ABSTRACT
TABLE OF CONTENTS
Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Approval Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Declaration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
Dimensionless Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv
Symbols and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii
CHAPTER ONE
GENERAL INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Governing Equations of MHD . . . . . . . . . . . . . . . 3
1.1.2 Continuity Equation . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Navier-Stokes Equation . . . . . . . . . . . . . . . . . . . 4
1.1.4 MHD Induction Equation . . . . . . . . . . . . . . . . . 5
1.1.5 Energy Equation . . . . . . . . . . . . . . . . . . . . . . 5
1.1.6 Maxwell0s Equations . . . . . . . . . . . . . . . . . . . . 5
1.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . 6
1.3 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 7
viii
1.3.1 General Objective . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Specic Objectives . . . . . . . . . . . . . . . . . . . . . 8
1.4 Signicance of the Study . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Research Methodology . . . . . . . . . . . . . . . . . . . . . . . 10
1.6.1 Laplace Transform . . . . . . . . . . . . . . . . . . . . . 11
1.6.2 Riemann Sum Approximation . . . . . . . . . . . . . . . 12
1.6.3 Implicit Finite Dierence Method . . . . . . . . . . . . . 13
1.6.4 Exact Solution . . . . . . . . . . . . . . . . . . . . . . . 15
1.7 Some Basic Denitions . . . . . . . . . . . . . . . . . . . . . . . 15
1.7.1 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7.2 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7.4 Mass and Volumetric Flow Rate . . . . . . . . . . . . . . 17
1.7.5 Compressible uid/Incompressible uid . . . . . . . . . . 17
1.7.6 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7.7 Boundary Layer Concept in Fluid Dynamics . . . . . . . 18
1.7.8 Skin-friction . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7.9 Suction/Injection . . . . . . . . . . . . . . . . . . . . . . 19
1.7.10 Couette Flow . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7.11 Current Density . . . . . . . . . . . . . . . . . . . . . . . 20
1.8 Dissertation Layout . . . . . . . . . . . . . . . . . . . . . . . . . 20
CHAPTER TWO
LITERATURE REVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Macro-channel Congurations: MHD Free Convective Flow in
Vertical Parallel Plate Channel . . . . . . . . . . . . . . . . . . 22
2.3 Suction-Injection Combination (SIC) on MHD Transient Free
Convective Flow in a Vertical Channel . . . . . . . . . . . . . . 27
ix
2.4 MHD Couette Flow in a Vertical Channel . . . . . . . . . . . . 29
2.5 Micro-channel Congurations: MHD Free-Convective Flow in
Annular Micro-channel . . . . . . . . . . . . . . . . . . . . . . . 32
CHAPTER THREE
MHD FREE CONVECTION FLOW IN VERTICAL
CHANNEL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Role of Induced Magnetic Field on Transient Natural Convection
ow in a vertical channel: The Riemann-sum Approximation
Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.2 Mathematical Analysis . . . . . . . . . . . . . . . . . . . 37
3.1.3 Skin Friction and Current density . . . . . . . . . . . . . 40
3.1.4 Validation of the Method: . . . . . . . . . . . . . . . . . 41
3.1.5 Results and Discussion . . . . . . . . . . . . . . . . . . . 43
3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Computational treatment of MHD transient natural convection
ow in a vertical channel due to symmetric heating in presence
of induced magnetic eld . . . . . . . . . . . . . . . . . . . . . . 58
3.3.1 Mathematical Analysis . . . . . . . . . . . . . . . . . . . 58
3.3.2 Skin Friction and Current Density . . . . . . . . . . . . . 61
3.3.3 Steady State Solution . . . . . . . . . . . . . . . . . . . . 62
3.3.4 Implicit Finite Dierence Method . . . . . . . . . . . . . 62
3.3.5 Results and Discussion . . . . . . . . . . . . . . . . . . . 64
x
CHAPTER FOUR
TIME DEPENDENT MHD FREE CONVECTIVE FLOW IN
VERTICAL CHANNEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1 Eects of suction-injection combination (SIC) on MHD transient
free-convective ow in a vertical channel in presence of
induced magnetic eld . . . . . . . . . . . . . . . . . . . . . . . 77
