THERMAL RESPONSE ON NANOFLUID FLOW IN

THE PRESENCE OF MAGNETIC FIELD WITH

VARIABLE STREAM CONDITIONS

VIBHU VIGNESH BALACHANDAR

UNIVERSITI TUN HUSSEIN ONN MALAYSIA

THERMAL RESPONSE ON NANOFLUID FLOW IN THE

PRESENCE OF MAGNETIC FIELD WITH VARIABLE STREAM

CONDITIONS

VIBHU VIGNESH BALACHANDAR

A thesis submitted in

fulfilment of the requirement for the award of the

Degree of Master in Mechanical Engineering

Faculty of Mechanical and Manufacturing Engineering

Universiti Tun Hussein Onn Malaysia

DECEMBER 2015

iii

I would like to dedicate this thesis to

ALMIGHTY “GOD”

(Who gave me strength, knowledge, patience and wisdom)

MY “PARENT”

(Their pure love, devotion, cares and prayers had helped me to achieve this target)

MY “SIBLINGS and FRIENDS”

(Their care, encouragement and motivation made me complete this valuable work)

iv

ACKNOWLEDGEMENT

I am grateful to Almighty GOD who is the most congenital, most sympathetic and

sustainer for the worlds for giving me the potency and the ability to do this research

work.

I would like to express my sincere thanks and cordial appreciation to my

academic supervisor Professor Dr. Sulaiman Bin Haji Hasan, his efforts and sincerity

enable my abilities to achieve this target. His support at every stage of study with

patience and unlimited guidance results the completion of this study within time. I

express my gratitude to my co-supervisor Professor Dr. R. Kandasamy who

supported for the research work, without his support it was impossible to complete

this study.

Sincere thanks to Universiti Tun Hussein Onn Malaysia who provide me

platform where I did this research work for my higher studies. It was impossible

without the financial support to complete this research. The university supported the

research by allocating Fundamental research grant scheme (FRGS) under Vot No.

1208 and Geran Insentif Penyelidik Siswazah (GIPS).

I am also thankful to my parents, siblings, friends and especially Engr. Qadir

Bakhsh Jamali, Engr. Shalini Sanmargaraja and Engr. Ashwin Kumar Erode Natrajan

for their moral support and motivation at every step of this research.

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ABSTRACT

Nanofluids are electrically conducting fluids which are the suspension of metallic,

non-metallic or polymeric nano-sized material in base liquid such as water, oil or air

are employed to perform tasks such as heat transfer and thermal conductivity. To

overcome limitations in heat transfer, an innovative new class of heat transfer fluid is

engineered as nanofluids by suspending metallic nanoparticles in conventional heat

transfer fluids, which expected to exhibit high thermal conductivities compared to

those of currently used heat transfer fluids, and they have the potential to enhance

heat transfer process. Enhancing the heat capacity of nanofluids in various

applications in industries and in most of the real life application as heat transfer is a

great challenge. This study proposes the analysis for thermal response of nanofluid

flow over the porous surface in the presence of magnetic field in various stream

conditions. The mechanical system of nanoparticle elements is suitable for

stagnation-point flow with convective boundary layer over a vertical porous

permeable surface is framed into a mathematical model. The governing nonlinear

partial differential equations are transformed into a system of coupled nonlinear

ordinary differential equations using similarity transformations and then the framed

mathematical equations are applied numerically using the fourth-fifth order Runge–

Kutta–Fehlberg method by using coded MAPLE 18 software. The work theoretically

investigate and analyse via simulation on the effects of various governing parameters

on flow field and heat transfer and nanoparticle volume concentration characteristics

of the convective boundary layer stagnation point flow of nanofluid towards a porous

stretching and shrinking permeable surface subjected to suction/ injection effect. The

result indicated that flow, heat transfer and nanoparticle concentration can be

controlled by changing the quantity of governing parameters.

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ABSTRAK

Nanofluids adalah cecair elektrik yang mengandungi bahan logam dan bukan logam

dan serbok polimerik yang bersaiz nano yang melaksanakan proses seperti

pemindahan haba dan kekonduksian therma. Untuk mengatasi kelemahan dalam

pemindahan haba, cecair inovatif kelas baru bagi pemindahan haba kejuruteraan,

nanofluids dengan campuran nanopartikel logam dalam cecair pemindahan haba

konvensional, yang dijangka mempamerkan keberaliran haba yang tinggi berbanding

dengan cecair yang ada dan cecair ini berpotensi untuk meningkatkan proses

pemindahan haba. Meningkatkan kapasiti haba nanofluids dalam pelbagai aplikasi

dalam industri dan kebanyakan aplikasi kehidupan sebagai pemindah haba masih ada

kelemahan yang perlu diselesaikan. Keperluan industri berorientasikan nanofluid

memerlukan kajian yang banyak. Kajian ini mencadangkan analisis kepada aliran

nanofluid atas permukaan berliang di dalam medan magnet mengikut pelbagai

keadaan aliran. Sistem mekanikal elemen nanopartikel sesuai untuk aliran stagnation

point dan aliran lapisan sempadan atas yang mengecut atau menegak dan permukaan

regangan dibingkaikan ke dalam model matematik. Persamaan separa pembezaan

tidak linear diubah menjadi satu sistem ditambah pula persamaan pembezaan tidak

linear biasa menggunakan transformasi persamaan dan kemudian persamaan

matematik dirangka digunakan secara berangka menggunakan order keempat dan

kelima kaedah Runge-Kutta-Fehlberg oleh dan dikodkan ke dalam perisian MAPLE

18. Kerja-kerja ini secara teori menyiasat melalui simulasi kesan pelbagai parameter

yang mengawal di padang aliran dan pemindahan haba dan ciri-ciri kepekatan jumlah

nanopartikel aliran titik stagnation point nanofluid ke arah regangan dan mengecut

aliran yang berliang tertakluk kepada kesan sedutan dan suntikan. Hasil kajian

menunjukkan aliran, pemindahan haba dan penumpuan nanopartikel boleh dikawal

dengan mengubah parameter kawalan.

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TABLE OF CONTENTS

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

LIST OF CONTENTS vii

LIST OF TABLES x

LIST OF FIGURES xi

LIST OF APPENDICES xiv

LIST OF PUBLICATIONS xv

LIST OF ABBREVIATIONS xvi

CHAPTER 1 INTRODUCTION 1

1.1 Background 1

1.2 Problem statement 3

1.3 Aim and objectives of the research 3

1.4 Scope of the research 4

1.5 Significance of study 4

1.6 Organization of thesis 5

CHAPTER 2 LITERATURE REVIEW 7

2.1 Introduction 7

2.2 Nanofluids and Nanoparticles 7

2.3 Experimental works on thermal conductivity of

nanofluids 8

2.4 Experimental works on viscosity of nanofluids 9

2.5 Nanofluid flow over stretching and shrinking surface 10

2.6 Classification of flow 11

2.6.1 Stagnation point flow over porous surface 11

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2.6.2 Stagnation point 12

2.7 Magnetohydrodynamic (MHD) flow over porous

surface 12

2.8 Conditions 14

2.8.1 Boundary layer 14

2.8.2 Porous medium, Porosity, Porous parameter 18

2.8.3 Suction pressure 19

2.8.4 Injection pressure 20

2.8.5 Prandtl number 21

2.8.6 Chemical reaction 22

2.8.7 Thermal conductivity 23

2.9 Stream Conditions 23

2.9.1 Magnetic effect 23

2.9.2 Thermal radiation 24

2.9.3 Brownian motion 25

2.9.4 Thermophoresis 26

2.9.5 Grashof number 27

2.9.6 Lewis number 28

2.9.7 Reynolds number 28

2.9.8 Thermal radiation 29

2.10 Computational fluid dynamic 30

2.10.1 Continuity equation 31

2.10.2 Momentum equation 32

2.10.3 Energy equation 32

2.11 MAPLE Software 33

2.12 Summary 35

CHAPTER 3 RESEARCH METHODOLOGY 36

3.1 The Algorithm 36

3.2 Research design 37

3.2.1 Stage 1: Identifying parameters 38

3.2.2 Stage 2: Framing code to MAPLE software 38

3.2.3 Stage 3: Optimization of program 39

3.2.4 Stage 4: Analysing the characteristics of nanofluid 39

3.3 Simulation conditions 40

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3.3.1 Validity and reliability of research tool 40

