ABSTRACT
A computational study on thermo-solutal convection in Magnetohydrodynamics (MHD) flow
with Dufour effect is the subject of this research work. A fully developed unsteady natural
convection flow of viscous, incompressible and electrically conducting fluid in a
microchannel formed by two vertical plates under the influence of transverse magnetic field
of uniform strength 0 B which is applied in the direction perpendicular to that of the flow is
considered. In this flow formation, temperature is influenced by concentration of the
introduced chemical species, leading to diffusion-thermo effect on the heat and mass transfer.
In an extended study, the flow is subjected to suction of the fluid from one porous plate and
at the same rate fluid is being injected through the other porous plate. Fluid motion is induced
by asymmetric heating of the channel plates taking into consideration velocity slip and
temperature as well as concentration jumps on the boundaries. Heat source parameter as well
as Dufour effects are also taken into cognizance. Partial differential equations that describe
the energy and concentration within the system follow the dual-phase-lag model for heat
conduction. Laplace transform approach is used to solve the resulting governing equations in
Laplace domain while the inversion into time domain is made possible through the Riemannsum
approximation technique. The response of mean chemical concentration, bulk fluid
temperature and mass flux within the channel to different flow parameters are investigated. It
is worth noting that regardless of temperature gradient or heat flux influence on the fluid
flow, the resultant mass flux decreases by increasing the Hartmann number (m) and also, it
increases with an increase in Biot number (Bi) as well as Dufour number (D). In addition, the
mean temperature rises following an increase in Dufour (D) number but decreases with
growing thermal retardation time ( T ). Furthermore, it is observed that heat generation ( )
increases temperature profile, velocity profile and also skin friction.
TABLE OF CONTENTS
Cover Page ……………………………………………………………………………………………………………………….. i
Fly Leaf …………………………………………………………………………………………………………………………….. i
Title Page ………………………………………………………………………………………………………………………… ii
DECLARATION …………………………………………………………………………………………………………………. iii
CERTIFICATION ………………………………………………………………………………………………………………… iv
DEDICATION …………………………………………………………………………………………………………………….. v
ACKNOWLEDGEMENT ………………………………………………………………………………………………………. vi
ABSTRACT ……………………………………………………………………………………………………………………….vii
TABLE OF CONTENT …………………………………………………………………………………………………………. viii
LIST OF TABLES …………………………………………………………………………………………………………………. x
LIST OF FIGURES ………………………………………………………………………………………………………………..xi
LIST OF APPENDICES ………………………………………………………………………………………………………… xiii
NOMENCLATURE AND GREEK LETTERS ……………………………………………………………………………….. xiv
CHAPTER ONE ………………………………………………………………………………………………………………….. 1
INTRODUCTION ………………………………………………………………………………………………………………… 1
1.1 Background of the study …………………………………………………………………………………….. 1
1.2 Statement of the problem ……………………………………………………………………………………. 3
1.3 Significance of the study……………………………………………………………………………………… 4
1.4 Aim and objectives …………………………………………………………………………………………….. 4
1.6 Basic Definitions ………………………………………………………………………………………………… 5
CHAPTER TWO …………………………………………………………………………………………………………………. 8
LITERATURE REVIEW………………………………………………………………………………………………………….. 8
2.1 Natural Convection ……………………………………………………………………………………………. 8
2.2 Magnetohydrodynamics ……………………………………………………………………………………… 9
2.3 Convective Boundary Condition ………………………………………………………………………… 10
2.4 Heat Generation ………………………………………………………………………………………………. 11
2.5 Dual-Phase-Lag ……………………………………………………………………………………………….. 11
2.6 Suction/Injection ……………………………………………………………………………………………… 13
2.7 Dufour effect …………………………………………………………………………………………………….. 13
CHAPTER THREE ……………………………………………………………………………………………………………… 15
MATHEMATICAL MODELS…………………………………………………………………………………………………. 15
3.1 Flow formation and Geometry ……………………………………………………………………………….. 15
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3.2 Dual-Phase-Lag Heat Conduction with Convective Boundary …………………………………….. 17
3.2.1 Dimensionless Analysis …………………………………………………………………………………….. 18
3.3 Dual-Phase-Lag Heat Conduction in a Vertical Porous Channel ………………………………… 20
3.3.1 Governing Equations in dimensionless form ……………………………………………………. 22
CHAPTER FOUR ………………………………………………………………………………………………………………. 23
SOLUTIONS TO THE PROBLEMS …………………………………………………………………………………………. 23
4.1 Solution to the Problem on Convective Boundary Condition ……………………………………… 23
4.1.1 Riemann Sum Approximation …………………………………………………………………………….. 24
