ABSTRACT
This dissertation presents a theoretical study on buoyancy driven flow of
viscous, electrically conducting and incompressible fluid in a vertical microchannel
formed by two infinite vertical parallel plates in the presence of Hall
effect. The problem is further extended to the case when the micro-channel
plates are porous to investigate the effect of continuous suction/injection on
the micro flow. Analytical solutions for energy and momentum equations are
obtained using the method of undetermined coefficient. The influence of each
governing parameter such as Hall current parameter, Hartmann number, suction/
injection parameter, rarefaction parameter, fluid wall interaction parameter
and wall-ambient temperature ratio on flow formation is analysed with
the aid of line graphs and thereafter discussed. Numerical comparison of the
present study with previously published results was established to validate the
accuracy of the current solution. The results reveal that the Hall current parameter
plays a significant role in enhancing the fluid flow, resulting to increase
in fluid velocity and volume flow rate in both primary and secondary direction.
In addition, it is interesting that the volume flow rate is higher in both flow
directions due to increase in buoyancy force by increasing the temperature of
the cooler wall. Also, the effect of the Hall current on velocity and volume flow
could be further magnified by increasing gas rarefaction or increasing cooler
wall temperature. Suction/injection on the other hand has a profound effect
on the boundary layer thickness in which the suction reduces the thermal
boundary layer thickness yielding a decrease in the convective current thereby
decreasing fluid velocity and also the skin friction whereas injection thickens
yielding an increase in fluid velocity.
vii
TABLE OF CONTENTS
lyleaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Certification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
CHAPTER ONE
INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . 2
1.3 Significance of the study . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Aim and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Research Methodology . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 4
CHAPTER TWO
LITERATURE REVIEW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Natural Convection in a Micro-Channel . . . . . . . . . . . . . . 7
2.2 Suction/Injection . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Hall Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
viii
CHAPTER THREE
MATHEMATICAL MODELS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 Hall Effect on MHD natural convection flow in a vertical microchannel.(
first case) . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Dimensionless Analysis . . . . . . . . . . . . . . . . . . . 13
3.1.2 Governing Equations in dimensionless form (first case) . 14
3.2 Fully developed MHD natural convection flow in a vertical microporous-
channel with Hall effects (second case) . . . . . . . . . . 16
3.2.1 Dimensionless Analysis . . . . . . . . . . . . . . . . . . . 17
3.2.2 Dimensionless form of problem (second case) . . . . . . . 17
CHAPTER FOUR
RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1 Hall Effect on MHD natural convection flow in a vertical microchannel
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Fully developed MHD natural convection flow in a vertical microporous-
channel with Hall effects . . . . . . . . . . . . . . . . . . 33
CHAPTER FIVE
SUMMARY, CONCLUSION AND RECOMMENDATION . . . . . 50
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3 Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . 52
ix
CHAPTER ONE
NTRODUCTION
1.1 General Introduction
Fluid flow induced by temperature gradients commonly called natural or free
convection has continued to attract a lot of attention from scientist and researchers
recently, due to its practical applications in many geophysical and
industrial/engineering applications. These include cooling of nuclear reactors,
the boundary layer control in aerodynamics, crystal growth, food processing
and cooling towers, thermal-energy storage system, crude oil extraction, and
cores of nuclear reactors. Convection is also seen to play an important role
in our weather system, oceanic currents, and sea-wind formation. In engineering
applications, convection is commonly visualized in the formation of
microstructures during the cooling of molten metals, and fluid flow around
shrouded heat-dissipation fins, and solar ponds. A very common industrial
application of natural convection is free air cooling without the aid of fans.
Natural convection through microchannels is gaining more interest recently.
This is due to the fact that, as the size of the devices decrease, as in electronic
devices, the amount of heat that needs to be dissipated per unit area increases.
The performance of these devices is directly related to the temperature; therefore,
it is highly desirable to understand the fluid flow and the heat transfer
characteristics of buoyancy-induced microheat exchangers and micropump in
microfluidic and thermal systems.
1
1.2 Statement of the Problem
Analysis for the fully developed natural convection in an open-ended vertical
parallel-plate microchannel with asymmetric wall temperature distributions
was investigated by Chen and Weng (2005) where they assumed the fluid
to be viscous incompressible but electrically non-conducting. This result was
extended by Jha et al (2015) by subjecting the flow under an applied transverse
magnetic field and neglecting the influence of the Hall current. However, if the
fluid considered is an ionized gas with low density, the electrical conductivity of
the fluid becomes tensor if the influence of the electromagnetic fields becomes
very strong yielding an effect known as the Hall current, in a direction normal
to both the electric and magnetic field. This dissertation will therefore fill
the observed vacuum in Jha et al. (2015) and Chen and Weng (2005) by
considering the fluid to be electrically conducting and also incorporating the
influence of the Hall currents by assuming the effect of the transversely applied
magnetic field to be strong. This dissertation will further consider the role of
suction/injection on the hydrodynamic behavior on the micro-fluidic flow.
