vii
TABLE OF CONTENTS
CHAPTER
ITEMS
TITLE PAGE
i
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xi
LIST OF SYMBOLS
xv
LIST OF APPENDICES
1
2
PAGE
xviii
INTRODUCTION
1.1
Introduction
1
1.2
Background of Research
2
1.3
Statement of Problem
5
1.4
Objectives and Scope of Research
5
1.5
Significance of Research
6
1.6
Outlines of Thesis
8
LITERATURE REVIEW
2.1
Introduction
10
2.2
Convective Heat Transfer of Nanofluids
10
2.3
Convection over a Continuous Stretching
23
Surface subjected to Suction or Injection
viii
2.4
Convection over a Stretching Surface in the
26
Presence of Magnetic Field
2.4
Convection over a Stretching Surface in the
31
Presence of Suction or Injection and
Magnetic Field
3
4
THE GOVERNING EQUATIONS
3.1
Introduction
34
3.2
Conservation of Mass
34
3.3
Conservation of Momentum
39
3.4
Conservation of Energy
41
3.4
Boundary Layer Scale Analysis
43
NATURAL CONVECTIVE BOUNDARY LAYER
FLOW OF A NANOFLUID PAST A VERTICAL
FLAT PLATE
5
4.1
Introduction
49
4.2
The Governing Equations
49
4.3
Similarity Transformation
51
4.4
Numerical Computation
54
4.5
Results and Discussion
56
4.6
Conclusion
64
FORCED CONVECTIVE BOUNDARY LAYER
FLOW OF A NANOFLUID PAST A
CONTINUOUS STRETCHING SURFACE
5.1
Introduction
66
5.2
The Governing Equations
66
5.3
Similarity Transformation
68
5.4
Results and Discussion
70
5.5
Conclusion
82
ix
6
FORCED CONVECTIVE BOUNDARY LAYER
FLOW OF A NANOFLUID PAST A
CONTINUOUS STRETCHING SURFACE WITH
SUCTION OR INJECTION IN THE PRESENCE
OF MAGNETIC FIELD
7
6.1
Introduction
83
6.2
Problem Formulation
83
6.3
Results and Discussion
86
6.4
Conclusion
99
CONCLUSIONS
7.1
Summary of Research
101
7.2
Suggestions for Future Research
102
REFERENCES
104-109
Appendices A-H
110-163
x
LIST OF TABLES
TABLE NO.
TITLE
PAGE
4.1
The values of reduced Nusselt number, −θ ′(0) and reduced
Sherwood number, −φ ′(0) for various values of Prandtl
number, Pr when Le = 2, Nb = 0.5, Nt = 0.5 and Nr = 0.5
64
4.2
The values of reduced Nusselt number, −θ ′(0) and reduced
Sherwood number, −φ ′(0) for various values of Lewis
number, Le when Pr = 10, Nb = 0.5, Nt = 0.5 and Nr = 0.5
64
5.1
Comparison of results for the reduced Nusselt number,
−θ ′(0) of a regular fluid for various values of Prandtl
number, Pr when m = 1, Nb = Nt = 10-4, Le = 10
71
5.2
Comparison of present results for the reduced Nusselt
number, −θ ′(0) with Khan and Pop (2010)
80
5.3
Comparison of present results for the reduced Sherwood
number, −φ ′(0) with Khan and Pop (2010)
81
6.1
Comparison of results for the reduced Nusselt number,
−θ ′(0) of a regular fluid for various values of Prandtl
number, Pr when m = 1, Nb = Nt = 10-4, Le = 10, d = 0 and
Mn = 0
87
6.2
Results of − f ′′(0) , −θ ′(0) and −φ ′(0) for various values of
suction or injection parameter, d when Pr = 2, Le = 2,
Nb = 0.5, Nt = 0.5, m = 1, and Mn = 0
99
6.3
Results of − f ′′(0) , −θ ′(0) and −φ ′(0) for various values of
magnetic parameter, Mn when Pr = 2, Le = 2, Nb = 0.5,
Nt = 0.5, m = 1, and d = 0
99
xi
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
3.1
Illustration of nanofluid control volume for the derivation
of continuity equation
35
3.2
Illustration of nanofluid control volume for the derivation
of energy equation
41
3.3
Illustration of similarity variables
44
4.1
Physical model and coordinate system for vertical flat
plate
50
4.2
Plots of the velocity f ′(η ) , temperature θ (η ) , and
nanoparticle fraction φ (η ) profiles for the case Pr = 2,
Le = 2, Nr = 0.5, Nb = 0.5, Nt = 0.5.
