two fluid electromagneto convective flow and heat … · poots [10] treated the open circuit case...

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –

6480(Print), ISSN 0976 – 6499(Online), Volume 6, Issue 4, April (2015), pp. 86-106 © IAEME

86

TWO FLUID ELECTROMAGNETO CONVECTIVE FLOW

AND HEAT TRANSFER BETWEEN VERTICAL WAVY

WALL AND A PARALLEL FLAT WALL

Mahadev M Biradar

Associate Professor, Department of Mathematics,

Basaveshwar Engineering College (Autonomous) Bagalkot Karnataka INDIA 587103

ABSTRACT

The mixture of viscous and magneto convective flow and heat transfer between a long

vertical wavy wall and a parallel flat wall in the presence of applied electric field parallel to gravity ,

magnetic field normal to gravity in the presence of source or sink is investigated. The non-linear

equations governing the flow are solved using the linearization technique. The effect of Grash of

number and width ratio is to promote the flow for both open and short circuits. The effect of

Hartmann number is to suppress the flow, the effect of source is to promote and the effect of sink is

to suppress the velocity for open and short circuits. Conducting ratio decreases the temperature

where as width ratio increases the temperature.

Keywords: Convection, Vertical Wavy, Electrically Conducting Fluid

1. INTRODUCTION

From a technological point of view the free convection study of fluids with known physical

properties, which are confined between two parallel walls, is always important, for it can reveal

hitherto-unknown properties of fluids of practical interest. In such buoyancy driven flows the exact

governing equations are unwieldy so that recourse to approximations such as Boussinesq’s is

generally called for. Assuming the linear density temperature variation (Boussinesq’s approximation)

Ostrach [1] studied the laminar natural convection heat transfer in fluids with and without heat

sources, in channels with constant wall temperature.

Previous studies of natural convection heat and mass transfer have focused mainly on a flat

plate or regular ducts. Botfemanne [2] has considered simultaneous heat and mass transfer by free

convection along a vertical flat plate only for steady state theoretical solutions with Pr = 0.71 and Sc

INTERNATIONAL JOURNAL OF ADVANCED RESEARCH IN ENGINEERING

AND TECHNOLOGY (IJARET)

ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 6, Issue 4, April (2015), pp. 86-106 © IAEME: www.iaeme.com/ IJARET.asp

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87

= 0.63. Callahan and Marner [3] studied the free convection with mass transfer on a vertical flat plate

with Pr = 1 and a realistic range of Schmidt number.

It is necessary to study the heat and mass transfer from an irregular surface because irregular

surfaces are often present in many applications. It is often encountered in heat transfer devices to

enhance heat transfer. For examples, flat-plate solar collectors and flat-plate condensers in

refrigerators. The natural convection heat transfer from an isothermal vertical wavy surface was first

studied by Yao [4, 5, 6] and using an extended Prantdl’s transposition theorem and a finite-difference

scheme. He proposed a simple transformation to study the natural convection heat transfer from

isothermal vertical wavy surfaces, such as Sinusoidal surface.

The fluid mechanics and the heat transfer characteristics of the generator channel or

significantly influenced by the presence of the magnetic field. Shail [7] studied the problem of

Hartmann flow of a conducting fluid in a horizontal channel of insulated plates with a layer of non-

conducting fluid over lying a conducting fluid in two fluid flow and predicted that the flow rate can

be increases to the order of 30% for suitable values of ratio of viscosities, width of the two fluids

with appropriate Hartmann number.

Hartmann [8] carried out the pioneer work on the study of steady magnetohydrodynamic

channel flow of a conducting fluid under a uniform magnetic field transverse to an electrically

insulated channel wall. Later Osterle and Young [9] investigated the effect of viscous and Joule

dissipation on hydromagnetic force convection flows and heat transfer between two vertical plates

with transverse magnetic field under short circuit condition. Poots [10] treated the open circuit case

with and without heat sources, the two bounding plates being maintained at different temperatures,

Umavathi [11] studied the effect of viscous and ohmic dissipation on magnetoconvection in a

vertical enclosure for open and short circuits, recently Malashetty et.al. [11] studied the

magnetoconvection of two-immiscible fluids in vertical enclosure.

Electric current in semi-conducting fluids such as glass and electrolyte generate Joulean

heating Song [12], other application include those involving exothermic and endothermic chemical

reaction and dealing with dissociating fluids Vajravelu and Nayefeh [13].

All the reference mentioned above dealt with vertical parallel walls. Since flow past wavy

boundaries are encountered in many situation such as the rippling melting surface, physiological

applications and many more as mention above, motivated us to study flow and heat transfer of the

mixture of viscous and electrically conducting fluid in the presence of applied electric and magnetic

fields between vertical wavy walls and a parallel flat wall.

2. MATHEMATICAL FORMULATION

Fig. 1. Physical Configuration



88

Consider the channel as shown in Fig. 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)( ε+−= 1 and the flat wall by )(

hY2= . The region

( )10 y h≤ ≤ is

occupied by viscous fluid of density 1ρ , viscosity 1µ , thermal conductivity 1k and the region

( )20h y≤ ≤ is occupied another viscous fluid of density 2ρ , viscosity 2µ , thermal conductivity 2k .