4.1.1 Mathematical Analysis . . . . . . . . . . . . . . . . . . . 78
4.1.2 Skin-friction and Current density . . . . . . . . . . . . . 81
4.1.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.1.4 Steady State Solution . . . . . . . . . . . . . . . . . . . . 82
4.1.5 Implicit Finite Dierence Method (IFDM) . . . . . . . . 82
4.1.6 Results and Discussion . . . . . . . . . . . . . . . . . . . 83
4.2 Unsteady MHD free-convective Couette ow in presence of Induced
Magnetic Field: Impulsive motion . . . . . . . . . . . . . 98
4.2.1 Mathematical Analysis . . . . . . . . . . . . . . . . . . . 98
4.2.2 Skin-friction . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . 104
4.2.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . 107
CHAPTER FIVE
MHD FREE CONVECTION FLOW IN VERTICAL
MICRO-CHANNEL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.1 MHD NATURAL CONVECTION FLOWIN VERTICAL MICROCONCENTRIC
ANNULI IN PRESENCE OF RADIAL MAGNETIC
FIELD: AN EXACT SOLUTION . . . . . . . . . . . . 120
5.1.1 Mathematical Analysis . . . . . . . . . . . . . . . . . . . 121
5.1.2 Results and Discussion . . . . . . . . . . . . . . . . . . . 124
5.1.3 Concluding Remark . . . . . . . . . . . . . . . . . . . . . 127
xi
5.2 FULLY DEVELOPED MHD NATURAL CONVECTION FLOW
IN A VERTICAL ANNULAR MICRO-CHANNEL: AN EXACT
SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.2.1 Mathematical Analysis . . . . . . . . . . . . . . . . . . . 134
5.2.2 Results and Discussion . . . . . . . . . . . . . . . . . . . 138
5.2.3 Concluding Remark . . . . . . . . . . . . . . . . . . . . . 141
CHAPTER SIX
MHD FREE CONVECTIVE FLOW IN ANNULAR
MICRO-CHANNEL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.1 TRANSIENT MHD FREE CONVECTIVE FLOW IN VERTICAL
MICRO-CONCENTRIC ANNULI . . . . . . . . . . . . . 147
6.1.1 Mathematical Analysis . . . . . . . . . . . . . . . . . . . 147
6.1.2 Transient Volume Flow Rate . . . . . . . . . . . . . . . . 151
6.1.3 Skin-Friction and Rate of Heat Transfer . . . . . . . . . 151
6.1.4 Validation of the Method . . . . . . . . . . . . . . . . . . 152
6.1.5 Results and Discussion . . . . . . . . . . . . . . . . . . . 153
CHAPTER SEVEN
CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . . . . . . 166
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . 167
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
xii
CHAPTER ONE
GENERAL INTRODUCTION
1.1 Introduction
Magnetohydrodynamics (MHD) can be regarded as a combination of uid mechanics
and electromagnetism, that is, the behaviour of electrically conducting
uids in the presence of magnetic and electric elds (Sutton and Sherman,
1965). MHD phenomenon basically, result from mutual interaction eect of
magnetic eld and electrically conducting uid owing across the channel and
these interactions generate a force called the Lorentz force. Examples of such
uids include plasmas (hot ionized gases), liquid metals, and salt water or electrolytes.
The word Magnetohydrodynamics (MHD) is derived from magneto-
meaning magnetic eld, hydro- meaning liquid (uids), and dynamics-meaning
movement (forces and laws of motion (Alfvén, 1942)). In 1832, Faraday had
tried vainly to measure the voltage induced across the river Thames by the
motion of the water in the earth’s magnetic eld (electromagnetic induction).
The principle behind this unsuccessful experiment laid the foundations of modern
MHD ow meters (Faraday, 1832). The rst appearance of a proposal
for applying electromagnetic induction phenomenon in technical devices with
electrically-conducting liquids and gases was seen in the beginning of the 20th
century. The description of MHD ows involves the Navier-Stokes equations
of uid dynamics and the Maxwell’s equations (equations of electrodynamics);
which are mutually coupled through the Lorentz force and Ohm’s law for moving
electrical conductors. Systematic studies of magnetohydrodynamic (MHD)
ows began in the 30s, through the pioneering works of Hartmann (1937) 1,
Hartmann and Lazarus (1937) who studied both theoretically and experimen-
1Hartmann,(1881 – 1951), was a leading Professor at the Technical University of Denmark,
in Copenhagen, where he founded the Laboratoriet for teknisk fysik, now Department of
Applied Physics.
1
tally, the ow of an electrically conducting uid between two parallel plates.
