combined heat and mass transfer in mhd three-dimensional porous flow with periodic permeability &...

7/30/2019 Combined Heat and Mass Transfer in Mhd Three-dimensional Porous Flow With Periodic Permeability & Heat Absor

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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 6340(Print),ISSN 0976 6359(Online) Volume 3, Issue 2, May-August (2012), IAEME

573

COMBINED HEAT AND MASS TRANSFER IN MHD THREE-DIMENSIONAL

POROUS FLOW WITH PERIODIC PERMEABILITY & HEAT ABSORPTION

Dr P.Ravinder Reddy1

Dr K.Srihari2

Dr S. Raji Reddy2

1Department of Mechanical Engineering, CBIT, Gandipet,Hyderabad, (A.P), India

500 075,+91-040-23518467,email:[email protected], fax:+91-08413-234155

2Department of Mathematics, Mahatma Gandhi institute of technology

Gandipet, Hyderabad, (A.P), India, 500075.

ABSTRACT

The paper analyzed the effects of mass transfer and heat sink on three-dimensional free

convective heat transfer flow through a highly porous medium with periodic permeability, in the presence

of transfers magnetic field. Assuming the free stream velocity to be uniform, solutions of governing

equations of motion are obtained, using finite deference technique, which is more economical from

computational view point. The results obtained for the velocity, temperature, concentration, skin friction,

rate of heat and mass transfer are discussed and analyzed through graphs, to observe the effects of various

flow parameters. It is found that the concentration of the species is higher for small values of Sc and

lower for larger values of Sc. Also it is found that heat sink and magnetic field reduces the velocity of the

fluid while heat transfer coefficient increases in the presence of heat absorption parameter.

Key words

Volumetric rate of heat absorption; Magnetic field; Porous medium; Periodic permeability; Finite

deference technique.

INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING ANDTECHNOLOGY (IJMET)

ISSN 0976 6340 (Print)

ISSN 0976 6359 (Online)

Volume 3, Issue 2, May-August (2012), pp. 573-593

IAEME: www.iaeme.com/ijmet.html

Journal Impact Factor (2012): 3.8071 (Calculated by GISI)

www.jifactor.com

IJMET

I A E M E


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574

INTRODUCTION

Many natural phenomena and engineering applications are susceptible to magneto hydrodynamics

(MHD) analysis. From technological point of view, magneto-hydrodynamic flow finds application in thefields of stellar and planetary magneto-spheres, aeronautics, meteorology, solar physics, cosmic fluid

dynamics, chemical engineering, electronics, and induction flow metry, MHD generators, MHDaccelerators, construction of turbine and other centrifugal machines. Due to its increasing importance invarious technical applications using magneto hydrodynamic effect, it is desirable to extend many of the

available hydrodynamic solutions for those cases when the viscous fluid is electrically conducting.

Also,in the recent years, the flows through porous medium are of principal interest because theseare quite prevalent in nature. Such Flows have many scientific and engineering applications, viz., in the

fields of agricultural engineering to study the under ground water resources, seepage of water in river beds;

in chemical engineering for filtration and purification processes In view of these applications, a series ofinvestigations have been made by Raptis et al. [1-3] in to the steady flow past a vertical wall. Raptis [4]

studied the unsteady flow through porous medium bounded by an infinite porous plate subjected to a

constant suction and variable temperature. Raptis and Perdikis [5] further studied the problem of freeconvective flow through a porous medium bounded by a vertical porous plate with constant suction where

the free stream velocity oscillates in time about a constant mean value.