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 PU V U g

X Y Xρ µ ρ

∂ ∂ ∂+ = − + ∇ −

∂ ∂ ∂ (1)

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

U V VX Y Y

ρ µ ∂ ∂ ∂

+ = − + ∇ ∂ ∂ ∂

(2)

011

=∂

∂+

∂

∂

Y

V

X

U)()(

(3)

( )(1) (1)

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

T TC U V k T Q

X Yρ

∂ ∂+ = ∇ +

∂ ∂ (4)

Region –II

( )2(2) (2)

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

0 0 0

U U PU V U g B U B E

X Y Xρ µ ρ σ σ

∂ ∂ ∂+ = − + ∇ − − −

∂ ∂ ∂ (5)

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

U V VX Y Y

ρ µ ∂ ∂ ∂

+ = − + ∇ ∂ ∂ ∂

(6)

022

=∂

∂+

∂

∂

Y

V

X

U)()(

(7)

( )(2) (2)

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

T TC U V k T Q

X Yρ

∂ ∂+ = ∇ +

∂ ∂ (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



89

The physical hydro dynamic conditions are

(1) 0U = (1) 0V = , at

(1) *Y h cosKXε= − +

(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

∂

∂+

∂

∂=

∂

∂+

∂

∂µµ at 0Y = (9)

The boundary and interface conditions on temperature are

(1)

WT T= at (1) *

Y h cosKXε= − +

(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 = (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 (1) to (8) are made dimensionless using the following transformations

( ) ( ) )1(

)1(

)1(Y,X

h

1y,x = ; ( ) ( ) )1(

)1(

)1()1(

V,Uh

v,uν

=

( ) ( ) )2(

)2(

)2(Y,X

h

1y,x = ; ( ) ( ) )2(

)2(

)2()2(

V,Uh

v,uν

=

( )( )SW

s

)1()1(

TT

TT

−

−=θ ;

( )( )SW

s

)2()2(

TT

TT

−

−=θ

(1)(1)

2(1)

(1)

PP

h

νρ

=

; (2)

(2)

2(2)

(2)

PP

h

νρ

=

(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 yθ

∂ ∂ ∂ ∂ ∂+ = − + + +

∂ ∂ ∂ ∂ ∂ (12)

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

(1) (1)

2 2

v v P v vu v

x y y x y

∂ ∂ ∂ ∂ ∂+ = − + +

∂ ∂ ∂ ∂ ∂ (13)



90

0y

v

x

u)1()1(

=∂

∂+

∂

∂ (14)

PryxPryv

xu

)()()()(

)()( αθθθθ

+

∂

∂+

∂

∂=

∂

∂+

∂

∂2

121

2

211

11 1

(15)

Region –II

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

(2) (2) 2 2 3 (2) 2 2 (2) 2 3 2

2 2

u u P u uu v m r h G M mh u m h rM E

x y x x yβ θ

∂ ∂ ∂ ∂ ∂+ = − + + + − −

∂ ∂ ∂ ∂ ∂

(16)

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

(2) (2)

2 2

v v P v vu v

x y y x y

∂ ∂ ∂ ∂ ∂+ = − + +

∂ ∂ ∂ ∂ ∂ (17)

0y

v

x

u)2()2(

=∂

∂+

∂

∂ (18)

Pr

Qmh

yxPr

km

yv

xu

)()()()(

)()( αθθθθ 2

2

222

2

222

22 +

∂

∂+

∂

∂=

∂

∂+

∂

∂ (19)

The dimensionless form of equation (9) and (10) using (11) become

(1) 0u = ; (1) 0v = at 1 cosy xε λ= − +

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

(1) (2)1u u

rmh= ; (1) (2)1

v vrmh

= at 0y =

( ) ( )2

22

1

x

v

y

u

hrm

1

x

v

y

u

∂

∂+

∂

∂=

∂

∂+

∂

∂ at 0y = (20)

(1) 1θ = at 1 cosy xε λ= − +

(2)θ θ= at 1y =

(1) (2)θ θ= at 0y = ( ) ( )21

xyh

k

xy

∂

∂+

∂

∂=

∂

∂+

∂

∂ θθθθ at 0y = (21)

3. SOLUTIONS

The governing equations (12) to (15), (16) to (19) and (24) to (25) along with boundary and

interface conditions (20) to (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



91

Region-I

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

1

)1(

0

)1( ε+=

( ) ( )y,xvy,xv )1(

1

)1( ε=

( ) ( )(1) (1)(1)0 1, ( ) ,P x y P x P x yε= +

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

1

)1(

0

)1( θεθθ += (22)

Region -II

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

1

)2(

0

)2( ε+=

( ) ( )y,xvy,xv )2(

1

)2( ε=

( ) ( )(2) (2)(2)0 1, ( ) ,P x y P x P x yε= +

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

1

)2(

0

)2( θεθθ += (23)

where the perturbations ( ) ( ) ( )1 1 1, ,

i i iu v P and

( )1

iθ for 1,2i = are small compared with the mean or the

zeroth order quantities. Using equation (5.3.1) and (5.3.2) in the equation (12) to (15), (16) to (19)

and (24) to (25) become

Zeroth order:

Region-I

)1(

02

)1(

0

2

Gdy

udθ−= (24)

αθ

−=2

1

0

2

dy

d)(

(25)

Region-II

)2(

0

322232)2(

0

22

2

)2(

0

2

θβ GhrmEMrhmuMmhdy

ud−=− (26)

k

hQ

dy

d)( 2

2

2

0

2 αθ −= (27)

First order:

Region-I

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

(1) (1) (1)01 1 1 1

0 1 12 2

duu P u uu v G

x dy x x yθ

∂ ∂ ∂ ∂+ = − + + +

∂ ∂ ∂ ∂ (28)

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

(1) 1 1 1 1

0 2 2

v P v vu

x y x y

∂ ∂ ∂ ∂= − + +

∂ ∂ ∂ ∂ (29)



92

0y

v

x

u2

)1(

1

2)1(

1 =∂

∂+

∂

∂ (30)

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

(1) (1) 01 1 1

0 1 2 2Pr

du v

x dy x y

θθ θ θ ∂ ∂ ∂+ = +

∂ ∂ ∂ (31)

Region-II

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

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

0 1 1 12 2

duu P u uu v m r h G mh M u

x dy x x yβ θ

∂ ∂ ∂ ∂+ = − + + + −

∂ ∂ ∂ ∂ (32)

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

(2) 1 1 1 1

0 2 2

v P v vu

x y x y

∂ ∂ ∂ ∂= − + +

∂ ∂ ∂ ∂ (33)

0y

v

x

u2

)2(

1

2)2(

1 =∂

∂+

∂

∂ (34)

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

(2) (2) 01 1 1

0 1 2 2Pr

d kmu v

x dy x y

θθ θ θ ∂ ∂ ∂+ = +

∂ ∂ ∂ (35)

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 (5.3.1) and (5.3.2) the boundary and interface conditions (20) to (21) become

( )(1)0 1 0u − = at 1−=y

( )(2)

0 1 0u = at 1y =

)()( urmh

u 2

0

1

0

1= at 0=y

(1) (2)

0 0

2 2

1du du

dy dyrm h= at 0=y (36)

( )(1)0 1 1θ − = at 1−=y

( )(2)

0 1θ θ= at 1y =

)()( 2

0

1

0 θθ = at 0=y

(1) (2)

0 0d dk

dy h dy

θ θ= at 0=y (37)

( )( )

( )1

(1) (1)0

1 11 ; 1 0du

u Cos x vdy

λ− = − − = ; at 1−=y

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



93

)()( urmh

u 2

1

1

1

1= ; )()( v

rmhv 2

1

1

1

1= ; at 0=y

∂

∂+

∂

∂=

∂

∂+

∂

∂

x

v

y

u

hrmx

v

y

u)()()()( 2

1

2

1

22

1

1

1

1 1 at 0=y (38)

( )( )1

(1) 0

1 1d

Cos xdy

θθ λ− = − at 1−=y

( )(2)1 1 0θ = at 1=y

)()( 2

1

1

1 θθ = at 0=y

∂

∂+

∂

∂=

∂

∂+

∂

∂

xyh

k

xy

)()()()( 2

1

2

1

1

1

1

1 θθθθ at 0=y (39)

Introducing the stream function 1ψ defined by

yu

)(

)(

∂

∂−=

1

11

1

ψ;

xv

)(

)(

∂

∂=

1

11

1

ψ (40)

and eliminating ( )1

1P and ( )2

1P from equation (5.3.7), (5.3.8) and (5.3.11), (5.3.12) we get

Region-I

2 (1)3 (1) 3 (1) (1) 4 (1) 4 (1) 4 (1) (1)

(1) 01 1 1 1 1 1 1

0 2 3 2 2 2 4 42

d uu G

x yx y x dy x y x y

ψ ψ ψ ψ ψ ψ θ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ − = + + −

∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ (41)

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

(1) 01 1 1 1

0 2 2Pr

du

x x dy x y

θθ ψ θ θ ∂ ∂ ∂ ∂+ = +

∂ ∂ ∂ ∂ (42)

Region-II

2 (2)3 (2) 3 (2) (2) 4 (2) 4 (2) 4 (2) (2) 2

(2) 2 2 3 2 201 1 1 1 1 1 1 10 2 3 2 2 2 4 4 2

2d u

u m r h G mh Mx y x x dy x y x y y y

ψ ψ ψ ψ ψ ψ θ ψβ

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ − = + + − −

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (43)

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

(2) 01 1 1 1

0 2 2Pr

d kmu

x x dy x y

θθ ψ θ θ ∂ ∂ ∂ ∂+ = +

∂ ∂ ∂ ∂ (43)

We assume the stream function in the form

( ) ( ),yey,x xi ψεψ λ=1 ( ) ( ),ytey,x xiλεθ =1 (44)

From which we infer

( ) ( ),yuey,xu xi

11

λε= ( ) ( ),yvey,xv xi

11

λε= (45)

and using (5.3.20) in (5.3.16) and (5.3.19) we get



94

Region-I

( ) ( )(1)

2 (1) (1)(1) 2 2 20

0 22iv d u dt

i u Gdydy

ψ λ ψ λ ψ ψ λ ψ λ ψ

′′ ′′− − − − − =

(46)