4.1.2 Sherwood Number ……………………………………………………………………………………………. 26
4.1.3 Nusselt Number ……………………………………………………………………………………………….. 26
4.1.4 Skin Friction ……………………………………………………………………………………………………. 27
4.1.5 Mass Flux ……………………………………………………………………………………………………….. 27
4.1.6 Mean Concentration ………………………………………………………………………………………….. 28
4.1.7 Mean Temperature ……………………………………………………………………………………………. 28
4.2 Solution to the Problem on Heat Generating Fluid in Porous Microchannel ……………….. 29
4.2.1 Sherwood Number ……………………………………………………………………………………………. 30
4.2.2 Nusselt Number ……………………………………………………………………………………………….. 30
4.2.3 Skin Friction ……………………………………………………………………………………………………. 31
4.2.4 Mass Flux ……………………………………………………………………………………………………….. 31
4.2.5 Mean Concentration ………………………………………………………………………………………….. 31
4.2.6 Mean Temperature ……………………………………………………………………………………………. 32
CHAPTER FIVE…………………………………………………………………………………………………………………. 33
RESULTS AND DISCUSSION ……………………………………………………………………………………………….. 33
5.1 Convective Boundary Condition …………………………………………………………………………….. 33
5.2 Heat Generating Fluid in Porous Microchannel ……………………………………………………….. 55
CHAPTER SIX…………………………………………………………………………………………………………………… 78
SUMMARY, CONCLUSION AND RECOMMENDATION……………………………………………………………… 78
6.1 Summary ……………………………………………………………………………………………………………… 78
6.2 Conclusion ………………………………………………………………………………………………………. 78
6.3 Recommendations …………………………………………………………………………………………………. 79
REFERENCES …………………………………………………………………………………………………………………… 80
CHAPTER ONE
INTRODUCTION
1.1 Background of the study
Fluid mechanics can be described as the study of fluids either in motion (fluid dynamics) or at rest (fluid statics) and the subsequent effects of the fluid upon the boundaries, which may be either solid surfaces or interfaces with other fluids. It is worth mentioning that both gases and liquids are classified as fluids, and the number of fluids engineering applications are enormous. These include: breathing, blood flow, swimming, pumps, fans, turbines, airplanes, ships, rivers, windmills, pipes, missiles, icebergs, engines, filters, jets, and sprinklers, just to name a few. In fact, there is no gain saying that almost everything on this planet either is a fluid or moves within or near a fluid.
A very key component of fluid mechanics is natural convection. Natural convection flows have in contemporary times attracted a great deal of attention due to its diverse applications in the field of engineering, industrial and heat transfer processes. Gebhart and Pera (1971), in their work explained that the phenomenon of natural convective flow is caused by density differences induced by temperature gradients, chemical composition gradients, and material composition. Convection is also seen to play a very vital role in defining our weather patterns, oceanic currents, and sea-wind formation. In engineering applications, convection is commonly visualized in the formation of micro-structures during the cooling of molten metals, and fluid flows 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: this can happen on small scales (computer chips) to large scale process equipment. Wubshet and Makinde (2013), asserted that there are many physical problems as well as engineering applications where heat
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transfer by natural convection can be found to happen more frequently. They include: heat
exchangers, geothermal systems, petroleum reservoirs and nuclear waste repositories, chemical
catalytic reactors, packed beds, fiber and granular insulation.
On the other hand, the study of magnetohydrodynamics (MHD) flows have inspired great
curiosity owing to its relevance in meteorology, cosmic fluid dynamics, solar physics and in the
motion of Earth‟s core. On a wider perspective, applications of MHD can be appreciated in areas
such as; engineering, astrophysics and geophysics. MHD therefore, is concerned with the mutual
interaction of fluid flow and magnetic fields. The fluids in question must be electrically
conducting and non-magnetic, which limits us to liquid metals, hot ionised gases (plasmas) and
strong electrolytes.
In the course of modelling fluid flow through a microchannel, one parameter which proves very
instrumental is the Knudsen number. It is defined as the ratio of the fluid‟s mean free path to
the characteristic length b of the physical system (i.e. Kn /b ). It can also be seen as channels
whose dimensions are less than 1 millimeter and greater than 1 micron. Above 1 millimeter the
flow exhibits behavior that is the same as most macroscopic flows. Current ly, microchannels
have characteristic dimensions anywhere from the submicron scale to hundreds of microns.
