magneto convective flowand heat transfer of two immiscible fluids between vertical wavy wall and a...

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 INDIA

587 102

Abstract Convective flow and heat transfer between vertical wavy wall and a parallel flat wall consisting of two regions, one filled with

electrically conducting and other with viscous fluid is analyzed. Governing equation of motion have been solved by linearization

technique. 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 suppression

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

temperature.

Keywords: Convection, vertical, wavy wall, immiscible

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1. INTRODUCTION

Many transport process exists in natural and industrial

applications in which the transfer of heat and mass occurs

simultaneously as a result of buoyancy effect of thermal

diffusion. Natural convection heat transfer plays an important

role in the electronic components cooling since it has desirable

characteristics in thermal equipments design; absence of

mechanical or electromagnetic noise; low energy

consumption, very important in portable computers; and

reliability, since it has no elements to fail.

The optimization of the heat transfer has increasingly

importance in electronic packaging due to the higher heat

densities and to the electronic components and equipments

miniaturization Sathe et.al. (1998). In spite of the abundant

results about natural convection in electronic packaging,

works dealing with heat transfer maximization is scarce in

literature 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 devices

employed for enhancing the heat transfer efficiency of

industrial transport process. The problem of viscous flow in

wavy channels was first treated analytically by Burns and

Parks (1967) who expressed the stream function as a Fourier

series 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 the

wavy-wall channel were only slightly larger than those for a

parallel-plate channel. The problem of natural or mixed

convection along a sinusoidal wavy surface extended previous

work to complex geometries Yao (1983), Moulic et.al.,

(1989,1989) and has received considerable attention due to its

relevance to real geometries. An example of such geometry is

a “roughened” surface that occurs often in problem involving

the enhancement of heat transfer.

The flow and heat transfer of electrically conducting fluids in

channels under the effect of a transverse magnetic field occurs

in MHD-generators, pumps, accelerators, nuclear-reactors,

filtration, geo-thermal system and others. Recently there are

experimental and theoretical studies on hydromagnetic aspects

of 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-phase

MHD flow and heat transfer in an inclined channel is

investigated by Malashetty and Umavathi (1997). Recently

Malashetty et.al, (2000, 2001) analyzed the problem of fully

developed two fluid magnetohydrodynamic flows with and

without applied electric field in an inclined channel.

Chamakha (2000) considered the steady, laminar flow of two

viscous, incompressible electrically conducting and heat

generating 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 of

the present work, is to study the problem of flow and heat

transfer between vertical wavy wall and a parallel flat wall

consisting of two regions, one filled with electrically

conducting 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 X

axis is taken vertically upwards and parallel to the flat wall

while the Y axis is taken perpendicular to it in such a way that

the wavy wall is represented by KXcos*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 1

k and the region 2

0h y is occupied

another viscous fluid of density 2

, viscosity 2

, thermal

conductivity 2

k . The wavy and flat walls are maintained at

constant and different temperatures Tw and T1 respectively.

We make the following assumptions:

(i) that all the fluid properties are constant except

the 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 comparison

with both the heat flow by conduction and the

wall 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-dimensional

flow 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 gX Y X

(2.1)

1(1) (1)

(1) (1) (1) (1) 2 (1)V V PU 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 TC 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 PU V U

X Y X

g B U

(2.5)

2(2) (2)

(2) (2) (2) (2) 2 (2)V V PU 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 TC U V k T Q

X Y

(2.8)

Where the superscript indicates the quantities for regions I and

II, respectively. To solve the above system of equations,

one needs proper boundary and interface conditions. We

assume 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)

WT 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 Tk k

Y X Y X

at 0Y (2.10)

The conditions on velocity represent the no-slip condition and

continuity of velocity and shear stress across the interface. The

conditions on temperature indicate that the plates are held at

constant but different temperatures and continuity of heat and

heat flux at the interface.

