC

TT

TT

ww2

2

0

*

22

0

*

;,,

.



numberandtltheisk

CP

DSc

p

r Pr

2

0

2

0

2

2

0 ,v

BMj

vj n

is magnetic parameter

2

00VU

TTgG wT

c

is the Grashof number

2

00VU

CCgG wc

r

is modified Grashof number

Further it is deduced that

2

11

2jj

A (9)

Where is the spin gradient viscosity, which is the relationship between the co-

efficient of viscosity and micro-inertia, β is the non dimensional ratio of viscosity and

is defined as

β = A/µ (10)

in which A is the co-efficient of vertex viscosity.

In view of the equations (7) to (10) the governing equations (2) to (5) reduce to

the following non-dimensional form

y

wuUMGG

y

u

dt

dU

y

ueA

t

urc

nt

2112

2

(11)

2

211

y

w

gdt

dU

y

weA

t

w

c

nt

(12)

Q

yPyeA

t r

nt

2

211 (13)

Where 2

0

*

V

QQ

is the internal heat generation

2

211

y

C

Sy

CeA

t

C

c

nt

(14)

Where 11 22),(2

jMkM n

The corresponding boundary conditions (5) and (6) are reduced to their

dimensionless form as ,

,0Uu nteA1 ,

y

weA nt ,1

2

2

y

u

at y = 0 (15)

yasUu ,0,0, (16)




Perturbation solution: In order to reduce the set of partial differential equations (2)

to (5) to a system of ordinary differential equations in non-dimensional form, we the

perturbation technique for linear temperature as

u = v0 (y) + ε ent

u1(y) + O (ε2 )+ ……. (17)

w = w0 (y) + ε ent

w1(y) + O (ε2 )+ ……. (18)

θ = θ 0 (y) + ε ent

θ 1(y) + O (ε2 )+ ……. (19)

φ = φ 0 (y) + ε ent

φ 1(y) + O (ε2 )+ ……. (20)

substituting (17) to (20) in equations (11) to (14)

and equating the periodic and non-periodic terms and neglecting the higher order

terms of O(ε2), we get the following pairs of equations for u0, u1, w0, w1, θ 0 , θ 1 and φ

0, φ1

00020200 21 WGrGckMnukMnuu (21)

111021211 21 WGrGcAunkMnunkMnuu (22)

000 WW (23)

0111 WAWnWW (24)

0PrPr 000 (25)

0111 PrPr An (26)

000 ScSc (27)

0111 ScAnSc (28)

The corresponding boundary conditions can be written as

01,1

,1,1,,,0,

11

00110010

yat

uWuWuUpu

(29)

yas

WWuu

0,0

,0,0,0,0,1,1

11

001010

(30)

Thus the solutions for all the equations of momentum, energy and mass

concentration from (21) to (28) satisfying the respective boundary conditions (29) and

(30) are found as

yScyyyh eAeAeAeAyu 43

Pr

21021 (31)

yyScyyhyhyhyhyh eBeBeBeBeBeBeBeByu 87

Pr

6543211543211

(32)

yeDyW 10 (33)

yyh eDn

AeDyW 121

1 (34)



yey Pr

0

(35)

yyhyh een

Aey Pr

144 Pr

(36)

yScey 0 (37)

yScyhyh eeScn

Aey

55

1

(38)

Where

nh

411

21

1411

12

12 Nh

1411

12

13 nNh

Pr

411

2

Pr4

nh

,

Sc

nSch

411

25

and

A1 = Up – 1 – A2 – A3 – A4 NScSc

GrA

N

GcA

23221

,PrPr1

1241

2D

NA

2

1

2

1

11

1

2D

nNhh

hB

12

2 An

hAB

1

12

1

1

2

1

2

31

2

1

13

2

31

2121

2

h

kk

h

nNhh

nNhhh

hB

nNhhn

AGB c

4

2

4

41

1Pr1

nNhhn

AScGB r

4

2

4

51

11

Nn

GcAB

PrPr1

1Pr26

NScScn

GrScAB

271

1

nN

Cn

AnAA

B

2

1

2

3

81

2




2

2

2

22

22

2

21 Pr121

11

AhhUphN

N

D

2

33

2

111

1

2

1hbhbk

hD

2

6

2

7

2

5

2

44

2

221

2

1 Pr ScBBBhBhBDA

k

8765423 1 BBBBBBk

With respect to equations (17), (18), (19) and (20) the solutions for stream wise

