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IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308

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Volume: 03 Special Issue: 03 | May-2014 | NCRIET-2014, Available @ http://www.ijret.org 777

MAGNETO-CONVECTIVE FLOWAND HEAT TRANSFER OF TWO

IMMISCIBLE FLUIDS BETWEEN VERTICAL WAVY WALL AND A

PARALLEL FLAT WALL

Mahadev M Biradar

Associate Professor, Dept. of Mathematics Basaveshwar Engineering College (Autonomous) Bagalkot, Karnataka INDIA587 102

AbstractConvective flow and heat transfer between vertical wavy wall and a parallel flat wall consisting of two regions, one filled withelectrically conducting and other with viscous fluid is analyzed. Governing equation of motion have been solved by linearizationtechnique. Results are presented for various parameters such as Hartmann number, Grashof number, viscosity ratio, width ratio,conductivity ratio and source or sink. The effect of all the parameters except the Hartmann number and source or sink remains same

for two viscous immiscible fluids. The effect of Hartmann number is to decrease the velocity at the wavy and flat wall. The s uppression

near the flat wall compared to wavy wall is insignificant. The velocity is large for source compared to sink for equal and different walltemperature.

Keywords: Convection, vertical, wavy wall, immiscible

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1. INTRODUCTION

Many transport process exists in natural and industrialapplications in which the transfer of heat and mass occurssimultaneously as a result of buoyancy effect of thermaldiffusion. Natural convection heat transfer plays an importantrole in the electronic components cooling since it has desirable

characteristics in thermal equipments design; absence ofmechanical or electromagnetic noise; low energyconsumption, very important in portable computers; andreliability, since it has no elements to fail.

The optimization of the heat transfer has increasinglyimportance in electronic packaging due to the higher heatdensities and to the electronic components and equipmentsminiaturization Sathe et.al. (1998). In spite of the abundantresults about natural convection in electronic packaging,works dealing with heat transfer maximization is scarce inliterature Landon et.al (1999) Da Silva et.al., (2004). This is,

probably, due to the non-linear nature and to the difficult of

natural convection simulation.

The corrugated wall channel is one of several devicesemployed for enhancing the heat transfer efficiency ofindustrial transport process. The problem of viscous flow inwavy channels was first treated analytically by Burns andParks (1967) who expressed the stream function as a Fourierseries under the assumption of stokes flow. Wang and Vanka(1995) determined the rates of heat transfer for flow through a

periodic array of wavy passages. They observed that in the

steady-flow regime, the average Nusselt numbers for thewavy-wall channel were only slightly larger than those for a

parallel-plate channel. The problem of natural or mixedconvection along a sinusoidal wavy surface extended previouswork to complex geometries Yao (1983), Moulic et.al.,(1989,1989) and has received considerable attention due to itsrelevance to real geometries. An example of such geometry isa “roughened” surface that occurs often in problem involvingthe enhancement of heat transfer.

The flow and heat transfer of electrically conducting fluids inchannels under the effect of a transverse magnetic field occursin MHD-generators, pumps, accelerators, nuclear-reactors,filtration, geo-thermal system and others. Recently there areexperimental and theoretical studies on hydromagnetic aspectsof two fluid flows available in literature. Lohrasbi and Sahai(1998) dealt with two-phase magnetohydrodynamic (MHD)flow and heat transfer in a parallel plate channel. Two-phaseMHD flow and heat transfer in an inclined channel isinvestigated by Malashetty and Umavathi (1997). Recently

Malashetty et.al, (2000, 2001) analyzed the problem of fullydeveloped two fluid magnetohydrodynamic flows with andwithout applied electric field in an inclined channel.Chamakha (2000) considered the steady, laminar flow of twoviscous, incompressible electrically conducting and heatgenerating or absorbing immiscible fluids in an infinitely –long, impermeable parallel-plate channel filled with a uniform

porous medium.




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In spite of the numerous applications of corrugated walls,much work is not seen in literature. Hence it is the objective ofthe present work, is to study the problem of flow and heattransfer between vertical wavy wall and a parallel flat wallconsisting of two regions, one filled with electricallyconducting fluid and second with electrically non-conducting

fluid.

