vii
TABLE OF CONTENTS
CHAPTER
1
2
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xii
LIST OF SYMBOLS
xx
LIST OF APPENDICES
xxv
INTRODUCTION
1
1.1
Research Background
1
1.2
Problem Statements
5
1.3
Objectives and Scope
6
1.4
Significance of Study
7
1.5
Thesis Outline
8
LITERATURE REVIEW
10
2.1
Introduction
10
2.2
Flow over a Stretching Sheet in a Micropolar Fluid
10
2.3
Flow over a Shrinking Sheet in a Micropolar Fluid
15
2.4
Micropolar Fluid in a Porous Medium
20
viii
3
4
MATHEMATICAL FORMULATION
23
3.1
Introduction
23
3.2
Governing Equations
23
3.3
Boussinesq Approximation
29
3.4
Non-dimensional Equations
31
3.5
Boundary Layer Approximation
34
3.6
Non-similar Transformation
36
3.7
Physical Quantities
38
THERMOPHORESIS AND SUCTION OR INJECTION
40
EFFECTS ON THE STAGNATION POINT FLOW
TOWARDS A HORIZONTAL SHEET
5
4.1
Introduction
40
4.2
Governing Equations
41
4.3
Non-dimensional Equations
43
4.4
Non-similar Transformation
43
4.5
Solution Procedure
45
4.6
Results and Discussions
46
THERMOPHORESIS AND SLIP EFFECTS ON THE
64
STAGNATION POINT FLOW TOWARDS A SHRINKING
SHEET
6
5.1
Introduction
64
5.2
Governing Equations
65
5.3
Non-similar Transformation
66
5.4
Results and Discussions
67
MIXED CONVECTION STAGNATION POINT FLOW
85
TOWARDS A PERMEABLE SHRINKING SHEET WITH
SLIP EFFECT
6.1
Introduction
85
6.2
Governing Equations
86
6.3
Non-dimensional Equations and Boundary Layer
Approximation
87
ix
7
6.4
Non-similar Transformation
88
6.5
Results and Discussions
90
MIXED CONVECTION STAGNATION POINT FLOW
107
TOWARDS A STRETCHING SHEET IN A POROUS
MEDIUM WITH SORET AND DUFOUR EFFECTS
8
7.1
Introduction
107
7.2
Governing Equations
108
7.3
Non-dimensional Equations and Boundary Layer
Approximation
109
7.4
Non-similar Transformation
111
7.5
Results and Discussions
113
CONCLUSION
132
8.1
Summary of Research
132
8.2
Suggestion for Future Research
136
REFERENCES
138
Appendices A-C
150-182
x
LIST OF TABLES
TABLE NO.
4.1
4.2
TITLE
PAGE
Comparison of values of the skin friction of final steady-state
flow (  1) for various values of f 0 when K  M  n  0
48
Results of the skin friction for various values of K  1, M  1,
f0  1, n  0.5,   0.6 with different step sizes.
49
4.3
The skin friction, the Nusselt number and the Sherwood number
for various values of f 0 and  when K  1, M  1, Pr  0.71,
49
Sc  0.94,  0.2, n  0.5
5.1
Comparison of the values of f ''(0) for stretching sheet in the
absence of concentration  , thermophoresis  and slip
parameter 
70
Comparison of the values of f ''(0) for shrinking sheet when
K  0 in the absence of concentration  , thermophoresis 
and slip parameter  .
71
The reduced skin friction, the reduced Nusselt number and the
reduced Sherwood number for various values of  and  when
K  1, M  1,Pr  0.71, Sc  0.94,  0.2, n  0,  0.2
71
The reduced skin friction, the reduced Nusselt number and the
reduced Sherwood number for various values of  and  when
K  1, M  1,Pr  0.71, Sc  0.94,  0.2, n  0,   0.75
72
Comparison of the values of f ''(0) for stretching sheet in the
absence of concentration  , magnetic parameter M and slip
parameter 
93
5.2
5.3
5.4
6.1
6.2
Comparison of the values of f ''(0) for shrinking sheet when
K  0 in the absence of concentration  , magnetic parameter
xi
6.3
6.4
7.1
7.2
7.3
M and slip parameter  .
94
The reduced skin friction, the reduced Nusselt number and
the reduced Sherwood number for various values of f 0 , 
and  when K  1, M  1,Pr  0.71, Sc  0.94, n  0,   0.2,
  1,   0.6
94
The reduced skin friction, the reduced Nusselt number and the
the reduced Sherwood number for various values of f 0 , 
and  when K  1, M  1,Pr  0.71, Sc  0.94, n  0,   0.75,
  1,   0.6
95
Comparison of the values of C fx Re1/x 2 and Nux / Re1/x 2 for
different values of Pr in absence of concentration  , Soret
S r and Dufour D f effects when   1 and   1
117
Comparison of the values of C fx Re1/x 2 and Nux / Re1/x 2 for
different values of Pr in absence of concentration  , Soret
S r and Dufour effects D f when   1 and   1
117
Values of C fx Re1/x 2 , Nux / Re1/x 2 and Shx / Re1/x 2 for different
values of  and MD when K  1, Pr  0.71, Sc  0.94, n  0,
  1, Sr  0.2, D f  0.2,   1,   0.6 for assisting    0 
and opposing    0  flow
7.4
118
Values of C fx Re1/x 2 , Nux / Re1/x 2 and Shx / Re1/x 2 for different
values of  and S r when K  1, MD  1.5, Pr  0.71, Sc  0.94,
n  0.5,   1, D f  0.2,   1,   0.6 for assisting    0 
and opposing    0  flow
7.5
119
Values of C fx Re1/x 2 , Nux / Re1/x 2 and Shx / Re1/x 2 for different
values of  and D f when K  1, MD  1.5, Pr  0.71, Sc  0.94,
