CHAPTER-3

Combined Free and Forced Convection Couette-Hartmann Flow in a Rotating System with Hall Effects*

3.1 Introduction

Interest in the theoretical/experimental study of hydromagnetic flow of an electrically

conducting fluid in a rotating medium is motivated by several important problems of

geophysical and astrophysical interest and fluid engineering e.g. the internal rotation rate of

the Sun, maintenance and secular variation of earth’s magnetic field, structure of rotating

magnetic stars, the planetary and solar dynamo problems and centrifugal machines, namely,

rotating MHD generator (Yantovisky and Tolmach, 1963), rotating-drum separator for

liquid-metal applications (Lenzo et al., 1978) etc. It is widely known that the terrestrial

magnetic field is maintained by the fluid motion in earth’s core. Keeping in view

importance of such study hydromagnetic Couette flow of a viscous, incompressible and

electrically conducting fluid in a rotating system is investigated by several researchers

during past few decades. Jana et al. (1977) studied steady hydromagnetic Couette flow of a

viscous, incompressible and electrically conducting fluid in a rotating system considering

stationary plate perfectly conducting and moving plate non-conducting. They also * Published in Journal of Nature Science and Sustainable Technology, 6 (No. 3), p. 125-150

(2012)

50

considered heat transfer characteristics of flow taking viscous and Joule dissipations into

account and found that rate of heat transfer at the stationary plate is unaffected by magnetic

field and rotation. Seth and Maiti (1982) considered the same problem taking both the

plates of the channel as non-conducting. It was found by them that induced electric field

had significant effect on the flow-field and heat transfer characteristics. Jana and Datta

(1980), Mandal et al. (1982), Mandal and Mandal (1983), Seth and Ahmad (1985) and

Ghosh (2002) investigated the effects of Hall current on steady MHD Couette flow and

heat transfer of a viscous, incompressible and electrically conducting fluid in a rotating

system taking viscous and Joule dissipations into account under different conditions. Nagy

and Demendy (1995) studied the effects of Hall current on Couette-Hartmann flow and

heat transfer in a rotating channel, under general wall conditions, taking viscous and Joule

dissipations into account. In their problem, fluid flow within the channel is induced due to

movement of upper plate of the channel and an applied pressure gradient acting along the

channel walls. In all these investigations, heat transfer characteristics of the fluid flow are

considered due to forced convection. It is well known that fluids with low Prandtl number

are electrically conducting and are more sensitive to gravitational field than that with high

Prandtl number. Hence the interplay of thermal buoyancy force with Coriolis and magnetic

forces determines the ultimate behavior of an electrically conducting fluid with low Prandtl

number in the presence of an applied magnetic field. Taking into consideration this fact

Mohan (1977), Prasada Rao et al. (1982), Seth and Banerjee (1996), Ghosh and

Bhattacharjee (2000), Seth and Singh (2008), Seth and Ansari (2009), Seth et al. (2010c)

and Ghosh et al. (2011) investigated combined free and forced convection Hartmann flow

of a viscous, incompressible and electrically conducting fluid in a rotating channel

considering different aspects of the problem. Seth and Ansari (2009) and Ghosh et al.

(2011) also considered the effects of Hall current on this fluid flow problem. As per our

knowledge, no attention has been given to study the combined effects of free and forced

convection Couette-Hartmann flow of a viscous, incompressible and electrically

conducting fluid in a rotating system. Such study seems to be important and useful partly

for gaining basic understanding of such fluid flow problems and partly for possible

51

applications to the geophysical and astrophysical problems of interest and fluid

engineering.

The present chapter deals with the study of combined free and forced convection

Couette-Hartmann flow of a viscous, incompressible and electrically conducting fluid in a

rotating system with Hall effects in the presence of a uniform transverse magnetic field.

Exact solution of the governing equations is obtained in closed form. Expressions for the

shear stress at both stationary and moving plates of the channel due to primary and

secondary flows and mass flow rates in the primary and secondary flow directions are also

derived. Asymptotic behavior of the solution for the fluid velocity and induced magnetic

field is analyzed for large values of rotation parameter 2K and magnetic parameter 2M to

gain some physical insight into the flow pattern. It is demonstrated that when rotation

parameter is large and magnetic parameter is of small order of magnitude, there arises a

thin boundary layer near the moving plate of the channel. This boundary layer may be

recognized as modified Ekman boundary layer and can be viewed as classical Ekman

boundary layer modified by Hall current and magnetic field. The thickness of this boundary

layer increases on increasing Hall current parameter whereas it decreases on increasing

either rotation parameter or magnetic parameter or both. Similar type of boundary layer

appears near the stationary plate of the channel. Outside the boundary layer region, the

velocity in the primary flow direction vanishes while it persists in the secondary flow

direction. In this region, fluid velocity is unaffected by Hall current and magnetic field and

varies linearly with . In the central core region, induced magnetic field persists in both

the primary and secondary flow directions due to Hall current, rotation and thermal

buoyancy force. It is unaffected by magnetic field and variation in it is non-linear with

due to presence of thermal buoyancy force. It is also noticed that when magnetic parameter

is large and rotation parameter is of small order of magnitude, there also appears a thin

boundary layer near the moving plate of the channel. This boundary layer may be identified

as modified Hartmann boundary layer and can be viewed as classical Hartmann boundary

layer modified by Hall current and rotation. The thickness of this boundary layer decreases

on increasing either rotation parameter or magnetic parameter or both whereas it increases

52

on increasing Hall current parameter. In the absence of Hall current it reduces to classical

