Chapter 1

Introduction

In this thesis we have analysed the analytical and numerical solutions of coupled

nonlinear differential equations arising in free, forced and mixed convection flow through

various geometries for Newtonian and non-Newtonian fluids by differential transform

method, finite difference method, Runge-Kutta shooting method and perturbation

method.

1.1 Objective and Scope of the Thesis

Most scientific problems in fluid mechanics are inherently nonlinear. All these problems

and phenomena are modeled by ordinary or partial differential equations. Solutions

of these differential equations are important in predicting the future states of the

phenomena under study. Such physical phenomena include the motion of planets,

nonlinear optics, oceanography, meteorology, projectiles, fluid dynamics and population

dynamics to mention just a few. Most of these equations are highly nonlinear and

exact solutions are not always possible. For those cases where exact solutions are not

possible, numerical methods often provide approximate solutions (Nayfeh, 1973). Both

numerical and analytical methods have their advantages and drawbacks. This study

sought to introduce new and improved semi-numerical-analytical technique known as

1

Introduction

DTM for solving nonlinear equations. This technique aims to combine the strength of

both numerical and analytical methods.

The DTM is a method for solving a wide range of problems whose mathematical

models yield equations or systems of equations involving algebraic, differential, integral

and integro-differential. Applications of the DTM to various problems in science and

engineering fields include solutions of the Blasius and difference equations, vibration

equations, singular two-point boundary value problems, convective straight fin problem

with temperature-dependent thermal conductivity, fractional modified Korteweg–de

Vries equation and other nonlinear differential equations.

In addition, flow and heat transfer continues to be a major field of interest to

engineering and scientific researchers, as well as designers, developers and manufacturers.

Considerable effort has been devoted to research in traditional applications such as

chemical processing, general manufacturing and energy devices including general power

systems, heat exchangers and high performance gas turbines. Also a significant number

of papers address topics that are at the frontiers of both fundamental research and

important emerging applications such as micro - channel flows, bioheat transfer,

electronics cooling, semiconductors and a number of natural phenomena ranging from

upwelling currents in the oceans to heat transport in stellar atmospheres. The vertical

channel geometry has been widely studied in heat transfer because of its fundamental

importance, and its many applications, including electronic cooling, and thermal

environmental control of dwelling. The interaction between buoyancy stemming from

one or more heated elements inside channel and the imposed forced flow forms the

topic of the current investigation.

Nanofluids are engineered by suspending nanoparticles with average sizes below

100nm in traditional heat transfer fluids such as water, oil, and ethylene glycol. A

very small amount of guest nanoparticles, when dispersed uniformly and suspended

stably in host fluids (base fluids), can provide dramatic improvements in the thermal

2

Introduction

properties of host fluids. Nanofluids (nanoparticle fluid suspensions) is the term coined

by Choi (1995) to describe this new class of nanotechnology-based heat transfer fluids

that exhibit thermal properties superior to those of their host fluids or conventional

particle fluid suspensions. Nanoparticles used in nanofluids have been made of various

materials, such as oxide ceramics (Al2O3, CuO), nitride ceramics (AlN, SiN), carbide

ceramics (SiC, TiC), metals (Cu, Ag, Au), semiconductors (TiO2, SiC), carbon

nanotubes, and composite materials such as alloyed nanoparticles or nanoparticle

core–polymer shell composites. In addition to nonmetallic, metallic, and other materials

for nanoparticles, completely new materials and structures, such as materials “doped”

with molecules in their solid-liquid interface structure, may also have desirable

characteristics. The goal of nanofluids is to achieve the highest possible thermal

properties at the smallest possible concentrations (preferably less than 1% by volume)

by uniform dispersion and stable suspension of nanoparticles in host fluids. To achieve

this goal it is vital to understand how nanoparticles enhance energy transport in liquids.

Theoretical and applied research in flow, heat, and mass transfer in porous media

has received increased attention during the past four decades. This is due to the

importance of this research area in many engineering applications. Significant advances

have been made in modeling fluid flow, heat, and mass transfer through a porous

medium including clarification of several important physical phenomena. For example,

the non-Darcy effects on momentum, energy, and mass transport in porous media have

been studied in depth for various geometrical configurations and boundary conditions.

Many of the research works in porous media for the past couple of decades utilize what

is now commonly known as the Brinkman–Forchheimer–extended Darcy model or the

generalized model. Literature concerning convective flow in porous media is abundant.