In all the studies mentioned above the permeability of the porous medium has been assumed asconstant. In fact, a porous material containing the fluid is a non-homogeneous medium and there can be

numerous in homogeneities present in a porous medium. Therefore, the permeability of the porous medium

may not necessarily be constant. Sing and Suresh Kumar [6] have analyzed a free convective twodimensional unsteady flow through a highly porous medium bounded by an infinite vertical porous plate

when the permeability of the medium fluctuates in time about a constant mean. Most of the investigators

have restricted themselves to two-dimensional flows only by assuming either constant or time dependent

permeability of the porous medium. However, there may arise situations where the flow field may beessentially three dimensional, for example, when variation of the permeability distribution is transverse to

the potential flow. The effect of such a transverse permeability distribution of the porous medium boundedby horizontal flat plate has been studied by Sing and Verma [7] and Singh et al [8]. Recently, Singh and

Sharma [9] studied the effect of transverse periodic variation of the permeability on the heat transfer and

the free-convective of a viscous incompressible fluid through a highly porous medium bounded by a

vertical porous plate. In addition to this, more recently, Jain et al [10] studied the effects of periodictemperature and periodic permeability on three-dimensional free convective flow through porous medium

in slip flow regime. But, in all the above mentioned three-dimensional studies, the effects of mass transfer

and heat sink in the presence of magnetic field have not been studied.

Coupled heat and mass transfer phenomenon in porous media is gaining attention due to itsinteresting applications. Processes involving in heat and mass transfer in porous media are oftenencountered in the chemical industry, in reservoir engineering in connection with chemical recovery

process, in the study dynamics of hot and salty springs of a sea. Underground spreading of chemical and

other pollutants, grain storage and evaporation cooling. Also, the propagation of thermal energy in the

presence of heat sink have great applications in various fields of energy, atomic pollutions, space scienceand in engineering and technology there are occasions where the heat sink is needed to maintain desired

heat transfer.


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575

In view of these applications, the aim of the present investigation is to study the effects of masstransfer and heat absorbing sink on three-dimensional free convective heat transfer flow with periodic

permeability, under the influence of transfers magnetic field. In the present work, the effects of different

flow parameters encountered in the equations were also studied. In the above stated three-dimensionalstudies, a series expansion method was employed to solve the fluid flow problem. But, the present

problem has been solved numerically, using finite difference method, because it is more economical

from computational view point.

Mathematical analysis:

We now consider the flow of a viscous fluid through a highly porous medium bounded by an infinite

vertical porous plate with constant suction. The plate is lying vertically on the x*-z

*plane with x

*-axis

taken along the plate in the upward direction. The y*-axis is taken normal to the plane of plate and

directed into the fluid flowing laminarly with a uniform free stream velocity U. A magnetic field ofuniform strength is applied normal to the flow, along *y -axis. The permeability of the porous medium is

assumed to be of the form.

)/cos1()(

*

*

0**

Lz

KzK

+= (1)

Where *0K is the mean permeability of the medium. L is the wavelength of the permeability

distribution and (


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576

Thus, denoting velocity components by *** ,, wvu in the directions of *** ,, zyx respectively and

the temperature by the T*

and concentration by C*, the flow through a highly porous medium is governed

by following equations:

0*

*

*

*

=

+

z

w

y

v(2)

*

2

0*

*2*

*2

2*

*2*****

*

**

*

** )()()( u

BUu

Kz

u

y

uCCgTTg

z

uw

y

uv

+

++=

+

(3)

*

*2*

*2

2*

*2

*

*

*

**

*

** 1

v

Kz

v

y

v

y

p

z

vw

y

vv

+

+

=

+

(4)

*

2

0*

*2*

*2

2*

*2

*

*

*

**

*

** 1 w

Bw

Kz

w

y

w

z

p

z

ww

y

wv

+

+

=

+

(5)

)(2*

*2

2*

*2

*

**

*

**

+

=

+

TTQ

z

T

y

T

C

k

z

Tw

y

Tv

p(6)


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577

+

=

+

2*

*2

2*

*2

*

**

*

**

z

C

y

CD

z

Cw

y

Cv (7)

The boundary conditions of the problem are:

********,,0,,0;0 ww CCTTwVvuy ======

********* ,,,0,; CCTTppwUuy (8)

where *wT and*

wC are the temperature and concentration of the plate,*

T and*

C are the temperature and

concentration of the fluid far away from the plate, *p is a constant pressure in the free stream and V>0

is a constant and the negative sign indicates that suction towards the plate.