3.4 Flow analysis 41

3.5 Summary 46

CHAPTER 4 RESULT AND DISCUSSION 47

4.1 Results and discussions 47

4.1.1 Nanofluid subjected to suction effect 49

4.1.2 Nanofluid subjected to injection effect 71

4.2 Findings for suction 93

4.3 Findings for injection 95

4.4 Discussion 97

4.4.1 Similarities between suction and injection process 97

4.4.2 Differences between suction and injection process 98

4.5 Summary 99

CHAPTER 5 CONCLUSION 100

5.1 Conclusion 100

5.2 Contributions to the research 101

5.3 Future recommendation 102

REFERENCES 103

APPENDIX A 111

APPENDIX B 120

VITA 128

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LIST OF TABLES

3. 2 Simulation conditions 41

4. 1 Comparison of the present results for �'' with published

works 48

4. 2 Parametric values of stream conditions 50

4. 3 Parametric values of stream conditions 72

4. 4 Differences between suction and injection process 99

xi

LIST OF FIGURES

2. 1 Boundary layer flow 15

2. 2 MAPLE 18 software screen 33

2. 3 MAPLE 18 software simulation screen. 34

3. 1 Research design flow chart 37

4. 1 Co-ordinate system and the physical model of porous

surface 42

4. 2 Brownian motion on temperature 48

4. 3 Brownian motion on nanoparticle volume concentration 49

4. 4 Porosity effects on the velocity distribution 53

4. 5 Porosity effects on the temperature distribution 53

4. 6 Porosity effects on the nanoparticle volume fraction 54

4. 7 Magnetic effects on the velocity distribution 55

4. 8 Magnetic effects on the temperature distribution 56

4. 9 Magnetic effects on the nanoparticle volume concentration 56

4. 10 Grashof number on the velocity distribution 58

4. 11 Grashof number on the temperature distribution 58

4. 12 Grashof number on the nanoparticle volume fraction 59

4. 13 Lewis number on the velocity distribution 60

4. 14 Lewis number on the temperature distribution 60

4. 15 Lewis number on the nanoparticle volume concentration 61

4. 16 Brownian motion effect on the velocity distribution 62

4. 17 Brownian motion effect on the temperature distribution 63

4. 18 Brownian motion effect on the nanoparticle volume

fraction 63

4. 19 Thermophoresis effect on the velocity distribution 65

4. 20 Thermophoresis effect on the temperature distribution 65

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4. 21 Thermophoresis effect on the nanoparticle volume

concentration 66

4. 22 Thermal radiation on the velocity distribution 67

4. 23 Thermal radiation on the temperature distribution 68

4. 24 Thermal radiation on the nanoparticle volume

concentration 68

4. 25 Suction effect on the velocity distribution 70

4. 26 Suction effect on the temperature distribution 70

4. 27 Suction effect on the nanoparticle volume concentration 71

4. 28 Porosity effects on the velocity distribution 75

4. 29 Porosity effects on the temperature distribution 75

4. 30 Porosity effects on the nanoparticle volume fraction 76

4. 31 Magnetic effects on the velocity distribution 77

4. 32 Magnetic effects on the temperature distribution 78

4. 33 Magnetic effects on the nanoparticle volume concentration 78

4. 34 Grashof number on the velocity distribution 80

4. 35 Grashof number on the temperature distribution 80

4. 36 Grashof number on the nanoparticle volume fraction 81

4. 37 Lewis number on the velocity distribution 82

4. 38 Lewis number on the temperature distribution 82

4. 39 Lewis number on the nanoparticle volume concentration 83

4. 40 Brownian motion effect on the velocity distribution 84

4. 41 Brownian motion effect on the temperature distribution 85

4. 42 Brownian motion effect on the nanoparticle

volumefraction 85

4. 43 Thermophoresis effect on the velocity distribution 87

4. 44 Thermophoresis effect on the temperature distribution 87

4. 45 Thermophoresis effect on the nanoparticle volume

concentration 88

4. 46 Thermal radiation on the velocity distribution 89

4. 47 Thermal radiation on the temperature distribution 90

4. 48 Thermal radiation on the nanoparticle volume

concentration 90

4. 49 Injection effect on the velocity distribution 92

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4. 50 Injection effect on the temperature distribution 92

4. 51 Injection effect on the nanoparticle volume concentration 93

xiv

LIST OF APPENDICES

APPENDIX TITLE PAGE

A MAPLE ALGORITHM 111

B NUMERICAL TRANSFORMATION 120

xv

LIST OF PUBLICATIONS

Journal Articles

1. Magnetohydrodynamic and heat transfer effects on stagnation flow of an

electrically conducting nanofluid flow past a porous vertical

shrinking/stretching sheet in the presence of variable stream conditions.

Vibhu Vignesh Balachandar, Sulaiman Bin Hasan, R. Kandasamy.

Journal of Applied Mechanics and Technical Physics. (In review)

2. Influence of injection and heat transfer effects on stagnation flow of an

electricaly conducting nanofluid flow over a porous shrinking/stretching

sheet in the presence of variable stream conditions

Vibhu Vignesh Balachandar, Sulaiman Bin Hasan, R. Kandasamy.

Internaational Journal of Applied Engineering Research. (In Press)

3. Thermal response of convective boundary layer stagnation flow of nanofluid

over shrinking surface influencing suction and variable stream conditions.

Vibhu Vignesh Balachandar, Sulaiman Bin Hasan, Ashwin kumar,

R. Kandasamy.

ARPN Journal of Engineering and Applied Sciences. (In Press)

4. Nanofluid thermal response through stagnation point flow over shrinking

surface influencing injection and variable stream conditions.

Vibhu Vignesh Balachandar, Sulaiman Bin Hasan, Ashwin kumar,

R. Kandasamy.

ARPN Journal of Engineering and Applied Sciences. (In Press)

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LIST OF ABBREVIATIONS

a Velocity component on porous surface B Uniform transverse magnetic field strength C Concentration of nanofluid Cw Concentration of nanofluid at wall C∞ Concentration of nanofluid at infinity

CFD Computational fluid dynamics

c Constants (+ve) cp Specific heat at constant pressure, DB Brownian diffusion coefficient DT Thermophoresis diffusion coefficient � Force � ' Dimensionless velocity � '' Dimensionless skin friction Gr Grashof number Gm Modified Grashof number � Gravitational force k Permeability of the porous medium k Mean absorption coefficient �� Lewis number � Length M Magnetic parameter

MHD Magnetohydrodynamics m Mass NB Brownian motion parameter Nt Thermophoresis parameter

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ODE Ordinary differential equations qr Radiative heat flux and S Suction / Injection parameter T Temperature of nanofluid Tw Temperature of nanofluid at wall T∞ Temperature of nanofluid at infinity U Velocity of nanofluid Uw Velocity of nanofluid at wall U∞ Velocity of nanofluid at infinity U x Free stream velocity u Velocity components along the x direction uT Tangential velocity v Velocity components along the y direction

2-D Two-dimensional

3-D Three-dimensional

Greek symbols αf Fluid thermal diffusivity αm Thermal diffusivity

η Similarity variable

Dimensionless temperature of the fluid

w Wall temperature excess ratio parameter ' Dimensionless heat transfer rate

Dynamic viscosity v Kinematic viscosity ρ Density ρf Density of base fluid σ Electrical conductivity of the fluid δ Stefan–Boltzmann constant τ Ratio of the effective heat capacity of the nanoparticle material to the

heat capacity of the ordinary fluid φ Nanoparticles volume fraction

ψ Stream function

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Porous medium v Mass flux

Superscripts ' Differentiate with respect to y, x, η correspondingly

Subscripts

� Fluid

s Solid

T Tangential

w Wall ∞ Free stream

CHAPTER 1

INTRODUCTION

1.1 Background

Heat transfer is indispensable to maintain the reliability of desired performance of a

wide variety of equipment’s in industries and consumer products. High-tech

industries face technical challenges day by day. Heat transfer is one of them

especially in industries such as manufacturing, metrology, microelectronics, nuclear

reactors, hybrid-powered engines, vehicle thermal management, domestic

refrigerator, chillers, heat exchanger, nuclear reactor coolant, grinding, machining,

space technology, ships, and boiler flue gas temperature reduction. These industries

faces heat transfer problems because of unprecedented heat loads and heat fluxes. Air

cooling, water cooling systems, coolant, fins, liquid nitrogen are commonly used as

coolants in industries. In optoelectronic devices in power electronics devices are also

found to have heat transfer problems. Increasing in heat flux necessitates the use of

liquid cooling technologies such as heat pipes, spray cooling for chips, direct

immersion cooling. In manufacturing industries conventional heat transfer fluids oil,

water, ethylene glycol, toluene are used for heat transfer and cooling purpose. But

these cooling systems are not able to satisfy the required heat transfer requirements

while dispersed solid particles with heat transfer fluid shows better cooling rate

compared with traditional cooling systems.

Heat transfer fluid having millimetre or micrometre sized particles have some

weakness. Rapid settling of millimetre or micrometre sized particle is the problem

and if the fluid is kept circulating to prevent particle settling, these particles would

2

wear out bearings, pumps and pipes. Furthermore such sized particles are not

applicable in microsystems because it can clog micro channels. Significantly this

type of cooling system results in pressure drop. To overcome disadvantages and

improve heat transfer characteristics an innovative idea of implementing nanometer

sized particle (Choi & Eastman, 1995) became a conventional method for cooling

purpose. Many research are still on going for nanofluids and its heat transfer

characteristics.

Nanofluid is a new kind of fluid containing small quantity of nano-sized

particles (usually less than 100nm) that are uniformly and stably suspended in a

liquid. The dispersion of a small amount of solid nanoparticles in conventional fluids

changes their thermal conductivity remarkably. Nanofluids are characterized by base

fluid like water, toluene, ethylene glycol or oil with nanoparticles in variety of types

like metals, oxides, carbides, carbon, nitrides and others. Some benefits of nanofluids

that make them useful are:

1. The nanoparticles surface to volume ratio is × times larger than

that of microparticles. Surface area of the particles conducts heat. Since

nanoparticles have larger surface area, it enhances the heat conduction of

nanofluids.