n  0.5,   1, Sr  0.2,   1,   0.6 for assisting    0  and
opposing    0  flow
120
xii
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
4.1
Physical model and coordinate system
4.2
The velocity profiles for various values of  when K  1, M  1,
n  0.5 (a) suction  f 0  1 and (b) injection  f 0  1
4.3
41
50
The microrotation profiles for various values of  when
K  1, M  1 , n  0.5 (a) suction  f 0  1 and (b) injection
f
4.4
0
 1
51
The temperature profiles for various values of  when K  1,
M  1, Pr  0.71, Sc  0.94, n  0.5 (a) suction  f 0  1 and
(b) injection  f 0  1
4.5
The concentration profiles for various values of  when
K  1, M  1, Pr  0.71, Sc  0.94,  0.2, n  0.5 (a) suction
f
4.6
0
 1 and (b) injection  f 0  1
53
The velocity profiles for various values of  when K  1,
M  1, n  0 (a) suction  f 0  1 and (b) injection  f 0  1
4.7
52
54
The microrotation profiles for various values of  when
K  1, M  1, n  0 (a) suction  f 0  1 and (b) injection
f
4.8
0
 1
55
The temperature profiles for various values of  when
K  1, M  1, Pr  0.71, Sc  0.94, n  0 (a) suction  f 0  1
and (b) injection  f 0  1
56
xiii
4.9
The concentration profiles for various values of  when
K  1, M  1, Pr  0.71, Sc  0.94,  0.2, n  0 (a) suction
f
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
5.1
0
 1 and (b) injection  f 0  1
57
The velocity profiles for various values of f 0 when K  1,
M  1, n  0,   0.4
58
The microrotation profiles for various values of f 0 when
K  1, M  1, n  0,   0.4
58
The temperature profiles for various values of f 0 when
K  1, M  1, Pr  0.71, Sc  0.94, n  0,   0.4
59
The concentration profiles for various values of f 0 when
K  1, M  1, Pr  0.71, Sc  0.94,  0.2, n  0,  0.4
59
The velocity profiles for various values of K for suction
or injection when M  1, n  0,   0.6
60
The microrotation profiles for various values of K for
suction or injection when M  1, n  0,   0.6
60
The concentration profiles for various values of  for
suction or injection when K  1, M  1, Pr  0.71,
Sc  0.94, n  0,   0.6
61
The velocity profiles for various values of M for suction
or injection when K  1, n  0,   0.6
61
The microrotation profiles for various values of M for suction
suction or injection when K  1, n  0,   0.6
62
Variation of the skin friction with  for different values of
suction or injection when K  1, M  1, n  0.5
62
Variation of the Nusselt number with  for different values of
suction or injection when K  1, M  1, Pr  0.71, Sc  0.94,
n  0.5
63
Variation of the Sherwood number with  for different values
of suction or injection when K  1, M  1, Pr  0.71, Sc  0.94,
  0.2, n  0.5
63
Physical model and coordinate system
65
xiv
5.2
The velocity profiles for various values of  when K  1, M  1,
73
n  0,   0.2,   0.8
5.3
The microrotation profiles for various values of  when K  1,
M  1, n  0,   0.2,   0.8
73
The temperature profiles for various values of  when K  1,
M  1, Pr  0.71, Sc  0.94, n  0,   0.2   0.8
74
The concentration profiles for various values of  when K  1,
M  1, Pr  0.71, Sc  0.94,  0.2, n  0,  0.2 ,   0.8
74
The velocity profiles for various values of  when K  1,
M  1, n  0,   0.75,   0.8
75
The microrotation profiles for various values of  when K  1,
M  1, n  0,   0.75,   0.8
75
The temperature profiles for various values of  when K  1,
M  1, Pr  0.71, Sc  0.94, n  0,   0.75,   0.8
76
The concentration profiles for various values of  when K  1,
M  1, Pr  0.71, Sc  0.94,  0.2, n  0,   0.75,   0.8
76
5.4
5.5
5.6
5.7
5.8
5.9
5.10
The velocity profiles for various values of  when K  1, M  1,
77
  0.75,   0.2 for (a) n  0.5 and (b) n  0
5.11
The microrotation profiles for various values of  when K  1,
M  1,   0.75,   0.2 for (a) n  0.5 and (b) n  0
78
The temperature profiles for various values of  when K  1,
M  1, Pr  0.71, Sc  0.94,   0.75,   0.2 for (a) n  0.5
and (b) n  0
79
The concentration profiles for various values of  when K  1,
M  1, Pr  0.71, Sc  0.94,  0.2,   0.75,  0.2
for (a) n  0.5 and (b) n  0
80
The concentration profiles for various values of  when K  1,
M  1, Pr  0.71, Sc  0.94, n  0,   0.75,   0.2,  0.8
81
5.12
5.13
5.14
5.15
Variation of the reduced Sherwood number with  for different
values of  when K  1, M  1, Pr  0.71, Sc  0.94, n  0,
  0.75,   0.2
81
xv
5.16
Variation of the reduced skin friction with  for different values
of  when K  1, M  1, n  0,   0.2
82
5.17
Variation of the reduced Nusselt number with  for different
values of  when K  1, M  1, Pr  0.71, Sc  0.94,  0.2,
n  0,   0.2
82
Variation of the reduced Sherwood number with  for different
values of  when K  1, M  1, Pr  0.71, Sc  0.94, n  0,