and angular velocities, temperature and concentration are obtained as follows

yyScyyhyhyhyhyht

yScyyyh

eBeBeBeBeBeBeBeBe

eAeAeAeAtyu

87

Pr

654321

43

Pr

21

54321

2

1

1,

(39)

yyhty eD

AeCeeCtyw

121

1),( (40)

yyhyhty ee

Aeeety PrPr 44 Pr),(

(41)

yScyhyhtScy eeSc

Aeeety 55),(

(42)

SKIN FRICTION

Skin friction at the wall of the plate is calculated by

000

y

ww

y

u

VeU

(43)

8765544332211

32222

Pr

Pr1

BScBBhBhBhBhBhB

eAhAhhUp t

(45)

NUSSELT NUMBER

Heat transfer co-efficient interms of Nusselt number is given by

TT

y

Tx

Nuw

wx

1

0

Re

x

y

Nuy

Pr1PrPr 4

2

A

hA

e t (46)

ye

y

t 10



SHREWOOD NUMBER (Sh) Mass transfer co-efficient interms of Shrewood

number is given by

1

0

xy

w

eRShy

C

CC

y

C

xSh

Sc

AhSc

AeSc t

15

2 (47)

010

y

t

ye

y

RESULTS AND DISCUSSION

The investigations of effect of magnetic field, porous media and suction velocity

varying exponentially with time about a constant mean (non-zero) on boundary layer

flow and heat transfer of an incompressible MHD polar fluid along a semi-infinite

vertical porous moving plate is studied. Numerous computations are carried out for

the velocity and temperature profiles for different numerical values of the flow and

encountered physical parameters in the study. In the present analysis y is

replaced by Ymax then velocity profile u approaches the appropriate free stream

velocity.

The distribution of velocity profiles across the boundary layer for a stationary

vertical porous plate in the presence of magnetic field and absence of magnetic field

are depicted in fig1 and 2.

In figure (3a) and (3b) the graphs of velocity and angular velocity profiles for

different values of viscosity ratio and fixed values of parameters ε, n, τ, A, Mn, k2, G,

Pr and Up.

From the figures we observe that velocity distribution is lower for β = 0. When β

takes the values greater than 0.5, the velocity distribution has a decelerating nature

near the plate. However the angular velocity profiles do not showing the consistency

with increase in viscosity ratio parameter β.

Figure 4a and 4b elucidate the behavior of variation is velocity and angular

velocity profiles for different values of plate moving velocity Up is the direction of

fluid flow. It is observed the figures that, as the plate moving velocity increase, the

peak velocity across the boundary layer decreases near the plate. However the angular

velocity distribution has an increment as an increment in plate moving velocity Up.

Figures 5a and 5b are drawn for velocity and angular velocity profiles for different

values of permeability parameter k2. It is obvious from the figures that the effect of

increasing values of permeability parameter results in a decreasing velocity

distribution and angular velocity distribution is also decreased near the plate as k2

increases.

Figure 6(a) and 6(b) are depicted to illustrate the effect of Grashof number on

velocity and angular velocity profiles respectively. From the figures it is observed that

increase in the values of G is to accelerate the velocity distribution across the

boundary layer but decelerates the angular velocity.

Figures 7 (a) and (b) shows the variation of velocity profiles against Y for

different values of Prandtl number Pr with constant values of all other parameters.




The result shows that increasing values of Prandtl number results in a decreasing

velocity and then approaches a constant value which is relevant to the free stream

velocity at the edge of the boundary layer the effect of Prandtl number decreases the

angular velocity distribution.

It has been observed that from figure 9 for a constant suction velocity parameter

A, the effect of increasing values of plate moving velocity Up results in a decreasing

skin friction on the plate. It is also evident from the figure that skin friction vanishes

near the plate velocity at Up = 1.7.

Figure 10 illustrate the graphs of surface heat transfer Nux versus suction velocity

parameter A for different values of Prandtl number. Pr. It is revealed from the figure

that as the magnitude of suction velocity A increases, the surface heat transfer from

the porous plate tends to decrease slightly.