Magnetohydrodynamics (MHD) is the science which deals with the motion of highly

conducting fluids in the presence of a magnetic field. The motion of the conducting

fluid across the magnetic field generates electric currents which change the magnetic

3

Introduction

field, and the action of the magnetic field on these currents gives rise to mechanical

forces which modify the flow of the fluid. The research in MHD flows was stimulated

by two problems: the protection of space vehicles from aerodynamic overheating and

destruction during the passage through the dense layers of the atmosphere; the

enhancement of the operational ability of the constructive elements of high temperature

MHD generators for direct transformation of heat energy into electric current. The

first problem showed that the influence of a magnetic field on ionized gases is a

convenient control method for mass, heat and hydrodynamic processes. The MHD

research extended gradually to new applied problems. The MHD devices for liquid

metals attracted the attention of metallurgist. It was shown that the effect of magnetic

field could be very helpful in the modernization of technological processes. One of the

ways of studying magnetohydrodynamic heat transfer field is the electro-magnetic field,

which is used to control the heat transfer as in the convection flows and aerodynamic

heating. Electroconducting fluid has been increasingly used in the manufacturing

processes of semiconducting material such as silicon crystal or gallium arsenide.

There are a variety of thermal boundary conditions in practical channel flow

applications. The typical thermal boundary condition could be either a uniform heat

flux or a uniform wall temperature. In practice, the reality lies somewhere in between

these two extremes.

The last two decades of the present century also witnessed a great surge in the use of

fluid systems involving non-Newtonian fluid as a working medium. This encompasses

in its realm complex liquids, e.g., polymeric suspensions, animal blood, liquid crystals,

which have very small-sized suspended particles of different shapes. These particles

may change shape, may shrink and expand, and moreover, they may execute rotation

independent of the rotation and movement of the liquid. Due to the addition of particles

the classical Navier-Stokes theory had to be re-examined. Most practical problems

involving these types of working liquids are non-isothermal and these thermally

4

Introduction

responding liquids have uncovered new application areas.

The dynamics of micropolar fluid has attracted considerable attention during the

last few decades because traditional Newtonian fluids cannot precisely describe the

characteristics of fluid flow with suspended particles. Eringen (1966) developed the

theory that the local effects arising from the microstructure and the intrinsic motion of

the fluid elements should be taken into account. The theory is expected to provide a

mathematical model for the Non - Newtonian fluid behavior observed in certain man -

made liquids such as polymers, lubricants, fluids with additives, paints, animal blood,

colloidal and suspension solutions, etc. The presence of dust or smoke, particularly in

a gas, may also be modeled using micropolar fluid dynamics. Micropolar fluids have

recently received considerable attention due to their potential application in many

industrial processes; for example, in continuous casting glass-fiber production, paper

production, metal extrusion, hot rolling, wire drawing, drawing of plastic films, metal

and polymer extrusion and metal spinning.

The micro-continuum theory of couple stress fluid proposed by Stokes (1966),

defines the rotational field in terms of the velocity field for setting up the constitutive

relationship between the stress and strain rate. Stokes micro-continuum theory is the

simplest generalization of the classical theory of fluids, which allows for polar effects

such as the presence of couple stresses, body couples and a non-symmetric stress tensor.

The theoretical study of blood flow indicates that some of the non-Newtonian flow

properties of blood may be explained by assuming the blood to be a fluid with couple

stress. The couple-stress fluid may be considered as a special case of a non-Newtonian

fluid which is intended to take into account the particle size effects. Moreover, the

couple stress fluid model is one of the numerous models that proposed to describe

response characteristics of non-Newtonian fluids. The constitutive equations in these

fluid models can be very complex and involving a number of parameters, also the

outcoming flow equations lead to boundary value problems in which the order of

5

Introduction

differential equations is higher than the Navier–Stokes equations.

With the above discussion in mind, the goal of this thesis is to provide a novel

technique to extend the DTM to analyze the flow and heat transfer characteristics in

different geometries. The governing differential equations along with the corresponding

boundary conditions of the following problems are solved by the DTM.

1. Mixed convective flow in a vertical channel filled with porous medium using

Differential Transform method.

2. Comparison study of Differential Transform method with Finite Difference method

and Perturbation method for magneto-convection in a vertical channel.

3. Combined effect of variable viscosity and thermal conductivity on free convection

flow of a viscous fluid in a vertical channel.

4. Convective heat transfer in a vertical channel filled with nanofluid.

5. Influence of viscous dissipation on non-Darcy mixed convection flow of a nanofluid

in a channel.

6. Effect of MHD on Jeffery-Hamel flow in nanofluids by Differential Transform

method.

7. Flow and heat transfer in a porous medium saturated by a micropolar fluid

between parallel permeable disks.