The basic equations (4.2.1) to (4.2.8) are made dimensionless

using the following transformations

)1(

)1(

)1(Y,X

h

1y,x ; )1(

)1(

)1()1(

V,Uh

v,u

)2(

)2(

)2(Y,X

h

1y,x ; )2(

)2(

)2()2(

V,Uh

v,u

SW

s

)1()1(

TT

TT

;

SW

s

)2()2(

TT

TT

(1)

(1)

2(1)

(1)

PP

h

; (2)

(2)

2(2)

(2)

PP

h

(2.11)

Where Ts 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 2

v v P v vu v

x y y x y

(2.13)

0y

v

x

u )1()1(

(2.14)

PryxPryv

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)

0y

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 cosy x

(2) 0u ;

(2) 0v at 1y

(1) (2)1u u

rmh ;

(1) (2)1v v

rmh at 0y

2

22

1

x

v

y

u

hrm

1

x

v

y

u

at 0y

(2.20)

(1) 1 at 1 cosy x

(2) at 1y

(1) (2) at 0y


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21

xyh

k

xy

at 0y (2.21)

3. SOLUTIONS

The governing equations (2.12) to (2.15), (2.16) to (2.19)

along with boundary and interface conditions (2.20) to (2.21)

are two dimensional, nonlinear and coupled and hence closed

form solutions can not be found. However approximate

solutions can be found using the method of regular

perturbation. We take flow field and the temperature field to

be

Region-I

y,xu)y(uy,xu )1(

1

)1(

0

)1(

y,xvy,xv )1(

1

)1(

(1) (1)(1)0 1, ( ) ,P x y P x P x y

y,x)y(y,x )1(

1

)1(

0

)1( (3.1)

Region -II

y,xu)y(uy,xu )2(

1

)2(

0

)2(

y,xvy,xv )2(

1

)2(

(2) (2)(2)0 1, ( ) ,P x y P x P x y

y,x)y(y,x )2(

1

)2(

0

)2( (3.2)

Where the perturbations 1 1 1, ,

i i iu v P and

1

i for

1,2i are small compared with the mean or the zeroth order

quantities. Using equation (4.3.1) and (4.3.2) in the equation

(2.12) to (2.15), (2.16) to (2.19) become

3.1 Zeroth Order

Region-I

)1(

02

)1(

0

2

Gdy

ud (3.3)

2

1

0

2

dy

d)(

(3.4)

Region-II

)2(

0

322)2(

0

22

2

)2(

0

2

GhrmuMmhdy

ud (3.5)

k

hQ

dy

d)( 2

2

2

0

2 (3.6)

3.2 First Order:

Region-I

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

(1) (1) (1)01 1 1 10 1 12 2

duu P u uu v G

x dy x x y

(3.7)

1(1) 2 (1) 2 (1)(1) 1 1 1 10 2 2

v P v vu

x y x y

(3.8)

0y

v

x

u2

)1(

1

2)1(

1

(3.9)

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

(1) (1) 01 1 1

0 1 2 2Pr

du v

x dy x y

(3.10)

Region-II

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

2 2 3 (2) 2 2 (2)

1 1

duu P u uu v

x dy x x y

m r h G mh M u

(3.11)

2(2) 2 (2) 2 (2)(2) 1 1 1 10 2 2

v P v vu

x y x y

(3.12)

0y

v

x

u2

)2(

1

2)2(

1

(3.13)

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

(2) (2) 01 1 1

0 1 2 2Pr

d kmu v

x dy x y

(3.14)


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Where 1

1 0 SP PC

x

and

2

2 0 SP PC

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 1y

(2)

0 1 0u at 1y

)()( urmh

u 2

0

1

0

1 at 0y

(1) (2)

0 0

2 2

1du du

dy dyrm h at 0y (3.15)

(1)0 1 1 at 1y

(2)

0 1 at 1y

)()( 2

0

1

0 at 0y

(1) (2)

0 0d dk

dy h dy

at 0y (3.16)

1

(1) (1)01 11 ; 1 0

duu Cos x v

dy at 1y

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

)()( urmh

u 2

1

1

1

1 ;

)()( vrmh

v 2

1

1

1

1 at 0y

x

v

y

u

hrmx

v

y

u )()()()( 2

1

2

1

22

1

1

1

1 1at 0y

(3.17)

1

(1) 0

1 1d

Cos xdy

at 1y

(2)1 1 0 at 1y

)()( 2

1

1

1 at 0y

xyh

k

xy

)()()()( 2

1

2

1

1

1

1

1 at 0y

(3.18)

Introducing the stream function 1 defined by

yu

)(

)(

1

11

1

;

xv

)(

)(

1

11

1

(3.19)