CONCLUSIONS

We have investigated the study of an unsteady incompressible polar fluid past a semi-

infinite porous moving plate whose velocity is constant and increased in saturated

porous medium and subjected to a transverse magnetic field. The method of small

perturbation approximation technique is applied to find the solution of all governing

equations of momentum, angular momentum and heat transfer. Numerous results are

obtained and examined to illustrate the details of the flow and heat transfer

characteristics.

It is concluded from the figures that the effect of increase of magnetic parameter is

to decrease the velocity and angular velocity distribution whereas the increase in

values of permeability parameter and Grashof number increases the distribution

velocity and angular velocity across the boundary layer.

For the further understanding of the thermal behavior of the work of Kim[16], the

present work is an extension of the same and all our results are well in argument with

the published results of Kim[16].

The effect of Schmidt number Sc on the velocity and concentration are shown in

figures 8(a) and (b). from the figures it is revealed the fact that as Sc increases, the

concentration across the boundary layer decreases. This causes the concentration

Buoyancy effects to decrease Yielding a reduction in the fluid velocity.

The concentration decreases as the Schmidt number increases. The Shrewood

number decreases as the Schmidt number increases.

From figure 11 is drawn to represent Shrewood number against the Suction

velocity parameter for different values of Sc and it is noticed from the figure that as

Sc increases Shrewood number decreases.



Graphs

U0x

t*

U0 *, T0

y

Porous

media

Tw

U0

Fig I physical configuration of the problem

|

1

|

2 |

3 |

4 | 5 |

6

0.5 -

1.0 -

1.5 -

2.0 -

u

0 y

k2= 0.5

Gr= 2

sc= 5

0

7.0Pr2

20

10

00

Up

M

GA

t

c

Fig 1a Velocity profiles across the boundary layer for a stationary vertical plate in presence of magnetic field when viscosity ratio is zero




|

1

|

2

|

3

|

4 |

5

|

6

0.5 -

1.0 -

1.5 -

2.0 -

u

0

y

k2= 0.5

Fig 2 Velocity distribution of across the boundary layer for a

stationary vertical porous plate in the absence of magnetic field.

ϵ=0 η=0

β=0 M=0

A=0

Fig 3a Velocity profiles against y-c0-ordinate for different

values of viscosity ratio β.

| 1 | 2 | 3 |

4 |

5

|

6

0.0 -

5.0 -

10.0 -

15.0 -

u

0

Y

β= 0.5

5,0

3Pr,0

5.0,0

1,0,0

2

ScUp

Mn

kA

t

= 0.0

= 3.0

= 1.0

= 0.7



| 1 | 2 | 3 | 4 | 5 | 6

-40.0 -

-20.0 -

20.0 -

W

0 y

β = 0.7

Fig 3b Angular velocity profiles versus Y-co-ordinate for different

values of viscosity ratio β

-60.0 -

40.0 -

=1.0

= 3.0

= 0.7

= 0.7

ϵ=0 k2=0.5 η=0 t=1

A=0 Sc=5 Mn=0

Pr=3 Up=0

| 7

2.5 -

= 1.0

| 1 |

2 | 3

|

4 | 5

|

6

0.5 -

1.0 -

1.5 -

2.0 -

u

0

y

Up= 0.25

= 0.0 =0. 5

Fig 4a Velocity profiles versus co-ordinate Y for various

values of plate moving velocity Up.




|

7

-15.0 -

15.0 -

= 1.0

|

1

|

2

|

3

|

4

|

5

|

6

-10.0 -

-5.0 -

5.0 -

10.0 -

w

0

y

Up= 1.5

= 0.0

=0. 5

Fig 4b Graph of angular Velocity profiles versus Y -

co-ordinate for different values of Up.