Introducing the following non dimensional quantities:

,*

L

yy = ,*

L

zz = ,*

U

uu = ,*

V

vv =

,*

V

ww = ,

2

*

U

pp

= ,

**

**

=

TT

TT

w

**

**

=

CC

CC

w

(9)

in Eq. (2) to (7)., the following equations are obtained:

0=

+

z

w

y

v(10)

uM

K

zu

z

u

y

uGmGr

z

uw

y

uv

ReRe

)cos1()1(

Re

1ReRe

2

02

2

2

2

+

+

++=

+

(11)

( )

02

2

2

2

Re

cos1

Re

1

K

vz

z

v

y

v

y

p

z

vw

y

vv

+

+

+

=

+

(12)

( )w

M

K

wz

z

w

y

w

z

p

z

ww

y

wv

ReRe

cos1

Re

12

02

2

2

2

+

+

+

=

+

(13)

Szyz

wy

v

+

=

+

2

2

2

2

PrRe1 (14)

+

=

+

2

2

2

2

Re

1

zySczw

yv (15)

where


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578

( )2

**

UV

TTgGr w

=

(Grashof number)

2

***

UV

CCgG w

m

=

(Modified Grashof number)

=

VLRe (Reynolds number),

k

Cp=Pr (Prandtl number)

DSc

= (Schmidt number),

2

*

00

L

KK = (Permeability parameter)

LBM 0= (Magnetic parameter),

2QLS = (Heat absorption parameter)

The corresponding boundary conditions reduce to

;0=y ,0=u ,1=v ,0=w ,1= 1= (16)

;y ,1u ,1w , pp ,0 0

In order to solve the problem we assume the solutions of the following form because the

amplitude ( )1 is very small:

( ) ( ) ( ) ( ) ...,,, 22

10 +++= zyuzyuyuzyu

( ) ( ) ( ) ( ) ...,,, 22

10 +++= zyvzyvyvzyu

( ) ( ) ( ) ( ) ...,,, 22

10 +++= zywzywywzyw (17)

( ) ( ) ( ) ( ) ...,,, 22

10 +++= zypzypypzyp

( ) ( ) ( ) ( ) ...,,, 22

10 +++= zyzyyzy

( ) ( ) ( ) ( ) ...,,, 22

10 +++= zyzyyzy

When ,0= the problem is reduced to the two-dimensional free convective flow through a porous

medium with constant permeability which is governed by following equations:

00 =dy

dv(18)

00

20

20

0

2002

02

1ReRe

1Re

KGmGru

KM

dy

duv

dy

ud=

+ (19)


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579

0PrRe 00

02

02

=

Sdy

dv

dy

d(20)

0Re

0

02

02

=

dy

d

Scvdy

d

(21)

The corresponding boundary conditions become

;0=y ,00 =u ,10 =v ,10 = 10 = (22)

;y ,10 u ,0 pp ,00 00

The solutions of Eq. (18) to (21). under the boundary conditions (22) are given by

( )

yScyryReGmeGreGmGru

Re

10100.11

++= (23)

rye

=0 (24)

ySce

Re0

= (25)

with

,10 =v 00 =w and = pp0 (26)

where

+

=

0

22

2

01

Re

Re

KMrr

,2

4PrRePrRe 22 Sr

++=

( )

+

=

0

22

2

11

1Re

Re

KMScSc

,0

22 1

4

Re

2

Re

KMR +++=

When ,0 substituting (17) in Eq. (10) to (15). and comparing the coefficients of identical power

of , neglecting the higher powers of , the following equations are obtained with the help of Eq.