And eliminating 1

1P and 2

1P 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

Gx y x y y

(3.20)

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

(1) 01 1 1 1

0 2 2Pr

du

x x dy x y

(3.21)

Region-II

2 (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 u

ux y x x dy x y

m r h G mh Mx y y y

(3.22)

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

(2) 01 1 1 1

0 2 2Pr

d kmu

x x dy x y

(3.23)

Assuming

1 , ,i ii xx y e y

1 ,i ii xx 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

dtG

dy

(3.26)

(1)

2 0

0Prd

t t i u tdy

(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

dtm r h G

dy

(2)

2 0

0

Pr dt 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 iiu and

1

iv for

1,2i

The boundary conditions (3.17) and (3.18) can be now written

in terms of

1(1)

01 ;du

Cos xy dy

(1)

1 0x

at 1y

(2)1 0;y

(2)1 0x

at 1y

yrmhy

)()(

2

1

1

1 1 ;

xrmhx

)()(

2

1

1

1 1

at 0y

2

2

1

2

2

2

1

2

222

1

1

2

2

1

1

21

xyhrmxy

)()()()(

at 0y (3.30)

1(1) 0d

tdy

at 1y

(2)

0t at 1y

1 2t t at 0y for 1i

1 2dt dt

dy dy at 0y (3.31)

If we consider small values of then substituting

20 1 2,

i i i iy

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 and

corresponding boundary and interface conditions

Region-I

4 (1) (1)

0 0

4

d dtG

dy dy

(3.33)

2 (1)

0

20

d t

dy (3.34)

(1)

2 24 (1) (1)

0 01 10 04 2 2

d d ud dti u G

dy dy dy dy

,(3.35)

(1)2 (1)

010 0 02

Prdd t

i u tdydy

(3.36)

Region-II

4 (2) 2 (2) (2)

2 2 2 2 30 0 0

4 2

d d dtM mh m r h G

dydy dy

(3.37)

2 (2)

0

20

d t

dy (3.38)


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(2)2 24 (2) 2 (2)

2 2 0 01 10 04 2 2 2

(2)2 2 3 1 (3.39)

d d ud dM mh i u

dy dy dy dy

dtm r h G

dy

(2)2 (2)

010 0 02

Pr dd ti u t

km dydy

(3.40)

and 1(1)

0 0d duCos x

dy dy

;

(1)0 0 at 1y

(2)0 0;

d

dy

(2)0 0 at 1y

(1) (2)

0 01d d

dy rmh dy

; )()(

rmh

2

0

1

0

1 at 0y

2 (1) 2 (2)

0 0

2 2 2 2

1d d

dy rm h dy

; )()(

hrm

2

022

1

0

1

at 0y (3.41)

(1)

1 0d

dy

;

(1)

1 0 at 1y

(2)1 0;

d

dy

(2)1 0 at 1y

(1) (2)1 11d d

dy rmh dy

; (1) (2)

1 1

1

rmh at 0y

2 (1) 2 (2)

1 1

2 2 2 2

1d d

dy rm h dy

; (1) (2)

1 12 2

1

rm h

at 0y (3.42)

1

1 0

0

dt

dy

at 1y

2

0 0t at 1y

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

(1) (2)0 0dt dtk

dy h dy at 0y (3.43)

(1)1 0t at 1y

(2)1 0t at 1y

1 2

1 1t t at 0y for 1i

1 2

1 1dt dt

dy dy at 0y (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) are

given by

Region-I

21

2

3

3

2

4

1

1

0 AyAylylylu )(

21

2

1

1

0 CyCyd)(

Region-II

(2) 2

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

(2) 2

0 2 3 4f y C y C

The solutions of zeroth and first order of are obtained by

solving the equation (3.33) to (3.40) using boundary and

interface conditions (3.41) and (3.42) and are given below

(1) 4 3 23 40 4 5 6

6 2

A Al y y y A y A

(2) 2

0 3 4 5 6 4cosh sinhB ny B ny B y B s y

(2)

0 7 8t C y C

(1) 10 9 8 7 6 5 4

1 20 21 22 23 24 25 26

3 27 89 10

6 2

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

A Ay y A y A

(1)