  0.2
83
Variation of the reduced skin friction with  for different
values of  when K  1, M  1, n  0,   0.75
83
Variation of the reduced Nusselt number with  for different
values of  when K  1, M  1, Pr  0.71, Sc  0.94, n  0,
  0.75
84
5.18
5.19
5.20
5.21
Variation of the reduced Sherwood number with  for different
values of  when K  1, M  1, Pr  0.71, Sc  0.94,  0.2,
n  0,   0.75
84
6.1
Physical model and coordinate system
87
6.2
The velocity profiles for various values of  when K  1, M  1,
Pr  0.71, Sc  0.94, n  0, f0  1,   0.75,   0.2,   1,
96
  0.8
6.3
The microrotation profiles for various values of  when
K  1, M  1, Pr  0.71, Sc  0.94, n  0, f0  1,   0.75,
  0.2,   1,   0.8
96
6.4
The temperature profiles for various values of  when K  1,
M  1, Pr  0.71, Sc  0.94, n  0, f0  1,   0.75,  0.2,   1,
97
  0.8
6.5
The concentration profiles for various values of  when
K  1, M  1, Pr  0.71, Sc  0.94, n  0, f0  1,   0.75,
  0.2,   1,   0.8
97
6.6
The velocity profiles for various values of  when K  1, M  1,
98
Pr  0.71, Sc  0.94, n  0, f0  1,   0.2,   1,   0.8
6.7
The microrotation profiles for various values of  when K  1,
xvi
6.8
6.9
6.10
M  1, Pr  0.71, Sc  0.94, n  0, f0  1,   0.2,   1,   0.8
98
The temperature profiles for various values of  when K  1,
M  1, Pr  0.71, Sc  0.94, n  0, f0  1,   0.2,   1,   0.8
99
The concentration profiles for various values of  when K  1,
M  1, Pr  0.71, Sc  0.94, n  0, f0  1,   0.2,   1,   0.8
99
The velocity profiles for various values of M when K  1,
Pr  0.71, Sc  0.94, n  0, f0  1,   0.75,   0.2,   1,
  0.8
100
6.11
The microrotation profiles for various values of M when K  1,
Pr  0.71, Sc  0.94, n  0, f0  1,   0.75,   0.2,   1,
100
  0.8
6.12
The temperature profiles for various values of M when K  1,
Pr  0.71, Sc  0.94, n  0, f0  1,   0.75,   0.2,   1,
  0.8
101
6.13
The concentration profiles for various values of M when K  1,
Pr  0.71, Sc  0.94, n  0, f0  1,   0.75,   0.2,   1,
101
  0.8
6.14
The velocity profiles for various values of K when M  1,
Pr  0.71, Sc  0.94, n  0, f0  1,   0.75,   0.2,   1,
  0.8
102
6.15
The microrotation profiles for various values of K when M  1,
Pr  0.71, Sc  0.94, n  0, f0  1,   0.75,   0.2,   1,
102
  0.8
6.16
The temperature profiles for various values of K when M  1,
Pr  0.71, Sc  0.94, n  0, f0  1,   0.75,   0.2,   1,
  0.8
103
6.17
The concentration profiles for various values of K when M  1,
Pr  0.71, Sc  0.94, n  0, f0  1,   0.75,   0.2,   1,
103
  0.8
6.18
Variation of the reduced skin friction with  for different values
of  and f 0 when K  1, M  1,Pr  0.71, Sc  0.94, n  0,   0.2,
104
  1,  0.6
xvii
6.19
Variation of the reduced Nusselt number with  for different
values of  and f 0 when K  1, M  1, Pr  0.71, Sc  0.94, n  0,
104
  0.2,   1,   0.6
6.20
Variation of the reduced Sherwood number with  for different
values of  and f 0 when K  1, M  1,Pr  0.71, Sc  0.94, n  0,
105
  0.2,   1,   0.6
6.21
Variation of the reduced skin friction with  for different values
of  and f 0 when K  1, M  1, Pr  0.71, Sc  0.94, n  0,
105
  0.75,   1,   0.6
6.22
Variation of the reduced Nusselt number with  for different
values of  and f 0 when K  1, M  1, Pr  0.71, Sc  0.94,
n  0,   0.75,   1,   0.6
106
6.23
Variation of the reduced Sherwood number with  for different
values of  and f 0 when K  1, M  1,Pr  0.71, Sc  0.94, n  0,
106
  0.75,   1,  0.6
7.1
Physical model and coordinate system
7.2
The velocity profiles for various values of  when K  1,
MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1, Sr  0.2, Df  0.2,
  1,   0.2
7.3
122
The velocity profiles for various values of S r when K  1,
MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1, Df  0.2,   1,
  0.8
7.7
122
The concentration profiles for various values of  when K  1,
MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1, Sr  0.2, Df  0.2,
  1,   0.2
7.6
121
The temperature profiles for various values of  when K  1,
MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1, Sr  0.2, Df  0.2,
  1,   0.2
7.5
121
The microrotation profiles for various values of  when K  1,
MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1, Sr  0.2, Df  0.2,
  1,   0.2
7.4
108
The microrotation profiles for various values of S r when K  1,
123
xviii
MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1, Df  0.2,   1,
7.8
7.9
7.10
  0.8
123
The temperature profiles for various values of S r when K  1,
MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1, Df  0.2,   1,
  0.8
124
The concentration profiles for various values of S r when K  1,
MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1, Df  0.2,   1,
  0.8
124
The velocity profiles for various values of D f when K  1,
MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1, Sr  0.2,   1,
  0.8
7.11
The microrotation profiles for various values of D f when K  1,
MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1, Sr  0.2,   1,
  0.8
7.12
7.15
126
The concentration profiles for various values of D f when K  1,
MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1, Sr  0.2,   1,
  0.8
7.14
125
The temperature profiles for various values of D f when K  1,
MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1, Sr  0.2,   1,
  0.8
7.13
125
126
Variation of the skin friction with  for different values of MD
when K  1, M  1, Pr  0.71, Sc  0.94, n  0.5,   1, Sr  0.2,
Df  0.2,   1,   0.6
127
Variation of the Nusselt number with  for different values of
MD when K  1, M  1, Pr  0.71, Sc  0.94, n  0.5,   1,
Sr  0.2, Df  0.2,   1,   0.6
127
7.16
Variation of the Sherwood number with  for different values of
MD when K  1, M  1, Pr  0.71, Sc  0.94, n  0.5,   1,
Sr  0.2, Df  0.2,   1,   0.6
128
7.17
Variation of the skin friction with  for different values of S r
when K  1, MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1,
Df  0.2,   1,   0.6
128
xix
7.18
Variation of the Nusselt number with  for different values of
S r when K  1, MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1,
Df  0.2,   1,   0.6
7.19
7.20
7.21
Variation of the Sherwood number with  for different values
of S r when K  1, MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1,
Df  0.2,   1,   0.6
129
Variation of the skin friction with  for different values of D f
when K  1, MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1,
Sr  0.2,   1,   0.6
130
Variation of the Nusselt number with  for different values of
D f when K  1, MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1,
Sr  0.2,   1,   0.6
7.22
129
130
Variation of the Sherwood number with  for different values of
D f when K  1, MD  1.5, Pr  0.71, Sc  0.94, n  0.5,   1,
Sr  0.2,   1,   0.6
131
xx
LIST OF SYMBOLS
Roman Letters
a, b
-
positive constants
b*
-
induced magnetic field
B
-
magnetic field vector
B0
-
externally imposed magnetic strength in the y  direction
cs
-
concentration susceptibility
C
-
fluid concentration
C
-
nondimensional fluid concentration
C
-
external concentration
Cw
-
surface concentration
Cp
-
specific heat at constant pressure
C fx
-
skin friction coefficient
C
-
concentration difference
Da 1 -
inverse Darcy number
Df
-
Dufour number
Dm
-
mass diffusivity
E
-
electric field vector
f
-
non-dimensional velocity
f0
-
suction or injection parameter
F
-
body force
Fx
-
scalar force in x  component
g
-
magnitude of the acceleration due to gravity
Gc
-
concentration Grashof number
xxi
Gr
-
thermal Grashof number
h
-
non-dimensional microrotation
j
-
microinertia density
J
-
electric current density vector
k
-
thermal conductivity
k1
-
permeability of porous medium
k1*
-
vortex viscosity
kT
-
thermal diffusion ratio
K
-
micropolar material parameter
L
-
characteristic length
L1
-
slip length
mw
-
non-dimensional mass flux from the surface of the wall
M
-
magnetic parameter
MD
-
effective Darcy number
n
-
ratio of the microrotation vector component to the fluid skin
friction at the wall
N
-
component of the microrotation vector normal to x  y plane
N
-
non-dimensional component of the microrotation vector
normal to x  y plane
Nu x
-
Nusselt number
O
-
order of magnitude
p
-
pressure
p
-
non-dimensional pressure
pd
-
dynamic pressure
ph
-
hydrostatic pressure
Pr
-
Prandtl number
qw
-
non-dimensional heat flux from the surface of the wall
Re
-
Reynolds number
Sc
-
Schmidt number
Shx
-
Sherwood number
xxii
Sr
-
Soret number
t
-
time
t
-
non-dimensional time
T
-
fluid temperature
T
-
non-dimensional fluid temperature
Tm
-
mean fluid temperature
Tr
-
reference temperature
Tw
-
surface temperature
T
-
external temperature
T
-
temperature difference
ue ( x ) -
dimensional external velocity
ue ( x) -
non-dimensional external velocity
uw ( x ) -
dimensional velocity along the sheet
uw ( x ) -
non-dimensional velocity along the sheet
u,v
-
velocity components along x , y axes
u, v
-
non-dimensional velocity components along x and y
U
-
reference velocity
V
-
dimensional velocity vector
VT
-
dimensional thermophoretic velocity
v0
-
velocity suction or injection
x, y
-
cartesian coordinates along the plate and normal to it,
respectively
x, y
-
non-dimensional cartesian coordinates along the wall and
normal to it, respectively