2. Nanoparticles stay suspended longer time compared to microparticles.

3. High stability and high thermal conductivity.

4. Nanoparticles reduce erosion, clogging and abrasion dramatically.

Thermal conductivity of nanofluid has been measured with several

nanoparticles volume fraction, material and dimension in several base fluids and all

findings showed that thermal conductivity of nanofluid is higher than the base fluids

and have great potential and more efficient for a cooling system.

The novel concept of nanofluids has been proposed as a route to analyse the

performance of heat transfer fluids currently available(Bachok et al., 2010). But the

characteristic of nanoparticle volume concentration, nanofluid temperature and flow

field velocity of nanofluid differs on the basis change in stream conditions. It is in a

great interest of this research to analyse the characteristics of fluid flow over vertical

porous medium in the presence of magnetic field with variable stream conditions.

3

1.2 Problem statement

Heat transfer is one of the challenges in industries such as manufacturing, metrology,

microelectronics, nuclear reactors, heat exchanger, space technology, ships, and

boiler flue gas temperature reduction. Heat transfer failure caused by unprecedented

increase in heat loads and heat flux caused challenges in high tech industries.

Conventional heat transfer methods like air cooling, water cooling systems, coolant,

fins and also conventional heat transfer liquids like water, toluene, ethylene glycol or

oil unable to fulfil the requirements of industries because of its low thermal

conductivity (Choi 1995) and (Eastman et al., 2001). To overcome the heat transfer

issues nanoparticles disperse with base fluids as coolant can achieve the desirable

performance due to high heat transfer capabilities (Gharagozloo & Goodson 2011).

Nanofluids ability has already provide but just that enough to enable nanofluid in real

life application. Analysing its thermal behaviour on various situation is important

(Bachok et al., 2010). Owing to this application, this research will try to find

solutions following questions arise.

1. What are nanofluid characteristics that provide optimum heat transfer with

the conditions involved?

2. How to formulate numerical model presenting the nanofluid characteristics?

3. How to optimize nanofluid model using simulation software?

4. What are the effects of various stream conditions towards the temperature,

velocity and nanoparticle volume concentration profiles?

5. How does the fluid flow characteristic differs between stretching and

shrinking surface?

6. How does the fluid flow characteristic differs between injection and

suction?

1.3 Aim and objectives of the research

The overall aim and objectives of present study is to analyse the characteristics of

heat transfer and mass transfer by constructing mathematical model and generating

the numerical algorithms on convective boundary layer stagnation point flow over

porous surface. The specific objectives are as follows:

4

1. To frame numerical algorithm of nanofluid model with various streaming

conditions involved.

2. To analyse the nanofluid model simulation using software MAPLE 18

software.

3. To analyse thermal response, velocity, concentration of nanofluid flow on

stretching/shrinking surface subjected to various physical effects in suction

process through simulation.

4. To analyse thermal response, velocity, concentration of nanofluid flow on

stretching/shrinking surface subjected to various physical effects in injection

process through simulation and comparing the results with the results of

objective 3.

1.4 Scope of the research

This work will be limited to the algorithm generated and its simulation using

MAPLE 18. The simulation will be focus on various thermal response, velocity,

concentration in suction and injection process. The simulation model will focus on

various streaming conditions and will study nanofluid response on velocity flow,

temperature distribution and concentration on stretching and shrinking surface

subjected to injection and suction process on convective boundary layer stagnation

point flow over porous surface.

1.5 Significance of study

Nanofluids in the presence of magnetic effect have great potential for heat transfer

enhancement and it is highly suited in application in heat transfer processes. Due to

magnetic field Lorentz force acts opposite to the flow, velocity of nanofluid

decreases, which tends to increase the thermal conductivity of nanofluid. Surface to

volume ratio of nanoparticles, stability, and are added advantages of using

nanofluids. This provides promising ways for engineers to develop highly compact

and effective heat transfer fluids. When addressing the nanofluids, it is foremost

important to establish its flow over surface. Nanofluids flow in porous surface play a

very important role on engineering applications. In diverse applications nanofluid

5

over porous surface has been an efficient active field. Nanofluids flow over porous

surface varies with respect to stream conditions. Brownian motion and

thermophoresis holds important handy in thermal conductivity and nanoparticle

volume fraction. Providing nanofluids with proper stream conditions can achieve

efficient heat transfer.

Heat transfer by convection in numerous examples of naturally occurring

fluid flow, such as wind, oceanic currents, and movements within the Earth's mantle.

Porous surface splits into stretching and shrinking surface, each one implies its

necessity on their particular applications. Flow over stretching/shrinking differs from

each other due to the characteristics of nanofluids and surface.

1.6 Organization of thesis

This thesis consists of an introductory chapter and four main chapters dealing with

the following problems.

Chapter 1 provides brief introduction of the subject, problems, physical

features involved in the study of stagnation flow of magnetohydrodynamic boundary

layer flow of nanofluids with Brownian motion, thermophoresis, thermal radiation,

magnetic effect, suction/ injection and various dimensionless numbers.

Nanofluids have high heat transfer ability to fulfil the necessity of heat

transfer fluid in industries. It is also a need to increase the efficiency of nanofluid, so

that performance will increase. The study of thermal response of nanofluid in the

presence of magnetic field is of great practical importance to engineers and scientists

because of its almost universal occurrence in many branches of science and

engineering. Nanofluids with magnetic effect change its response to heat with

variable stream conditions. Therefore, Chapter 2 presents the literature review that

leads to overall study in this thesis.

Chapter 3 presents an overview of the research methodology involved in

choosing appropriate steps and procedures to give the solution of problem to achieve

the objectives of study. In order to ensure the effectiveness of nanofluid in heat

transfer applications.

Chapter 4 presents the effects of thermal response of nanofluid flow in the

presence of magnetic field and suction/injection effect with variable stream

6

conditions on a vertical porous surface. The main objective is to investigate the

thermal response of nanofluid flow on various physical effects, nanofluids energy

transmission regarding its volume concentration and to analyse stability of nanofluid.

Here the stagnation flow over porous surface (stretching/shrinking surface) on

vertical plate is to investigate numerically and analyse using MAPLE software

(version 18). The physical aspects of fluid flow are governed with governing

equations of Computational fluid dynamics. Governing nonlinear boundary-layer

equations are converted by similarity transformation to coupled higher order

nonlinear ordinary differential equation. The output nanoparticle volume

concentration, temperature and velocity profiles show the clear objectives of thesis

with flow and characteristics when subjected to various physical effects.

Finally, concluded the thesis with Chapter 5 which consists of two main

parts: contributions and future work.

CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

A number of researchs relating to effects of nanoparticles usage in heat transfer

fluids have been carried out by a number of researchers recently. This chapter will

discuss their findings. The chapter will also include parameters and factors that

relates to heat transfer characteristics and properties that have been studied. The

literatures presented here have influenced the work and methodology used.

2.2 Nanofluids and Nanoparticles

Choi (1995) introduced the term nanofluid. It refers to the fluids with suspended

nanoparticles. He engineered and innovated suspending metallic nanoparticles in

conventional heat transfer fluids. The results showed that the addition of a very small

amount of nanoparticles to convectional heat transfer fluids, increased the thermal

conductivity of the fluid up to approximately two times. The reason was the high

surface area of nanoparticles enhances the heat conduction of nanofluids since heat

conduction occurs on the surface of the particle. The surface to volume ratio of

nanoparticles is 1000 times larger than that of microparticles so heat transfer rate is

also high.

Eastman et al., (2001) indicated that a small amount (<1% volume fraction)

of Cu nanoparticles or carbon nanotubes dispersed in ethylene glycol or oil is

reported to increase the thermal conductivity of a liquid by 40% and 150%

8

respectively. However, a convectional solid particle-liquid suspensions with larger

size and higher density requires more concentrations (>10%) of particle to achieve

such enhancement. This amount of concentration induces a problem of settling, flow

resistance and possible erosion. Masuda et al., (1993) observed a phenomenon that

nanofluids has a characteristic feature of thermal conductivity enhancement, which

indicates the possibility of using nanofluids in advanced nuclear systems due to high

stability and good conductivity. The research therefore will study this phenomenon

through simulation and study the characteristics of heat conduction when certain

parameters are changed.

2.3 Experimental works on thermal conductivity of nanofluids

From Eastman et al., (1997) report, nanoparticles induced in water made a promising

turning point of using nanofluids in heat transfer process. The article showed that 5%

volume of nanocrystalline copper oxide CuO particles suspended in water resulted in

an improvement in thermal conductivity of almost 60% compared to water without

nanoparticles. Murshed et al., (2005) found that nanofluids prepared by dispersing

5% volume fraction Titanium Dioxide TiO2 nanoparticles in deionized water, the

thermal conductivity enhancement is observed through hot wire technique to be

nearly 33% respectively over the base fluid. Putnam et al., (2006) used an optical

beam deflection technique to measure the thermal conductivity of ethanol-water

mixtures. The enhancement was in a factor of 2 larger than predicted by effective

medium theory.