0 5 6t C y C


__________________________________________________________________________________________


(1) 7 6 5 4 3 2

1 8 9 10 11 12 13

6 8

Prt i d y d y d y d y d y d y

q y q

(2)

1 7 8 7 10

3 3 2

29 30 31

2

32 33 34

6 5

35 36 37 38

4 3 2

39 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

Prcosh 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

xcosxsinu ri 1

xcosxsinv ir 1

xsintxcost ir 1

Region-I

1 11

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

1 1(1)

1 0 1 0 1cos sini i r rv x x

1 1(1)

1 0 1 0 1cos sinr r i ix 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 rv x x

2 2(2)

1 0 1 0 1cos sinr r i ix 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 suppression

is more effective near the flat wall as M increases as shown in

figure 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 observed

that heat source promote the flow, sink suppress the flow, and

the 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 on

zeroth order velocity as shown in figure 4. The effect of free

convection parameter G, viscosity ratio m, width ratio h, on

zeroth order velocity and the effect of width ratio h,

conductivity ratio k, on zeroth order temperature remain the

same as explained in chapter-III

The effect of free convection parameter G on first order

velocity is shown in figure.5. As G increases u1 increases near

the 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 u1 shows that as m increases

first order velocity increases near the wavy and flat wall, but

the magnitude is very large near flat wall . At the interface

velocity decreases as m increases and the suppression is

significant towards the flat wall as seen in figure 6 for

0, 1 . The effect of width ratio h on first order velocity

u1 shows that u1 remains almost same for h<1 but is more

effective for h>1. For h = 2, u1 increases near the wavy and

flat wall and drops at the interface, for 0, 1 as seen in

figure 7.

The effect of Hartmann number M on u1 shows that as M

increases velocity decreases at the wavy wall and the flat wall

but the suppression near the flat wall compared to wavy wall

is insignificant and as M increases u1 increases in magnitude

at the interface for 0, 1 as seen in figure 8. The effect

of on u1 is shown in figure 9, which shows that velocity is

large near the wavy and flat wall for heat source 5 and is

less for heat sink 5 . Similar result is obtained at the

interface but for negative values of u1. Here also we observe

that the magnitude is very large near the wavy wall compared

to flat wall.


__________________________________________________________________________________________


The effect of convection parameter G, viscosity ratio m, width

ratio h, thermal conductivity ratio k on first order velocity v1 is

similar as explained in chapter III. The effect of Hartmann

number M is to increase the velocity near the wavy wall and

decrease velocity near the flat wall for 0, 1 , whose

results are applicable to flow reversal problems as shown in

figure.10. The effect of source or sink on first order velocity v1

is shown in figure 11. For heat source, v1 is less near wavy

wall and more for flat wall where as we obtained the opposite

result for sink i.e. v1 is maximum near wavy wall and

minimum near the flat wall, for 0 the profiles lie in

between 5


ratio h, thermal conductivity ratio k, Hartmann number and

source and sink parameter on first order temperature are

shown 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 .


ratio h, and thermal conductivity ratio k, on total velocity

remains the same as explained in chapter III. The effect of

heat source or sink on total velocity shows that U is very large

for 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 that

increasing G and m suppress the total temperature but the

supression for m is negligible as seen in figures.19 and 20.

The effects of width ratio h and conductivity ratio k is same as

explained in chapter-III. The effect of source or sink

parameter on total temperature remains the same as that on

total velocity as seen figure.21.

Where

1 / 2d ; 2

1 /f Qh k ; 2 1 / 2f f

hk

hfdC

21

3

1 ; 234 fCC ;

42 CC 121 1 dCC ;

12

11

Gdl ;

6

12

GCl ;

2

23

GCl

2

2

322

1n

Gfhrms

;

2

3

322

2n

GChrms

n

GChrm

n

Gfhrms

2

4

322

4

2

322

3

2

24

54

CGl ;

2 2 3

74 2

.m r h G Cs

n

22

2

5

6

hrm

nl

;

rmh

nl

46 ;

rmhl

47 ;

228

6

hrml

32122

4419 468

622 lll

hrm

slAl

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 4

6

Ad l C l C C l

1 344 3 5 2 8

2 6

C AAd 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 1f B C B f ;

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

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

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

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


__________________________________________________________________________________________


10 3 8 3 6f s C C B ; 2

11 3 /f f n ;