(1)

2 0

0Prd

t t i u tdy

θλ λ ψ

′′ − = +

(47)

Region-II

( ) ( )(2)

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

0 22iv d u t

i u mh M m r h Gydy

ψ λ ψ λ ψ ψ ψ λ ψ λ ψ β ∂

′′ ′′ ′′− − − − − − = ∂

(48)

(2)

2 0

0

Pr dt t i u t

km dy

θλ λ ψ

′′ − = +

(49)

Below we restrict our attention to the real parts of the solution for the perturbed quantities ( ) ( ) ( )1 1 1, ,

i iiuψ θ and

( )1

iv for 1,2i =

The boundary conditions (5.3.17) and (5.3.18) can be now written in terms of ( )1

iψ as

( )1(1)

01 ;du

Cos xy dy

ψλ

∂= −

∂

(1)

1 0x

ψ∂=

∂; at 1−=y

(2)1 0;y

ψ∂=

∂

(2)1 0x

ψ∂=

∂; at 1=y

yrmhy

)()(

∂

∂=

∂

∂ 2

1

1

1 1 ψψ;

xrmhx

)()(

∂

∂=

∂

∂ 2

1

1

1 1 ψψ; at 0=y

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

1 1 1 1

2 2 2 2 2 2

1

x y rm h x y

ψ ψ ψ ψ ∂ ∂ ∂ ∂− = −

∂ ∂ ∂ ∂ at 0=y

( ) ( ) ( )

( ) ( ) ( )

2 (1) (1) 3 (1) 3 (1)1 1 11 1 1 1

0 0 12 3

2 (2) (2) 3 (2) 3 (2)2 2 23 2 21 1 1 1

0 0 12 3 2 3

1

y

y

u u Grxy x x y y

u u Gr h m rrm h xy x x y y

ψ ψ ψ ψθ

ψ ψ ψ ψβ θ

∂ ∂ ∂ ∂− + + + −

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂= − + + + −

∂ ∂ ∂ ∂

(50)

( )( )1

(1) 0cosd

t xdy

θλ= − at 1−=y

(2) 0t = at 1=y

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

( ) ( ) ( ) ( )1 1 2 22 2 2 2

2 2 2 2

d t d t d t d tk

hdx dy dx dy

+ = +

at 0=y (51)



95

If we consider small values of λ then substituting

( ) ( ) ( ) ( ) ( )2

0 1 2,i i i i

yψ λ ψ λψ λ ψ= + + + − − − −

( ) ( ) ( ) ( ) ( )20 1 2,

i i i it y t t tλ λ λ= + + + − − − − for 1, 2i = (52)

In to (5.3.26) to (5.3.31) gives, to order of λ , the following sets of ordinary differential

equations and corresponding boundary and conditions

Region-I

4 (1) (1)

0 0

4

d dtG

dydy

ψ= (53)

2 (1)

0

20

d t

dy= (54)

(1)2 24 (1) (1)

0 01 1

0 04 2 2

d d ud dti u G

dydy dy dy

ψψψ

− − =

, (55)

(1)2 (1)

01

0 0 02Pr

dd ti u t

dydy

θψ

= +

(56)

Region-II

4 (2) (2)

2 2 30 0

4

d dtm r h G

dydy

ψβ= (57)

2 (2)

0

20

d t

dy= (58)

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

2 2 2 2 30 01 1 1

0 04 2 2 2

d d ud d dtM mh i u m r h G

dydy dy dy dy

ψψ ψψ β

− = − +

(59)

(2)2 (2)01

0 0 02

Pr dd ti u t

km dydy

θψ

= +

(60)

with boundary conditions

( )1(1)

0 0d duCos x

dy dy

ψλ= ;

(1)0 0ψ = ; at 1−=y

(2)0 0;

d

dy

ψ=

(2)0 0ψ = at 1=y

(1) (2)

0 01d d

dy rmh dy

ψ ψ= ; )()(

rmh

2

0

1

0

1ψψ = at 0=y

2 (1) 2 (2)0 0

2 2 2 2

1d d

dy rm h dy

ψ ψ= ;

)()(

hrm

2

022

1

0

1ψψ = at 0=y (61)



96

(1)

1 0d

dy

ψ= ; (1)

1 0ψ = at 1−=y

(2)1 0;

d

dy

ψ=

(2)1 0ψ = at 1=y

(1) (2)1 11d d

dy rmh dy

ψ ψ= ; (1) (2)

1 1

1

rmhψ ψ= at 0=y

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

1 22 21 11 12 2 2 2

1d d

dy rm h dy

ψ ψλ ψ λ ψ+ = + ;

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )( )

1 11 1 1 1(1) 20 0

2 22 2 2 2(2) 2 2 2 30 02 3

1

y y y yyy

y y y yyy

i u i u Gt

i u i u G m r h trm h

λ ψ λψ λ ψ ψ

λ ψ λψ λ ψ ψ β

− + − + −

= − + − + − at 0=y (62)

( )( )1

1 0

0

dt

dy

θ= − at 1−=y

( )2

0 0t = at 1=y

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

( ) ( )(1) (2)

1 20 0dt dtki t i t

dy h dyλ λ

+ = +

at 0=y (63)