Hartmann boundary layer. Similar type of boundary layer arises adjacent to the stationary

plate of the channel. Outside the boundary layer region, fluid flows in both the primary and

secondary flow directions. Fluid flow in both the primary and secondary flow directions are

affected by thermal buoyancy force and magnetic field and varies linearly with . In the

absence of Hall current fluid flow persists in the primary flow direction only. Induced

magnetic field persists in both the primary and secondary flow directions. Primary induced

magnetic field is affected by Hall current, rotation, thermal buoyancy force, pressure

gradient and magnetic field and the variation in it is non-linear with due to presence of

thermal buoyancy force. Secondary induced magnetic field is unaffected by thermal

buoyancy force and pressure gradient and it varies linearly with . Heat transfer

characteristics of the fluid flow are considered taking viscous and Joule dissipations into

account. Numerical solution of the energy equation and numerical values of rate of heat

transfer at both stationary and moving plates of the channel are computed with the help of

MATLAB software. The profiles of fluid velocity, induced magnetic field and fluid

temperature are depicted graphically versus channel width variable for various values of

pertinent flow parameters while numerical values of shear stress at both stationary and

moving plates of the channel due to primary and secondary flows, mass flow rates in the

primary and secondary flow directions and rate of heat transfer at both the stationary and

moving plates of the channel are presented in tabular form for various values of pertinent

flow parameters.

3.2 Formulation of the Problem and its Solution

Consider steady flow of a viscous, incompressible and electrically conducting fluid

between two parallel electrically non-conducting plates 0 andz z L in the presence of

a uniform transverse magnetic field 0H applied parallel to z-axis. Fluid and channel rotate

in anti-clockwise direction with uniform angular velocity about z-axis. Fluid flow within

the channel is induced due to an applied uniform pressure gradient along x-direction as well

as the movement of upper plate z L with uniform velocity 0U along x-direction. The

53

plates of channel are heated or cooled by a uniform temperature gradient acting along x-

direction so that temperature varies linearly along the plates. Since plates of the channel are

of infinite extent in x and y directions and flow is steady and fully developed so all physical

quantities, except pressure and temperature, depend on z only. Geometry of the problem is

presented in figure 3.1.

Fig. 3.1 Geometry of the problem

Consistent with the above assumptions, fluid velocity q

and magnetic field H

are given

by

0( , ,0); ( , , )x y x yq u u H H H H

, (3.1)

which are compatible with the fundamental equations of Magnetohydrodynamics in a

rotating frame of reference.

Equations of motion and induction equation for magnetic field in rotating frame of

reference for this fluid flow problem assume the following form

0U

POROUS PLATE ( )z L

z y Primary Flow

0H

Secondary Flow x o

LOWER PLATE ( 0)z

y

1 px

UPPER PLATE ( )z L

54

20

2

12 x e xy

d u H dHpux dz dz

, (3.2)

20

22 y yex

d u dHHudz dz

, (3.3)

01 1 0ye x

x y

dHdHp g T T H Hz dz dz

, (3.4)

22

0 2 2 0yx xm m

d Hdu d HH mdz dz dz

, (3.5)

2 2

0 2 2 0y y xm m

du d H d HH mdz dz dz

, (3.6)

where 0, , , , , 1/ , , , , , , ande m e e e e ep g T T m are modified pressure

including fluid pressure, centrifugal force and magnetohydrodynamic pressure, electrical

conductivity, magnetic permeability, density, kinematic coefficient of viscosity, magnetic

viscosity, acceleration due to gravity, coefficient of thermal expansion, fluid temperature,

temperature in the reference state, Hall current parameter, cyclotron frequency and electron

collision time respectively.

Formulation of the problem will be complete if we specify boundary conditions for the

fluid velocity and induced magnetic field. Boundary conditions for the fluid velocity and

induced magnetic field are given by

0, 0 at 0 ; 0, 0 at 0x y x yu u z H H z , (3.7)

0, 0 at ; 0, 0 atx y x yu U u z L H H z L . (3.8)

Since plates of the channel are heated or cooled by a uniform temperature gradient acting

along x-direction so fluid temperature is assumed in the following form

0 1 1( )T T A x z , (3.9)

55

where 1 1and ( )A z are, respectively, uniform temperature gradient acting along x-

direction and an arbitrary function of z.

Integrating equation (3.4), we obtain

2 2 20 0 12

ex y

p gz H H H g T T dz K x

, (3.10)

where 1K is uniform pressure gradient acting along x-direction.

Making use of equations (3.9) and (3.10) in equation (3.2), we obtain

20

1 1 22 x e xy

u H Hu g A z Kz z

. (3.11)

Representing equations (3.11), (3.3), (3.5) and (3.6), in non-dimensional form, we obtain

22 2

22 xdhd uK v G R Md d

, (3.12)

22 2

22 ydhd vK u Md d

, (3.13)

22

2 2 0yxd hd hdu m

d d d , (3.14)

2 2

2 2 0y xd h d hdv m

d d d , (3.15)

where

0 0 0 0, / , / , / and /x y x x m y y mz L u u U v u U h H H R h H H R . (3.16)

Here 0m eR U L is magnetic Reynolds number, 3

1

0

g A LGU

is Grashof number,

21

0

K LRU

is non-dimensional pressure gradient, 2 2K L is rotation parameter which is

56

reciprocal of Ekman number and 2 2 20M B L is magnetic parameter which is square

of Hartmann number.

It may be noted from equation (3.9) that positive or negative value of 1A corresponds to

heating or cooling of the channel walls along x-direction. Therefore, it follows from the

definition of G that 0 or 0G G according as the channel walls are heated or cooled in

axial direction.