8. Flow and heat transfer of a micropolar fluids separated by viscous fluid layer.

9. Mixed convective flow of a couple stress fluid with Robin boundary conditions.

10. Flow and heat transfer of couple stress fluid in a vertical channel in the presence

of heat source/sink.

6

Introduction

11. Free convection flow of an electrically conducting micropolar fluid between parallel

porous vertical plates.

The following section reviews the literature in the aforementioned areas.

1.2 Literature Review

The study of free, forced and mixed convection in a vertical/horizontal channel has

received much attention and has been studied extensively because of their importance

in many braches of science and engineering.

In this section, therefore, we review some of the important works related to the

above mentioned problem in brief with the motive of paving way for further study.

1.2.1 History of fluid mechanics

Before proceeding with our study of fluid mechanics, we should pause for a moment

to consider the history of this important engineering science. As is true of all basic

scientific and engineering disciplines, their actual beginnings are only faintly visible

through the haze of early antiquity. But we know that interest in fluid behavior dates

back to the ancient civilizations. Through necessity there was a practical concern

about the manner in which spears and arrows could be propelled through the air, in

the development of water supply and irrigation systems, and in the design of boats and

ships. These developments were, of course, based on trial-and-error procedures without

any knowledge of mathematics or mechanics. However, it was the accumulation of

such empirical knowledge that formed the basis for further development during the

emergence of the ancient Greek civilization and the subsequent rise of the Roman

Empire. Some of the earliest writings that pertain to modern fluid mechanics are

those of Archimedes (287–212 B.C.), a Greek mathematician and inventor who first

expressed the principles of hydrostatics and flotation. Elaborate water supply systems

7

Introduction

were built by the Romans during the period from the fourth century B.C. through the

early Christian period, and Sextus Julius Frontinus (A.D. 40–103), a Roman engineer,

described these systems in detail. However, for the next 1000 years during the Middle

Ages (also referred to as the Dark Ages), there appears to have been little added to

further understanding of fluid behavior.

1200 1300 1400 1500 1600 1700 1800 1900 2000

Geoffrey Taylor

Theodor von Karman

Ludwig Prandtl

Osborne Reynolds

Ernst Mach

George Stokes

Jean Poiseuille

Louis Navier

Leonhard Euler

Daniel Bernoulli

Isaac Newton

Galileo Galolei

Year

Leonardo da Vinci

Figure 1.1: Time line of some contributors to the science of fluid mechanics.

As shown in Fig. 1.1, beginning with the Renaissance period (about the fifteenth

century) a rather continuous series of contributions began that forms the basis of what

we consider to be the science of fluid mechanics. Leonardo da Vinci (1452–1519)

described through sketches and writings many different types of flow phenomena. The

work of Galileo Galilei (1564–1642) marked the beginning of experimental mechanics.

Following the early Renaissance period and during the seventeenth and eighteenth

centuries, numerous significant contributions were made. These include theoretical

and mathematical advances associated with the famous names of Newton, Bernoulli,

Euler, and d’Alembert. Experimental aspects of fluid mechanics were also advanced

8

Introduction

during this period, but unfortunately the two different approaches, theoretical and

experimental, developed along separate paths. Hydrodynamics was the term associated

with the theoretical or mathematical study of idealized, frictionless fluid behavior,

with the term hydraulics being used to describe the applied or experimental aspects

of real fluid behavior, particularly the behavior of water. Further contributions and

refinements were made to both theoretical hydrodynamics and experimental hydraulics

during the nineteenth century, with the general differential equations describing fluid

motions that are used in modern fluid mechanics being developed in this period.

Experimental hydraulics became more of a science, and many of the results of

experiments performed during the nineteenth century are still used today.

In addition to practical hydromechanics, analytical fluid mechanics developed in

the nineteenth century, in order to solve analytically manageable problems. Euler and

his colleagues, the next significant analytical advance was the addition of frictional-

resistance terms to Euler’s inviscid equations. This was done, with varying degrees of

elegance, by Navier in 1827, Cauchy in 1828, Poisson in 1829, St. Venant in 1843 and

George Gabriel Stokes in 1845 made analytical contributions to the fluid mechanics of

viscous media, especially to wave mechanics and to the viscous resistance of bodies,

and formulated Stokes’ law for spheres falling in fluids. The first four wrote their

equations in terms of an unknown molecular function, whereas Stokes was the first to

use the coefficient of viscosity µ. Today these equations, which are fundamental to

the subject, are called the Navier-Stokes relations. Osborne Reynolds (1832–1912), a

new chapter in fluid mechanics was opened. He carried out pioneering experiments

in many areas of fluid mechanics, especially basic investigations on different turbulent

flows. He demonstrated that it is possible to formulate the Navier–Stokes equations

in a time-averaged form, in order to describe turbulent transport processes in this

way. Essential work in this area by Ludwig Prandtl (1875–1953) followed, providing

fundamental insights into flows in the field of the boundary layer theory. In 1904

9

Introduction

Prandtl’s idea was that for flow next to a solid boundary a thin fluid layer (boundary

layer) develops in which friction is very important, but outside this layer the fluid

behaves very much like a frictionless fluid. This relatively simple concept provided

the necessary impetus for the resolution of the conflict between the hydrodynamicists

and the hydraulicists. Prandtl is generally accepted as the founder of modern fluid

mechanics (see Young et al., 2011).