55
4.3
Plots of dimensionless similarity functions
f (η ), f ′(η ), θ (η ), φ (η ) for the case Pr = 10, Le = 10,
Nr = 0.5, Nb = 0.5, Nt = 0.5
58
4.4
Effect of Prandtl number, Pr on the velocity profiles
58
4.5
Effect of Prandtl number, Pr on the temperature profiles
59
4.6
Effect of Prandtl number, Pr on the nanoparticle fraction
profiles
59
4.7
Effect of Lewis number, Le on the temperature profiles
60
4.8
Effect of Lewis number, Le on the nanoparticle fraction
profiles
60
4.9
Effect of Brownian motion parameter, Nb on the
temperature profiles
61
4.10
Effect of Brownian motion parameter, Nb on the
nanoparticle fraction profiles
61
xii
4.11
Effect of thermophoresis parameter, Nt on the temperature
profiles
62
4.12
Effect of thermophoresis parameter, Nt on the nanoparticle
fraction profiles
62
4.13
Effect of buoyancy ratio parameter, Nr on the temperature
profiles
63
4.14
Effect of buoyancy ratio parameter, Nr on the nanoparticle
fraction profiles
63
5.1
Physical model and coordinate system for stretching
surface
67
5.2
Effect of velocity stretching parameter, m on the temperature
profiles
74
5.3
Effect of velocity stretching parameter, m on the nanoparticle
fraction profiles
74
5.4
Effect of Prandtl number, Pr on the temperature profiles
75
5.5
Effects of Lewis number, Le on the temperature profiles
75
5.6
Effects of Lewis number, Le on the nanoparticle fraction
profiles
76
5.7
Effects of Brownian motion parameter, Nb on the
temperature profiles
76
5.8
Effects of Brownian motion parameter, Nb on the
nanoparticle fraction profiles
77
5.9
Effects of thermophoresis parameter, Nt on the
temperature profiles
77
5.10
Effects of thermophoresis parameter, Nt on the
nanoparticle fraction profiles
78
5.11
Effects of Brownian motion parameter, Nb and
thermophoresis parameter, Nt on the reduced Nusselt
number, −θ ′(0)
78
5.12
Effects of Brownian motion parameter, Nb and
thermophoresis parameter, Nt on the reduced Sherwood
number, −φ ′(0)
79
6.1
Effect of suction or injection parameter, d on the velocity
profiles
90
xiii
6.2
Effect of suction or injection parameter, d on the
temperature profiles
90
6.3
Effect of suction or injection parameter, d on the
nanoparticle fraction profiles
91
6.4
Effect of Brownian motion parameter, Nb on the
temperature in the presence of suction or injection
91
6.5
Effect of Brownian motion parameter, Nb on the
nanoparticle fraction profiles in the presence of suction or
injection
92
6.6
Effect of thermophoresis parameter, Nt on the temperature
profiles in the presence of suction or injection
92
6.7
Effect of thermophoresis parameter, Nt on the nanoparticle
fraction profiles in the presence of suction or injection
93
6.8
Effect of magnetic parameter, Mn on the velocity profiles
93
6.9
Effect of magnetic parameter, Mn on the temperature
profiles
94
6.10
Effect of magnetic parameter, Mn on the nanoparticle
fraction profiles
94
6.11
Effect of Brownian motion parameter, Nb on the
temperature profiles in the presence of magnetic field
95
6.12
Effect of Brownian motion parameter, Nb on the
nanoparticle fraction profiles in the presence of magnetic
field
95
6.13
Effect of thermophoresis parameter, Nt on the temperature
profiles in the presence of magnetic field
96
6.14
Effect of thermophoresis parameter, Nt on the nanoparticle
fraction profiles in the presence of magnetic field
96
6.15
Effect of magnetic parameter, Mn on the reduced Nusselt
number, −θ ′(0) for various values of Brownian motion
parameter, Nb
97
6.16
Effect of magnetic parameter, Mn on the reduced
Sherwood number, −φ ′(0) for various values of Brownian
motion parameter, Nb
97
xiv
6.17
Effect of magnetic parameter, Mn on the reduced Nusselt