Boundary conditions (3.7) and (3.8), in non-dimensional form, become

0, 0 at 0 ; 0, 0 at 0x yu v H H , (3.17)

1, 0 at 1 ; 0, 0 at 1x yu v H H . (3.18)

Combining equations (3.12) and (3.14) with the equations (3.13) and (3.15) respectively,

we obtain

22 2

22 d F dHiK F G R Md d

, (3.19)

2

2(1 ) 0dF d Himd d

, (3.20)

where , x yF u iv H h ih .

Boundary conditions (3.17) and (3.18), in compact form, become

0, 0 at 0 ; 1, 0 at 1F H F H . (3.21)

Solution of equations (3.19) and (3.20) subject to the boundary conditions (3.21) is given

by

1 21 sinh (1 cosh ) GF c c

, (3.22)

2 2 2 2

1 2 221 2 2(1 cosh ) sinh

2iK G iKH c c c R

M

, (3.23)

57

where

2 2

1 2 2 2

2 1 cosh1 sinh 1 coshsinh 22 2 1 cosh sinh

iKc G

iK i K

2 2

2 2

sinh 1 cosh2

R GiK

, (3.24a)

2 2

2 2 2 2

2 sinh 1 cosh22 2 1 cosh sinh

iKc G

iK i K

2

2 2

sinh 1 cosh2

RiK

, (3.24b)

i , (3.24c)

1/21/224 2 2 2 2

2

1, 2 12 1

M K m mM Mm

. (3.24d)

3.3 Shear Stress at the Plates

Non-dimensional shear stress components x and y , due to primary and secondary flows

respectively, at stationary and moving plates of the channel, are given by

10

1 ,x yGi c

(3.25)

1 21cosh sinhx y

Gi c c

. (3.26)

3.4 Mass Flow Rates

Non-dimensional mass flow rates xQ and yQ , in the primary and secondary flow directions

respectively, are given by

58

1 2 12

cosh sinh12x y

c c c GQ iQ c

. (3.27)

3.5 Asymptotic Solutions

We shall now analyze asymptotic behavior of the solution for large values of 2 2andK M

to gain some physical insight into the flow pattern.

Case-I: 2 2K 1 and M O(1)

When 2K is large and 2M is of small order of magnitude, boundary layer type flow is

expected near plates of the channel. For boundary layer flow near moving plate 1 ,

introducing boundary layer variable 1 , we obtain the expressions for fluid velocity

and induced magnetic field from the solution (3.22) to (3.24) which are presented in the

following form

12 1 3 1cos sinu e K K , (3.28)

13 1 2 12 cos sin

2Gv e K KK

, (3.29)

1

2

4 4 1 5 12 2cos sin

4 1xmGh K e K K

K m

, (3.30)

1

2

5 5 1 4 12 2cos sin

4 1yGh K e K K

K m

, (3.31)

where

2 2

1 1 22 2 2 2

2

3 4 52 2 2 2 2

1, 1 , 1 ,

4 (1 ) 4 (1 )

1 1, , K .4 (1 ) 2 2 1 2 1

M m mMK KK m K m

M R G m mK KK m K K m K m

(3.32)

59

It is evident from the expressions in (3.28) to (3.32) that there arises a thin boundary layer

of thickness 11O near moving plate of the channel. This boundary layer may be

recognized as modified Ekman boundary layer and can be viewed as classical Ekman

boundary layer modified by Hall current and magnetic field. The thickness of this boundary

layer increases on increasing m whereas it decreases on increasing either 2 2orK M or

both. Similar type of boundary layer appears near stationary plate of the channel. The

exponential terms in the expressions in (3.28) to (3.31) damp out quickly as increases.

When 11/ i.e. in the central core region, (3.28) to (3.31) reduce to

20, Gu vK , (3.33)

2 2

4 52 2 2 2,

4 1 4 1x ymG Gh K h K

K m K m

. (3.34)

It is revealed from the expressions in (3.33) and (3.34) that, in the central core region, i.e.

outside the boundary layer region, velocity in the primary flow direction vanishes while it

persists in the secondary flow direction. In this region, the fluid velocity is unaffected by

Hall current and magnetic field and varies linearly with . In the central core region,

induced magnetic field persists in both the primary and secondary flow directions due to

Hall current, rotation and thermal buoyancy force. It is unaffected by magnetic field and

variation in it is non-linear with due to presence of thermal buoyancy force.

Case-II: 2 2M 1 and K O(1)

In this case also boundary layer type flow is expected adjacent to plates of the channel. For

the boundary layer flow near the moving plate 1 , the expressions for fluid velocity and

induced magnetic field are obtained from solution (3.22) to (3.24) and are expressed in the

following form

21 2 2 1 2 2cos sinGu e aM bM bM aM

M , (3.35)

60

21 2 2 1 2 2cos sinmGv e bM aM aM bM

M , (3.36)

2

2 2

2 22 22 2 2

2 1 cos sin2 1 1

xG K m Rh e a bM MM m M m

, (3.37)

2

2

2 22 2 2

2 1 cos sin1 1

yKh e b a

M m M m

, (3.38)

where

2 2 2 2 2 22 22 2

1/22

2 2

1 22 2 2 2

1 1( ) , ( )(1 ) (1 )

1, 1 1 ,2

, .1 1 1 1

a M mK bK b M mK aKM m M m

a b m

K am b K a mbGa Ma Gb MbM MM Mm M m m M m

(3.39)

The expressions in (3.35) to (3.39) demonstrate the existence of a thin boundary layer of

thickness 12O near moving plate of the channel. This boundary layer may be identified

as modified Hartmann boundary layer and can be viewed as classical Hartmann boundary

layer modified by Hall current and rotation. The thickness of this boundary layer decreases

on increasing either 2 2orK M or both whereas it increases on increasing m . In the

absence of Hall current it reduces to Classical Hartmann boundary layer of thickness

1O M . Similar type of boundary layer arises adjacent to stationary plate of the channel.

