vii
TABLE OF CONTENTS
CHAPTER
1
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF FIGURES
xi
LIST OF TABLES
xvii
LIST OF SYMBOLS
xix
LIST OF APPENDICES
xxii
INTRODUCTION
1
1.1
Research Background
1
1.1.1
Peristalsis
1
1.1.2
Heat and Mass Transfer
3
1.1.3
Non-Newtonian fluids
4
1.1.4
Slip Condition
6
1.2
Problem Statement
7
1.3
Objectives of the Research
7
1.4
Scope of the Research
8
1.5
Significance of the Research
8
1.6
Research Methodology
9
1.6.1
Problem Formulation in Laboratory
Frame
9
viii
1.6.2
2
Laboratory Frame into Wave Frame
Transformations
10
1.6.3
Non-dimensionalization
10
1.6.4
Stream Function
10
1.6.5
Linearization
11
1.6.6
Regular Perturbation Method
11
1.7
Dimensionless Parameters
13
1.8
Thesis Outline
14
LITERATURE REVIEW
17
2.1
Introduction
17
2.2
Peristaltic Flow of Viscous Fluid
17
2.3
Peristaltic Flow of Non-Newtonian Fluid with Heat
and Mass Transfer
2.4
2.5
3
22
Peristaltic Flow of Non-Newtonian Fluid with Heat
Transfer and Slip Condition
24
Studies on Peristaltic Flow of Sisko Fluid
27
HEAT AND MASS TRANSFER ON MHD
29
PERISTALTIC FLOW IN A POROUS
ASYMMETRIC CHANNEL WITH SLIP
4
3.1
Introduction
29
3.2
Formulation of the Problem
30
3.3
Governing Equations
32
3.4
Solution of the Problem
45
3.5
Different Wave Forms
46
3.6
Results and Discussion
48
3.7
Conclusions
64
HEAT AND MASS TRANSFER ON
65
PERISTALTIC FLOW OF WALTER’S B FLUID
IN AN ASYMMETRIC CHANNEL
4.1
Introduction
65
4.2
Formulation of the Problem
66
ix
5
4.3
Governing Equations
67
4.4
Solution of the Problem
70
4.4.1
Zeroth Order System
70
4.4.2
First Order System
72
4.5
Results and Discussion
73
4.6
Conclusions
86
HEAT TRANSFER ON MHD PERISTALTIC
87
FLOW OF FOURTH GRADE FLUID IN AN
INCLINED ASYMMETRIC CHANNEL WITH
SLIP
6
5.1
Introduction
87
5.2
Formulation of the Problem
88
5.3
Governing Equations
89
5.4
Solution of the Problem
94
5.4.1
Zeroth Order System
95
5.4.2
First Order System
95
5.5
Results and Discussion
5.6
Conclusions
HEAT TRANSFER ON PERISTALTIC FLOW OF
97
109
110
SISKO FLUID IN AN ASYMMETRIC CHANNEL
WITH SLIP
7
6.1
Introduction
110
6.2
Formulation of the Problem
111
6.3
Governing Equations
111
6.4
Solution of the Problem
114
6.4.1
Zeroth Order System
115
6.4.2
First Order System
115
6.5
Results and Discussion
117
6.6
Conclusions
130
CONCLUSION
131
7.1
131
Summary of Research
x
7.2
Suggestions for Future Research
134
REFERENCES
136
Appendices A-H
146-180
xi
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
1.1
Structure of GIT
2
1.2
Cross section of GIT
2
3.1
Sketch of the physical model
31
3.2
(i) Triangular waves (equation (3.99)) (ii) Square waves
(equation (3.100)) (iii) Trapezoidal waves (equation
(3.101))
48
3.3
Comparison of pressure rise per wavelength p (equation
(3.98)) against  when d  2, a  0.7, b  1.2, K  ,
M  0,   0
52
3.4
Pressure rise p (equation (3.98)) versus  for (i)
different  with d  2, a  0.7, b  1.2,    / 4, K  1,
M  1, (ii) different wave forms when d  1, a  0.5,
b  0.5,   0, K  1,   0, M  1
53
3.5