2. MATHEMATICAL FORMULATION

Fig-1: Physical Configuration

Consider the channel as shown in figure 4.1, in which the Xaxis is taken vertically upwards and parallel to the flat wallwhile the Y axis is taken perpendicular to it in such a way thatthe wavy wall is represented by KX cos*hY )( 1 and

the flat wall by )( hY 2 . The region 10 y h is occupied

by viscous fluid of density 1 , viscosity 1 , thermal

conductivity 1k and the region 2 0h y is occupied

another viscous fluid of density 2 , viscosity 2 , thermal

conductivity 2k . The wavy and flat walls are maintained atconstant and different temperatures T w and T 1 respectively.We make the following assumptions:

(i) that all the fluid properties are constant exceptthe density in the buoyancy-force term;

(ii) that the flow is laminar, steady and two-dimensional;

(iii) that the viscous dissipation and the work done by pressure are sufficiently small in comparisonwith both the heat flow by conduction and thewall temperature;

(iv) that the wavelength of the wavy wall, which is proportional to 1/K, is large.

Under these assumptions, the equations of momentum,continuity and energy which govern steady two-dimensionalflow and heat transfer of viscous incompressible fluids are

Region – I 1(1) (1)

(1) (1) (1) (1) 2 (1) (1)U U P U V U g

X Y X

(2.1)

1(1) (1)(1) (1) (1) (1) 2 (1)V V P U V V

X Y Y

(2.2)

011

Y V

X U )( )(

(2.3)

(1) (1)

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

T T C U V k T Q

X Y

(2.4)

Region – II 2(2) (2)

(2) (2) (2) (2) 2 (2)

(2) 2 (2)0

U U P U V U X Y X

g B U

(2.5)

2(2) (2)(2) (2) (2) (2) 2 (2)V V P

U V V X Y Y

(2.6)

022

Y V

X U )( )(

(2.7)

(2) (2)

2 2(2) (2) (2) (2) 2 (2) p

T T C U V k T Q

X Y

(2.8)

Where the superscript indicates the quantities for regions I andII, respectively. To solve the above system of equations,one needs proper boundary and interface conditions. Weassume 1

pC = 2 pC

The physical hydro dynamic conditions are

(1) 0U (1) 0V , at (1) *Y h cosKX

(2) 0U (2) 0,V at (2)Y h

(1) (2)U U (1) (2) ,V V at 0Y

2 )2(

1 )1(

X V

Y U

X V

Y U

at 0Y (2.9)




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The boundary and interface conditions on temperature are

(1)W T T at (1) *Y h cosKX

(2)1T T at (2)Y h

(1) (2)T T at 0Y

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

k k Y X Y X

at 0Y (2.10)

The conditions on velocity represent the no-slip condition andcontinuity of velocity and shear stress across the interface. Theconditions on temperature indicate that the plates are held atconstant but different temperatures and continuity of heat andheat flux at the interface.

The basic equations (4.2.1) to (4.2.8) are made dimensionlessusing the following transformations

)1( )1(

)1( Y , X h

1 y , x ; )1(

)1(

)1( )1( V ,U

hv ,u

)2(

)2(

)2( Y , X h

1 y , x ; )2(

)2(

)2( )2( V ,U

hv ,u

S W

s )1(

)1(

T T T T

; S W

s )2(

)2(

T T T T

(1)(1)

2(1)

(1)

P P

h

;(2)

(2)2(2)

(2)

P P

h

(2.11)

Where T s is the fluid temperature in static conditions.

Region – I (1)1(1) (1) 2 2 (1)

(1) (1) (1)2 2

u u P u uu v G

x y x x y

(2.12)

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

(1) (1)2 2v v P v vu v

x y y x y (2.13)

0 y

v x

u )1( )1(

(2.14)

Pr y x Pr yv

xu

)( )( )( )(

)( )(

2

121

2

211

11 1

(2.15)

Region – II

(2 )2(2) (2) 2 2 (2)(2) (2)

2 2

2 2 3 (2) 2 2 (2)

u u P u uu v

x y x x y

m r h G M mh u

(2.16)

(2)2(2) (2) 2 2 (2)(2) (2)

2 2

v v P v vu v

x y y x y

(2.17)

0 y

v x

u )2( )2(

(2.18)

(2 )(2) (2) 2 2 (2)(2) (2)