However, these studies on the ow of an electrically conducting incompressible
uid between two parallel horizontal plates lead to the well-known Hartmann
ow and had since then, laid the foundation for MHD investigations. In 1942,
Celebrated Nobel laureate, Swedish electrical engineer and plasma physicist
Hannes Olof Góasta Alfvén 2, had combined classical electromagnetism with
uid mechanics and developed the theory of magnetohydrodynamics and was
the rst to introduce the term Magnetohydrodynamics (MHD).
The interests of many researches have been acknowledged in the study of behaviours
of unsteady laminar free convection ow for some decade ago. However,
this area continue to draw much attention due to its vast scientic and
technological applications, such as in the early stage of melting and in transient
heating of insulating air gap by heat input at the start-up of furnace, solar
heating, ventilating passive systems and electric equipment in construction of
vertical circuit boards which can be modelled by parallel heated plates with
upward ow in the intervening space as reported by Jha et al (2011). Also,
unsteady laminar free convection is used in energy storage system, combustion
in underground reservoir for enhancement of oil recovery, storage of grain,
food and vegetables, industrial and agricultural water distribution as identi-
ed by Singh et al (2011). It is generally accepted that prior to this decade,
heat transfer characterised by laminar free-convection were studied with deepseated
scientic and engineering curiosity. Hence the literature continues to
grow. This is evident when one realizes that the Industrial Revolution has been
largely based on man’s increasing ability to convert heat to useful work, and
this inevitably involved the ability to design systems to transfer a predictable
amount of heat per unit of time. The interest in this area stems largely to its
important applications, for instance, the requirement of obtaining high rates
of heat extraction from reactors with irreversible temperature drop as little as
2Winner of the 1970 Nobel Prize in Physics.
2
possible in nuclear engineering, the need for eective blade-cooling systems in
gas turbines. However, combined heat and mass transfer problems are important
in many processes and have therefore received a considerable amount of
attention. In many mass transfer processes, heat transfer considerations arise
due to the very nature of the process. In processes such as drying, evaporation
at the surface of a water body, energy transfer in a wet cooling tower and the
ow in a desert cooler, heat and mass transfer occur simultaneously.
1.1.1 Governing Equations of MHD
The set of equations describing MHD are set of equations comprising the
Navier-Stokes equations of uid dynamics and the Maxwell’s equations of electromagnetism.
The solutions of these dierential equations are obtain simultaneously,
either analytically or numerically.
1.1.2 Continuity Equation
Continuity Equation or Conservation of mass is a basic tool in the study of
uid ow. It expresses the fact that, in a uid particle, per unit volume, V
, the total sum of all masses, m, owing in and out per unit time t, must be
equal to the change in mass due to change in density, , per unit time (Dalton,
2009). For unsteady ows of a general uid, this yields,
@
@t
+ r (V ) = 0 (1.1.1)
which is the mass continuity equation. However, the term r(V ) in (eq.1.1.1)
is called the divergence of (V ) and it is written in the case of 3D-uid ow
as;
r (V ) =
@
@x
(u) +
@
@y
(v) +
@
@z
(w) (1.1.2)
3
For a constant in the case of incompressible uid ow, which is our concern
in this study. The mass continuity equation reduces ( @
@t = 0) to;
r V = 0 (1.1.3)
and for V = (u; v;w) is the velocity vector having components u in the x-
direction, v in the y-direction and w in the z-direction, therefore, the continuity
equation is given as
@u
@x
+
@v
@y
+
@w
@z
= 0 (1.1.4)
In the dissertation, we have considered one dimensional ow so that the continuity
equation is
@u
@x
= 0 (1.1.5)
1.1.3 Navier-Stokes Equation
Navier-Stokes equations described the motion of uid substances. They arise
from applying Newton0s second law to uid motion, assuming that the uid
stress is the sum of a diusing viscous term (proportional to the gradient of
velocity) and a pressure term. They are named after Claude-Louis Navier and
George Gabriel Stokes. Their Solution is called a velocity eld or ow eld,
which is a description of the velocity of the uid at a given point in space and
time is given by.
@V
@t
+ (V r)V + rp = r2V + g + J B (1.1.6)
In the absence of gravity and considering incompressibility (i.e g = 0 and
r V = 0, the Navier- Stokes Equation is
@V
@t
+ (V r)V + rp = r2V (1.1.7)
which is the normal Navier-Stokes equation for incompressible uid.