|

7

2.5 -

= 8.0

|

1

|

2

|

3

|

4

|

5

|

6

0.5 -

1.0 -

1.5 -

2.0 -

u

0

y

k 2 = 0.0

= 2.0

= 4.0

Fig 5a Distribution of velocity and angular velocity profiles versus

y- co-ordinate for various values of permeability parameter k2



|

7

-6.0 -

-10.0

-

4.0 -

= 6.0

|

1

|

2

|

3

|

4

|

5

|

6

-8.0

-

-4.0 -

-2.0

-

2.0 -

w

0.0

y

K2= 8.0

= 2.0

=4.0

Fig 5b

2.5 -

= -2.0

|

1

|

2

|

3

|

4

|

5

|

6

0.5 -

1.0 -

1.5 -

2.0 -

u

0

y

Gc = 4.0

= 2.0

= 0.0

Fig 6a Distribution of velocity and angular velocity profiles versus

y- co-ordinate for various values of permeability parameter k2




0.0 -

-6.0 -

= 0.0

|

1 |

2

|

3 |

4 |

5 |

6

-8.0 -

-4.0 -

-2.0 -

w

0.0

y

Gc = 4.0

= - 2.0

= 2.0

Fig 6a and 6b Graphs of velocity and angular velocity profiles against Y

various values of Grashof number Gc with ε = 0.1, β = 0.2, n = 1, t

= 1, A = 0.5, M = 0.5, k2 = 2, Pr = 3, Up = 0.5, Gr = 2, Sc = 5

Fig 7 Variation of velocity and temperature distribution versus Y for various values of Prndtl

number Pr with ε = 0.1, β = 0.2, n = 1, t = 1, A = 0, k2 = 0.5, Gr = 2, Up = 0.5, Gc = 1, Mn =

0. 5

2.5

-

= 5.0

|

1 |

2

|

3 |

4

|

5

| 6

0.5 -

1.0 -

1.5 -

2.0

-

u

0

y

Pr = 0.7

= 1.0

= 3.0

2.5

-

= 5.0

|

1 |

2

|

3 |

4

|

5

| 6

0.5 -

1.0 -

1.5 -

2.0 -

T

0

y

Pr = 0.7

= 1.0

= 3.0



|

7

3.5 -

3.0 -

2.5 -

= 0.5

|

1

|

2

|

3

|

4

|

5

|

6

0.5 -

1.0 -

1.5 -

2.0 -

U

0

y

= 2. 0

= 1.5

= 1.0

1.2 -

1.0 -

= 0.5

|

1

|

2 |

3

|

4

|

5 |

6

0.2 -

0.4 -

0.6 -

0.8 -

C

0

y

Sc = 2.0

= 1.5

= 1.0

Fig 8a and 8b Variation of velocity concentration distribution versus Y co-ordinate for various values of

Schimdt number Sc with ε = 0.1, β = 0.2, n = 1, t = 1, A = 0, k2 = 0.5, Gr = 2, Up = 0.5, Gc = 1, Mn = 0. 5

-15.0 -

-5.0 -

0

A = 0.5

|1

|2 |

3

|

4

|5 |

6

-10.0 -

0.0 -

5.0 -

τ

10.0 -

Up

A = 1.0

A= - 0.5

A= 0.0

Fig 9 Graph of skin friction co-efficient versus plate moving velocity for various values of suction parameter A and with ε = 0.1,

β = 0.2, n = 1, t = 1, A = 0.5, M = 0.5, k2 = 0.5, Pr = 1, Up = 0.5, Gr =

2, Sc = 3




-8.0 -

-6.0 -

0

-2.0 -

-1.0 -

Pr = 1

|1

|2

|3

|4

|5

|6

-7.0 -

-5.0 -

-4.0 -

NUx

-3.0 -

A

Pr = 0.7

Pr = 5

Pr= 3

Fig 10 Graph of surface heat transfer Vs suction velocity

parameter for various values of Prandtl number Pr A and with ε = 0.1, β = 0.2, n = 1, t = 1, Mn = 0.5, k2 = 0.5,

Up = 0.5, Gr = 2, Gc = 1

-8.0

0

-6.0

-

Sc = 1.5

-2.0

-

-1.0 -

|

1 |

2

|

3 |

4 |

5

|

6

-7.0

-

-5.0

-

-4.0 -

Sh Rex-1

-3.0

-

A

Sc = 2. 0

Sc =

0.5

Sc = 1.0

Fig 11 Graph of surface mass transfer Vs suction velocity parameter A for various

values of Schmidt number Sc with ε = 0.1, β = 0.2, n = 1, t = 1, Mn = 0.5, k2 = 0.5,

Up = 0.5, Gr = 2, Gc = 1

|

7



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