2 2

2

Pr

(2.19)Pr

kmu v

x y x y

mh Q

The dimensionless form of equation (2.9) and (2.10) using

(2.11) become(1) 0u ; (1) 0v at 1 cos y x

(2) 0u ; (2) 0v at 1 y

(1) (2)1u u

rmh; (1) (2)1v v

rmh at 0 y

2

22

1

xv

yu

hrm1

xv

yu

at 0 y

(2.20)

(1) 1 at 1 cos y x

(2) at 1 y

(1) (2) at 0 y






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Where 11 0 S P P

C x

and 2

2 0 S P P C

x

taken equal to zero (see Ostrach, 1952). With the help of (3.1)and (3.2) the boundary and interface conditions (2.20) to(2.21) become

(1)0 1 0u at 1 y

(2)0 1 0u at 1 y

)( )( urmh

u 20

10

1 at 0 y

(1) (2)0 0

2 2

1du dudy dyrm h

at 0 y (3.15)

(1)0 1 1 at 1 y

(2)0 1 at 1 y

)( )( 20

10 at 0 y

(1) (2)0 0d d k

dy h dy

at 0 y (3.16)

1

(1) (1)01 11 ; 1 0duu Cos x v

dy at 1 y

(2) (2)1 11 0; 1 0u v at 1 y

)( )( urmh

u 21

11

1; )( )( v

rmhv 2

11

11

at 0 y

xv

yu

hrm xv

yu )( )( )( )( 2

12

122

11

11 1

at 0 y

(3.17)

1

(1) 01 1

d Cos x

dy

at 1 y

(2)1 1 0 at 1 y

)( )( 21

11 at 0 y

x yhk

x y

)( )( )( )( 21

21

11

11

at 0 y

(3.18)

Introducing the stream function 1 defined by

yu

)( )(

111

1

;

xv

)( )(

111

1

(3.19)

And eliminating 11 P and 2

1 P from equation (3.7), (3.8) and

(3.11), (3.12) we get

Region-I

2 (1)3 (1) 3 (1) (1)(1) 01 1 10 2 3 2

4 (1) 4 (1) 4 (1) (1)1 1 1 1

2 2 4 42

d uu

x y x x dy

G x y x y y

(3.20)

(1)(1) (1) 2 (1) 2 (1)(1) 01 1 1 10 2 2Pr

d u

x x dy x y

(3.21)

Region-II2 ( 2)3 (2) 3 (2) (2) 4 (2)

(2) 01 1 1 10 2 3 2 2 2

4 (2) 4 (2) (2) 22 2 3 2 21 1 1 1

4 4 2

2d uu x y x x dy x y

m r h G mh M x y y y

(3.22)

(2 )(2) (2) 2 (1) 2 (1)(2) 01 1 1 10 2 2Pr

d kmu

x x dy x y

(3.23)Assuming

1 , ,i ii x

x y e y

1 ,i ii x

x y e t y

for i = 1,2 (3.24)

From which we infer

1 1, ,i ii xu x y e u y 1 1,i ii xv x y e v y

for i = 1,2 (3.25)




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And using (3.24) in (3.20) to (3.23), we get

Region-I

(1)2 (1)(1) 2 00 2

(1)2 22

iv d ui u

dy

dt G

dy

(3.26)

(1)

2 00Pr d

t t i u t dy

(3.27)

Region-II

(2)2 ( 2)(2) 2 2 200 2

(2)2 2 2 2 32 (3.28)

iv d ui u mh M

dy

dt m r h Gdy

(2 )

2 00

Pr d t t i u t

km dy

(3.29)

Below we restrict our attention to the real parts of the solution

for the perturbed quantities 1 1 1, ,i ii u and

1

iv for

1,2i

The boundary conditions (3.17) and (3.18) can be now writtenin terms of

1(1)01 ;

duCos x

y dy

(1)1 0

x

at 1 y

(2)1 0;

y

(2)1 0

x

at 1 y

yrmh y

)( )( 21

11 1

; xrmh x

)( )( 21

11 1

at 0 y

2

21

2

2

21

2

222

11

2

2

11

2 1 x yhrm x y

)( )( )( )(

at 0 y (3.30)

1(1) 0d

t dy

at 1 y

(2) 0t at 1 y

1 2t t at 0 y for 1i

1 2dt dt

dy dy at 0 y (3.31)