Microchannels can be fabricated in many materials-glass, polymers, silicon, metals-using various
processes including surface micromachining, bulk micromachining, molding, embossing, and
conventional machining with microcutters. Microchannels are used to transport biological
materials such as (in order of size) proteins,DNA, cells, and embryos or to transport chemical
samples and analytes. For a given microchannel, when Kn<0.001, the flow domain is considered
as a continuum and as such modelled by the Navier–Stokes equation with the no-slip wall
boundary condition. However, in the interval 0.001<Kn<0.1, the no-slip boundary condition
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becomes invalid thereby giving way to the slip flow regime. Fluid flow through microchannel has in recent years become a hunting ground for researchers.Chen and Weng (2005)analytically studied the fully developed natural convection in open-ended vertical parallel plate micro-channel with asymmetric wall temperature distribution in which the effect of rarefaction and fluid wall interaction was shown to increase the volume flow rate and decrease the heat transfer.In another study by Khadrawi et al (2005) on the transient hydrodynamics and thermal behaviours of fluid flow in a vertical parallel-plate microchannel, under the effect of the hyperbolic-heat-conduction model,it was observed that increasing Kn results in an increase in the slip hydrodynamics and thermal boundary condition. Therefore, in the light of the background above, this research work shall be investigating the Dual-phase-lag and diffusion-thermo effects on thermosolutal convection in MHD flow both in a microchannel and micro-porous channel.
1.2 Statement of the problem
Jha and Ajibade (2010, 2011) investigated the effect of diffusion-thermo on natural convection heat and mass transfer under symmetric boundary conditions as well as asymmetric boundary conditions. These works were further extended by Ajibade (2014) when he investigated the combined effects of diffusion-thermo (Dufour) and dual-phase-lags (DPL) on unsteady double diffusive convection flow in a vertical microchannel filled with porous material. None of these studies mentioned above has been able to examine what effects magnetohydrodynamics (MHD), convective heat exchange as well as heat generation parameters could have on the fluid flow. Therefore, this work will attempt to fill the vacuum in Ajibade (2014) by considering the impacts of the parameters earlier mentioned on an electrically conducting fluid. In addition, this dissertation will also take into account the role of suction/injection on the hydrodynamic behaviour of the micro-fluidic flow.
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1.3 Significance of the study
This research work derived and presented a computational study on thermo-solutal convection
inMHD flow with Dufour effect which is subject to asymmetric wall heating conditions. Its
physical applications can be found in microelectronic devices, MHD marine propulsion, MHD
stirring of molten metal, electronic packages, thermal insulation, petroleum reservoirs, magneticlevitation
casting and exothermic reaction in packaged reactors, transportation cooling and
drying of materials. More importantly, it is hoped that the results obtained in this research work
will not only serve as an improvement on the previous studies related to such flows but also
provide useful information for the advancement of industrial or engineering applications.
1.4 Aim and objectives
The aim of this research is to carry out a computational study on thermo-solutal convection in
MHD flow with Dufour effect.
The objectives to attain the set aim are to:
i. Examine thermal and hydrodynamic responses of the fluid to convective heat
exchange parameter and also diffusion-thermo effects;
ii. Examine the effect of externally applied transverse magnetic field on natural
convection flow in both microchannel and micro-porous channel;
iii. Observe the impact of heat generation parameter ( ) on temperature, velocity profile
and skin-friction of the fluid;
iv. Investigate the impact of suction/injection on fluid velocity and skin-friction on
micro-porous channel flow formation.
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1.5 Research Methodology To successfully attain the above set goals, a well-known Laplace transform technique is employed to solve the time dependent governing concentration, momentum and energy equations and then the method of Riemann-sum approximation is applied to invert the Laplace domain analytical solutions into time domain in order to obtain the velocity, temperature, skin-friction and rate of heat transfer. Using a computer package, MATLAB (R2012b), the results obtained are 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.
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 the 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 caused by temperature gradients, concentration gradients or both. 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
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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.
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. Microchannel: is a channel with hydraulic diameter less than 1mm.
12. Hartmann Number: is the ratio of electromagnetic force to the viscous force.
13. Magnetic Field: is the magnetic effect of electric currents and magnetic materials.
14. Biot Number: is the ratio of conductive heat resistance within the object to the convective heat transfer resistance across the object‟s boundary.
15. Schmidt number: is a dimensionless number defined as the ratio of momentum diffusivity (viscocity) and mass diffusivity, and is used to characterize fluid flow in which there are simultaneous momentum and mass diffusion convection process.
16. Sherwood number: this is a dimensionless number which represents the ratio of convective mass transfer to the rate of diffusive mass transport.
17. Thermo-solutal: is a phenomenon which describes convection in a liquid caused by a combination of heat and varying concentration of solute.
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18. Dufour effect: refers to the energy flux due to a mass concentration gradient occurring as
a coupled effect of irreversible process.
19. MHD: implies Magnetohydrodynamics which is the study of the magnetic properties of
electrically conducting fluids. 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 as follows:
Maxwell’s equations:
t
B
E
B j 0
B 0
e K E 0
Conservation of Mass:
( ) 0
t
Equation of motion:
p j B
Dt
D
Conservation of energy:
s
s s s E j K T V h p
Dt
D
Dt
De
( ) ( ) ( )
2
1 2
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