(26).:


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580

011 =

+

z

w

y

v(27)

( )

1

2

0

10

2

12

2

12

111

10

1 ReRe

cos1

Re

1ReRe u

M

K

uzu

z

u

y

uGG

y

u

y

uv

+

+

++=

(28)

( )

0

121

2

21

211

Re

cos

Re

1

K

zv

z

v

y

v

y

p

y

v

+

+

=

(29)

1

2

0

121

2

21

211

ReReRe

1w

M

K

w

z

w

y

w

z

p

y

w

+

+

=

(30)

12

12

2

12

101

PrRe

1

S

zyyy

v

+

=

(31)

+

=

21

2

21

210

1Re

1

zyScyyv (32)

The corresponding boundary conditions are:

;0=y ,01 =u ,01 =v ,01 =w ,01 = 01 = (33)

;y ,01 u ,01 w ,01 p ,01 01

Eq. (27) to (32). are the partial differential equations, which describe free convective three-

dimensional flow. In order to solve these equations we shall first consider (27), (29) and (30), being

independent of the main flow component 1u , temperature field 1 and concentration field 1 . In the

following form, 11 , wv and 1p are assumed:

( ) ( ) zyvzyv cos, 111 = (34)

( ) ( ) zyvzyw

sin1

, 111 = (35)

( ) ( ) zypzyp cos, 111 = (36)

Where the prime in )(11 yv denotes the differentiation with respect to .y Expressions for ( )zyv ,1

and ( )zyw ,1 have been chosen so that the equation of continuity (28) is satisfied. Substituting the

expressions (34), (35) & (36) in (29) and (30), the following differential equations can be obtained:


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581

( ) ( )1Re

1

Re

111

01111

21111 +++= v

Kvvpv (37)

11

2

0

1111

1111

11

ReReRe

1v

M

K

vv

vp

v

+=

(38)

Eliminating the terms 1111 ,pp in Eq. (37) and (38)., the following is obtained.

0Re21

Re0

2

110

24

112

112

0

21111 =

+

++

+++

Kv

Kvv

KMvv iv (39)

The corresponding boundary conditions become

0:

0,0:0

11

1111

=

===

vy

vvy. (40)

In order to solve the differential Eq. (28), (31) and (32). for 11 ,u and 1 respectively,

the following are assumed

zyuzyu cos)(),( 111 = (41)

zyzy cos)(),( 111 = (42)

zyzy cos)(),( 111 = . (43)

Substituting the above equations in (28), (31) and (32), the following equations can be obtained:

0011

2111

201111

2

0

21111

1ReReRe

1Re K

uGGuvuKMuu

+=

+++ (44)

011112

1111 PrRePrRe =+ v (45)

011112

1111 ReRe =+ vScSc (46)

with corresponding boundary conditions

0,0,0:0 111111 ==== uy (47)

0,0,0: 111111 uy .

Substituting the following finite difference formulae

h

iviviv

2

)1()1()( 111111

+=

2

11111111

)1()(2)1()(

h

iviviviv

++=


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582

3

1111111111

2

)2()1(2)1(2)2()(

h

iviviviviv

+++=

4

111111111111

)2()1(4)(6)1(4)2()(

h

iviviviviviv iv

++++=

in Eq. (39)., we get

02)2()1()()1()2(0

42

115114113112111 =+++++K

hivAivAivAivAivA

(48)

where

hA Re21 +=

232

0

222 Re2

12Re28 +

++++= h

KMhhA

++

+++=

0

2442

0

223 22

1412

Kh

KMhA

232

0

224 Re2

12Re28

+++= h

KMhhA

hA Re25 = .