If we consider small values of then substituting

20 1 2,i i i i y

20 1 2,i i i it y t t t

for 1,2i (3.32) in to (3.26) to (3.31) gives, to order of the following sets of ordinary differential equations andcorresponding boundary and interface conditions

Region-I

4 (1) (1)0 04

d dt G

dy dy

(3.33)

2 (1)

02 0d t

dy (3.34)

(1)2 24 (1) (1)0 01 1

0 04 2 2

d d ud dt i u G

dy dy dy dy

,(3.35)

(1)2 (1)01

0 0 02 Pr d d t

i u t dydy

(3.36)

Region-II4 (2) 2 (2) (2)

2 2 2 2 30 0 04 2

d d dt M mh m r h Gdydy dy

(3.37)

2 (2)02 0

d t

dy (3.38)




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(2)2 24 (2) 2 (2)2 2 0 01 1

0 04 2 2 2

(2)2 2 3 1 (3.39)

d d ud d M mh i u

dy dy dy dy

dt m r h G

dy

(2)2 (2)01

0 0 02Pr d d t

i u t km dydy

(3.40)

and 1(1)

0 0d duCos x

dy dy

; (1)0 0 at 1 y

(2)0 0;

d dy (2)

0 0 at 1 y

(1) (2)0 01d d

dy rmh dy

; )( )(

rmh

2

0

1

0

1 at 0 y

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

1d d

dy rm h dy

; )( )(

hrm2

0221

01

at 0 y (3.41)

(1)1 0

d dy

; (1)1 0 at 1 y

(2)1 0;

d

dy

(2)

1 0 at 1 y

(1) (2)1 11d d

dy rmh dy ; (1) (2)

1 11

rmh at 0 y

2 (1) 2 (2)1 12 2 2 2

1d d dy rm h dy

; (1) (2)1 12 2

1

rm h

at 0 y (3.42)

1

1 00

d t

dy

at 1 y

20 0t at 1 y

(1) (2)0 0t t at 0 y

(1) (2)0 0dt dt k

dy h dy at 0 y (3.43)

(1)1 0t at 1 y

(2)1 0t at 1 y

1 21 1t t at 0 y for 1i

1 21 1dt dt

dy dy at 0 y (3.44)

3.3 Zeroth-Order Solution (Mean Part)

The solutions to zeroth order differential Eqs. (3.3) to (3.6)

using boundary and interface conditions (3.15) and (3.16) aregiven by

Region-I

212

33

24

11

0 A y A yl yl yl u )(

212

11

0 C yC yd )(

Region-II

(2) 2

0 1 2 1 2 3cosh sinhu B ny B ny s y s y s

(2) 20 2 3 4 f y C y C

The solutions of zeroth and first order of are obtained bysolving the equation (3.33) to (3.40) using boundary andinterface conditions (3.41) and (3.42) and are given below

(1) 4 3 23 40 4 5 66 2

A Al y y y A y A

(2) 20 3 4 5 6 4cosh sinh B ny B ny B y B s y

(2)0 7 8t C y C

(1) 10 9 8 7 6 5 41 20 21 22 23 24 25 26

3 27 89 106 2

i l y l y l y l y l y l y l y

A A y y A y A

(1)0 5 6t C y C




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(1) 7 6 5 4 3 21 8 9 10 11 12 13

6 8

Pr t i d y d y d y d y d y d y

q y q

(2 )1 7 8 7 10

3 3 229 30 31

232 33 34

6 535 36 37 38

4 3 239 40 41

cosh sinh

cosh sinh cosh

sinh cosh sinh

cosh sinh

B ny B ny B y B

i s y ny s y ny s y ny

s y ny s y ny s y ny

s ny s ny S y S y

S y S y S y

(2)1 11 12 13

5 4 3 214 15 16 17 18

5 7

Pr cosh sinh cosh

sinh

t i f y ny f y ny f nykm

f ny f y f y f y f y

q y q

The first order quantities can be put in the forms

xcos x sinu r i 1

xcos x sinv ir 1

x sint xcost ir 1

Region-I

1 111 0 1 0 1cos sinr r i iu x x

1 1(1)1 0 1 0 1cos sini i r r v x x

1 1(1)1 0 1 0 1cos sinr r i i x t t x t t

Region-II

2 2(2)1 0 1 0 1cos sinr r i iu x x

2 2(2)1 0 1 0 1cos sini i r r v x x

2 2(2)1 0 1 0 1cos sinr r i i x t t x t t

4. RESULTS AND DISCUSSION

4.1 Discussion of the Zeroth Order Solution:

The effect of Hartmann number M on zeroth order velocity is

to decrease the velocity for 0, 1 , but the suppressionis more effective near the flat wall as M increases as shown infigure 2.