4
1.1.4 MHD Induction Equation
The MHD induction equation is given as (Davidson, 2001)
@B
@t
= r (V B)
1
e
r
r B
r
B
e
(r B) B
(1.1.8)
or
@B
@t
= r (V B) + r2B (1.1.9)
where is the magnetic diusivity which is given as 1
e
1.1.5 Energy Equation
The energy equation is written as (Davidson, 2001)
cp
@T
@t
+ (qr)T
= r2T + +
J2
(1.1.10)
1.1.6 Maxwell0s Equations
Maxwell0s equations are a set of partial dierential equations that, together
with the Lorentz force law, form the foundation of classical electrodynamics,
classical optics, and electric circuits. These elds in turn underlie modern
electrical and communications technologies. These equations describe the behaviour
of electric E and magnetic B elds and are named after a Scottish
mathematical physicist, James Clerk Maxwell (Fleisch, 2008). Therefore, the
simplied form of these equations are;
Faraday0s Law r E =
@B
@t
(1.1.11)
Ampere0s CircuitalLaw r B = eJ (1.1.12)
Solenoidal Constrant r B = 0 (1.1.13)
Charge Conservation r J = 0 (1.1.14)
5
Ohm0s Law J = [E + V B] (1.1.15)
Lorentz force F = J B (1.1.16)
Note that total magnetic eld B is related to the magnetic eld intensity (or
magnetic eld strength, H) as B = 0H if the magnetization is proportional to
the magnetic eld. Also, it is assumed accurate for liquid metals and gases in
most MHD problems, although it is not true in general for all types of ows.
When E 6= 0, the polarization eects are non-negligible and the magnetic
Reynold’s number is large, thereby considering the induced magnetic eld.
These are some of the assumptions used to reduce the Maxwell’s equations to
an MHD approximation form.
1.2 Statement of the Problem
In recent years studies on macro and micro channels have been on the increase
due to their wider application in many technological and engineering processes.
Also, the ow analysis of MHD natural convection in both macro- and microchannels
has been an important area of investigation due to its practical application
in industry. This conguration exists in several practical systems such
as electronic equipment, furnace and heat exchangers (Jha, 1998). Considering
this enormous increase in practical applications as enunciated in Singh et al.
(2010) and Weng and Chen (2009), it is imperative that this present study
is undertaken. Therefore, the need in this study, to investigate the ow behaviour
of an electrically conducting uid in macro and micro-channel under
dierent geometries and the following problems are formulated;
1. Role of induced magnetic eld on transient natural convection ow in a
vertical channel: The Riemann-sum Approximation Approach3
3Int. J. of Applied Mechanics and Engineering, vol. 20, No.1, (2015), pp.65-
85(DOI: 10.1515/ijame-2015-0005) Poland
6
2. Computational treatment of MHD transient natural convection ow in
a vertical channel due to symmetric heating in presence of induced magnetic
eld4
3. Eects of suction-injection combination (SIC) on MHD transient freeconvective
ow in a vertical channel in presence of induced magnetic
eld
4. Unsteady MHD free-convective Couette ow in presence of Induced Magnetic
Field: Impulsive motion
5. MHD natural convection ow in vertical micro-concentric annuli in presence
of radial magnetic eld: An exact solution5
6. Fully Developed MHD Natural Convection ow in a Vertical annular
Microchannel: An exact solution6
7. Transient MHD Free Convective Flow in Vertical Micro-concentric Annuli
7
1.3 Research Objectives
1.3.1 General Objective
The general objective of this dissertation is to study the ow behaviour of an
electrically conducting, viscous and incompressible uid owing in a macrochannel
formed by two vertical parallel plates and also in vertical annular
micro-channel formed by two concentric cylinders with a magnetic eld applied
4Journal of the Physical Society of Japan(JPSJ) 82 (2013) 084401 (IF = 2:087)
5Ain Shams Engineering Journal (2015) Accepted (Elsevier)
6Journal of King Saud University – Science vol. 27, (2015), 253 – 259 (IF =
0:787)
7 J. Nanoengineering and Nanosystems 1 – 12 IMechE 2015 DOI:
10.1177/1740349915578956 (SAGE)UK
7
perpendicular to the ow direction, subjected to dierent boundary conditions
and uid properties. In addition, the inuence of dierent parameters that
govern the ows shall be discussed.