Penas et al., (2008) conducted an experiment using the multi current hot wire

technique by dispersing nanoparticles of silica (SiO2) and Copper Oxide (CuO) in

water and ethylene glycol at various concentrations up to 5% in mass fraction. Good

agreement was found in recommended and published articles about the enhancement

in the thermal conductivity of the fluids due to presence of dispersed nanoparticles

using this technique. Han (2008) reported that 52% enhancement in the thermal

conductivity was found in water-Fluorocarbons (FC72) nanofluids. At room

temperature, the thermal conductivity of FC72 is 0.057 W/m.K, only one tenth of the

thermal conductivity of water (0.58 W/m.K). This water FC72 nanoemulsion fluid

9

was prepared by using high- intensity ultrasonic homogenizer to emulsify water

droplets (<10nm in radius) into FC72.

A benchmark study on the thermal conductivity of nanofluids was made by

(Buongiorno et al., (2009). The nanofluids tested were comprised of aqueous and

nanoaqueous basefluids, metal and metal oxide particles, near-spherical and

elongated particles, at low and high particle concentrations. The thermal conductivity

of the nanofluids was found to increase with particle concentration and aspect ratio

through a variety of experimental approaches, including the transient hot wire

method, steady state methods, and optical methods. The resulted thermal

conductivities were then validated through the effective medium theory developed

for dispersed particles by Maxwell in 1881 and generalized by (Nan et al., 1997).

Evans et al., (2006) have demonstrated that based on kinetic theory analysis of heat

flow in fluid suspensions of solid nanoparticles in nanofluids, the hydrodynamics

effects associated with Brownian motion have only a minor effect on the thermal

conductivity of the nanofluid which was also discussed in (Gharagozloo & Goodson

2011). All the above research have shown that thermal conductivity of nanofluids are

much higher than normal fluids.

2.4 Experimental works on viscosity of nanofluids

Einstein was the first to calculate the effective viscosity of a suspension of spherical

solids using phenomenological hydrodynamic equations. A benchmark study on the

viscosity of nanofluids was made by (Venerus et al., 2010). The viscosity

measurements were on the colloidal dispersions (nanofluids) for heat transfer

applications. They examined the influence of particle shape and concentration on the

viscosity of the same nanofluids tested by (Buongiorno et al., 2009). They compared

data to prediction from classical theories on suspension rheology. The result showed

that the lower concentration ( = . ) fluid is Newtonian, while the higher

concentration ( = . ) fluid is non-Newtonian. The addition of solid particles to a

liquid can significantly alter its rheological behaviour.

A study on viscosity was conducted by Duan et al., (2011) two weeks after

aluminium oxide Al2O3-water nanofluids were prepared at the volume concentration

of 1-5%. A higher nanoparticle aggregation had been observed in the nanofluids

10

resembled non-Newtonian fluids. After ultrasonic agitation treatment, the nanofluids

resumed as a Newtonian fluids. The relative viscosity increases about 60% as the

volume concentrations increases to 5% in comparison with the base fluid. These

experiments have shown that relative viscosity of nanofluids increases as the results

of the addition of nanoparticles.

2.5 Nanofluid flow over stretching and shrinking surface

Nazar et al., (2004) studied unsteady two-dimensional stagnation point flow of an

incompressible viscous fluid over a deformable sheet. He discussed the analysis

when the flow is started impulsively from rest and the sheet is suddenly stretched in

its own plane with a velocity proportional to the distance from the stagnation point.

Wang (2008) studied two dimensional stagnation point flow on a two dimensional

shrinking sheet and axisymmetric stagnation point flow on an axisymmetric

shrinking sheet. It was noted that the line of stagnation flow is perpendicular to the

stretching surface.

Pop et al., (2004) presented radiation effects on the flow near the stagnation

point of a stretching sheet. They show that a boundary layer is formed and its

thickness increases with the radiation, velocity and temperature parameters and

decreases when the Prandtl number is increased. Ishak et al., (2006) analysed steady

two dimensional stagnation point flow of an incompressible viscous and electrically

conduction fluid subjected to a uniform magnetic field, over a vertical stretching

surface, he stated velocity of nanofluid decreases as Lorentz force acting opposite to

the flow, which tends to increase the thermal conductivity of nanofluid

proportionally.

Dulal (2011) numerically analysed the flow and heat transfer in laminar flow

of an incompressible Newtonian fluid past an unsteady stretching sheet in the

presence of a non-uniform heat source/sink and thermal radiation. The time-

dependent stretching velocity and surface temperature resulted to the unsteadiness in

the flow and temperature fields. Hamad et al., (2012) studied heat and mass transfer

for boundary layer stagnation-point flow over a stretching sheet in a porous medium

saturated by nanofluid with internal heat generation/absorption and suction/injection.

He stated that suction tends to stabilize the boundary layer flow and injection can

11

reduce the friction drag. Therefore, one can conclude that there is some effect on

flow of nanofluids over stretching and shrinking surfaces. This work will also take

the nanofluid flow over stretching and shrinking surface into considerations.

2.6 Classification of flow

Flow are generally classified into uniform and non-uniform flow, compressible and

incompressible flow, steady and unsteady flow, and laminar and turbulent flow. A

steady flow is one in which the conditions (velocity, pressure and cross-section) may

differ from point to point but do not change with time (Shaughnessy et al., 2005). If

at any point in the fluid, the conditions change with time, the flow is described as

unsteady. Density of all fluids will change if pressure changes. Liquids are quite

difficult to compress, so under most steady conditions they are treated as

incompressible. In some unsteady conditions very high pressure differences can

occur and it is necessary to take these into account.

2.6.1 Stagnation point flow over porous surface

The proposed algorithm will also investigate the thermal characteristics of nanofluid

at stagnation point over porous surface. Mahapatra & Gupta (2001) numerically

studied boundary-layer and magnetohydrodynamics stagnation point flow towards a

stretching sheet. Their analysis showed that velocity at a point increase with an

increase in the magnetic field when the free stream velocity is greater than the

stretching velocity. Mahapatra et al. (2002) extend their investigation to a power-law

fluid and studied the magnetohydrodynamic stagnation point flow of a power-law

fluid towards a stretching surface. The result signifies that for a given magnetic

parameter, the dimensionless shear stress coefficient increases in magnitude with an

increase in power-law index when the values of the ratio of free stream velocity and

stretching velocity are close to 1.

Makinde et al., (2013) analysed the effects of buoyancy force, magnetic field

and convective heating on stagnation-point flow and heat transfer due to nanofluid

flow towards a stretching sheet. The combined effects of Brownian motion and

thermophoresis on nanofluid over a stretching sheet were investigated. He concluded

12

response of nanofluid depends mainly on Brownian motion and thermophoresis.

These findings will also be trailed through simulation. Porous surface and stagnation

point will be taken into consideration.

2.6.2 Stagnation point

In general, there are steady and unsteady flow, laminar and turbulent flow. In steady

flow, the flow properties at any given point in space are constant in time such as

velocity, pressure and cross-section. In unsteady flow, the flow properties at any

given point in space change with time. As for laminar flow, fluid particles move in

smooth, layered fashion (no substantial mixing of fluid occurs) whereas for turbulent

flow, the fluid particles move in a chaotic, tangled fashion (significant mixing of

fluid occurs). There is another classification of flow available namely compressible

and incompressible flow. In in compressible flow, the volume of a given fluid

particle does not change which implies that the density is constant everywhere. In

compressible flow, the volume of a given fluid particle can change with position

which implies that the density will vary throughout the field.

In fluid dynamics, a stagnation point is a point in a flow field where the local

velocity of the fluid is zero. In the flow field at the surface of objects stagnation

points exist, where the object brought fluid to rest. When the velocity is zero the

static pressure will be highest says Bernoulli equation and hence at stagnation points

static pressure is at its maximum value. This static pressure is called as stagnation

pressure. The Bernoulli equation applicable to incompressible flow shows that

addition of dynamic pressure and static pressure is equal to the stagnation pressure.

So in incompressible flows, total pressure is equal to stagnation pressure. Since

addition of dynamic pressure and static pressure is equal to the total pressure

providing the fluid entering the stagnation point is brought to rest. This phenomenon

is also considered in this work.

2.7 Magnetohydrodynamic (MHD) flow over porous surface

The word magnetohydridynamics (MHD) is derived from magneto which means

magnetic field, hydro means liquid and dynamics which mean movement. In

13

addition, MHD nanofluid flow is the study of electrically conducting nanofluids

movement. The field of MHD was initiated by Hannes Alfven, an astrophysicist,

who received the Noble Prize in physics in 1970. He found that a magnetic field line

can transmit transverse initial wave (Davidson, 2001).

Prior to this, an engineer called J.Hartmann invented an electromagnetic

pump in 1918, with this pump he investigated the flow of mercury, a conducting

liquid in a magnetic field (Davidson, 2001). It was during the 1960s that the

development of MHD came into the field of engineering. The result was three

technological innovations: (i) a liquid sodium pump cooler for fast-breeder reactor;

(ii) a controlled thermonuclear fusion that require the hot plasma separated from

material surface by magnetic force; and (iii) a MHD power generator that use the

magnetic field to propel the ionized gas to improve power station efficiencies

(Davidson, 2001).

The set of equations which describe MHD are a combination of Navier-

Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism.