Pressure gradient dp / dx (equation (3.91)) against x for
different  with fixed d  2, a  0.7, b  0.8,    / 4,
M  1, K  1,   1
54
3.6
Shear stress S xy ( h1 ) (equation (3.92)) against x for different
54
Frictional forces (equation (3.98)) F1 (i) and F 2 (ii) for
different  with fixed d  2, a  0.7, b  1.2,    / 2,
K  1, M  1
55
 with fixed d  1, a  0.5, b  0.5,    / 2, K  1,
M  1,   0.5
3.7
xii
3.8
Velocity profile u (equation (3.93)) against y for different
 when d  1, a  0.5, b  0.5,   4.5, K  1, x  0.5,
M  1, (i)   0 (ii)    / 6
56
3.9
Temperature  (equation (3.94)) against y when d  2,
b  1.2, x  0.5, M  1,   2, Br  4, (i) a  0.7,
   / 6, K  2,   0.03, (ii) a  0.9,    / 4,
  0.01, K  1
57
3.10
Concentration  (equation (3.96)) against y for
d  2, a  0.7, b  1.2,   2, x  0.2,    / 4, Sr  1,
(i)   0.03,   0.03, K  1,   0.03, Sc  1, M  1,
(ii)   0.03,   0.03, Br  2,   0.03, Sr  1, M  1,
(iii)   0.03,   0.03, Br  1,   0.03, K  1, M  1,
(iv)   0.03,   0.03, Br  1, K  1, Sc  1, M  1, (v)
  0.03,   0.03, Br  1,   0.03, K  1, Sc  1, (vi)
Br  1,   0.03, M  1,   0.03, K  1, Sc  1
58
3.11
Streamlines (equation (3.89)) with fixed a  0.5, b  0.5,
d  1,   0,   1.65, M  1, K  1 and sinusoidal wave
form (equation (3.54)) (symmetric channel) for  (a) 0.00,
(b) 0.05, (c) 0.09
59
3.12
Streamlines (equation (3.89)) with fixed a  0.5, b  0.5,
d  1,   0,   1.65, M  1, K  1 and triangular wave
form (equation (3.99)) (symmetric channel) for  (a) 0.00,
(b) 0.03, (c) 0.06
60
3.13
Streamlines (equation (3.89)) with fixed a  0.5, b  0.5,
d  1,   0,   1.70, M  1, K  1 and square wave form
(equation (3.100)) (symmetric channel) for  (a) 0.00,
(b) 0.10, (c) 0.22
61
3.14
Streamlines (equation (3.89)) with fixed a  0.5, b  0.5,
d  1,   0,   1.70, M  1, K  1 and trapezoidal wave
form (equation (3.101)) (symmetric channel) for
 (a) 0.00, (b) 0.10, (c) 0.22
62
3.15
Streamlines (equation (3.89)) with fixed a  0.5, b  0.5,
d  1, M  1,   1.70,    / 2, K  1 and sinusoidal
wave form (equation (3.54)) (asymmetric channel) for
63
xiii
 (a) 0.00, (b) 0.05, (c) 0.09
4.1
Effects of  on (i) S xy ( h ) (ii) S xy ( h ) (equations (4.32) and
1
2
76
(4.48)) against x when a  0.7, b  1.2, d  2,    / 6,
  1,   0.02, Re  1
4.2
Effects of Re on (i) S xy ( h ) (ii) S xy ( h ) (equations (4.32) and
(4.48)) against x when a  0.7, b  1.2, d  2,    / 2,
  1,   0.02,   2
77
4.3
Temperature  (equations (4.34) and (4.50)) for fixed
a  0.5, b  1.2, d  1,    / 2,   1,   0.01,   0.1,
Re  5, x  0.5 (i) Pr = 5 (ii) Er = 4
78
4.4
Temperature  (equations (4.34) and (4.50)) for fixed
a  0.5, b  1.2, d  1,    / 2,   0.01,   0.1, Re  5,
x  0.5, Pr  1, Er  4 (i)   1 (ii)   1
79
4.5
Concentration  (equations (4.35) and (4.51)) for fixed
a  0.5, b  1.2, d  1,   1,    / 2,   0.01,   0.1,
Re  5, x  0.5, Sc  1, Sr  4 (i) Pr  4 (ii) Er  4
80
4.6
Concentration  (equations (4.35) and (4.51)) for fixed
a  0.5, b  1.2, d  1,   1,    / 2,   0.01,   1,
Re  5, x  0.5, Pr  1, Er  4 (i) Sc  1 (ii) Sr  1
81
4.7