When 21/ i.e. in the central core region, (3.35) to (3.38) assume the form

, ,G mGu vM M (3.40)

2 2 2

2 22 2 2 2

2 2,2 1 1

x yG K m R Kh hM MM m M m

. (3.41)

It is noticed from the expressions in (3.40) and (3.41) that, in the central core region, i.e.

outside the boundary layer region, fluid flows in both the primary and secondary flow

61

directions. Fluid flow in both the primary and secondary flow directions are affected by

thermal buoyancy force and magnetic field and varies linearly with . In the absence of

Hall current fluid flow persists in the primary flow direction only. Induced magnetic field

persists in both the primary and secondary flow directions. Primary induced magnetic field

xh is affected by Hall current, rotation, thermal buoyancy force, pressure gradient and

magnetic field and the variation in it is non-linear with due to presence of thermal

buoyancy force. Secondary induced magnetic field yh is unaffected by thermal buoyancy

force and pressure gradient and it varies linearly with .

3.6 Heat Transfer Characteristics

Energy equation for steady fully developed flow of a viscous, incompressible and

electrically conducting fluid taking viscous and Joule dissipations into account is given by

22222*0 0

2

( ) ( ) 1y yx xx

p p

u HT T T T u Hux z C z z C z z

, (3.42)

where * / pk C is thermal diffusivity of fluid. and pk C are, respectively, thermal

conductivity of fluid and specific heat at constant pressure.

The plates 0 andz z L are cooled or heated by a uniform temperature gradient 1A

acting along x-direction, therefore, the boundary conditions for temperature field are given

by

10 1 1 at 0T T A x z , (3.43)

20 1 1 atT T A x z L , (3.44)

where 1 21 1and are constants.

Representing equation (3.42), in non-dimensional form with the help of (3.9) and (3.16),

we obtain

62

22

2 . .r r rd dF d F dH d HGPu P E Md d d d d

, (3.45)

where 1

31 1 *2

( ), / andr r

p

g L g LP EC

are non-dimensional fluid

temperature, Prandtl number and Eckert number respectively. andF H are complex

conjugates of F and H respectively.

The boundary conditions (3.43) and (3.44), in non-dimensional form, become

2 1

31 1

02

( )(0) 0 and (1) (say)

g L

. (3.46)

Making use of (3.22) to (3.24) in equation (3.45) and solving the resulting equation subject

to the boundary conditions (3.46) by MATLAB software we have obtained numerical

solution for the fluid temperature . The numerical values of rate of heat transfer at

stationary and moving plates of the channel are also obtained with the help of MATLAB

software.

3.7 Results and Discussion

To study the effects of Hall current, magnetic field, rotation and thermal buoyancy force on

flow-field, the numerical values of fluid velocity and induced magnetic field, computed

from the analytical solution (3.22) to (3.24), are depicted graphically versus channel width

variable in figures 3.2 to 3.9 for various values of Hall current parameter m , magnetic

parameter 2M , rotation parameter 2K and Grashof number G taking 1R . It is evident

from figure 3.2 that primary velocity u decreases in the region 0 0.7 whereas it

increases in the region 0.7 1 on increasing m. Secondary velocity v decreases in

magnitude in the region 0 0.55 and it increases in magnitude in the region

0.55 1 on increasing m . This implies that Hall current tends to retard fluid flow in the

primary flow direction in the region 0 0.7 and it tends to retard fluid flow in the

secondary flow direction in the region 0 0.55 . It has reverse effect on the fluid flow

63

in the primary flow direction in the region 0.7 1 and on the fluid flow in the

secondary flow direction in the region 0.55 1 . It is revealed from figure 3.3 that

primary velocity u increases in the region 0 0.69 whereas it decreases in the region

0.69 1 on increasing 2M . Secondary velocity v decreases in magnitude on

increasing 2M throughout the channel. This implies that magnetic field tends to retard

fluid flow in the secondary flow direction throughout the channel whereas it tends to

accelerate fluid flow in the primary flow direction in the region 0 0.69 . Magnetic

field has reverse effect on the fluid flow in primary flow direction in the region

0.69 1 . It is noticed from figure 3.4 that primary velocity u decreases whereas

secondary velocity v increases in magnitude on increasing 2K . This implies that rotation

tends to accelerate fluid flow in the secondary flow direction whereas it has reverse effect

on the fluid flow in primary flow direction. It is observed from figure 3.5 that primary

velocity u decreases and secondary velocity v decreases in magnitude on increasing

( 0)G whereas primary velocity increases and secondary velocity increases in magnitude

on increasing ( 0)G . This implies that thermal buoyancy force tends to retard fluid flow

in both the directions when walls of the channel are heated whereas it has reverse effect on

fluid flow in both the directions when walls of the channel are cooled. It is evident from

figure 3.6 that primary induced magnetic field xh decreases whereas secondary induced

magnetic yh increases on increasing m . This implies that Hall current tends to reduce

primary induced magnetic field whereas it has reverse effect on the secondary induced

magnetic field. It is noticed from figure 3.7 that both the primary and secondary induced

magnetic fields decrease on increasing 2M . This implies that magnetic field tends to

reduce both the primary and secondary induced magnetic fields. It is revealed from figure

3.8 that primary induced magnetic field xh decreases in the lower half of the channel

whereas secondary induced magnetic yh decreases in the region 0 0.75 on increasing

2K . The nature of both the induced magnetic fields is changed with respect to 2K near the

moving plate. This implies that rotation tends to reduce both the induced magnetic fields in

64

lower half of the channel whereas it has reverse effect on both the induced magnetic fields

in the region near to moving plate. It is found from figure 3.9 that primary induced

magnetic field xh decreases on increasing ( 0)G and it increases on increasing ( 0)G

in the region 0 0.65 whereas secondary induced magnetic yh decreases on increasing

( 0)G and it increases on increasing ( 0)G in the region 0.3 1 . This implies that

thermal buoyancy force tends to reduce primary induced magnetic field in the region

0 0.65 when walls of the channel are heated whereas it tends to enhance primary

induced magnetic field in the region 0 0.65 when walls of the channel are cooled.