2

12 4 /f f n 5413 3 2

2ff

fn n

;

3 6

14 3 22

f ff

n n

15 7 / 20f f

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

18 10 / 2f f

1 8 9 10 11 12 3Prq d d d d d d

11 13 12 14

2

15 16 17 18

cosh sinhPr f f n f f nq

km f f f f

3 13

Prq f

km ; 4 14 11

Prkq f n f

h km

1 2 3 45

q q q q hq

k h

; 5

6 4

kqq q

h ;

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

13 Pr6

dl G ;

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

5

dl 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

4

dl 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 Pr3 3

l l A l A A l l A

A dl A l G

17 3 4 1 3 2 4 1 6

62 5 3 4

12 12

6 Pr2

l l A A A A l l A

dl A l A G

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

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

1320

5040

ll

; 1421

3024

ll ; 15

221680

ll ;

1623

840

ll ; 17

24360

ll ; 18

25120

ll ;

1926

24

ll ;

2 2

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

2 2

6 1 4 2 4s s B n B s n ;

2 2 2 2 37 2 3 1 5 12

Prs s B n B B n m r h f

km

2 2 2 2 38 2 4 2 5 11

Prs s B n B B n m r h f

km

2 2 2

9 3 3 1 6 1 4 1 3

2 2 3

11 14 1 4

2

Pr2

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

m r h f f n B skm

2 2 2

10 3 4 2 6 2 4 1 4

2 2 3

12 13 2 4

2

Pr2

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

m r h f f n B skm

2 2 311 15

Pr5s f m r h G

km ; 2 2 3

12 16

Pr4s f m r h G

km

2 2 313 17

Pr3s f m r h G

km ;

2 2 3

14 18 1 5 2 4

Pr2 2 2s f m r h G s B s s

km

2 2 3

15 1 6 1 4 5 3 42 2 2s s B s s m r h Gq s s ; 616

6

ss

n ;

517

6

ss

n ; 5 8

18 2 44

s ss

nn ; 6 7

19 2 44

s ss

nn ;

6 7 1020 3 2 24 4

s s ss

nn n

5 8 921 3 2 24 4

s s ss

nn n ; 5 8

22 4 38 8

s ss

n n ;

6 723 4 38 8

s ss

n n ;

1124 2

ss

n ;

1225 2

ss

n ;


__________________________________________________________________________________________


131126 4 2

12 sss

n n ; 12 14

27 4 2

6 s ss

n n

13 151128 6 4 2

224 s sss

n n n ;

1629 2

ss

n ;

1730 2

ss

n ; 17 17 18

31 3 4 2

3 3s s ss

n n n ;

16 16 1932 3 4 2

3 3s s ss

n n n

16 16 17 19 2033 4 5 4 3 2

6 12 6 4s s s s ss

n n n n n ;

16 17 17 18 2134 4 5 4 3 2

6 12 6 4s s s s ss

n n n n n

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

12 6 18 6 2s s s s s ss

n n n n n n

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

12 6 18 6 2s s s s s ss

n n n n n n

2437

30

ss ; 25

3820

ss ; 26

3912

ss ; 27

406

ss

2841

2

ss

-1.0 -0.5 0.0 0.5 1.00

2

4

6

8

10

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

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


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


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

u1

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

u1


-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

u1

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

u1

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

u1

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 1.0-0.5

-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

0.1

0.1

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 First 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

6

6

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 G

-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


__________________________________________________________________________________________


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developments in same practical aspects of air-cooled

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laminar natural convective cavities containing air and

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pp 203-214.

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[5]. Wang G. Vanka P. (1995) Convective heat transfer in

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[6]. Yao L.S., (1983) Natural convection along a vertical

wavy surface, ASME J. Heat Transfer vol. 105 pp 465-

468.

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

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[8]. Lohrasbi. J and Sahai.V (1988) Magnetohydrodynamic

Heat Transfer in two-phase flow between parallel

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[9]. Malashetty M.S. and Umavathi J.C., (1997) Two-

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an inclined channel, Int. J. Multiphase Flow, vol. 22, pp

545-560.

[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-

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[11]. Malashetty M. S, Umavathi J. C, Prathap Kumar J.

(2001) Convection magnetohydrodynamic two fluid

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