These differential equations have to be solved simultaneously, either analytically or

numerically because MHD is a fluid theory, it cannot treat kinetic phenomena. There

are many studies on MHD flow such as Kafoussias & Nanoousis (1997) suction or

injection pressure on MHD laminar boundary layer flow and ( Chamkha & Khaled.,

2000) on hydromagnetic combined heat and mass transfer by natural convection

from a permeable surface embedded in a fluid saturation porous medium while

(Chamkha et al. 2003) on thermal radiation effect on MHD forced convection flow

in the presence of heat source or sink. Ishak et al., (2008) have investigated MHD

flow and heat transfer on stretching surface and stated the flow of nanofluids. It can

be argued that similar conditions are applicable for nanofluids on shrinking surface.

Consequently, the studies had entered a new phase where nanofluid was

included. Chamkha & Aly (2010) have analysed MHD free convection flow of a

nanofluid past a vertical plate in the presence of heat generation or absorption

effects. Kandasamy et al., (2011) investigated MHD boundary layer flow of a

nanofluid past a vertical stretching surface in the presence of suction and injection.

Nourazar et al., (2011) have studied MHD nanofluid flow over a horizontal

stretching surface. Hamad & Pop (2011) have examined the MHD free convection

flow past a vertical permeable flat plate in rotating frame of reference with constant

14

heat source in a nanofluid. All the above research has shown that MHD flow over

porous surface is must in considering the study of thermal response of nanofluid.

2.8 Conditions

2.8.1 Boundary layer

Ludwig Prandtl in 1904 pointed out that the flow of liquid with a small viscosity in

the neighbourhood of fixed body can be divided into two regions. Acheson (1990)

stated the region of very thin layer near the boundary layer, where viscosity forces

can be neglected. The influences of viscosity forces in the boundary layer can be

explained by the liquid adhering to the surface of the body, and the adhesion of the

liquid cause on account of friction a dragging of thin layer adjoining the walls. In this

thin layer, the velocity of the flow past the body at rest changes abruptly increasing

from zero to the value of the velocity in the external current (Acheson, 1990;

Anderson, 2005).

Prandtl set up an equation satisfied in the first approximation by the velocity

of flow of fluid in the boundary layer. These equations are called prandlt’s system.

They form the basis of boundary layer theory. He theorized that an effect of friction

was to cause the fluid immediately adjacent to the surface to stick to the surface. In

other words, he assumed the no slip condition at the surface. Outside the boundary

layer, the flow was essentially the inviscid flow (Achenson, 1990; Anderson, 2005).

The concept of boundary layer is sketched in Figure 2.1 are velocity, thermal,

velocity, concentration boundary layers respectively.

A thermal boundary layer will develop if the surface temperature and free

stream temperature are different. In Figure 2.1, at the leading edge, the temperature

profile is uniform with T = T∞. The wall is maintained the thermal boundary layer

thickness in x-direction. The fluid particles coming into contact with the surface

exchange thermal energy with those in the neighbouring layers and a thermal

gradient is setup. With increasing distance, y, from the surface, the fluid temperature

approaches the free stream temperature. The effects of heat transfer penetrate further

into free stream resulting in the growth of thermal boundary layer thickness. Like the

15

velocity boundary, the thermal boundary will also be defined as laminar or turbulent

depending upon the critical value of Reynolds number (Kahar, 2014).

In addition to the lack of consensus on physical properties there is not yet an

accepted model for nanofluid flow. A number of models and physical mechanisms

are framed is described (Das et al., 2003). This research focus on a particular form of

model shown in Figure 2.1. In the real world situation, the heat exchanger for an

example, the fluid is in contact with the walls and tubes of the exchanger and the

bulk of the fluid moves relative to these surfaces. As the fluid moves, the layer of the

fluid in contact with the metal surface is in fact stationary. Away from the metal

surface the fluid begins to move faster and faster. This flow is known as laminar flow

and is characterised by the fluid moving parallel to the surface of exchanger. Further

into the fluid there comes a point where the flow is in longer laminar and is

composed of an ever increasing amount of turbulence. This flow is known as

turbulent flow.

Figure 2. 1 Boundary layer flow (Kandasamy et al., 2011)

where,

Velocity of nanofluid

Velocity of nanofluid at wall

16

∞ Velocity of nanofluid at infinity

Temperature of nanofluid

Temperature of nanofluid at wall

∞ Temperature of nanofluid at infinity

Concentration of nanofluid

Concentration of nanofluid at wall

∞ Concentration of nanofluid at infinity

These regions of laminar and turbulent flow are extremely important in heat

transfer. The layer of fluid that exhibits laminar flow is termed the boundary layer.

There is a critical velocity of the fluid at which the laminar boundary layer becomes

turbulent. As the fluid velocity increases, the change from laminar to turbulent flow

occurs closer and closer to the metal surface. In other words, the boundary layer

becomes thinner as the fluid velocity increases (Mahapatra et al., 2011). The equation

developed takes steady laminar flow as an important part of the work. The boundary

layer shows the general flow and response of temperature and velocity in the Figure

2.1. The above condition is taken into consideration in the theoretical flow analysis

of this work.

2.8.1.1 Convective boundary layer flow over porous surface

The study of convective flow, heat transfer in porous surface has been an active field

of research as it plays a crucial role in diverse applications, such as thermal

insulation, extraction of crude oil and chemical catalytic reactors. More over our

research is with convective boundary layer flow. Numerous models and group theory

methods have been proposed by different authors to study convective boundary layer

flows of fluid, which are (Birkfoff 1998, 1960; Yurusoy & Pakdemirli1997, 1999a,

1999b; Yurusoy et al., 2001).

17

Kuznetsov & Nield (2010) have examined the influence of nanoparticles on natural

convection boundary layer flow past a vertical plate, using a model in which

Brownian motion and thermophoresis are accounted. In this study it is assumed the

simplest possible boundary conditions, namely those in which both the temperature

and the nanoparticle fraction are constant along the wall. Nield & Kuznetsov (2009b)

have analysed the effect of nanoparticles on natural convective boundary layer flow

in a porous medium past a vertical plate and employed the Darcy model for the

momentum equation. Bachok, et al., (2010) have studied theoretically the problem of

steady boundary layer flow of a nanofluid past a moving semi-infinite flat plate in a

uniform free stream flow move in the opposite directions. The problem of laminar

fluid flow resulting from the stretching of a flat surface in a nanofluid has been

investigated numerically (Khan & Pop, 2010). Mankinde & Aziz (2011) studied the

boundary layer flow of a nanofluid past a stretching sheet with a convective heating

boundary condition. Kuznetsov & Neild (2011) studied the double diffusive natural

convective boundary layer flow of a binary nanofluid past a vertical surface

incorporated it with the effects of Brownian motion and thermophoresis.

Maiga et al., (2005) investigated the problem of laminar forced convection

flow of nanofluids for two geometrical configurations, namely a uniformly heated

tube and a system of parallel, coaxial and heated disks. For the case of tube flow, the

heat transfer enhancement increases considerably with an augmentation of the flow

Reynolds number. For the case of radial flow, both the Reynolds number and the

distance separating the disks do not seem to considerably affect in one way or

another the heat transfer enhancement of nanofluids (i.e., when compared to the base

fluid at the same Reynolds number and distance). Hamad & Ferdows (2012) have

investigated a two dimensional laminar forced convection flow over a permeable

stretching surface in a porous medium saturated by a nanofluid. Evans et al., (2006)

have demonstrated that the Brownian motion have only minor effect on the thermal

conductivity of the nanofluid which was also shown in (Gharagozloo & Goodson,

2011).

Fully developed laminar mixed convection of a nanofluid consisting of water

and Al2O3 in a horizontal curved tube has been studied numerically (Akbarinia &

Behzadmehr, 2007). For a given Reynolds number, Buoyancy force has a negative

effect on the Nusselt number while the nanoparticles concentration has a positive

effect on the heat transfer enhancement and also on a skin friction reduction. Ahmad

18

& Pop (2011) studied a steady mixed convection boundary layer flow past a vertical

plate embedded in a porous medium filled with nanofluids of Cu, Al2O3 and TiO2. It

is shown that the solution has two branches in a certain range of the parameters.

Gorla et al., (2011) presented a boundary layer analysis of mixed convection past a

vertical plate in a porous medium saturated with a nanofluid. The equation in this

study will take into consideration of all these findings.

2.8.2 Porous medium, Porosity, Porous parameter

Porous materials are encountered literally everywhere in everyday life, in technology

and in nature. With the exception of metals, some dense rocks, and some plastics,

virtually all solids and semi-solid materials are porous to varying degrees. Porous

medium is characterized by a very large surface area to a volume ratio. This peculiar

feature of the porous media can be utilized to either distribute heat energy uniformly

or to enhance the heat transfer in heat exchange systems (Seigen, 1972). Porous

material contain relatively small spaces, so called pores or voids, free of solids,

imbedded in the solid or semi-solid matrix. The pores usually contain some fluid,

such as air, water, oil. etc., or a mixture of different fluids. Pores material are

permeable to a variety of fluids, that is fluids should be able to penetrate through one

face of septum made of the material and emerge on the other side (Bear & Bachmat,

1990). In this research the study of transport phenomena in porous media have

primarily been initiated by the research activity in geophysical and chemical

engineering. The study of transport phenomena in porous medium has attracted

considerable attentions and has been motivated by broad range of engineering

applications.