Thermal buoyancy force has reverse effect on the primary induced magnetic field near the

moving plate. Thermal buoyancy force has same effect on the secondary induced magnetic

field in the region 0.3 1 as it has on primary induced magnetic field in the region

0 0.65 and it has reverse effect on the secondary induced magnetic field near the

stationary plate.

The numerical values of fluid temperature, computed from energy equation (3.45) with

the help of MATLAB software, are displayed graphically versus channel width variable

in figures 3.10 to 3.15 for various values of 2 2, , , , andr rm M K G P E taking

0 1 and 1R . It is evident from figures 3.10 and 3.12 that fluid temperature increases

on increasing either 2orm K . This implies that Hall current and rotation tend to enhance

fluid temperature. It is noticed from figure 3.11 that fluid temperature decreases on

increasing 2M . This implies that magnetic field tends to reduce fluid temperature. It is

evident from figure 3.13 that fluid temperature decreases on increasing ( 0)G whereas

it increases on increasing ( 0)G . This implies that thermal buoyancy force tends to

reduce fluid temperature when walls of the channel are heated and it has reverse effect on

fluid temperature when walls of the channel are cooled. It is noticed from figures 3.14 and

3.15 that fluid temperature increases on increasing either orr rP E . This implies that

thermal diffusion tends to reduce fluid temperature whereas viscous dissipation has reverse

effect on it.

65

The numerical values of primary and secondary shear stress components at the

stationary and moving plates and those of mass flow rates in the primary and secondary

flow directions, computed from the expressions (3.25) to (3.27), are presented in tabular

form in tables 3.1 to 3.9 for various values of 2 2, , andm M K G taking 1R . It is found

from table 3.1 that primary shear stress at the stationary plate i.e. 0x

increases on

increasing 2M whereas it decreases on increasing 2K . Secondary shear stress at the

stationary plate i.e. 0y

increases on increasing 2M when 2 5K and it increases on

increasing 2K . This implies that magnetic field tends to enhance primary shear stress at the

stationary plate and it tends to enhance secondary shear stress at the stationary plate only

when 2 5K . Rotation tends to reduce primary shear stress at the stationary plate whereas

it has reverse effect on the secondary shear stress at the stationary plate. It is evident from

table 3.2 that primary shear stress at the moving plate i.e. 1x

and secondary shear stress

at the moving plate i.e. 1y

increase on increasing either 2 2orM K . This implies that

magnetic field and rotation tend to enhance both the primary and secondary shear stress at

the moving plate. It is observed from table 3.3 that 0x

decreases on increasing either

or ( 0)m G and it increases on increasing ( 0)G . This implies that Hall current tends to

reduce primary shear stress at the stationary plate. Thermal buoyancy force tends to reduce

primary shear stress at the stationary plate when walls of the channel are heated and it has

reverse effect on the primary shear stress at the stationary plate when walls of the channel

are cooled. It is noticed from table 3.4 that 0y

decreases on increasing m when ( 0)G

and it decreases, attains a minimum and then increases in magnitude on increasing m when

( 0)G . 0y

increases in magnitude on increasing ( 0)G when 1m and it increases

on increasing ( 0)G . This implies that Hall current tends to reduce secondary shear stress

at the stationary plate when walls of the channel are cooled. Thermal buoyancy force tends

to enhance secondary shear stress at the stationary plate when walls of the channel are

cooled and it has same effect on the secondary shear stress at the stationary plate when the

66

channel walls are heated and 1m . Also there exists flow separation at the stationary plate

in the secondary flow direction on increasing Hall current parameter m when walls of the

channel are heated. It is found from table 3.5 that 1x

decreases on increasing m . 1x

increases on increasing ( 0)G and it decreases on increasing ( 0)G . This implies that

Hall current tends to reduce primary shear stress at the moving plate. Thermal buoyancy

force tends to enhance primary shear stress at the moving plate when walls of the channel

are heated and it has reverse effect on primary shear stress at the moving plate when walls

of the channel are cooled. It is observed from table 3.6 that 1y

increases on increasing

m . 1y

decreases on increasing ( 0)G and it increases on increasing ( 0)G . This

implies that Hall current tends to enhance secondary shear stress at the moving plate.

Thermal buoyancy force tends to reduce secondary shear stress at the moving plate when

walls of the channel are heated and it has reverse effect on secondary shear stress at the

moving plate when walls of the channel are cooled. It is evident from table 3.7 that primary

mass flow rate i.e. xQ increases on increasing 2M whereas it decreases on increasing 2K .

Secondary mass flow rate i.e. yQ decreases on increasing 2M when 2 5K whereas it

increases on increasing 2K . This implies that magnetic field tends to enhance mass flow

rate in the primary flow direction and it has reverse effect on the mass flow rate in the

secondary flow direction when 2 5K . Rotation tends to reduce mass flow rate in the

primary flow direction and it has reverse effect on the mass flow rate in the secondary flow

direction. It is also noticed from table 3.7 that there is no variation in the secondary mass

flow rate on increasing 2M when 2 27 and 36K M . It is found from tables 3.8 and 3.9

that xQ decreases on increasing m . With an increase in m , yQ decreases when 0G and

it increases when 2G . andx yQ Q decrease on increasing ( 0)G whereas it increase

on increasing ( 0)G . This implies that Hall current tends to reduce mass flow rate in the

primary flow direction. It tends to reduce mass flow rate in the secondary flow direction

when walls of the channel are heated whereas it has reverse effect on the mass flow rate in

the secondary flow direction when walls of the channel are cooled and 2G . Thermal

67

buoyancy force tends to reduce mass flow rates in both the directions when walls of the

channel are heated whereas it has reverse effect on mass flow rates in both the directions

when walls of the channel are cooled.