The effect of heat source 0 or sink 0 and in the

absence of heat source or sink 0 , on zeroth order

velocity is shown in figure 3 for 0, 1 . It is observedthat heat source promote the flow, sink suppress the flow, andthe velocity profiles lie in between source or sink for 0 We also observe that the magnitude of zeroth order velocity is

optimum for 1 and minimal for 1 , and profiles

lies between 1 for 0 . The effect of source or sink parameter on zeroth order temperature is similar to that onzeroth order velocity as shown in figure 4. The effect of freeconvection parameter G, viscosity ratio m, width ratio h, onzeroth order velocity and the effect of width ratio h,conductivity ratio k, on zeroth order temperature remain thesame as explained in chapter-III

The effect of free convection parameter G on first ordervelocity is shown in figure.5. As G increases u 1 increases nearthe wavy and flat wall where as it deceases at the interface and

the suppression is effective near the flat wall for 0, 1 The effect of viscosity ratio m on u 1 shows that as m increasesfirst order velocity increases near the wavy and flat wall, but

the magnitude is very large near flat wall . At the interfacevelocity decreases as m increases and the suppression issignificant towards the flat wall as seen in figure 6 for

0, 1 . The effect of width ratio h on first order velocityu1 shows that u 1 remains almost same for h<1 but is moreeffective for h>1. For h = 2, u 1 increases near the wavy and

flat wall and drops at the interface, for 0, 1 as seen infigure 7.

The effect of Hartmann number M on u 1 shows that as Mincreases velocity decreases at the wavy wall and the flat wall

but the suppression near the flat wall compared to wavy wallis insignificant and as M increases u 1 increases in magnitude

at the interface for 0, 1 as seen in figure 8. The effectof on u 1 is shown in figure 9, which shows that velocity islarge near the wavy and flat wall for heat source 5 and isless for heat sink 5 . Similar result is obtained at theinterface but for negative values of u 1. Here also we observethat the magnitude is very large near the wavy wall comparedto flat wall.




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The effect of convection parameter G, viscosity ratio m, widthratio h, thermal conductivity ratio k on first order velocity v 1 issimilar as explained in chapter III. The effect of Hartmannnumber M is to increase the velocity near the wavy wall and

decrease velocity near the flat wall for 0, 1 , whoseresults are applicable to flow reversal problems as shown infigure.10. The effect of source or sink on first order velocity v 1 is shown in figure 11. For heat source, v 1 is less near wavywall and more for flat wall where as we obtained the oppositeresult for sink i.e. v 1 is maximum near wavy wall andminimum near the flat wall, for 0 the profiles lie in

between 5

The effect of convection parameter G, viscosity ratio m, widthratio h, thermal conductivity ratio k, Hartmann number andsource and sink parameter on first order temperature areshown in figures 12 to 17. It is seen that G, m, h, and k

increases in magnitude for values of 0, 1 . It is seen that

from figure 16 that as M increases the magnitude of 1

decreases for 0, 1 . Figure 17 shows that the magnitude

of is large for heat source and is less for sink whereas 1

remains invariant for 0 .

The effect of convection parameter G, viscosity ratio m, widthratio h, and thermal conductivity ratio k, on total velocityremains the same as explained in chapter III. The effect ofheat source or sink on total velocity shows that U is very largefor 5 compared to 5 and is almost invariant for

0 as seen figure 18, for all values of .

The effect of Grashof number G, viscosity ratio m, shows thatincreasing G and m suppress the total temperature but thesupression for m is negligible as seen in figures.19 and 20.The effects of width ratio h and conductivity ratio k is same asexplained in chapter-III. The effect of source or sink

parameter on total temperature remains the same as that ontotal velocity as seen figure.21.