(1)1 0t = at 1−=y

(2)1 0t = at 1=y

( ) ( )1 2

1 1t t= at 0=y

( ) ( )1 2

1 1dt dt

dy dy= at 0=y (64)

Zeroth-order solution (mean part)

The solutions to zeroth order differential Eqs. (5.3.3) to (5.3.6) using boundary and interface

conditions (5.3.15) and (5.3.16) are given by

Region-I

21

2

2

3

1

)1(

0 AyAylylu +++=

21

)1(

0 CyC +=θ

Region-II

(2) 3

0 1 2 1 1 2cosh sinhu B ny B ny s y s y s= + + + +

43

)2(

0 CyC +=θ



97

The solutions of zeroth and first order of λ are obtained by solving the equation (5.3.33) to

(5.3.40) using boundary and interface conditions (5.3.41) and (5.3.42) and are given below

(1) 4 3 23 4

0 4 5 66 2

A Al y y y A y Aψ = + + + +

(2) 2

0 3 4 5 6 4cosh sinhB ny B ny B y B s yψ = + + + +

(2)

0 7 8t C y C= +

( )(1) 10 9 8 7 6 5 4 3 27 81 20 21 22 23 24 25 26 9 10

6 2

A Ai l y l y l y l y l y l y l y y y A y Aψ = + + + + + + + + + +

( )(1) 7 6 5 4 3 2

1 8 9 10 11 12 13 6 8Prt i d y d y d y d y d y d y q y q= + + + + + + +

(

)

(2) 3 3 2

1 7 8 7 10 29 30 31

2

32 33 34 35 36

6 5 4 3 2

37 38 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 hny s ny

S y S y S y S y S y

ψ = + + + + + +

+ + + + + +

+ + + +

(

)

(2) 5 411 12 13 14 15 161

3 217 18 5 7

Prcosh sinh cosh sinht i f y ny f y ny f ny f ny f y f y

km

f y f y q y q

= + + + + +

+ + + +

The first order quantities can be put in the forms

xcosxsinu ri λψλψ ′−′=1

xcosxsinv ir λψλλψλ ′−′=1

xsintxcost ir λλθ −=1

Region-I

( ) ( )( ) ( )( )1 11

1 0 1 0 1cos sinr r i iu x xλ ψ λψ λ ψ λψ′ ′ ′ ′= − + + +

( )( ) ( )( )1 1(1)

1 0 1 0 1cos sini i r rv x xλ λ ψ λψ λ λ ψ λψ= − + − +

( )( ) ( )( )1 1(1)

1 0 1 0 1cos sinr r i ix t t x t tθ λ λ λ λ= + − +

Region-II

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

1 0 1 0 1cos sinr r i iu x xλ ψ λψ λ ψ λψ′ ′ ′ ′= − + + +

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

1 0 1 0 1cos sini i r rv x xλ λ ψ λψ λ λ ψ λψ= − + − +

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

1 0 1 0 1cos sinr r i ix t t x t tθ λ λ λ λ= + − +

(1)

0 5 6t C y C= +



98

Skin friction and Nusselt number

The shearing stress yxτ at any point in the fluid is given in nondimensional form, by

=

= yxyx

hτ

υρτ

2

2

( ) ( ) ( )̀yveiyueyuxixi

110

λλ λεε +′+′=

yxτ at the wavy wall ( )xcoshy )( λε+−= 1 and at the flat wall 1=y are given by

( ) ( )( )11(1)

0 10 Rei x

e u uλτ τ ε ′′ ′= + +

( ) ( )22(2)10 Re

i xe u

λτ τ ε ′= +

Where Re represent the real part of

( )1

1 2 3 10 4 3 2l l l Aτ = − + − +

( )2

1 2 1 20 sinh cosh 2B n n B n n s sτ = + + +

( ) ( ) ( ) ( )( )11 1 1

0 0 0 11 cos cos sinr iu x u x xτ λ ψ λ λψ λ′ ′′ ′′ ′′= + + − +

( ) ( ) ( )( )22 2

0 0 11 cos sinr iu x xτ ψ λ λψ λ′ ′′ ′′= + − +

The nondimensional Nusselt number is given by

( ) ( )0 1

i xdNu y e y

dy

λθθ ε θ′ ′= = +

At the wavy wall and flat fall wall Nu takes the form

( ) ( )( )11(1)

0 10 Rei x

Nu Nu e tλε θ ′′ ′= + +

( ) ( )22(2)10 Re

i xNu Nu e t

λε ′= +

where

( )1

1 10 2Nu d C= − +

( )2

2 30 2Nu f C= +

Pressure drop:

The fluid pressure ( ),P x y ( ( )0P x = constant) at any point ( ),x y as

( ),P P

P x y dP dx dyx y

∫ ∫ ∂ ∂

= = + ∂ ∂



99

( )( )

( )exp

. , Reii x

i e P x y L i z yλ

ελ

− =

(65)

where L is an arbitrary constant and

( ) ( ) ( )20 0Z y i u u G tψ λ ψ λ ψ ψ′′′ ′ ′ ′= − − − −

( ) ( ) ( ) ( ) ( )( ) ( )20 1 0 1 0 0 1 0 0 1 0 1Z y i u u G t tψ λψ λ ψ λψ λ ψ λψ ψ λψ λ′′′ ′′′ ′ ′ ′ ′ ′= + − + − + − + − +