1.3.2 Specic Objectives
The targeted specic objectives are to;
1. investigate the role of Induced Magnetic Field on Transient Natural Convection
Flow in a Vertical Channel:The Riemann Sum Approximation
Approach
2. present a computational treatment of MHD transient natural convection
ow in a vertical channel due to symmetric heating in presence of induced
magnetic eld
3. investigate the eects of suction-injection combination (SIC) on MHD
transient free-convective ow in a vertical channel in presence of induced
magnetic eld
4. present a semi-analytical solution to unsteady MHD free-convective Couette
ow in presence of Induced Magnetic Field: Impulsive motion
5. discuss transient MHD free convective ow in vertical micro-concentric
annuli
6. present a solution of fully developed MHD natural convection ow in a
vertical annular micro-channel: An Exact Solution
7. investigate the inuence of wall surface curvature on steady fully developed
MHD natural convection ow of viscous, incompressible, electrically
conducting uid in micro-concentric annuli in presence of radial magnetic
eld
8
1.4 Signicance of the Study
Natural convection in real uids has been extensively studied due to its widespread
applications in industry and geophysics. Most of the studies on natural convective
heat and mass transfer phenomena under dierent physical geometries
in the presence of a magnetic eld have been limited to the case when the
induced magnetic eld is not taken into consideration, despite its vital role in
many physical situations (Singh et al., 2010). Moreover, the progress in science
and technology in recent years have promoted a rapid development of dierent
microuidic devices in physical, chemical, biological, medical, engineering, and
energy-related elds. Due to the diculty in making precise measurements,
microscale gas ow and heat transfer modeling was emphasized in modern micro
uidic applications, such as microelectrochemical cell transport, microheat
exchanging, and microchip cooling, etc. (Weng and Chen, 2013). Therefore,
the results obtained in the present investigations will no doubt nd its application
in these and similar areas. Most importantly, this study will form
the basis for further mathematical investigations in macro and micro- channel
ows.
1.5 Research Questions
The stated objectives above will be achieved by answering the following questions:
1. which mathematical model(s) and the necessary boundary conditions can
be formulated for Macro- and Micro-channel problems considered for the
present study?
2. what shall be the role of Induced Magnetic Field on MHD transient
natural convection ow in a vertical channel due to asymmetric and
9
symmetric heating of the channel walls?
3. what are the eects of suction/injection combination (SIC) on MHD
transient free-convective ow in a vertical channel in presence of induced
magnetic eld?
4. is the exact solution of fully developed MHD natural convection ow in
a vertical annular micro-channel possible?
5. what are the eects of wall surface curvature on MHD natural convective
ow in micro-concentric annuli in presence of radial magnetic eld ?
1.6 Research Methodology
The methodology to be employ in this investigation involves;
1. formulation and modication of existing mathematical models and boundary
conditions (Singh et al. (2010), Jha and Apere (2010) and Weng and
Chen (2009)).
2. obtain analytical solutions through Laplace transform technique where
possible, otherwise we use Riemann sum approximation method.
3. validate the obtained results using nite dierence method
4. generate line graphs for the numerical values using MATLAB
5. interpret, discuss and draw conclusions.
The mathematical methods used in obtaining several solutions in mathematics,
mathematical physics and engineering problems as part of the methodology
found worthy for the present study are as follows;
10
1.6.1 Laplace Transform
This is an integral transform introduced by Laplace8 during his studies of
probability. The English mathematician and engineer Oliver Heaviside had
extensively used the methods of Laplace transforms in the name of operational
calculus in solving linear dierential equations arising from electrical
networks. The Laplace transform, in its common form, applies to functions
that are causal. This limitation, compared to the Fourier transform, is more
than compensated by the fact that the Laplace transform can be applied to exponentially
growing functions. These two features make the Laplace transform
ideal for time-dependent functions, which are zero for t < 0 and bounded by
an exponentially growing function. The main idea behind the Laplace Transform
is that, it is a technique that stood the test of time that convert system
of dierential equations to a purely algebraic equations. These equations are
solved and their solutions are inverted back to the function solving the initial
value problem. This makes the problem much easier to solve.