The numerical values of rate of heat transfer at both the stationary and moving plates are

computed with the help of MATLAB software and are presented in tables 3.10 to 3.13 for

various values of 2 2, , , , andr rm M K G P E taking 0 1 and 1R . It is revealed from

table 3.10 that rate of heat transfer at the stationary plate i.e. 0

dd

and that at moving

plate i.e. 1

dd

increase on increasing either 2 2orM K . This implies that magnetic

field and rotation have tendency to enhance rate of heat transfer at both stationary and

moving plates. It is found from table 3.11 that 0

dd

decreases on increasing m and it

decreases on increasing ( 0)G whereas it increases on increasing ( 0)G . This implies

that Hall current tends to reduce rate of heat transfer at the stationary plate. Thermal

buoyancy force tends to reduce rate of heat transfer at the stationary plate when walls of the

channel are heated and it has reverse effect on the rate of heat transfer at the stationary

plate when walls of the channel are cooled. It is observed from table 3.12 that 1

dd

increases on increasing either ( 0) or ( 0)G G . 1

dd

decreases on increasing m when

2G and it increases on increasing m when 3G . This implies that Hall current tends

to reduce rate of heat transfer at the moving plate when 2G and it has reverse effect on

the rate of heat transfer at the moving plate when 3G . Thermal buoyancy force tends to

enhance rate of heat transfer at the moving plate when walls of the channel are either

heated or cooled. It is evident from table 3.13 that 0

dd

increases on increasing either

68

orr rP E . 1

dd

decreases, attains a minimum and then increases in magnitude on

increasing rP . 1

dd

decreases on increasing rE when 0.3rP and it increases in

magnitude on increasing rE when 3rP . This implies that thermal diffusion tends to

reduce rate of heat transfer at the stationary plate. Viscous dissipation tends to enhance rate

of heat transfer at the stationary plate, it tends to reduce rate of heat transfer at the moving

plate when 0.3rP and it has reverse effect on rate of heat transfer at the moving plate

when 3rP .

3.8 Conclusions

This study presents a theoretical investigation of combined free and forced convection

Couette-Hartmann flow of a viscous, incompressible and electrically conducting fluid in a

rotating system with Hall effects in the presence of a uniform transverse magnetic field.

The significant results are summarized below:

1. For large values of rotation parameter 2K there arises thin modified Ekman

boundary layer near walls of the channel whereas for large values of magnetic

parameter 2M there appears thin modified Hartmann boundary layer near walls of

the channel.

2. Hall current tends to retard fluid flow in the primary flow direction in the region

0 0.7 and it retards fluid flow in the secondary flow direction in the region

0 0.55 . It has reverse effect on the fluid flow in the primary flow direction in

the region 0.7 1 and on the fluid flow in the secondary flow direction in the

region 0.55 1 . Magnetic field tends to retard fluid flow in the secondary flow

direction throughout the channel whereas it tends to accelerate fluid flow in the

primary flow direction in the region 0 0.69 . Magnetic field has reverse effect

on the fluid flow in the primary flow direction in the region 0.69 1 . Rotation

69

tends to accelerate fluid flow in the secondary flow direction whereas it has reverse

effect on the fluid flow in the primary flow direction. Thermal buoyancy force tends

to retard fluid flow in both the directions when walls of the channel are heated

whereas it has reverse effect on fluid flow in both the directions when walls of the

channel are cooled.

3. Hall current tends to reduce primary induced magnetic field whereas it has reverse

effect on the secondary induced magnetic field. Magnetic field tends to reduce both

the primary and secondary induced magnetic fields. Rotation tends to reduce both the

induced magnetic fields in the lower half of the channel whereas it has reverse effect

on both the induced magnetic fields in the region near to the moving plate. Thermal

buoyancy force tends to reduce primary induced magnetic field in the region

0 0.65 when walls of the channel are heated whereas it tends to enhance

primary induced magnetic field in the region 0 0.65 when walls of the channel

are cooled. Thermal buoyancy force has reverse effect on the primary induced

magnetic field near the moving plate. Thermal buoyancy force has same effect on the

secondary induced magnetic field in the region 0.3 1 as it has on primary

induced magnetic field in the region 0 0.65 and it has reverse effect on the

secondary induced magnetic field near the stationary plate.

4. Hall current and rotation tend to enhance fluid temperature. Magnetic field tends to

reduce fluid temperature. Thermal buoyancy force tends to reduce fluid temperature

when walls of the channel are heated and it has reverse effect on fluid temperature

when walls of the channel are cooled. Thermal diffusion tends to reduce fluid

temperature whereas viscous dissipation has reverse effect on it.

5. Magnetic field tends to enhance primary shear stress at the stationary plate and it

tends to enhance secondary shear stress at the stationary plate when 2 5K . Rotation

tends to reduce primary shear stress at the stationary plate whereas it has reverse

effect on the secondary shear stress at the stationary plate. Magnetic field and rotation

tend to enhance both the primary and secondary shear stress at the moving plate. Hall

70

current tends to reduce primary shear stress at the stationary plate. Thermal buoyancy

force tends to reduce primary shear stress at the stationary plate when walls of the

channel are heated and it has reverse effect on the primary shear stress at the

stationary plate when walls of the channel are cooled. Hall current tends to reduce

secondary shear stress at the stationary plate when walls of the channel are cooled.