Equation (5.3.45) can be rewritten as

( ) ( ) ( ) ( ) ( )( )^

, ,1 / Re 1i xP P x y P x i e z y zλε λ = − = −

( ) ( ) ( )( )Re 1P i Cos x iSin x z y zε

λ λλ

= + −

( ) ( ) ( )( )Re 1i Cos x Sin x z y zε

λ λλ

= − − (66)

where ^

P has been named the pressure drop since it indicates the difference between the present at

any point y in the flow field and that at the flat wall, with x fixed. The pressure drops ^

P at

10

2x andλ π= have been named

^

0P and ^

2Pπ and their numerical values for several sets of values of

the non-dimensional parameters G have been evaluated. In what follows we record the qualitative

differences in the behavior of the various flow and heat-transfer characteristics which show clearly

the effects of the wavy wall of the channel under consideration.

4. RESULTS AND DISCUSSION

Discussion of the Zeroth Order Solution

The effect of Hartmann number M on zeroth order velocity is to decrease the velocity for

0, 1= ±θ , but the suppression is more effective near the flat wall as M increases as shown in figure

2. The effect of heat source ( )0α > or sink ( )0α < and in the absence of heat source or sink ( )0α = ,

on zeroth order velocity is shown in figure 3 for 0, 1θ = ± . 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=α . We

also observe that the magnitude of zeroth order velocity is optimum for 1θ = and minimal for

1θ = − , and profiles lies between 1= ±θ for 0θ = . The effect of source or sink parameter α 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, 1θ = ± . 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, 1θ = ± . The effect of width ratio h on first



100

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, 1θ = ± 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, 1θ = ± as seen in figure 8. The

effect of α on u1 is shown in figure.9, which shows that velocity is large near the wavy and flat wall

for heat source 5α = and is less for heat sink 5α = − . 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. The effect of Hartmann number M is to increase the

velocity near the wavy wall and decrease velocity near the flat wall for 0, 1θ = ± , 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 0α = the profiles lie in between 5α = ±

The effect of convection parameter G, viscosity ratio m, width ratio h, thermal conductivity

ratio k, Hartmann number and source and sink parameter α 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, 1θ = ± . It is

seen that from figure 16 that as M increases the magnitude of 1θ decreases for 0, 1θ = ± . Figure17

shows that the magnitude of α is large for heat source and is less for sink where as 1θ remains

invariant for 0α = .

The effect of convection parameter G, viscosity ratio m, width ratio h, and thermal

conductivity ratio k, on total velocity remains the same . The effect of heat source or sink on total

velocity shows that U is very large for 5α = compared to 5α = − and is almost invariant for 0α =

as seen figure 18, for all values ofθ .

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. The effect of source or sink parameter on total

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

-1.0 -0.5 0.0 0.5 1.0

0

5

10

15

20

25

30Gr = 15.0

Gr = 10.0

Gr = 5.0

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

y

E = - 1.0

E = 0.0

E = 1.0

Fig 2 Zeroth order solution of velocity profiles for different values of Grashoff number Gr

u

0

-1.0 -0.5 0.0 0.5 1.0

0

2

4

6

8

10 m = 1.0

m = 0.5

m = 0.1


y

E = - 1.0

E = 0.0

E = 1.0

Fig 3 Zeroth order solution of velocity profiles for different values of viscosity ratio m

u

0

-1.0 -0.5 0.0 0.5 1.0

0

15

30

45

60h = 0.1 ,0.5,2.0


y

E = - 1.0

E = 0.0

E = 1.0

Fig 4 Zeroth order solution of velocity profiles for different values of width ratio h

u

0



101

-1.0 -0.5 0.0 0.5 1.0

0

10

20

30

k = 0.5

k = 1.0

k = 0.1 Region-II (flat wall)Region-I (wavy wall)

y

E = - 1.0

E = 0.0

E = 1.0

Fig 5 Zeroth order solution of velocity profiles for different values of k

u

0

-1.0 -0.5 0.0 0.5 1.0

0

4

8

M = 6.0

M = 4.0

M = 2.0

θR = -1.0

θR = 0.0

θR = 1.0

u0

Fig. 6 Zeroth order solution of velocity profiles for different values of Hartmann number M


-1.0 -0.5 0.0 0.5 1.0

-10

-5

0

5

10

α = -5.0

α = 5.0

α = 0.0

θR = -1.0

θR = 0.0

θR = 1.0

u0

Fig 7 Zeroth order solution of velocity profiles for different values of α


-1.0 -0.5 0.0 0.5 1.0

0

10

20

h = 2.0

h = 0.5

h = 0.1


y

E = - 1.0

E = 0.0

E = 1.0

Fig 8 Zeroth order solution of velocity profiles for different values of width ratio h