Where

1 / 2d ; 21 / f Qh k ; 2 1 / 2 f f

hk

h f d C 213

1; 234 f C C ;

42 C C 121 1 d C C ;12

11

Gd l ;

61

2

GC l ;

22

3

GC l

22

322

1 nGf hr m

s ;

23

322

2 nGC hr m

s

n

GC hr mn

Gf hr m s 2

4322

42

322

32

245

4

C Gl ;

2 2 37

4 2

.m r h G s

n

22

2

56

hrmn

l

;rmh

nl

46

rmhl 47 ; 228 6

hrml

321224

419 4686

22 l l l hrm

sl Al

10 7 6

7 5

cosh sinh cosh

sinh sinh cosh

l n n n l n n l

l n n l n n n

11 7 6 8cosh sinh cosh l l n n l l n n n

12 9 4 7 6sinh cosh coshl l n n n s l n n l

2 1 5 4d l C l ; 33 2 5 1 6 1 46

Ad l C l C C l

1 344 3 5 2 8 2 6

C A Ad l C l C

;

1 45 1 5 3 6 4 2

C Ad A C l C A

6 2 5 1 6 6 1 5d A C AC A C A ;

7 2 6 1 6d A C C A 8 2 / 42d d ;

9 3 / 30d d ; 10 4 / 20d d

11 5 /12d d ; 12 6 / 6d d ;

13 7 / 2d d 3 1 7 3 1 f B C B f ;

4 2 7 4 1 f B C B f ; 5 1 8 3 3 f B C B C

6 2 8 4 3 f B C B C ; 7 1 7 4 1 f s C s C

8 2 7 1 8 5 1 4 3 f s C s C B f s C ;

9 3 7 2 8 6 1 5 3 f s C s C B f B C




__________________________________________________________________________________________


10 3 8 3 6 f s C C B ; 211 3 / f f n ;

212 4 / f f n 54

13 3 22 f f

f n n

;

3 614 3 22 f f f

n n

15 7 / 20 f f

16 8 /12 f f ; 17 9 / 6 f f ;

18 10 / 2 f f

1 8 9 10 11 12 3Pr q d d d d d d

11 13 12 142

15 16 17 18

cosh sinhPr f f n f f nq

km f f f f

3 13Pr

q f km

; 4 14 11Pr k

q f n f h km

1 2 3 45

q q q q hq

k h

; 56 4

kqq q

h ;

7 5 2q q q ; 8 6 1q q q ; 213 Pr

6d

l G ;

314 1 3 2 4 1 3 2 412 2 6 Pr 5

d l l A l l l A l l G

415 1 4 2 3 3 4 1 4 2 3 3 412 6 2 Pr

4d

l l A l A l l l A l A l l G

16 2 4 3 3 1 4 1 5

3 52 4 3

12 12

3 Pr 3 3

l l A l A Al l A

A d l A l G

17 3 4 1 3 2 4 1 6

62 5 3 4

12 12

6 Pr 2

l l A A A A l l A

d l A l A G

18 1 4 2 3 2 6 3 5 76 2 Pr l A A A A l A l A G d ;

19 2 4 3 6 62l A A l A Gq

1320 5040

l l ; 14

21 3024l

l ; 1522 1680

l l ;

1623 840

l l ; 17

24 360

l l ; 18

25 120l

l ;

1926 24

l l ; 2 2

5 1 3 1 4 s s B n B s n ;

2 26 1 4 2 4 s s B n B s n ;

2 2 2 2 37 2 3 1 5 12

Pr s s B n B B n m r h f

km

2 2 2 2 38 2 4 2 5 11

Pr s s B n B B n m r h f

km

2 2 29 3 3 1 6 1 4 1 3

2 2 311 14 1 4

2

Pr 2

s s B n B B n B s n s B

m r h f f n B skm

2 2 210 3 4 2 6 2 4 1 4

2 2 312 13 2 4

2

Pr 2

s s B n B B n B s n s B

m r h f f n B skm

2 2 311 15

Pr 5 s f m r h G

km ; 2 2 3

12 16Pr

4 s f m r h Gkm

2 2 313 17

Pr 3 s f m r h G

km ;

2 2 314 18 1 5 2 4

Pr 2 2 2 s f m r h G s B s s

km

2 2 315 1 6 1 4 5 3 42 2 2 s s B s s m r h Gq s s ; 6

16 6

s s

n;