Exponential Order
A function f(t) is said to be of exponential order as t ! 1 if there exists a
real number % and positive constants M and T such that
jf(t)j < Me%t 8t 2 T (1.6.1)
Denition
Let f(t) be continuous and single valued function of the real variable t for all
0 < t < 1, and is of exponential order. Then the Laplace transform of f(t) is
8French mathematician Pierre Simon de Laplace (1749 – 1827)
11
dened as a function f(s) denoted by the integral
Ljf(t)j = f(s) =
Z 1
0
f(t)estdt; s > 0 (1.6.2)
over that range of values of s for which the integral exist. Here s is a parameter
which may be real or complex. Obviously, L(f(t)) is a function of s. Thus,
Lj(f(t))j = f(s) (1.6.3)
f(t) = L1[ f(s)] =
1
2
Z 1+i1
1i1
f(s)estds (1.6.4)
where L is the operator which transforms f(t) into f(s), called the Laplace
transform operator, and L1 is the inverse Laplace transform operator.
Theorem
If the causal function f(t) is piecewise-continuous on [0;1] and is of exponential
order, with abscissa of convergence %c,then the Laplace transform F(s) of
f(t) exists, with region of convergence <e(s) > %c in the s domain; that is
Lff(t)g = F(s) =
Z 1
0
estf(t)dt; (1.6.5)
for Re(s) > %c.
1.6.2 Riemann Sum Approximation
Although, numerical inverse Laplace transform is generally an ill posed problem,
it become necessary when Laplace transform cannot be inverted analytically
by manipulating tabled formulas of special cases. There is no universal
method which works well for all problems (Quanrong and Hongbin, 2015).
In this study, we have chosen Riemann-sum approximation method which is
found suitable. Simplied Fourier-series by the Poisson summation formula
is applied by replacing the integral with a series which corresponds to the
12
trapezoidal rule with a specied discretization error. The Fourier integral thus
obtained can be approximated by its Riemann-sum. The Fourier-series method
for the numerical inversion of Laplace transforms might be rst proposed by
(Dubner and Abate, 1968) and to speed up the convergence, Simon et al Simon
et al. (1972) rstly introduced the Euler summation and proposed a model as
follows
f(y; t) =
exp t
t
[
1
2
f(y; ) + <e
XN
k=1
f(y; +
ik
t
)(1)k] (1.6.6)
Where <e refers to the real part of the complex number, i =
p
1 is imaginary
number, N is the number of terms used in the Riemann-sum approximation
and is the real part of the Bromwich contour that is used in inverting Laplace
transforms. The Riemann-sum approximation for the Laplace inversion involves
a single summation for the numerical process. Its accuracy depends
on the value of and the truncation error dictated by N. According to Tzou
(1997), the value of must be selected so that the Bromwich contour encloses
all the branch point. For faster convergence the quantity t = 4:7 gives the
most satisfactory results.
1.6.3 Implicit Finite Dierence Method
Implicit Finite dierence method (IFDM) works by replacing the region over
which the independent variables in the PDE are dened by a nite grid of
points at which the dependent variable is approximated. The partial derivatives
in the PDE at each grid point are approximated based on Taylor’s theorem.
Basically, three steps are involve in obtaining nite dierence solution;
a. Divide the solution into grids of nodes
b. Approximate the given dierential equation by nite dierence equivalence
that relates the solutions to grid points
13
c. Solve the dierence equations subject to the prescribed initial and boundary
conditions
Derivatives in the partial dierential equation are approximated by linear combinations
of function values at the grid points. By Taylor’s series expansion
@u
@t
(yi; tj)
u (yi; tj) u (yi; tj1)
t
+ O
(t)2
(1.6.7)
@h
@t
(yi; tj)
h (yi; tj) h (yi; tj1)
t
+ O
(t)2
(1.6.8)
@
@t
(yi; tj)
(yi; tj) (yi; tj1)
t
+ O
(t)2
(1.6.9)
while the rst and second order space derivatives to be approximated by the
central dierence formula.
@u
@y
(yi; tj)
u (yi+1; tj) u (yi1; tj)
2 (y)
+ O
(y)2
(1.6.10)
@h
@y
(yi; tj)
h (yi+1; tj) h (yi1; tj)
2 (y)
+ O
(y)2
(1.6.11)
@
@y
(yi; tj)
(yi+1; tj) (yi1; tj)
2 (y)
+ O
(y)2
(1.6.12)
@2u
@y2 (yi; tj)
u (yi+1; tj) 2u (yi; tj) + u (yi1; tj)
(y)2 + O
(y)2
(1.6.13)
@2h
@y2 (yi; tj)
h (yi+1; tj) 2h (yi; tj) + h (yi1; tj)
(y)2 + O
(y)2
(1.6.14)
@2
@y2 (yi; tj)
(yi+1; tj) 2 (yi; tj) + (yi1; tj)
(y)2 + O
(y)2
(1.6.15)
Thus, the transport equations at the grid points will be constructed such that
the iterative system will not restrict the time steps, which is reach, and less
expensive compared to some implicit schemes. Customised codes will be implemented
using Fortran and MATLAB software packages to obtain the numerical
14
values, which will be presented in gures (line graphs) and tables to illustrate
physical situations of the problems under study and draw valid conclusions.