Thermal buoyancy force tends to enhance secondary shear stress at the stationary

plate when walls of the channel are cooled and it has same effect on the secondary

shear stress at the stationary plate when the channel walls are heated and 1m . Also

there exists flow separation at the stationary plate in the secondary flow direction on

increasing Hall current parameter m when walls of the channel are heated. Hall

current tends to reduce primary shear stress at the moving plate. Thermal buoyancy

force tends to enhance primary shear stress at the moving plate when walls of the

channel are heated and it has reverse effect on primary shear stress at the moving

plate when walls of the channel are cooled. Hall current tends to enhance secondary

shear stress at the moving plate. Thermal buoyancy force tends to reduce secondary

shear stress at the moving plate when walls of the channel are heated and it has

reverse effect on secondary shear stress at the moving plate when walls of the

channel are cooled.

6. Magnetic field tends to enhance mass flow rate in the primary flow direction and it

has reverse effect on the mass flow rate in the secondary flow direction when 2 5.K

Rotation tends to reduce mass flow rate in the primary flow direction and it has

reverse effect on the mass flow rate in the secondary flow direction. There is no

variation in the secondary mass flow rate on increasing 2M when 2 7K and 2 36M . Hall current tends to reduce mass flow rate in the primary flow direction. It

tends to reduce mass flow rate in the secondary flow direction when walls of the

channel are heated whereas it has reverse effect on the mass flow rate in the

secondary flow direction when walls of the channel are cooled and 2G . Thermal

buoyancy force tends to reduce mass flow rates in both the directions when walls of

71

the channel are heated whereas it has reverse effect on mass flow rates in both the

directions when walls of the channel are cooled.

7. Magnetic field and rotation have tendency to enhance rate of heat transfer at both the

stationary and moving plates. Hall current tends to reduce rate of heat transfer at the

stationary plate. Thermal buoyancy force tends to reduce rate of heat transfer at the

stationary plate when walls of the channel are heated and it has reverse effect on the

rate of heat transfer at the stationary plate when walls of the channel are cooled. Hall

current tends to reduce rate of heat transfer at the moving plate when 2G and it has

reverse effect on the rate of heat transfer at the moving plate when 3G . Thermal

buoyancy force tends to enhance rate of heat transfer at the moving plate when walls

of the channel are either heated or cooled. Thermal diffusion tends to reduce rate of

heat transfer at the stationary plate. Viscous dissipation tends to enhance rate of heat

transfer at the stationary plate, it tends to reduce rate of heat transfer at the moving

plate when 0.3rP and it has reverse effect on rate of heat transfer at the moving

plate when 3rP .

72

Fig. 3.2 Velocity profiles when 2 2M = 25, K = 3 and G = 1

Fig. 3.3 Velocity profiles when 2m = 0.5, K = 3 and G = 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

u,v m=0.5, 1, 1.5

m=0.5, 1, 1.5

m=0.5, 1, 1.5

m=0.5, 1, 1.5

u

v

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

u, v

M2=25, 36, 49

M2=25, 36, 49

M2=25, 36, 49

v

u

73

Fig. 3.4 Velocity profiles when 2m = 0.5, M = 25 and G = 1

Fig. 3.5 Velocity profiles when 2 2m = 0.5, M = 25 and K = 3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

u, v

K2=3, 5,7 v u

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

u, v

G=-3, -2, -1, 0, 1, 2, 3

G=-3, -2, -1, 0, 1, 2, 3

u

v

74

Fig. 3.6 Induced magnetic field profiles when 2 2M = 25, K = 3 and G = 1

Fig. 3.7 Induced magnetic field profiles when 2m = 0.5, K = 3 and G = 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

h x, hy

m=0.5, 1, 1.5

m=0.5, 1, 1.5

hx

hy

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

h x, hy

M2=25, 36, 49 hx

hy

75

Fig. 3.8 Induced magnetic field profiles when 2m = 0.5, M = 25 and G = 1

Fig. 3.9 Induced magnetic field profiles when 2 2m = 0.5, M = 25 and K = 3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

h x, hy

K2=3, 5, 7

K2=3, 5, 7

hy

hx

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

h x, hy

G=-3, -2, -1, 0, 1, 2, 3

G=-3, -2, -1, 0, 1, 2, 3

hx

hy

76

Fig. 3.10 Temperature profiles when 2 2

r rM = 25, K = 3, G = 1, P = 0.71 and E = 2

Fig. 3.11 Temperature profiles when 2

r rm = 0.5, K = 3, G = 1, P = 0.71 and E = 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

m=0.5, 1, 1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

M2=25, 36, 49

77

Fig. 3.12 Temperature profiles when 2

r rm = 0.5, M = 25, G = 1, P = 0.71 and E = 2

Fig. 3.13 Temperature profiles when 2 2

r rm = 0.5, M = 25, K = 3, P = 0.71 and E = 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

K2=3, 5, 7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

G=-3, -2, -1, 0, 1, 2, 3

78

Fig. 3.14 Temperature profiles when 2 2

rm = 0.5, M = 25, K = 3, G = 1 and E = 2

Fig. 3.15 Temperature profiles when 2 2

rm = 0.5, M = 25, K = 3, G = 1 and P = 0.71

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

Pr=0.02, 0.3, 0.71, 3, 7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Er=0.5, 1, 1.5, 2

79

Table 3.1 Primary and secondary shear stress at the stationary plate when m = 0.5 and G = 1

2M 2K

0x 0y

3 5 7 3 5 7 25 1.3858 1.0593 0.7765 0.2531 0.4768 0.5398 36 1.8327 1.4963 1.1813 0.2072 0.4981 0.6194 49 2.2909 1.9489 1.6079 0.1564 0.5085 0.6865