517 6

s s

n; 5 8

18 2 44

s s s

nn ; 6 7

19 2 44

s s s

nn ;

6 7 1020 3 2 24 4

s s s s

nn n

5 8 921 3 2 24 4

s s s s

nn n ; 5 8

22 4 38 8

s s s

n n ;

6 723 4 38 8

s s s

n n ; 11

24 2

s s

n ; 12

25 2

s s

n ;




__________________________________________________________________________________________


131126 4 2

12 s s s

n n ; 12 14

27 4 2

6 s s s

n n

13 151128 6 4 2

224 s s s s

n n n ; 16

29 2

s s

n;

1730 2

s s

n; 17 17 18

31 3 4 2

3 3 s s s s

n n n ;

16 16 1932 3 4 2

3 3 s s s s

n n n

16 16 17 19 2033 4 5 4 3 2

6 12 6 4 s s s s s s

n n n n n ;

16 17 17 18 2134 4 5 4 3 2

6 12 6 4 s s s s s

s n n n n n

16 17 17 18 21 2235 5 5 6 4 3 2

12 6 18 6 2 s s s s s s s

n n n n n n

17 16 16 19 20 2336 5 5 6 4 3 2

12 6 18 6 2 s s s s s s s

n n n n n n

2437 30

s s ; 25

38 20 s

s ; 2639 12

s s ; 27

40 6

s s

2841 2

s s

-1.0 -0.5 0.0 0.5 1.00

2

4

6

8

10

Region-II (flat wall)Region-I (wavywall )

M = 6

M = 4

M = 2

y

= -1.0= 0.0= 1.0

Fig. 2 Zeroth order velocity profiles for different values of Hartmann number M

u 0

-1.0 -0.5 0.0 0.5 1.0-10

-5

0

5

10

= 5

= 0

= -5

Region-II (flat wall)Region-I (wavywall)

y

= -1.0= 0.0= 1.0

Fig. 3 Zeroth order velocity profiles for different values of

u 0

-1.0 -0.5 0.0 0.5 1.0-9

-6

-3

0

3

6

9

= 5

= 0

= -5

Region-II (flat wall)Region-I (wavy wall)

y

= -1.0= 0.0= 1.0

Fig. 4 Zeroth order temperature profiles for different values of

-1.0 -0.5 0.0 0.5 1.0

-0.16

-0.08

0.00

0.08

0.16

G = 15

G = 10

G = 5 u

1

Fig. 5 First order velocity profiles for different values of Grashof number G


= -1.0= 0.0= 1.0

y




__________________________________________________________________________________________


-1.0 -0.5 0.0 0.5 1.0

-0.04

0.00

0.04

0.08

y

= -1.0= 0.0= 1.0

2.0

2.0

m = 2.0

m = 1.0

m=0.1

Fig. 6 First order velocity profiles for different values of viscosity ratio m

u 1


-1.0 -0.5 0.0 0.5 1.0

-0.6

-0.3

0.0

0.3

0.6

y

= -1.0= 0.0= 1.0

h = 2.0

u 1

Fig. 7 First order velocity profiles for different values of width ratio h


-1.0 -0.5 0.0 0.5 1.0-0.04

-0.02

0.00

0.02

y

= -1.0= 0.0= 1.0

M = 6

M = 4

M = 2

u 1

Fig. 8 First order velocity profiles for different values of M


-1.0 -0.5 0.0 0.5 1.0

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

= -1.0= 0.0= 1.0

= 5

= -5

= 0

y

u 1

Fig. 9 First order velocity profiles for different values of


-1.0 -0.5 0.0 0.5 1.0-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

y

= -1.0= 0.0= 1.0

M = 6

M = 4

M = 2

Fig. 10 First order velocity profiles for different values of M

v 1


-1.0 -0.5 0.0 0.5 1.0

-0.015

-0.010

-0.005

0.000

0.005

0.010

y

= -1.0= 0.0= 1.0

Fig. 11 First order velocity profiles for different values of

v 1


-1.0 -0.5 0.0 0.5 1-0.5

-0.4

-0.3

-0.2

-0.1

0.0

= -1.0= 0.0= 1.0

15

15

10

10

G = 5

G = 10

G = 15

Fig. 12 First order temperature profiles for different values of G


y

-1.0 -0.5 0.0 0.5 1.0

-0.16

-0.12

-0.08

-0.04

0.00

y

= -1.0= 0.0= 1.0

2.0

2.0

1.0

0.10.1

m = 0.1

m = 1.0

m = 2.0

Fig. 13 First order temperature profiles for different values of m





__________________________________________________________________________________________


-1.0 -0.5 0.0 0.5 1.0

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

y

h = 2.0

= -1.0= 0.0= 1.0

h = 0.1,0.5

Fig. 14 First order temperature profiles for different values of h

1


-1.0 -0.5 0.0 0.5 1.0-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0 k = 1