Heat transfer coefficients (i) Z h at upper wall (equations
(4.37) and (4.53)) (ii) Z h at lower wall (equations (4.38)
82
1
2
1
2
and (4.54)) for different values of Er with fixed a  0.4,
b  1.2, d  1.5,   0.5,    /12,   0.01,   0.1,
Re  1, Pr  1
4.8
Heat transfer coefficients (i) Z h at upper wall (equations
(4.37) and (4.53)) (ii) Z h at lower wall (equations (4.38)
1
83
2
and (4.54)) for different values of Pr with fixed a  0.4,
b  1.2, d  1.5,   0.5,    /12,   0.01,   1,
Re  1, Er  2
4.9
Heat transfer coefficients (i) Z h at upper wall (equations
(4.37) and (4.53)) (ii) Z h at lower wall (equations (4.38)
1
2
84
xiv
and (4.54)) for different values of  with fixed a  0.4,
b  1.2, d  1.5,   0.5,    /12,   0.01, Pr  1,
Re  1, Er  2
4.10
Heat transfer coefficients (i) Z h at upper wall (equations
(4.37) and (4.53)) (ii) Z h at lower wall (equations (4.38)
1
85
2
and (4.54)) for different values of  with fixed a  0.4,
b  1.2, d  1.5,   0.5,   1,   0.01, Pr  1,
Re  1, Er  1
5.1
Schematic diagram of the problem
89
5.2
Comparison of Pressure rise p for different values of M
with fixed a  0.2, b  0.4, d  0.7,   0.01,    / 6,
   / 4, Fr  2, Re  10,    / 6,   0
100
5.3
Pressure rise p for different values of  with fixed
a  0.2, b  0.4, d  0.7,   0.01,    / 6,    / 4,
Fr  2, Re  10,    / 6, M  2
101
5.4
Pressure gradient dp / dx (equation (5.41)) for different
values of  with fixed a  0.7, b  0.9, d  2,   0.01,
   / 6,    / 4, Fr  2, Re  10,    / 6, M  2,
  0.1
101
5.5
Variation of axial velocity u (equation (5.42)) for different
values of  when a  0.5, b  0.5,   0.1,    / 4,
M  2,   1, x  0.5, (i)   0, d  1, (ii)    / 4,
d  1.5
102
5.6
Frictional forces (i) F1 and (ii) F 2 for different values of
 when a  0.2, b  0.4, d  0.7,   0.01,    / 6,
M  1,    / 4, Fr  2, Re  10,    / 6
103
5.7
Streamlines (equation (5.40)) in symmetric channel (  0)
for different values of  (a) 0.0 (b) 0.5 (c) 1.5 (d) 2 with
fixed a  0.5, b  0.5, d  1,   0.001,    / 4,
M  0.5,   1.88
105
5.8
Streamlines (equation (5.40)) in asymmetric channel
(   / 6) for different values of  (a) 0.0 (b) 0.5 (c) 1.5
106
xv
(d) 2 with fixed a  0.5, b  0.5, d  1,   0.001,
   / 4, M  0.5,   1.88
5.9
Temperature  (equation (5.43)) for fixed b  1.2, d  1,
  1, (i) a  0.9,   0.03,    / 4, M  1,    / 4,
Br  4, x  1, (ii) a  0.7,   0.01,    / 4,    / 4,
Br  2, x  0.2,   0.01, (iii) a  0.7,   0.03, M  1,
   / 4,    / 2, x  1,   0.03, (iv) a  0.7,    / 4,
M  1,    / 4, Br  4, x  1,   0.03, (v) a  0.9,
  0.01, M  1,    / 4, Br  4, x  1,   0.03
107
6.1
Comparison of pressure rise p for different values of the
flow rate  when a  0.7, b  1.2, d  2, n  1, bs  0,
 0
120
6.2
Temperature profiles  (equation (6.31)) for fixed a  0.7,
b  1.2, d  2,   0, x  0.25,   0.02,   0.02,
(i) bs  0.3,   1.2, Br  5 (ii) bs  0.02,   1.5, n  2
121
6.3
Temperature profiles  (equation (6.31)) for fixed a  0.7,
b  1.2, d  2,   0, x  0.25,   0.02,   0.02,
  1.3, Br  5 (i) n  0, (ii) n  2
122
6.4
Temperature profiles  (equation (6.31)) for fixed a  0.7,