1.3 Significance of the study
This research work derived and presented an analytical solution for buoyancy
driven flow of conducting fluid in a vertical micro-channel with Hall effect subject
to symmetric/asymmetric wall heating conditions. This research work is
applicable in cooling device in computer microchips, Microelectromechanical
system (MEMS), micro heat exchangers, micropump, microfuel cells, Hall accelerators,
Hall sensors etc. Further, it is hoped that the results obtained will
not only provide useful information for industrial or engineering applications
but also serve as an improvement on the previous studies related to such flows.
2
1.4 Aim and Objectives
The aim of this work is to carry out an analytical study on buoyancy driven flow
of viscous and electrically conducting fluid in a vertical micro-channel/microporous
channel in presence of Hall Effect.
This aim will be achieved through the following objectives: To;
i examine the effect of varying the Hall current parameter on fluid velocity,
skin-friction and volume flow rate on micro-channel/micro-porous
channel flow behavior.
ii investigate the impact of suction/injection on fluid velocity, skin-friction
and volume flow rate on micro-porous channel flow formation.
iii examine the effect of externally applied transverse magnetic field on natural
convection flow in both microchannel and micro-porous channel.
iv investigate fluid flow behavior for different micro-channel dimensions dictated
by vKn
1.5 Research Methodology
To attain the above set objectives, we review existing literatures on natural
convection flow in a micro-channel under different physical situations. Solution
to the governing momentum and energy equations is presented analytically
using the method of undetermined coefficient for both problems. Using
a computer package, MATLAB (R2012b), the results obtained is presented
graphically to investigate the influence of each governing parameter. Numerical
comparison of the result obtained and previously published results is presented
to establish the accuracy of the current solution.
3
1.6 Basic Definitions
1. Compressible and incompressible fluid: A compressible flow is a
flow which the fluid density varies significantly within the flow field.
While in an incompressible flow the density does not vary within he flow
field.
2. Free or Natural Convection: Free or Natural convection is a mechanism
in which fluid motion is generated only by density differences in
fluid cause by temperature gradients. It is a type of heat transport in
which the fluid motion is not generated by an external source (pump,
fan, suction devices).
3. Conducting and non-conducting fluid: Materials that conducts electricity
have free ions (charged particle) thus there is flow of current while
Materials that do not have free ions (charged particle) to help in the flow
of current are known as Non-conducting fluid.
4. Boussinessq approximation: Boussinessq approximation is the assumption
that the fluid flow is considered under little variations of temperature
and density.
5. Skin friction: Is a component of drag, the force resisting the motion
of a solid body through a fluid. It arises from the friction of the fluid
against the skin of the object that is moving through it. Skin friction
follows the drag equation and rises with the square of the velocity. Skin
friction can be reduced by shaping the moving body so that smooth flow
is possible.
6. Injection: Is a force that causes a fluid to be drawn out of an interior
space or to adhere to a surface because of the difference between the
external and internal pressures.
4
7. Suction: Is a force that causes a fluid to be drawn into an interior space
or to adhere to a surface because of the difference between the external
and internal pressures
8. Symmetric/asymmetric heating: This is a type of heating of the
boundary layers in which equal/unequal amount of constant heating is
applied on boundary surfaces of the system.
9. Dimensionless quantity: is a quantity without an associated physical
dimension.
10. Magnetic field: is the magnetic effect of electric currents and magnetic
material.
11. MHD: Implies Magnetohydrodynamics which is the study of the interaction
between electric and magnetic forces in a flow. Since the magnetohydrodynamic
equations combine the full complexity of Maxwell’s
equations with the fluid dynamics equations, it is clear that they will be
extremely difficult to solve in their general form. However, by adopting
some approximations, the general equations are simplified. these are:
Maxwell’s equations:
r E = @B
@t
r B = 0j
r B = 0
K0r E = e
Conservation of mass:
@
@t
+ r (v) = 0
Equation of motion:
Dv
Dt
= ô€€€rp + r2v + + j B
5
Conservation of energy:
De
Dt
+
1
2
Dv2
Dt
= E j +r (KrT)ô€€€
X
s
r (Vsshs)ô€€€r (pv)+
12. Hall Effect: is the production of a voltage difference across an electrical
conductor, transverse to an electric current in the conductor and a
magnetic field perpendicular to the current.
From the generalised Ohms law in which ion slip and and pressure diffusion
effects are neglected,
j = [E + v B ô€€€ (j B)]
where = 1
nee , = nee2=me
since the electrical conductivity in compressible gaseous plasmas are not
not constant, the above expression can be written as follows:
jx =
1 + (! )2 [Ex ô€€€ ! (Ey ô€€€ UB)]
jy =
1 + (! )2 [Ey ô€€€ UB + !Ex]
where B = ! . From the above we observe that as a consequence of
the Hall effect, axial current can flow
13. Microchannel: is a channel with hydraulic diameter below 1mm.
6
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