Porous media are proven to operate in most of the corresponding free fluid

ranges. They can be used as an insulator for all temperature ranges and can be used

as a heat transfer promoter for either sensible or latent heat transfer. Different

transport models are used to model energy and momentum transport in porous media.

These models are phonologically based upon governing equations which are

inherited from the corresponding free-fluid flow. Porosity is the factor often

characterizes the porous medium. Porosity is a fraction between 0 and 1. pores

structure and the media porosity with respective properties can sometimes be derive

19

some of the other properties of porous medium like tensile strength, permeability

and electrical conductivity, but such a derivation is usually complex. Porous medium

the porosity is defined as the fraction of the total volume of the medium that is

occupied by void space. It is shown that under conditions convective flow may occur

in fluid which permeates a porous medium and is subject to a vertical temperature

gradient, on the assumption that flow obeys Darcy’s law. This phenomenon is also

considered in this work and the equation (2.2) is an integral part of the algorithm. � = (2.2)

where, � Permeability of the porous medium � Porous medium (dimensionless)

Kinematic viscosity

� Velocity component on porous surface

2.8.3 Suction pressure

Flow of a fluid into a low pressure region, or a partial vacuum region is called as

suction pressure. The regions differ of pressure gradient between them and the

ambient pressure will pull liquid toward area of low pressure. Mostly suction is

considered as an attractive effect or vacuum effect. The production of a vacuum or

partial vacuum in a cavity or over a surface so that the external atmospheric pressure

forces the surrounding fluid, particulate solid, etc. into the cavity or causes

something to adhere to the surface. Vacuum cannot attract matter but the atmospheric

high pressure of the surrounding fluid can propel liquid into a vacuum area (Calvert

& James, 2000).

Typically pumps have both an inlet as well as an outlet. The inlet location is

said to be at the suction side where the fluid enters the pump and the outlet location

is said to be at the discharge side of the pump where the fluid comes out. Operation

differs on the basis of pump. Pump creates a low pressure region (suction pressure) at

the inlet side so that fluid can enter the pump, at the discharge side pump operation

20

causes higher pressure by forcing the fluid out at the outlet (Calvert & James, 2000).

Under normal conditions of atmospheric pressure suction can draw pure water up to

a maximum height of approximately 10.3 m. This is the same as the maximum height

of a siphon, which operates by the same principle. = √ (2.3)

where,

is the suction parameter >

Mass flux ∗

� Velocity component on porous surface

Kinematic viscosity

Suction effect is given in one part of the work and following with injection

effect on another part of the work. Both played major role in this work. The suction

parameter equation (2.3) is included in this work.

2.8.4 Injection pressure

Flow of a fluid into a high pressure region is called as injection pressure. The regions

differ of pressure gradient between them and the ambient pressure will force liquid

toward area of high pressure. Mostly injection is considered as a blowing effect. In

injection, with the help of external force the fluid is forced into high pressure region

(Calvert & James, 2000).

A similar pump characteristic as discussed in section 2.9.3 is applicable.

Forcing a fluid to a region is termed as injection or blowing (Calvert & James, 2000).

Injection force is given in the later part of the work by replacing suction effect. As in

equation 2.3, similar equation can be used here. The difference is in value which is

now an injection parameter < .

21

= − √ (2.4)

where,

Injection parameter <

Mass flux ∗

� Velocity component on porous surface

Kinematic viscosity

2.8.5 Prandtl number

The Prandtl number is a dimensionless number, defined as the ratio of momentum

diffusivity (kinematic viscosity) to thermal diffusivity. That is, the Prandtl number is

given as:

� = � = � � � �ℎ � � (2.5)

where,

Kinematic viscosity

Thermal diffusivity

� means thermal diffusivity dominates � means momentum

diffusivity dominates. In heat transfer problems, the Prandtl number controls the

relative thickness of the momentum and thermal boundary layers. When Pr is small,

it means that the heat diffuses very quickly compared to the velocity (momentum).

This means that for liquid metals the thickness of the thermal boundary layer is much

bigger than the velocity boundary layer (Sherman, 1990). The dimensionless Prandtl

22

number is used as the integral part of the algorithm. The equation (2.5) is used in the

algorithm

Gases typically have Prandtl numbers in the range 0.7 -1 , while the Prandtl

number for most liquids is much larger than unity. The Prandtl number for water

ranges from 5-10, while that for an oil might be of the order of 50-100. It is not

uncommon to encounter Prandtl numbers for viscous liquids that are of the order of

several thousand or even larger. One exception to the general rule for liquids is a

liquid metal. Liquid metals conduct heat very efficiently, and therefore have a

relatively large value of the thermal diffusivity when compared with ordinary liquids.

On the other hand, viscosities are about the same as that of ordinary liquids. As a

consequence, the Prandtl number of a liquid metal is typically of the order of 10-2

(White, 2006; Hewitt et al., 1994). Since many researchers like Gururaj & Devi

(2014), Pal et al. (2014) and Rohni et al. (2012) have chosen Prandtl number around

7 for water nanofluids at 20℃. In this research Prandtl number is chosen as 6.2 for

nanofluids as most researchers use 6.2, the author has decided to use Prandtl number

as 6.2.

2.8.6 Chemical reaction

A chemical reaction is a process involving one or more substances called reactants,

characterised by a chemical reaction change and yielding one or more products

which or different from the reactants. A chemical change is defined as molecules

attaching to each other form large molecules, molecules breaking apart to form two

or more, smaller molecules or rearrangement of atoms within molecules (The

Encyclopaedia of Earth, 2011). The study of heat and mass transfer with chemical

reaction in the presence of nanofluids is of considerable importance in chemical and

hydrometallurgical industries. Chemical reaction can be codified as either

heterogeneous or homogeneous process. This depends on whether they occur at an

interface or as single phase volume reaction.

Homogenous reaction or chemical reaction in which the reactants are in the

same phase, while heterogeneous reactions have reactants in two or more phases.

Reactions that take place on the surface of a catalyst of a different phase are also

heterogeneous. A reaction between two gases, two liquids or two solids is

23

homogenous. A reaction between a gas and a liquid, a gas and a solid or a liquid and

a solid is heterogeneous. Practical application of heterogeneous reaction are in

catalytic converter, fuel cells and chemical vapour deposition among others, recently,

manufacturing engineers have surface reactions for synthesis of micro and nanoscale

in biomedical devices. In this research water is considered as a base fluid so chemical

reaction is not considered. If other base fluids like oil, glycol, and petrol are used

then chemical reaction should be considered.

2.8.7 Thermal conductivity

Choi & Eastman (1995) states thermal conductivity is the property of a material to

conduct heat. It is the quantity of heat transmitted through a unit thickness in a

direction normal to a surface of unit area, due to a unit temperature gradient under

steady state conditions. The ratio of nanofluid thermal conductivity � to the base

fluid thermal conductivity � (Buongiorno, 2009).

2.9 Stream Conditions

There are various other factors including stream conditions that influenced the

characteristics of the boundary layer flow over porous surfaces. Stream conditions

have some dimensionless parameters. Several physical parameters, which

characterised the flow often called as dimensionless numbers arise in the study of

nanofluid flows (Vandiver & Marcollo, 2003). In this section, few of these

dimensionless parameters are presented which is more importantly used in this

research. In this thesis, nanofluid flow velocity, temperature and nanoparticle volume

concentration on porous vertical stretching and shrinking surface are analysed. Some

of the factors to be considered while analysing nanofluid flow. Some of the factors

and stream conditions are as follows:

2.9.1 Magnetic effect

Magnetic field has two important mathematical properties that relate it to its sources.

The sources are currents and changing electric fields. These two properties, along

24

with the two corresponding properties of the electric field, make up Maxwell's

Equations. Maxwell's Equations together with the Lorentz force law form a complete

description of classical electrodynamics including both electricity and magnetism

(Huray & Paul, 2009). Magnetic field tends to produce Lorentz force to the flow

field. Lorentz force induced by the magnetic field plays a major role in flow field. In

this work, magnetic effect is given to observe the electrically conducting nanofluids

flow in the presence of magnetic field. The equation (2.6) is an integral part of

algorithm. = �� (2.6)

where,

Magnetic parameter Ω

� Electrical conductivity of the fluid Ω ∗

� Density of base fluid

Uniform transverse magnetic field strength �

� Velocity component on porous surface

2.9.2 Thermal radiation

Thermal radiation is electromagnetic radiation generated by the thermal motion of

charged particles in matter. All matter with a temperature greater than absolute zero

emits thermal radiation. When the temperature of the body is greater than absolute

zero, interatomic collisions cause the kinetic energy of the atoms or molecules to

change. This results in charge-acceleration and/or dipole oscillation which produces

electromagnetic radiation, and the wide spectrum of radiation reflects the wide

spectrum of energies and accelerations that occur even at a single temperature

(Sciuto, 2012). Heat and power can be harvested from the solar radiation or radiation

or from sun. this term is denoted by radiation. Unlike heat transfer methods

REFERENCES

Akbarinia, A. & Behzadmehr, A. (2007). Numerical study of laminar mixed

convection of a nanofluid in horizontal curved tubes. Applied Thermal

Engineering, 27(8), 1327-1337.