b  1.2, d  2,   0, x  0.25, n  2, bs  0.02,   1.5,
Br  4 (i)   0.02, (ii)   0.02
123
6.5
Pressure rise per wavelength p for different values of 
when a  0.7, b  1.2, d  2, n  2, bs  0.02,    / 6
125
6.6
Axial velocity profiles u (equation (6.30)) for different
values of  when a  0.5, b  0.5, d  1, n  2, bs  0.01,
   / 4,   2, x  0.25
126
6.7
Pressure gradient dp / dx (equation (6.28)) for different
values of  when a  0.7, b  1.2, d  2, n  2, bs  0.02,
  0,   1.8
126
6.8
Shear stress profiles (equation (6.29)) at upper wall S xy ( h1 )
127
for different values of  when a  0.7, b  1.2, d  2,
xvi
n  2, bs  0.02,   0,   2
6.9
Streamlines (equation (6.27)) in symmetric channel for
different values of  (a) 0.00 (b) 0.05 (c) 0.10 (d) 0.20
with fixed a  0.5, b  0.5, d  1,   0, n  2, bs  0.02,
  1.6
128
6.10
Streamlines (equation (6.27)) in asymmetric channel for
different values of  (a) 0.00 (b) 0.05 (c) 0.10 (d) 0.20
with fixed a  0.5, b  0.5, d  1,    / 2, n  2,
bs  0.02,   1.6
129
xvii
LIST OF TABLES
TABLE NO.
TITLE
4.1
Comparison of pressure rise per wavelength p for different
values of the flow rate  when a  0.7, b  1.2, d  2,   0,
  0, Re  0
5.1
Maximum pressure rise p (max) for different values of a, 
PAGE
74
104
and  with fixed b  0.4, d  1.5,    / 4, Fr  2,
Re  10,    / 6,    / 6, M  2
5.2
Maximum pressure rise p (max) for different values of  , M
and  with fixed a  0.2, b  0.4, d  1.5, Fr  2, R e  10,
   / 6,    / 6,   0.01
104
5.3
Maximum pressure rise p (max) for different values of d , 
and  with fixed a  0.2, b  0.4,    / 4, Fr  2, Re  10,
   / 6,   0.01, M  2
104
5.4
Heat transfer coefficient Z h1 (equation (5.44)) at different
108
Heat transfer coefficient Z h1 (equation (5.44)) at different
108
cross sections for different values of  with fixed a  0.25,
b  1.2, d  1.5,   0.03,   0.5, Br  2, M  2,    / 4,
   /12
5.5
cross sections for different values of Br with fixed a  0.4,
b  1.2, d  1.5,   0.03,   0.5,   0.03, M  1,
   / 4,    /12
5.6
Heat transfer coefficient Z h1 (equation (5.44)) at different
cross sections for different values of  with fixed a  0.25,
b  1.2, d  1.5,   0.02,   0.5, Br  2, M  2,    / 4,
108
xviii
   /12
6.1
Heat transfer coefficient at upper wall Z h1 (equation (6.32))
at different cross sections for n with fixed a  0.25, b  1.2,
d  1.5,    /12,   0.02,   0.02,   0.5, Br  2,
bs  0.03
124
6.2
Heat transfer coefficient at upper wall Z h1 (equation (6.32))
124
at different cross sections for bs with fixed a  0.25, b  1.2,
d  1.5,    /12,   0.02,   0.02,   0.5, Br  2,
n2
6.3
Heat transfer coefficient at upper wall Z h1 (equation (6.32))
124
at different cross sections for Br with fixed a  0.25,
b  1.2, d  1.5,    /12,   0.02,   0.02,   0.5,
bs  0.03, n  2
6.4
Heat transfer coefficient at upper wall Z h1 (equation (6.32))
125
Heat transfer coefficient at upper wall Z h1 (equation (6.32))
125
at different cross sections for  with fixed a  0.25, b  1.2,
d  1.5,    /12, Br  2,   0.02,   0.5, bs  0.03,