The rapid progress that has been achieved in the last few decades in the field of

numerical fluid mechanics should also be mentioned. Considerable developments in

applied mathematics took place to solve partial differential equations numerically. In

parallel, great improvements in the computational performance of modern high-speed

computers occurred and computer programs became available that allow one to solve

practical flow problems numerically. Numerical fluid mechanics has therefore also

become an important sub-domain of the entire field of fluid mechanics. Its significance

will increase further in the future.

Mixed convection in vertical or inclined channels has been widely studied due to

its importance for several engineering applications that range from cooling devices for

electronic equipments to the design of passive solar systems for energy conversion.

A wide literature on this subject has appeared in the last decades, including several

theoretical studies where analytical solutions of the governing balance equations

have been presented (see, Al-Nimr and El-Shaarawi, 1995; Aung and Worku, 1986b;

Barletta and Zanchini, 1999; Barletta et al., 2005; Boulama and Galanis, 2004; Buhler,

2003; Chamkha, 2002; Hamadah and Wirtz, 1991; Kumar et al., 2011b; Lavine, 1988;

Mahmood and Merkin, 1989; Umavathi and Shekar, 2011, 2013a; Umavathi et al., 2010c,

2012c,e; Weidman, 2006).

10

Introduction

1.2.2 Differential Transform method

The Differential Transform method was first proposed by Zhou (1986) for solving

linear and non-linear initial value problems in electrical circuit analysis. Based on

Taylor’s series expansion, the DTM provides an effective and simple means of solving

linear and non-linear differential equations. By using DTM, Chen and Ho (1996)

investigated the eigenvalues of Strum-Liouville problem. Chen and Ho (1999a) also

used this method to study the transverse vibration of rotating twisted Timoshenko

beams under axial loading. Chen and Ho (1999) also developed DTM for partial

differential equations and obtained closed form series solutions for linear and nonlinear

initial value problems. Jang et al. (2001) applied the two-dimensional DTM to the

solution of partial differential equations. Abdel-Halim Hassan (2002) adopted the DTM

to solve some eigen value problems. Chen and Chen (2004) studied the free vibration

of a conservative oscillator with inertia and static cubic non-linearity and pointed out

that the DTM has the inherent ability to deal with non-linear problems and it can be

employed for the solutions of both ordinary and partial differential equations and Ayaz

(2004) applied it to the system of differential equations. Kumaz and Oturanc (2005)

applied DTM for solution of system of ordinary differential equations. Kumaz et al.

(2005) generalized the DTM to n-dimensional cases for solving partial differential

equations with n-variables. It was shown that the DTM is a feasible tool to solve

n-dimensional linear or nonlinear partial differential equations. Arikhoglu and Ozkol

(2006) employed DTM on differential-difference equations. Furthermore, the method

may be employed for the solution of partial differential equations. Kanth and Aruna

(2008a) propose a reliable algorithm to develop exact and approximate solutions for

the linear and non-linear systems of partial differential equations. Kanth and Aruna

(2008b) also find the solutions of singular two-point boundary value problems using

DTM. Mei (2008) utilized the DTM to analyze the free vibration of a centrifugally

stiffened beam. Odibat and Momani (2008) using generalized DTM find the solutions

11

Introduction

for linear partial differential equations of fractional order. Shahed (2008) applied the

DTM to non-linear oscillatory systems of equations. Erturk et al. (2008) applied

the generalized differential transform method to multi-order fractional differential

equations. Chu and Chen (2008) presented a hybrid Differential Transformation and

finite difference method to analyze nonlinear transient heat conduction problems.