Acheson, D.J. (1990). Elementary fluid dynamics. NewYork: Oxford University

Press.

Anderson, J.D. (2005). Ludwig Prandtl’s boundary layer. Physics Today, 58(12), 42-

48.

Anderson, J.D. (2009). Basic Philosophy of Computational Fluid Dynamics. Berlin

Heidelberg: Springer. 3-14. doi: 10.1007/978-3-540-85056-4_1

Bachok, N., Ishak, A. & Pop, I. (2010). Boundary layer flow of nanofluids over a

porous surface in a flowing fluid. International Journal of Thermal Sciences,

49(9), 1663-1668.

Bitwise magazine. (2005). Power of two. Retrieved April 2, 2015, from

http://www.bitwisemag.com/copy/reviews/software/maths/maple10_mathema

tica52.html

Beer, J. & Bachmat, Y. (1990). Introduction to modelling of transport phenomena in

porous media. 1st

ed. Netherlands: Springer. doi: 10.1007/978-94-009-1926-6

Bouery, C. (2012). Contribution to algorithmic strategies for solving coupled

thermo-mechanical problems by an energy-consistent vibrational approach.

Ecole Centrale de Nantes: Ph.D. Thesis.

Brikhoff, G. (1948). Lattice theory. 25th

ed. New York: American Mathematical

Society.

Brikhoff, G. (1960). Hydrodynamics. New Jersey : Princeton University Press.

Buongiorno, J. (2006). Convective transport in nanofluids. Journal of Heat Transfer,

128(3), 240–250.

104

Buongiorno, J., Venerus, D.C., Prabhat, N., McKrell, T., Townsend, J., Christianson,

R. & Leong, K.C. (2009). A benchmark study on the thermal conductivity of

nanofluids. Journal of Applied Physics, 106(9), 094312-094312.

Chamkha, A.J. & Khaled, A.R.A. (2000). Hydromagnetic combined heat and mass

transfer by natural convection from a permeable surface embedded in a fluid-

saturated porous medium. International Journal of Numerical Methods for

Heat and Fluid Flow, 10(5), 455-477.

Chamkha, A.J., Mujtaba, M., Quadri, A. & Iss, C. (2003). Thermal radiation effects

on MHD forced convection flow adjacent to a non-isothermal wedge in the

presence of heat source or sink. Heat and Mass Transfer, 39(4), 425-441.

Chamkha, A.J., & Aly, A.M. (2010). MHD free convection flow of a nanofluid past

a vertical plate in the presence of heat generation or absorption effects.

Chemical Engineering Communications, 198(3), 2040-2044.

Chiam, T.C. (1994). Stagnation-point flow towards a stretching plate. Journal of

Physics Society, 63(6), 2443–2444.

Choi., S.U.S., (1995). Enhancing thermal conductivity of fluids with nanoparticles.

Fluids Engineering Division, 231, 99-103.

Choi, S.U.S., Zhang, Z.G., Yu, W., Lockwoow F.E. & Grulke, E.A. (2001).

Anomalous thermal conductivities enhancement on nanotube suspension.

Applied Physics Letters, 79(10), 2252–2254.

Das, S., Putra, N., Thiesen, P. & Roetzel, W. (2003). Temperature dependence of

thermal conductivity enhancement for nanofluids. Journal of Heat Transfer,

125(4), 567–574

Davidson, P.A. (2001). An Introduction to magnetohydrodynamics.1st ed. New York:

Cambridge University Press.

Duan. F., Kwek, D. & Crivoi, A. (2011). Viscosity affected by nanoparticle

aggregation in Al2O3-water nanofluids. Nanoscale Research Letters, 6(1), 1-

5.

Dulal, P. (2011). Combined effects of non-uniform heat source/sink and thermal

radiation on heat transfer over an unsteady stretching permeable surface,

Communications in Nonlinear Science and Numerical Simulation, 16(4),

1890–1904.

Eastman, J.A., Choi, S.U.S., Li, S. & Thompson, L.J. (1997). Enhanced thermal

conductivity through the development of nanofluids. Proc. of the fifth

105

Symposium on Nanophase and Nanocomposite Materials II, 457. USA.

Materials Research Society. 3–11.

Eastman, J.A., Choi, S.U.S., Yu, W. & Thompson, L.J. (2001). Anomalously

increased effective thermal conductivity of ethylene glycol-based nanofluids

containing copper nanoparticles. Applied Physics Letters, 78(6), 718–720.

Evans, W., Fish, J. & Keblinski, P. (2006). Role of Brownian motion hydrodynamic

on nanofluid thermal conductivity. Applied Physics Letters, 88(9), 5-8. doi:

10.1063/1.2179118.

Fang, T.G., & Zhang, J. (2010). Thermal boundary layers over a shrinking sheet. An

Analytical Solution of Mechanical, 209, 325–343.

Gharagozloo, P.E. & Goodson, K.E. (2011). Temperature-dependent aggregation and

diffusion in nanofluids. International Journal of Heat and Mass Transfer,

54(4), 797-806.

Gururaj, A.D.M. & Devi, S.P.A. (2014). MHD boundary layer flow with forced

convection past a nonlinearly stretching surface with variable temperature and

nonlinear radiation effects. International Journal of Development Research,

4(1), 75-80.

Hamad, M.A.A. (2011). Analytical solution of natural convection flow of a nanofluid

over a linearly stretching sheet in the presence of magnetic field.

International Communication in Heat and Mass Transfer, 38(4), 487-492.

Hamad, M.A.A. & Ferdows, M. (2012). Similarity solution of boundary layer

stagnation-point flow towards a heated porous stretching sheet saturated with

a nanofluid with heat absorption/generation and suction/blowing: A lie group

analysis. Communications in Nonlinear Science and Numerical Simulation.

17(1), 132–140. doi: 10.1016/j.cnsns.2011.02.024

Han, Z. (2008). Nanofluids with enhanced thermal transport properties. University

of Maryland: Ph.D. Thesis.

Hewitt, G. F., Shires, G. L. & Bott, T. R. (1994). Heat Transfer. Florida: CRC Press.

Huray, P.G. (2009). Maxwell’s Equation. 2nd ed. Hoboken: Wiley-IEEE Press.

Ishak, A., Nazar, R. & Pop, I. (2006). Magnaetohydrodynamic stagnation point flow

towards a stretching vertical sheet. Magnaetohydrodynamic, 42(1), 17-30.

Ishak, A., Nazar, R. & Pop, I. (2008). Magnetohydrodynamic flow and heat transfer

due to stretching surface. Energy Conversation and Management, 49(11),

3265-3269.

106

Incropera, F.P., Lavine, A.S. & DeWitt, D.P. (2011). Fundamentals of heat and mass

transfer. Danvers: John Wiley & Sons Incorporated.

Kafoussias, N.G. & Nanousis, N.D. (1997). Magnetohydrodynamic laminar

boundary layer flow over a wedge with suction or injection. Canadian

Journal of Physics, 75(10), 733-741.

Kandasamy, R., Loganathan, P. & Arasu, P.P. (2011). Scaling group transformation

for MHD boundary-layer flow of a nanofluid past a vertical stretching surface

in the presence of suction/injection. Nuclear Engineering and Design, 241,

2053–2059.

Kandasamy, R., Muhaimin, I. & Mohamad, R. (2013). Thermophoresis and

Brownian motion effects on MHD boundary-layer flow of a nanofluid in the

presence of thermal stratification due to solar radiation. International Journal

of Mechanical Sciences, 10(3), 760-771.

Kahar, A.R. (2011). Scaling group transformation for boundary-layer flow of a

nanofluid past a porous vertical stretching surface in the presence of chemical

reaction with heat radiation. Computers and Fluids, 52 (1).15-21.

Kahar, A.R. (2014). Heat transfer of nanofluid flow over porous surfaces with

variable stream conditions. Universiti Tun Hussein Onn Malaysia: Ph.D.

Thesis.

Khan, W.A. & Pop, I. (2010). Boundary layer flow of a nanofluid past a stretching

sheet. International Journal of Heat and Mass Transfer, 53(11-12), 2477-

2483.

Khan, M.S., Karim, I., Ali, LE. & Islam, A. (2012). Unsteady MHD free convection

boundary-layer flow of a nanofluid along a stretching sheet with thermal

radiation and viscous dissipation effects. International Nano Letter, 24(2),

760-771. doi:10.1186/2228-5326-2-24.

Kuznetsov, A.V. & Nield, D.A. (2010). Natural convective boundary-layer flow of a

nanofluid past a vertical plate. International Journal of Thermal Science,

49(2), 243–247.

Kuznetsov, A.V. & Nield, D.A. (2011). Double diffusive natural convective

boundary-layer flow of a nanofluid past a vertical plate. International Journal

of Thermal Science, 50(5), 712-717.

107

Lapwood, E.R. (1948). Convection of fluid in a porous medium. Mathematical

Proceedings of the Cambridge Philosophical Society, 44(4), 508-521. New

York: Cambridge University Press.

Lok, Y.Y., Ishak, A. & Pop, I. (2011). MHD stagnation point flow with suction

towards a shrinking sheet. Sains Malaysia, 40(10), 1179–1186.