Substitution of similar finite difference formulae in Eq. (44) to (46)., the following equations are

obtained:

)()1()()1( 115111111 iBiuAiuBiuA =++ (49)

)()1()()1( 321 iDiDiDiD =++ (50)

)()1()()1( 321 iEiEiDiE =++ (51)

where RandAA 1051 ,,, have already been defined and

hD PrRe21 +=

22

2 24 hD +=

hD PrRe23 =

ihr eivPhiD

PrRe11

2 )()Re(2)( =

SchE Re21 +=

hScE Re23 =


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583

ihSceivSchiE Re112 )()Re(2)( =

+++= 2

0

22

1

124

KMhB

( ) )(2

)()(Re)(2)()()( 30

2

12

211 iBK

hiGiGhiBivRiB e ++=

where ( ) ihRihScih eGGRSceGeGiB ++= 1RePrRe)( 110Re

11

PrRe

02

( ) ihRihScih eGGeGeGiB ++= 1)( 110Re

11

PrRe

03 .

Eq. (48), (49), (50) and (51). have been solved by Gauss-seidel iteration methodfor velocity,

temperature and concentration. Also, numerical solutions for these equations have been obtained, using

C-Program. To prove convergence of finite difference scheme, the computation is carried out for

slightly changed value of h , running same program. Negligible change is observed in the values

andu, and also after each cycle of iteration the convergence checking is performed, i.e.

81 10+


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584

( )0

**

*

=

=

=

ypwp

uyLVC

k

TTVC

qN

0

110

cosPrRe

1

=

+

= yz

dy

d

dy

d

(53)

Sherwood - number

Knowing the concentration field, the expression for the rate of mass transfer in terms of modified

Nusselt number is given by

( )0

**

*

1

=

=

=

yw

uyVL

D

CCV

DqN

0

110

cosRe

1

=

+

=y

zdy

d

dy

d

Sc (54)

Results and discussion:

In order to get the physical insight of the problem, numerical calculations are carried out for

different flow parameters such as Heat absorption Parameter S, Grashof number Gr, Modified Grashof

number Gm, Magnetic parameter M, Permeability Parameter K0, Reynolds number Re, Prandtl number

(Pr) and Schmidt number (Sc) are studied. During the course of numerical calculations, the values of Pr

are chosen to be 0.71 &1.0 corresponding to air and electrolytic solution. But the propagation of

thermal energy through electrolytic solution in the presence of heat absorbing sink and magnetic field

has wide range of applications in chemical, aeronautical engineering and atomic propulsion space

science.

Fig. 1. shows the effect of free convection parameter Gr on velocity field u for cooling of the

plate both in the presence and absence of heat absorption parameter. It is observed that the velocity

increases due to greater cooling of the channel (as Gr increases). Further, it is interesting to note that the

velocity of fluid decreases in the presence of heat absorption parameter. In Fig. 2. the effects of Gm,

K on velocity field u has been exhibited by the curves both in presence and absence of heat absorption.

It is observed that an increase in Gm and K leads to an increase in the velocity, but it decreases in the

presence of heat absorption. This is in good agreement with the physical fact that heat sink decreases

the velocity of the fluid.


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585

Fig. 3. and Fig. 4. are drawn for various values S and M respectively; on velocity field u. it is

observed from these figures that the velocity of the fluid decreases with the increase of heat absorption

and magnetic parameter. Fig. 5. reveals that an increase in Pr and Sc decreases the velocity when in

the presence of heat absorption parameter.

Fig. 6. display the effects of Re and Pr on temperature profile in the presence of heat

absorption. From this it is evident that the fluid temperature decreases due to increase in the Pr. This is

in agreement with the physical fact that the thermal boundary layer thickness decreases with increase in

Pr. The reason underlying such a behavior is that the high prandtl number fluid has a low thermal

conductivity. This results in the reduction of the thermal boundary layer thickness. This figure also

shows that an increase in heat absorption parameter and Reynolds number leads to decrease in the

temperature of the fluid. Further, it is interesting to note that the effect of heat absorption on

temperature is more significant than in the case of velocity field.