Table 3.2 Primary and secondary shear stress at the moving plate when m = 0.5 and G = 1

2M 2K

1x 1y 3 5 7 3 5 7

25 3.5328 3.9697 4.3788 1.8256 2.4106 2.8173 36 3.9448 4.3761 4.7968 1.9485 2.5531 2.9757 49 4.3695 4.7929 5.2228 2.0702 2.6974 3.1419

Table 3.3 Primary shear stress at the stationary plate when 2 2M = 25 and K = 3

m G

0x -3 -2 -1 0 1 2 3

0.5 1.9783 1.8302 1.6820 1.5339 1.3858 1.2376 1.0895 1 1.6384 1.4998 1.3612 1.2226 1.0841 0.9455 0.8069

1.5 1.3420 1.2110 1.0800 0.9490 0.8180 0.6870 0.5560

Table 3.4 Secondary shear stress at the stationary plate when 2 2M = 25 and K = 3

m G

0y

-3 -2 -1 0 1 2 3 0.5 0.5093 0.4453 0.3812 0.3172 0.2531 0.1891 0.1251 1 0.2192 0.1591 0.0990 0.0389 -0.0212 -0.0812 -0.1413

1.5 0.1387 0.0785 0.0183 -0.0419 -0.1021 -0.1622 -0.2224

80

Table 3.5 Primary shear stress at the moving plate when 2 2M = 25 and K = 3

m G

1x -3 -2 -1 0 1 2 3

0.5 2.4442 2.7163 2.9885 3.2607 3.5328 3.8050 4.0772 1 2.1176 2.3871 2.6565 2.9260 3.1954 3.4648 3.7343

1.5 1.8507 2.1197 2.3887 2.6577 2.9267 3.1957 3.4647

Table 3.6 Secondary shear stress at the moving plate when 2 2M = 25 and K = 3

m G

1y -3 -2 -1 0 1 2 3

0.5 2.1602 2.0765 1.9929 1.9092 1.8256 1.7419 1.6583 1 2.3768 2.2890 2.2012 2.1134 2.0256 1.9378 1.8500

1.5 2.4128 2.3215 2.2302 2.1390 2.0477 1.9564 1.8652

Table 3.7 Mass flow rates in the primary and secondary flow directions when m = 0.5 and G = 1

2M 2K

xQ yQ 3 5 7 3 5 7

25 0.3465 0.2887 0.2398 0.1078 0.1410 0.1502 36 0.3593 0.3051 0.2568 0.1020 0.1380 0.1511 49 0.3711 0.3206 0.2735 0.0964 0.1344 0.1511

Table 3.8 Mass flow rate in the primary flow direction when 2 2M = 25 and K = 3

m G

xQ -3 -2 -1 0 1 2 3

0.5 0.4449 0.4203 0.3957 0.3711 0.3465 0.3218 0.2972 1 0.4327 0.4080 0.3834 0.3587 0.3341 0.3094 0.2848

1.5 0.4252 0.4000 0.3748 0.3495 0.3243 0.2990 0.2738

81

Table 3.9 Mass flow rate in the secondary flow direction when 2 2M = 25 and K = 3

m G

yQ -3 -2 -1 0 1 2 3

0.5 0.1610 0.1477 0.1344 0.1211 0.1078 0.0946 0.0813 1 0.1632 0.1479 0.1326 0.1172 0.1019 0.0866 0.0712

1.5 0.1681 0.1515 0.1348 0.1181 0.1015 0.0848 0.0681

Table 3.10 Rate of heat transfer at the stationary and moving plates when

r rm = 0.5, G = 1, P = 0.71 and E = 2

2M 2K

0

dd

1

dd

3 5 7 3 5 7 25 1.6335 1.7088 1.7612 1.0762 1.8345 2.5074 36 1.6762 1.7245 1.7678 1.1717 1.9135 2.6032 49 1.7521 1.7703 1.7957 1.2855 2.0052 2.7025

Table 3.11 Rate of heat transfer at the stationary plate when 2 2r rM = 25, K = 3, P = 0.71 and E = 2

m G 0

dd

-3 -2 -1 0 1 2 3 0.5 2.3825 2.1476 1.9445 1.7731 1.6335 1.5256 1.4494 1 2.2490 2.0313 1.8450 1.6900 1.5665 1.4743 1.4136

1.5 2.1692 1.9620 1.7866 1.6433 1.5318 1.4523 1.4048

Table 3.12 Rate of heat transfer at the moving plate when 2 2r rM = 25, K = 3, P = 0.71 and E = 2

m G 1

dd

-3 -2 -1 0 1 2 3 0.5 1.3463 1.2034 1.1108 1.0684 1.0762 1.1342 1.2425 1 1.4014 1.2478 1.1437 1.0891 1.0841 1.1285 1.2225

1.5 1.4074 1.2447 1.1321 1.0695 1.0569 1.0943 1.1818

82

Table 3.13 Rate of heat transfer at the stationary and moving plates when 2 2m = 0.5, M = 25, K = 3 and G = 1

rP rE 0

dd

1

dd

0.5 1 1.5 2 0.5 1 1.5 2 0.02 1.0028 1.0078 1.0128 1.0178 0.9889 0.9731 0.9573 0.9415 0.3 1.0417 1.1170 1.1923 1.2677 0.8334 0.5965 0.3596 0.1227 0.71 1.0987 1.2770 1.4552 1.6335 0.6057 0.0451 -0.5156 -1.0762

3 1.4170 2.1702 2.9235 3.6768 -0.6659 -3.0349 -5.4038 -7.7728 7 1.9729 3.7305 5.4882 7.2458 -2.8871 -8.4147 -13.9422 -19.4698

*****