number, −θ ′(0) for various values of thermophoresis
parameter, Nt
98
6.18
Effect of magnetic parameter, Mn on the reduced
Sherwood number, −φ ′(0) for various values of
thermophoresis parameter, Nt
98
C.1
The grid for difference approximations
128
D.1
Plots of the velocity f ′(η ) , temperature θ (η ) , and
nanoparticle fraction φ (η ) profiles for the case Pr = 2,
Le = 2, Nb = 0.5, Nt = 0.5, m = 1, d = 0, Mn = 3
149
xv
LIST OF SYMBOLS
Roman Letters
B
-
magnetic field vector
B( x)
-
magnetic field in the y-direction
B0
-
constant
cp
-
nanoparticle specific heat
C
-
nanoparticle volume fraction
Cf
-
skin-friction coefficient
Cw
-
nanoparticle volume fraction at the surface
C∞
-
ambient nanoparticle volume fraction
d
-
suction/injection parameter
DB
-
Brownian diffusion coefficient
DT
-
thermophoretic diffusion coefficient
f
-
dimensionless stream function
F
-
body forces vector
g
-
gravity acceleration vector
hp
-
nanoparticle specific enthalpy
I
-
identity vector
jp
-
nanoparticle mass flux
j p, B
-
mass flux due to Brownian diffusion
jp,T
-
mass flux due to thermophoresis
J
-
electrical current vector
k
-
thermal conductivity
L
-
characteristics length of plate
xvi
Le
-
Lewis number
m
-
velocity stretching parameter
Μ
-
mass
M
-
mass generation rate
Mn
-
magnetic parameter
n
-
unit normal
Nb
-
Brownian motion parameter
Nr
-
bouyancy-ratio parameter
Nt
-
thermophoresis parameter
Nu
-
Nusselt number
Pr
-
Prandtl number
p
-
pressure
q
-
energy flux
qm
-
wall mass flux
qw
-
wall heat flux
Ra x
-
local Rayleigh number
Re x
-
local Reynolds number
S
-
surface
Sh
-
local Sherwood number
t
-
time
T
-
fluid temperature
Tw
-
temperature at the surface
T∞
-
ambient temperature
U0
-
constant
u, v
-
velocity components along x- and y-axes
uw
-
velocity of the stretching sheet
vw
-
wall mass suction velocity
V
-
arbitrary control volume
x, y
-
Cartesian coordinates measured along the wall and normal to it,
respectively
xvii
Greek Letters
α
-
thermal diffusivity
β
-
volumetric expansion coefficient of the nanofluid
δ
-
boundary layer thickness
η
-
similarity variable
θ
-
dimensionless temperature
µ
-
dynamic viscosity
ν
-
kinematic viscosity
ρf
-
fluid density
ρp
-
nanoparticle mass density
( ρc) f -
heat capacity of the fluid
( ρc) p -
effective heat capacity of the nanoparticle material
τ
-
stress vector
τ
-
heat capacity ratio
τw
-
wall shear stress
φ
-
rescaled nanoparticle volume fraction
ψ
-
stream function
σ
-
electric conductivity of the fluid
Subscripts
w
-
condition at the surface
∞
-
condition at ambient medium
xviii
LIST OF APPENDICES
APPENDIX
TITLE
PAGE
A
Mathematical Formulation of the Natural Convective
Boundary Layer Flow of a Nanofluid Past a Vertical
Flat Plate
110
B
Mathematical Formulation of the Forced Convective
Boundary Layer Flow of a Nanofluid Past a Continuous
Stretching Surface with Suction or Injection in the
Presence of Magnetic Field
118
C
Numerical Formulation of the Natural Convective
Boundary Layer Flow of a Nanofluid Past a Vertical
Flat Plate
127
D
Numerical Formulation of the Forced Convective
Boundary Layer Flow of a Nanofluid Past a Continuous
Stretching Surface with Suction or Injection in the
Presence of Magnetic Field
138
E
The Matlab® Program for Natural Convective
Boundary Layer Flow of a Nanofluid Past a Vertical
Flat Plate
150
F
The Matlab® Program for Forced Convective Boundary
Layer Flow of a Nanofluid Past a Continuous Stretching
Surface
154
G
The Matlab® Program for Forced Convective Boundary
Layer Flow of a Nanofluid Past a Continuous Stretching
Surface with Suction or Injection in the Presence of
Magnetic Field
158
H
Academic Activities
163