Fig. 7. depicts the species concentration for different gases like Hydrogen (H2: Sc=0.22),

Oxygen (O2: Sc=0.66) and methanol (Sc=1.0) at a temperature 250C and 1 atmospheric pressure. The

values of Schmidt number (Sc) are chosen to represent the most common diffusing chemical species

which are of interest. A comparison of the curves in the figure shows a decrease in concentration

distribution with an increase in Schmidt number because the smaller values of Sc are equalent to

increasing chemical molecular diffusivity (D). Hence, the concentration of the species is higher for

small values of Sc and lower for larger values of Sc. This figure also shows that an increase in Re,

decreases the concentration field.

In Fig. 8. the non-dimensional skin-friction coefficient plotted against the Reynolds number for

different values of Gr, Gm and S. it is evident from this figure that the skin-friction increases with the

increase of Gr and Gm while it decreases in the presence of heat absorption parameter S. Fig. 9. is

plotted for non-dimensional heat transfer coefficient versus Reynolds number for different values of S

and Pr .It is observed that the heat transfer coefficient increases in the presence of heat absorption but, it

decreases with the increase of Pr. In Fig. 10. variation of the non-dimensional mass transfer


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586

coefficient is plotted against the Reynolds number for various values of Sc. It has been observed that

the mass transfer coefficient decreases as the value of Sc decreases.

CONCLUSIONS

The following conclusions have been drawn from the above results:

1. The effect of a heat absorbing sink on steady incompressible three-dimensional fluid flow

through a highly porous medium is to suppress the velocity and temperature fields, which is turn,

causes the enhancement of the heat transfer coefficient.

2. The velocity of a fluid decreases in the presence magnetic field. This due to the fact that

magnetic field reduces the velocity field.

3. The effect of heat absorption parameter on temperature field is more significant than in the case

of velocity field.

4. The concentration of the species is higher for small values of Sc and lower for larger values of

Sc.

5. This problem has been solved numerically, using finite difference technique and the results

obtained are in good agreement with the experimental results, as valid in the literature.

Nomenclature

G Acceleration due to gravity

Coefficient of volumetric thermal expansion

* Coefficient of mass expansion

*p Pressure

Density

Kinematics viscosity

Viscosity


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587

k

Thermal conductivity

Cp Specific heat at constant pressure

D Concentration diffusivity

*wC Concentration of the plate

*

wT Temperature of the plate,

*

T Temperature of the fluid far away from the plate

*

C Concentration of the fluid far away from the plate

Gr Grashof number

Gm Modified Grashof number

Re Reynolds numberB0 Magnetic field component

Sc Schmidt number

K0 Permeability parameter

M Magnetic parameter

Q Volumetric rate of Heat absorption

S Heat absorption parameter


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588

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2

____ S=0.0

------- S=2.0Gr=15.0

Gr=1.0

Gr=10.0

Gr=5.0

Fig.1- Effect of Gr on velocity field u when Gm=1.0,

Re=5.0, M=1.0, KO=1.0, Pr=0.71, Sc=0.66, =0.1 andZ=0.0

Fig.2-Effects of Gm, K0 and S on velocity field u when

Gr=1.0, M=1.0, Re=5.0, Pr=0.71, Sc=0.66, =0.1 and Z=0.0

Gm K0 S

1) 1.0 1.0 0.0

2) 1.0 1.0 2.0

3) 1.0 4.0 0.0

4) 3.0 1.0 2.0

1

4

3

2


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589

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2

Fig.3- Effect of S on velocity field u when Gr=1.0, Gm=1.0,

Re=5.0, K0=1.0, M=1.0, Pr=0.71, Sc=0.66, =0.1 and Z=0.0

Fig.4-Effect of M on velocity field u when Gr=1.0, Gm=1.0,

Re=5.0, K0 =1.0, Pr=0.71, Sc=0.66, S=1.0, =0.1 andZ=0.0

S=2.0

S=3.0

S=1.0S=0.0

M=0.0

M=1.0

M=2.0

M=3.0


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590

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4.