1.6.4 Exact Solution
This is a mathematical model that have solution which does not involve any
approximation. The analytical solution of a given problem is referred to as
the exact solution if it satises the dierential equation and the boundary
conditions. This can be veried by substituting the solution function into
the dierential equation and the boundary conditions. Distinction should be
made between an actual real-world problem and the mathematical model that
is an idealized representation of it. The solutions we get are the solutions
of mathematical models, and the degree of applicability of these solutions
to the actual physical problems depends on the accuracy of the model. An
‘approximate’ solution of a realistic model of a physical problem is usually more
accurate than the ‘exact’ solution of a crude mathematical model (Cengel,
2003). Since, the derived equations in this thesis do not necessarily vanish
identically, the solution of these equations proves challenging and therefore,
only solution of two cases are obtained.
1.7 Some Basic Denitions
The macro-channels referred in this dissertation are those channels that function
in the continuum domain, with no velocity slip and temperature jump,
whose ow and heat transfer behaviour can be predicted from the continuum
theory, while micro-channels are those for which this approach failed. Since,
MHD free convection ow of viscous incompressible uids between vertical
innite parallel plates and annular macro-channel and micro-channel is the
15
theme of this dissertation, the ow is subjected to dierent physical situations,
dierent boundary conditions and dierent uid properties. We give
here a brief overviews/denitions of some of the dierent ow situations on
which the dissertation is based
1.7.1 Pressure
This is one of the basic properties of all uids. Pressure is the force exerted
on or by the uid on a unit of surface area. Mathematically expressed;
pressure =
Force
surface area
(1.7.1)
The basic unit of pressure is Pascal (Pa). When a uid exerts a force of 1N
over an area of 1m2, the pressure equals one Pascal, i.e., 1Pa = 1N=m2.
1.7.2 Density
This is another basic uid property. Symbolically represented by and is
dened as mass (m) of a unit of volume (V ). Its basic unit is kg=m3. Mathematically
it is expressed as;
=
m
V
(1.7.2)
1.7.3 Viscosity
This is a measure of the uid’s resistance to ow due to its internal friction.
Viscosity is measured in two ways: dynamic (absolute) and kinematic. These
two parameters are related since the kinematic viscosity may be obtained by
dividing the dynamic viscosity by density. Symbolically =
. Dynamic
(absolute) viscosity () is the measure of the tangential force needed to shear
one parallel plane of uid over another parallel plane of uid. The thicker
16
or more viscous the uid, the larger the area of contact, and the larger the
velocity change between the layers of the uid, the larger the tangential force.
The basic unit is the pascal second (Pas). In a nutshell one can think of
terms like “thickness” or “internal friction” in the case of water which is thin
compared with a honey which is thick.
1.7.4 Mass and Volumetric Flow Rate
These are two types of ow rate, since uid quantity can be expressed as
volume or mass. Volumetric ow rate (V ) is the volume of the uid passing
through a given cross section in a unit of time. The basic unit is m3=s. It is a
function of the uid velocity (u) and the cross section of the channel through
which the uid is owing. While the Mass ow rate (m) is the mass of uid
passing through a given cross section in a unit of time. Its basic unit is kg=s.
1.7.5 Compressible uid/Incompressible uid
This is the uid ow where the volume of the uid particle changes with position.
This implies that density, , will vary throughout the ow. Furthermore,
compressible ows are classied according to the value of the dimensionless
Mach number, M, which is given by
M =
u
c
(1.7.3)
where, u, is the velocity of the uid and, c, is the speed of sound. Whereas the
uid ow where the volume of the given uid particle does not change with
position. This implies that the density, , of the uid is constant everywhere
and this is essentially valid for all liquids (McDonough, 2009).
17
1.7.6 Validation
This is to check on whether the mathematical model is a good representation
for the real life problem being studied. It is also a means of comparing numerical
results with established results from similar physical problem. Physically
results may emerge either from real life or small-scale laboratory experiments.