-
gradient
Greek Letters

-
thermal diffusivity

-
coefficient of thermal expansion
xxiii
*
-
coefficient of concentration expansion

-
partial derivative

-
spin gradient viscosity
 , ,  -
transformed coordinate

-
slip parameter
*
-
boundary layer thickness

-
non-dimensionl temperature

-
thermal conductivity
*
-
thermophoretic coefficient

-
dynamic viscosity
m
-
magnetic permeability

-
kinematic viscosity

-
density

-
fluid density in the ambient medium

-
shrinking parameter
*
-
convergence tolerance

-
non-dimensionl concentration

-
stream function

-
thermophoretic parameter
w
-
non-dimensional wall shear stress or skin friction
*
-
concentration buoyancy parameter

-
mixed convection parameter

-
electrical conductivity

-
velocity ratio parameter

-
buoyancy ratio
Subscripts
,e
-
far field or free stream condition
w
-
wall condition
d
-
dynamic pressure
xxiv
p
-
constant pressure condition
Superscripts
k
-
number of iteration
'
-
differentiation with respect to 
xxv
LIST OF APPENDICES
APPENDIX
TITLE
A
The Keller-Box Method
B
FORTRAN Program for the Problem of the
PAGE
150
Thermophoresis and Suction or Injection
Effects on the Stagnation Point Flow towards a
C
Horizontal Sheet
164
Status of Publications
180