Pr Sc S

1) 0.71 0.66 0.0

2) 0.71 0.66 1.0

3) 7.0 0.66 1.0

4) 0.71 2.62 1.0

Fig.5-Effects of Pr and Sc on velocity field u when Gr=1.0,

Gm=1.0, Re=5.0, K0=1.0, M=1.0, =0.1 and Z=0.0

Fig.6-Effects of Re, Pr and S on Temperature field

when M=1.0, K0=1.0, =0.1 and Z=0.0

1

2

34

1

2

3

5

4

Re Pr S

1) 2.0 0.71 0.0

2) 2.0 0.71 2.03) 5.0 0.71 2.0

4) 5.0 1.0 2.0

5) 10.0 0.71 2.0


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591

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4

Re Sc

1) 5.0 0.22

2) 5.0 0.663) 5.0 1.0

4) 10.0 0.22

0

1

2

3

4

5

6

7

8

9

0 5 10 15 20 25 30

Skin-frictio

n

Gr=1.0,Gm=1.0,S=0.0

Gr=1.0,Gm=1.0,S=2.0

Gr=5.0,Gm=1.0,S=2.0

Gr=1.0,Gm=5.0,S=2.0

Fig.7- Effects of Re and Sc on Concentration field

when M=1.0, Ko=1.0, =0.1 and Z=0.0

Fig.8-Effect of S, Gr and Gm on Skin-friction Coefficient

when M=1.0,Ko=1.0, Pr=0.71, Sc=0.66, =0.1 and Z=0.0

1

2

3

4


20/21


592

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

S=0.0

S=2.0

Pr=0.71

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

Sc=0.22

Sc=0.66

Sc=1.0

Fig.9-Effect of S and Pr on Nusselt number Nu

when M=1.0 Ko=1.0 =0.1 and Z=0.0

Fig.10-Effect of Sc on Sherwood number Sh

when Re=5.0, M=1.0, Ko=1.0, =0.1 and Z=0.0


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593

REFERENCES

1. A. Raptis, G. Perdikis and G. Tzirandis, 1981a, "Free convection flow through a porous

medium bounded by a vertical surface,J. Phys. D. Appl. Phys. 14, pp. 99-102.

2. A. Raptis, G. Tzirandis and N. Kafousias, 1981b, Free convection and mass transfer flow

through a porous medium bounded by an infinite vertical limiting surface with constant

suction, LettersHeat Mass Transfer,8, pp. 417-24.

3. A. Raptis, N. Kafousis and C. Massalas, 1982, Free convection and mass transfer flow

through a porous medium bounded by an infinite vertical porous plate with constant heat

flux,ZAMM, 62, pp.489-91.

4. A.Raptis, 1983, Unsteady free convective flow through a porous medium,Int.J. Engng.

Sci. 21, pp.345-48.

5. A.Raptis and C.P.Perdikis, 1985, Oscillatory flow through a porous medium by the

presence of free convective flow,Int. J. Engng Sci.23, pp. 51-55.

6. K.D. Singh and Suresh Kumar, 1993, Two dimensional unsteady free convective flow

through a porous medium bounded by an infinite vertical porous plate with periodic

permeability,J. Math. Phys. Sci. 27, pp. 141-48.

7. K.D. Singh and G.N. Verma, 1995, Three dimensional oscillatory flow thorough a

porous medium with periodic permeability,ZAMM75, pp. 599-604.

8. K.D. Singh, Khem Chand and G.N. Verma, 1995, Heat transfer in a three dimensional

flow through a porous medium with periodic permeability,ZAMM75, pp. 950-52.

9. K.D. Singh and Rakesh Sharma, 2002, Three dimensional free convective flow and heat

transfer through a porous medium with periodic permeability,Indian J. pure appl. Math.

33(6), pp. 941-949.

10. N.C.Jain, B. Sharma and D.K. Vijay, 2006, Three dimensional free convective flow heat

transfer flow through a porous medium with periodic permeability in slip Flow regime,

J.Energy,Heat &Mass transfer, 28(1), pp. 29-44.

combined heat and mass transfer in mhd three-dimensional porous flow with periodic permeability &...

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