θ

0

-1.0 -0.5 0.0 0.5 1.0

0

10

20

30

40

k = 0.5

k = 1.0

k = 0.1


y

E = - 1.0

E = 0.0

E = 1.0

Fig 9 Zeroth order solution of velocity profiles for different values of k

θ

0

-1.0 -0.5 0.0 0.5 1.0

-5

0

5

α = -5.0

α = 5.0

α = 0.0

θR = -1.0

θR = 0.0

θR = 1.0

θ0

Fig 11 Zeroth order solution of velocity profiles for different values of α


-1.0 -0.5 0.0 0.5 1.0

-2

-1

0

1

2Gr = 15.0

Gr = 10.0

Gr = 5.0


y

E = - 1.0

E = 0.0

E = 1.0

Fig 12 First order solution of velocity profiles for different values of Grashoff number Gr

U

1

-1.0 -0.5 0.0 0.5 1.0

-2

-1

0

1

h = 0.1 , 0.5,2.0


y

E = - 1.0

E = 0.0

E = 1.0

Fig 14 First order solution of velocity profiles for different values of width ratio h

u

1

-1.0 -0.5 0.0 0.5 1.0-30

-20

-10

0

10

20

30

Fig 15 First order solution of velocity profiles for different values of k

k = 0.5

k = 1.0

k = 0.1 Region-II (flat wall)

Region-I (wavy wall)

y

E = - 1.0

E = 0.0

E = 1.0

u

1



102

-1.0 -0.5 0.0 0.5 1.0

-0.2

0.0

0.2

M = 6.0M = 4.0

M = 2.0θR = -1.0

θR = 0.0

θR = 1.0

u1

Fig 16 First order solution of velocity profiles for different values of Hartmann number M


-1.0 -0.5 0.0 0.5 1.0

-0.2

0.0

0.2

α = -5.0

α = 5.0

α = 0.0

θR = -1.0

θR = 0.0

θR = 1.0

u1

Fig 17 First order solution of velocity profiles for different values of α


-1.0 -0.5 0.0 0.5 1.0-0.04

-0.02

0.00

0.02

Fig 21 First order solution of velocity profiles for different values of k

k = 0.5

k = 1.0

k = 0.1


y

E = - 1.0

E = 0.0

E = 1.0

v

1

-1.0 -0.5 0.0 0.5 1.0

-0.01

0.00

0.01

Fig 22 First order solution of velocity profiles for different values of Hartmann number M

M = 6.0

M = 4.0

M = 2.0

E = -1.0

E = 0.0

E = 1.0

v1


-1.0 -0.5 0.0 0.5 1.0

-0.01

0.00

0.01

α = -5.0

α = 5.0

α = 0.0

E = -1.0

E = 0.0

E = 1.0

V1

Fig 23 First order solution of velocity profiles for different values of α


-1.0 -0.5 0.0 0.5 1.0-3

-2

-1

0

k = 0.5

k = 1.0

k = 0.1


y

E = - 1.0

E = 0.0

E = 1.0

Fig 27 First order solution of temperature profiles for different values of k

θ

1

-1.0 -0.5 0.0 0.5 1.0

-0.12

-0.06

0.00

Fig 28 First order solution of temperature profiles for different values of Hartmann number M

M = 6.0

M = 4.0

M = 2.0

E = -1.0

E = 0.0

E = 1.0

θ1


-1.0 -0.5 0.0 0.5 1.0

-0.15

-0.10

-0.05

0.00

α = -5.0

α = 5.0

α = 0.0

E = -1.0

E = 0.0

E = 1.0

θ1

Fig 29 First order solution of temperature profiles for different values of α


-1.0 -0.5 0.0 0.5 1.0

0

5

10

15

20

25

30

Gr = 15.0

Gr = 10.0

Gr = 5.0


y

E = - 1.0 E = 0.0

E = 1.0

Fig 30 Total solution of velocity profiles for different values of Grashoff number Gr

U



103

-1.0 -0.5 0.0 0.5 1.0

-12

0

12

24

36

m = 1.0

m = 0.5

m = 0.1


y

E = - 1.0

E = 0.0

E = 1.0

Fig 31 Total solution of velocity profiles for different values of viscosity ratio m

U

-1.0 -0.5 0.0 0.5 1.0

0

15

30

45

60

Fig 32 Total solution of velocity profiles for different values of width ratio h


y

E = - 1.0

E = 0.0

E = 1.0

U

-1.0 -0.5 0.0 0.5 1.0-15

0

15

30

45

k = 0.5

k = 1.0



y

E = - 1.0

E = 0.0

E = 1.0

Fig 33 Total solution of velocity profiles for different values of k

U

-1.0 -0.5 0.0 0.5 1.0

0

4

8

M = 6.0

M = 4.0

M = 2.0

E = -1.0

E = 0.0

E = 1.0

U

Fig 34 Total solution of velocity profiles for different values of Hartmann number M


-1.0 -0.5 0.0 0.5 1.0

-10

-5

0

5

10

α = -5.0

α = 5.0

α = 0.0

E = -1.0

E = 0.0

E = 1.0

U

Fig 35 Total solution of velocity profiles for different values of α


-1.0 -0.5 0.0 0.5 1.0-2

0

2

4

6

8Gr = 15.0

Gr = 10.0

Gr = 5.0


y

E = - 1.0

E = 0.0 E = 1.0

Fig 36 Total solution of temperature profiles for different values of Grashoff number Gr