In either way errors are bound to occur, therefore the failure of the model is
likely to occur or may not capture all the underlying physics. In this situation
the numerical and the physical results may not agree and the user is at liberty
to decide what is ‘close enough’.
1.7.7 Boundary Layer Concept in Fluid Dynamics
In a lecture delivered titled ‘On uid ow with very little friction’ by Ludwig
Prandtl in 1904 at the International Mathematical Congress in Heidelberg,
Germany, explained that the viscosity of a uid plays a role in a thin layer
adjacent to the surface, which he called ‘boundary layer’ or ‘shear layer’. In
this lecture, the understanding of uid ow was signicantly increased. As a
result of this, the d’Alembert’s Paradox, which states that a body placed in a
potential ow does not experience a force which is clearly in conict with every
– day experience was resolved (Experimentally, it is proved that drag force
exists when a body is immersed in a uid which is an apparent contradiction
between theory and experiment). He rst proposed the interactively coupled,
two layer solution which properly models many ow problems. As an object
moves through a uid, or as a uid moves past an object, the molecules of the
uid near the object are disturbed and move around the object. Subsequently,
it could be explained further, why birds, air planes can y (Veldman, 2011).
18
1.7.8 Skin-friction
Skin friction is a component of drag, the force resisting the motion of solid
body through a uid. It arises from the friction of the uid against the skin of
the object that is moving through it. Skin friction arises from the interaction
between the uid and the skin of the body, and is directly related to the area
of the surface of the body that is in contact with the uid. Skin friction follows
the drag equation and rises with the square of the velocity (Jibril, 2012).
1.7.9 Suction/Injection
The simplest active ow control system is surface suction/injection as they
signicantly aect the ow eld. Controlling uid ow within vertical channel
with suction and injection is an interesting exercise as both activities are carried
out simultaneously on opposite channel plates. The inuence of suction
on rate of heat transfer and skin friction is opposite to the inuence of injection
(Ajibade, 2009).
1.7.10 Couette Flow
This is a ow between two parallel plates where one plate is moving while
the other is stationary. It is of two type, the plane Couette ow and the
generalized Couette ow. In the case of plane Couette ow, the pressure
gradient is constant (zero) and the only force on the uid is the force due to
the moving plate. While in the later case the pressure gradient is imposed in
the direction parallel to the plates (for detail see Schlichting (1960)).
19
1.7.11 Current Density
In this dissertation, the ow of electrically conducting uid is considered, in
either vertical parallel plates channel or between two concentric cylinders. The
charges are in continuous random motion with a velocity distribution. Therefore,
from the Amphere’s law (1.1.12), we have
J =
r B
e
(1.7.4)
where J is the total current density in a medium, while for a moving charges,
the current density is given by the sum of products of the charge on each particle
and its velocity eq.(1.1.15). Charged particles executing helical or cylindrical
motion motion, their velocity can be decomposed in a circular motion
plus the motion of the centre of curvature called the guiding-centre velocity
(drift velocity) (Sutton and Sherman, 1965).
1.8 Dissertation Layout
This dissertation is divided into eight chapters. Chapter One deals with the
general introduction and provides the background information on the basic
equations describing the MHD ow behaviour. Chapter Two provides the
available related literature consulted in this study and all acknowledged. In
Chapter Three, we begin the study by looking at the computational treatments
of time-dependent MHD transient natural convection ow in vertical channel
due to asymmetric and symmetric heating of the channel walls in presence of
induced magnetic eld viz; Role of induced magnetic eld on transient natural
convection ow in a vertical channel: The Riemann Sum Approximation
Approach and Computational treatment of MHD transient natural convection
ow in a vertical channel due to symmetric heating in presence of induced
magnetic eld are examined. The Eects of suction-injection combination
20
(SIC) on MHD transient free-convective ow in a vertical channel in presence
of induced magnetic eld and the time dependent MHD Couette ow of an
incompressible, viscous and electrically conducting uid between two vertical
parallel plates were presented in Chapter Four. Chapters Five and Six, discussed
the steady/unsteady fully developed MHD free-convective ow in an
annular micro-channel. The problems considered here includes; Fully developed
MHD natural convection ow in a vertical annular micro-channel: An
exact solution. MHD Natural Convection ow in a vertical Micro-concentric
Annuli in presence of Radial Magnetic Field: An exact solution and Transient
MHD free convective o in vertical micro-concentric annuli. Finally, conclusions
and recommendations are given in Chapter Seven.
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