k = 0.5

k = 0.1

y

= -1.0= 0.0= 1.0

Fig. 15 Fir st order temperature profiles for different values of k


-1.0 -0.5 0.0 0.5 1.0

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

= -1.0= 0.0= 1.0

y

66

4

4

2

2

M = 6

M = 4

M = 2

Fig. 16 First order temperature profiles for different values of M


-1.0 -0.5 0.0 0.5 1.0

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

y

= -1.0= 0.0= 1.0

Fig. 17 First order temperature profiles for different values of


-1.0 -0.5 0.0 0.5 1.0

-8

-4

0

4

8

12

= -1.0= 0.0= 1.0

= 5

= 0

= -5.0


y

Fig. 18 Total solution of velocity profiles for different values of

U

-1.0 -0.5 0.0 0.5 1.0-2

0

2

4

6

8

10

= -1.0= 0.0= 1.0

m = 0.1,1.0 ,2.0

Fig. 20 Total solution of temperature profiles for different values of m


y

-1.0 -0.5 0.0 0.5 1.0

0

2

4

6

8

G = 5, 10, 15

= -1.0= 0.0= 1.0


y

Fig. 19 Total solution of temperature profiles for different values of Grashof number

-1.0 -0.5 0.0 0.5 1.0-10

-8

-6

-4

-2

0

2

4

6

8

10

= 5

= 0

= -5


y

= -1.0= 0.0= 1.0

Fig. 21 Total solution of temperature profiles for different values of




__________________________________________________________________________________________


REFERENCES:

[1]. Sathe S., Sammkia B., (1998) A review of recentdevelopments in same practical aspects of air-cooledelectronic packages, Trans. ASME. J. Heat Transfervol. 120 pp 830-839.

[2]. London M.D., Campo A., (1999) Optimal shape forlaminar natural convective cavities containing air andheated from the side, Int. Comm. Heat mass Transfervol. 26, pp 389-398.

[3]. Da silva A.K, Lorente S, Bejan A,(2004) Optimaldistribution of discrete heat sources on a wall withnatural convection, Int. J. Heat Mass Transfer vol. 47

pp 203-214.[4]. Burns J.C, Parks T, J. (1967) Fluid Mech.29 405-416[5]. Wang G. Vanka P. (1995) Convective heat transfer in

periodic wavy passages. Int. J. Heat Mass Transfer vol.38(17) pp 3219

[6]. Yao L.S., (1983) Natural convection along a verticalwavy surface, ASME J. Heat Transfer vol. 105 pp 465-

468.[7]. Moulic S. G., Yao L.S., (1989) Mixed convection

along a wavy surface, ASME J. Heat transfer vol.111 pp 974-979.

[8]. Lohrasbi. J and Sahai.V (1988) MagnetohydrodynamicHeat Transfer in two-phase flow between parallel

plates. Applied scientific Research vol. 45 pp. 53-66.[9]. Malashetty M.S. and Umavathi J.C., (1997) Two-

phase Magnetohydrodynamic flow and heat transfer inan inclined channel, Int. J. Multiphase Flow, vol. 22, pp545-560.

[10]. Malashetty M. S, Umavathi J. C. and Prathap Kumar(2000) Two-fluid magneto convection flow in n

inclined channel” I.J. Trans phenomena, vol. 3. Pp 73-84.[11]. Malashetty M. S, Umavathi J. C, Prathap Kumar J.

(2001) Convection magnetohydrodynamic two fluidflow and heat transfer, vol.37, pp 259-264.

[12]. Chamkha Ali. J, (2000) Flow of two- immiscible fluidsin porous and nonporous channels, Journal of FluidsEngineering, vol. 122, pp 117-124.

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