n2
6.5
at different cross sections for  with fixed a  0.25, b  1.2,
d  1.5,    /12, Br  2,   0.02,   0.5, bs  0.03,
n2
xix
LIST OF SYMBOLS
Roman Letters
a
-
Amplitude ratio at upper wall
a1
-
Wave amplitude at upper wall
a2
-
Wave amplitude at lower wall
b
-
Amplitude ratio at lower wall
b
-
Body force per unit volume
B0
-
Uniform applied magnetic field
Br
-
Brinkman number
c
-
Wave speed
C
-
Fluid concentration
C0
-
Concentration at upper wall
C1
-
Concentration at lower wall
d
-
Channel width ratio
d1
-
Upper channel width
d2
-
Lower channel width
D
-
Coefficient of mass diffusivity
D
Dt
-
Substantial derivative
dp
dx
-
Axial pressure gradient
e
-
Rate of strain tensor
Er
-
Eckert number
F
-
Dimensionless time mean flow rate in wave frame
Fr
-
Froude number
g
-
Acceleration due to gravity
xx
H1
-
Shape of upper wall in laboratory frame
H2
-
Shape of lower wall in laboratory frame
h1
-
Shape of upper wall in wave frame
h2
-
Shape of lower wall in wave frame
h1
-
Dimensionless shape of upper wall
h2
-
Dimensionless shape of lower wall
I
-
Identity tensor
k
-
Thermal conductivity
K
-
Dimensionless permeability parameter
KT
-
Thermal diffusion ratio
k0
-
Short memory coefficient
M
-
Hartmann number
P
-
Pressure in laboratory frame
p
-
Pressure in wave frame
p
-
Dimensionless pressure
Pr
-
Prandtl number
Re
-
Reynolds number
S
-
Extra stress tensor
Sr
-
Soret number
Sc
-
Schmidt number
t
-
Time
T
-
Fluid temperature
T
-
Cauchy stress tensor
Tm
-
Mean temperature
T0
-
Temperature at upper wall
T1
-
Temperature at lower wall
U
-
Axial velocity component in laboratory frame
u
-
Axial velocity component in wave frame
u
-
Dimensionless axial velocity component
V
-
Fluid velocity
V
-
Transverse velocity component in laboratory frame
xxi
v
-
Transverse velocity component in wave frame
v
-
Dimensionless transverse velocity component
Z h1
-
Heat transfer coefficient at upper wall
Z h2
-
Heat transfer coefficient at lower wall

-
Channel inclination

-
Velocity slip parameter

-
Electrical conductivity

-
Wave number

-
Phase difference

-
Dimensionless concentration

-
Thermal slip parameter

-
Dimensionless temperature
0
-
Limiting viscosity at small shear rates

-
Viscoelastic parameter

-
Wave length
F1
-
Frictional force at upper wall
F 2
-
Frictional force at lower wall

-
Dynamic viscosity

-
Dimensionless time mean flow rate in laboratory frame

-
Fluid density

-
Concentration slip parameter

-
Magnetic field inclination

-
Stream function

-
Specific heat at constant volume

-
Deborah number
p
-
Pressure rise per wavelength
Greek Letters
xxii
LIST OF APPENDICES
APPENDIX
TITLE
PAGE
A
Solution Coefficients (Chapter 3)
146
B
Derivation of Governing Equations (Chapter 4)
148
C
Solution Coefficients (Chapter 4)
153
D
Derivation of Governing Equations (Chapter 5)
161
E
Solution Coefficients (Chapter 5)
171
F
Derivation of Governing Equations (Chapter 6)
175
G
Solution Coefficients (Chapter 6)
177
H
List of Publications
179