Mahapatra, T.R. & Gupta, A.G. (2002). Heat transfer in stagnation point flow

towards a stretching sheet. Heat Mass Transfer, 38(6), 517–521.

Mahapatra, T.R. & Gupta, A.S. (2001). Magnetohydrodynamics stagnation-point

flow towards a stretching sheet. Acta Mechanica, 152(1-4), 191–196.

Mahapatra, T.R., Nandy, S.K. & Gupta, A.S. (2009). Magnetohydrodynamic

stagnation point flow of a power-law fluid towards a stretching sheet.

International Journal of Non-Linear Mechanic, 44(2), 124–129.

Mahapatra, T.R. & Nandy, S.K. (2011). unsteady stagnation-point flow and heat

transfer over an unsteady shrinking sheet. International Journal of Applied

Mathematics & Mechanics, 16(7), 11-26.

Mahapatra, T.R. & Nandy, S.K. (2013). Stability of dual solutions in stagnation-

point flow and heat transfer over a porous shrinking sheet with thermal

radiation. Meccanica, 48, 23-32.

Maiga, S. E. B., Nguyen, C. T., Galanis, N. & Roy, G. (2004), "Heat transfer

behaviour of nanofluids in uniformly heated tube", Superlattice and

Microstructure 35(3-6), 543-557.

Roy, G., S.J. Palm, S.J. & Nguyen, C.T. (2005). Heat Transfer enhancement by

using nanofluids in forced convection flows. International Journal of Heat

and Fluid Flow, 26(4), 530-546.

Makinde, O., Khan, W. & Khan, Z. (2013). Buoyancy effects on MHD stagnation

point flow and heat transfer of a nanofluid past a convectively heated

stretching/shrinking sheet. International Journal of Heat and Mass Transfer,

62(0), 526-533.

Maplesoft. (2015). Maplesoft Services. Retrieved on June 24th

, 2014, from

http://www.maplesoft.com/brochures/PDFs/maple18/Maplesoft_SolutionsEng

ineers.pdf

Masuda, H., Ebata A., Teramae K. & Hishinuma N. (1993). Alteration of thermal

conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu

Bussei, 7(4), 227–233.

108

Mankinde, O.D. & Aziz, A. (2011). Boundary layer flow of a nanofluid past a

stretching sheet with convective boundary conditions. International Journal

on Thermal Science, 50(5), 1326-1332.

Miklavcic, M. & Wang, C.Y. (2006). Viscous flow due to a shrinking sheet. Applied

Mathematics, 64(2), 283-290.

Mori, H. (1965). Transport, collective motion, and Brownian motion. Progress of

Theoretical Physics, 33(3), 423-455.

Murthy, P.V. & Singh, P. (1997). Effect of viscous dissipation on a non-Darcy

natural convection regime. International Journal of Heat and Mass Transfer,

40(6), 1251-1260.

Murshed, S.M.S., Leong, K.C. & Yang, C. (2005). Enhanced thermal conductivity of

TiO2-water based nanofluids. International Journal of Thermal Sciences,

40(4), 367-373.

Murshed, S.M.S., Leong, K.C. & Yang, C. (2009). A combined model for the

effective thermal conductivity of nanofluids. Applied Thermal Engineering,

29(11), 2477-2483.

Nan, C.W., Birringer, R., Clark, D.R. & Glieter, H. (1997). Effective thermal

conductivity of particulate composite with interfacial thermal resistance.

Journal of Applied Physics, 81(5), 6692 -6699.

Nazar, R., Amin, N., Filip, D. & Pop, I. (2004). Unsteady boundary layer flow in the

region of the stagnation point on a stretching sheet. International Journal of

Engineering and Science, 42, 1241–1253.

Nield, D.A. & Kuznwtsov, A.V. (2009). Thermal instability of porous medium layer

saturated by a nanofluid. International Journal of Heat and Mass Transfer,

52(25), 5796-5801.

Nourazar, S.S., Habibi, M. M. & Simiari, M. (2011). The HPM applied to MHD

nanofluid flow over a horizontal stretching plate. Journal of Applied

Mathematics, 1-17. doi:10.1155/2011/876437.

Nguyen, C.T., Desgranges, F., Roy, G., Galanis, N., Mare, T., Boucher, S. & Angue,

M. H. (2007). Temperature and particle-size dependent viscosity data for

water based nanofluids-hysteresis phenomenon. International Journal of Heat

and Fluid Flow, 28(6), 1492-1506.

Oosthuizen, P.H. & Naylor, D. (1999). An introduction to convective heat transfer

analysis. New York: WCB/McGraw Hill.

109

Pal, D., Mandal, G. & Vajravelu, K. (2014). Flow and heat transfer of nanofluids at a

stagnation point flow over a stretching/shrinking surface in a porous medium

with thermal radiation. Applied Mathematics and Computation, 238, 208–

224.

Penas, J.R.V., Ortiz de Zarare, J.M. & Khayet, M. (2008). Measurement of the

thermal conductivity of nanofluids by the multi current hotwire method.

Journal of Applied Physics, 104(4), 044314.

Pop, S.R., Grosan, T. & Pop, I. (2004). Radiation effects on the flow near the

stagnation point of a stretching sheet. Technische Mechanik, 25(2), 100-106.

Purcell, E.M. (1997). Life at low Reynolds number. American Journal of Physics,

45(1), 3-11.

Putnam, S.A., Cahill, D.G., Braun, P.V., Ge, Z. & Shimmim, R.G. (2006). Thermal

conductivity of nanoparticle suspensions. Journal of Applied Physics, 99(8),

084308-084308.

Rohni, A.M., Ahmad, S. & Pop, I. (2012). Flow and heat transfer over an unsteady

shrinking sheet with suction in nanofluids. International Journal of Heat and

Mass Transfer, 55(7–8), 1888–1895.

Sanders, C.J. & Holman, J.P. (1972). Franz Grashof and Grashof number.

International Journal of Heat and Mass Transfer, 15(3), 562-563.

Sandnes, B. (2003). Energy efficient production, storage and distribution of solar

energy. University of Oslo: Ph.D. Thesis.

Samir, K. N. & Ioan Pop, (2014). Effects of magnetic field and thermal radiation on

stagnation flow and heat transfer of nanofluid over a shrinking surface.

International Communication in Heat and Mass Transfer, 53(4), 50-55.

Sciuto, G. (2012). Innovative latent heat thermal storage elements design based on

nanotechnologies. University of Trieste: Ph.D Thesis.

Siegel, R. & Goldstein, M.E. (1972) Theory of heat transfer in a two-dimensional

porous cooled medium and application to an eccentric annular region. Journal

of Heat Transfer, 94(4), 425-431.

Shaughnessy, E.J., J., Katz, I. M., & Schaffer, J. P. (2005). Introduction to Fluid

Mechanics. New York: Oxford University Press.

Sherman, F.S. (1990). Viscous Flow. New York: McGraw-Hill, Inc.

Sparrow, E.M. & Cess, R.D. (1978). Radiation Heat Transfer. Washington:

Hemisphere.

110

The Encyclopaedia of Earth. (2011). Chemical reaction. Retrieved on January 10th

,

2015, from http://www.eoearth.org/view/article/171508/

Vandiver, J.K., & Marcollo, A. (2003). High mode number VIV experiments. In

IUTAM symposium on integrated modelling of fully coupled fluid structure

interactions using analysis, computations and experiments, 75, 211-231.

Venerus, D., Buongiorno, J., Christianson, R., Townsend, J., Bang, I.C., Chen, G. &

Zhou, S. (2010). Viscosity measurements on colloidal dispersions

(nanofluids) for heat transfer applications. Journal of Applied Rheology,

20(4), 44582. doi: 10.3933/ApplRheol-20-44582.

Wang. C.Y. (2008). Stagnation point flow towards a shrinking sheet. International

Journal of Non-Linear Mechanical systems, 43, 377–382.

Welty, J.R., Wicks, C.E., Rorrer, G. & Wilson, R.E. (2009). Fundamentals of

Momentum, Heat and Mass Transfer. Hoboken: John Wiley &Sons.

White, F. M. (2006). Viscous Fluid Flow. 3rd

ed. New York: McGraw-Hill.

Xuan, Y. & Li, Q. (2000). Heat transfer enhancement of nanofluids. International

Journal of Heat and Fluid Flow, 21(1), 58-64.

Yurusoy, M. & Pakdemirili, M. (1997). Symmetry reductions of steady two

dimensional boundary layers of some non-Newtonian Fluids. International

Journal of Engineering Science, 35(2), 731-740.

Yurusoy, M. & Pakdemirili, M. (1999a). Group classification of a non-Newtonian

fluid model using classical approach and equivalence transformations.

International Journal of Non-linear Mechanics, 34(2), 341-346.

Yurusoy, M. & Pakdemirili, M. (1999b). Exact solutions of boundary layer equations

of a special non-Newtonian fluid over a stretching surface. Mechanics

Research communications, 26(1), 171-175.

Yurusoy, M., Pakdemirili, M. & Noyan. (2001). Lie Group analysis of creeping flow

of a second grade nanofluid. International Journal of Non-Linear Mechanics,

36(6), 955-960.