θ

-1.0 -0.5 0.0 0.5 1.0-2

0

2

4

6

8Region-II (flat wall)


y

E = - 1.0

E = 0.0

E = 1.0

Fig 37 Total solution of temperature profiles for different values of viscosity ratio m

θ

-1.0 -0.5 0.0 0.5 1.0

0

10

20

Fig 38 Total solution of temperature profiles for different values of width ratio h


y

E = - 1.0

E = 0.0

E = 1.0

θ

-1.0 -0.5 0.0 0.5 1.0

0

10

20

30

40

Fig 39 Total solution of temperaturre profiles for different values of k

k = 0.5

k = 1.0



y

E = - 1.0

E = 0.0

E = 1.0

θ



104

-1.0 -0.5 0.0 0.5 1.0-2

0

2

4

6

8

M = 6.0M = 4.0M = 2.0

E = -1.0

E = 0.0

E = 1.0

θ

Fig 40 Total solution of temperature profiles for different values of Hartmann number M


-1.0 -0.5 0.0 0.5 1.0-10

-5

0

5

α = -5.0

α = 5.0

α = 0.0

E = -1.0

E = 0.0

E = 1.0

θ

Fig 41 Total solution of temperature profiles for different values of α


-4 -2 0 2 4 6

-6

-3

0

3

6

NuW

0

NuW

0

Nu1

0

Nu1

0

Fig . First order Nusselt numbder at y = -1.0 and y = 1.0

τ

-4 -2 0 2 4 6

-100

-50

0

50

100

Fig . Zeroth order Nusselt number y = -1 and y = 1.0

K = 0.1, 0.5

Nu

-4 -2 0 2 4 6-25

-20

-15

-10

-5

0

5

10

15

20

Gr = 5.0

Fig . Zeroth order Nusselt numbder at y = -1.0 and y = 1.0

Nu1

0

NuW

0

Nu

-4 -2 0 2 4 6

-100

-50

0

50

100

150

K = 0.1, 0.5

Nu

0

Fig . Zeroth order Nusselt number y = -1 and y = 1.0

-3 0 3 6

-6

-3

0

3

6

h=0.5

Nu1

0

Nu1

0

NuW

0

NuW

0

Fig . Zeroth order Nusselt numbder at y = -1.0 and y = 1.0

-4 -2 0 2 4 6

-15

-10

-5

0

5

10

15

20

τ

τ0

w

τ0

1

E = -1.0

E = 1.0

Fig . First order skin friction profiles for different values of m

-4 -2 0 2 4 6-50

-40

-30

-20

-10

0

10

20

30

40

50

τ

τ0

w

τ0

w

τ0

w

τ0

1

τ0

1

τ0

1

E = -1.0

E = 1.0

Fig . First order skin friction profiles for different values of Gr



105

-4 -2 0 2 4 6-60

-40

-20

0

20

40

60

τ0

1

τ0

1

τ0

1

τ0

w

τ0

w

E = -1.0

E = 1.0

Fig . Zeroth order skin friction profiles for different values of Gr

τ0

w

τ0

-4 -2 0 2 4 6

-6

-3

0

3

6

9

τ

τ0

w

τ0

1

E = -1.0

E = 1.0

Fig . First order skin friction profiles for different values of h

-4 -2 0 2 4 6-6

-4

-2

0

2

4

6

τ0

τ0

w

τ0

w

τ0

1

τ0

1

E = -1.0

E = 1.0

Fig . Zeroth order skin friction profiles for different values of h

-4 -2 0 2 4 6

-15

-10

-5

0

5

10

τ0

1

τ0

w

τ0

wE = -1.0

E = 1.0

Fig . Zeroth order skin friction profiles for different values of m

τ0

w

τ0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

-30

-25

-20

-15

0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8

10

Gr = 15.0

Gr = 10.0

Gr = 5.0

Gr = 15.0

Gr = 10.0

Gr = 5.0

P^ π

/2 X

10

2P

^ π/2

X 1

02

E = -1.0

E = 1.0

Fig . Pressure drop profiles for different values of Gr

-1.0 -0.8 -0.6 -0.4 -0.2 0.0-20

-10

0

10

20

0.0 0.2 0.4 0.6 0.8 1.0-30

-15

0

15

30

α = -5.0

α = 0.0

α = 5.0

τ

Fig . Pressure drpo profiles for different values of α

α = -5.0

α = 0.0

α = 5.0

E = -1.0

E = 1.0

REFERENCE

1. S. Ostrach, Laminar natural convection flow and heat transfer of fluids with and without Heat

sources in channels with constant wall temperature. NACATN, 2863 (1952)

2. F.A Bottemanne., Theoretical solution of simultaneous heat and mass transfer by free

convection about a vertical flat plate, Appl. scient. Res. 25, 137 (1971)

3. G.D Callahan,. W.J Marner, Transient free convection with mass transfer on an isothermal

vertical flat plate, Int. J. Heat Mass Transfer 19 165 (1976)

4. L.S Yao, Natural convection along a wavy surface, ASME J. Heat Transfer 105, 465(1983).

5. L.S Yao, A note on Prandtl transposition theorem, ASME J. Heat transfer 110, 503(1988).

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two fluid electromagneto convective flow and heat … · poots [10] treated the open circuit case...

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