Chen and Chen (2009) investigated the free vibration behavior of a strongly non-linear

oscillator with fifth-order non-linearities and showed that the DTM is a powerful

tool for solving non-linear problems. They also pointed out that the DTM provides

an accurate and efficient way for solving differential equations with high-order

nonlinearities. Balkaya et al. (2009) adopted the DTM to analyze the vibration

of an elastic beam supported on elastic soil. Kanth and Aruna (2009b) propose a

reliable algorithm to develop exact and approximate solutions for the linear and

nonlinear Schrodinger equations. Kanth and Aruna (2009a) also solve the nonlinear

Klein-Gordon equation using DTM. Lo and Chen (2009) proposed an alternative

numerical method to investigate hyperbolic heat conduction problems using the hybrid

differential transfer/control-volume method. Joneidi et al. (2009a) obtained analytical

solution of fin efficiency of convective straight fins with temperature-dependent thermal

conductivity by DTM. Rashidi and Erfani (2009) used the DTM to solve Burgers’ and

nonlinear heat transfer equations. Jang et al. (2010) investigated and characterized

a two-dimensional thermal conductive boundary value problem with discontinuous

boundary and initial conditions. Rashidi et al. (2010) applied DTM to find the analytic

solution for the problem of mixed convection about an inclined flat plate embedded

in a porous medium. Rashidi and Pour (2010) applied DTM to steady flow over

a rotating disk in porous medium with heat transfer. Reza and Borhanifar (2010)

studied the Burgers’ and coupled Burgers’ equations by DTM. Borhanifar and Abazari

(2010) studied the nonlinear Schrodinger and coupled Schrodinger equations by using

Differential Transform method. Keskin and Oturanc (2010) presented the reduced

12

Introduction

DTM for standard and fractional partial differential equations. Ni et al. (2011) using

DTM analyzed the free vibration problem of pipes conveying fluid with several typical

boundary conditions. Yaghoobi and Torabi (2011) solved the two nonlinear heat

transfer equations by considering variable specific heat coefficient. Ghafoori et al.

(2011) find the solutions of nonlinear oscillation equation using DTM and compare

the results with variation iteration method and homotopy perturbation method.

Keimanesh et al. (2011) studied the third grade non-Newtonian fluid flow between

two parallel plates using the multi-step DTM. Biazar et al. (2012) employed DTM to

solve special kinds of systems of integral equations. Villafuerte and Chen-Charpentier

(2012) analyzed a DTM based on the mean fourth calculus to solve random differential

equations. Umavathi et al. (2012d) studied the convective heat transfer in a vertical

channel filled with a nanofluid using DTM and compared their results with finite

difference and regular perturbation method. Aruna and Kanth (2013) obtained

the approximate solutions of non-linear fractional Schrodinger equation using DTM

and modified DTM. Umavathi et al. (2013) compared results of DTM with finite

difference method and perturbation method for magnetoconvection in a vertical

channel. Caponetto and Fazzino (2013) found an application of DTM, to the area of

fractional differential equation for calculating Lyapunov exponents of fractional order

systems. Deka and Paul (2013) studied the stability of Taylor-Couette and Dean Flow

using DTM. Merdan (2013) obtained the numerical solution of the fractional-order

Vallis systems using multi-step DTM. Umavathi and Shekar (2013c) analyze the mixed

convective flow in a vertical channel filled with porous medium using DTM.

In a latest literature, a generalized DTM (GDTM), i.e., the Differential Transform-

Pade technique, mixed by DTM and Pade approximation, was proposed and used

to solve differential-difference equation successfully by Zou et al. (2009). Chen et al.

(2009) studied the natural frequencies and mode shapes of marine risers with different

boundary conditions by using the DTM. Odibat et al. (2010) proposed a reliable new

13

Introduction

algorithm of DTM, namely multi – step DTM, which will increase the interval of

convergence for the series solution, and applied the multi-step DTM to study the

non-chaotic and chaotic dynamics of Lotka – Volterra, Chen and Lorenz systems.

Rashidi and Erfani (2011) obtained the similarity solution for the MHD Hiemenz flow

against a flat plate with variable wall temperature in a porous medium by using a

novel analytical method called DTM-Pade technique. Gokdogana et al. (2012) propose

a fast and effective adaptive algorithm for the multi-step DTM. Kanth and Aruna

(2012) compare the solutions of two dimensional DTM and GDTM for time-dependent

Emden-Fowler type equations. Reza and Malek (2012) solve the generalized Hirota-

Satsuma coupled Korteweg–de Vries equation using by two methods, DTM and reduced

form of DTM.

These works confirmed the fact that this methodology is reliable, efficient and

promising as well as having a wider applicability.

1.2.3 Flow through porous media

A porous medium or a porous material is a solid permeated by an interconnected

network of pores (voids) filled with a fluid (liquid or gas). In most of the investigations

the medium is assumed to be continuous so as to form two interpenetrating continua

such as sponge. Porous media can be classified as consolidated porous media and

unconsolidated-porous media. Examples of consolidated media are charcoal, human

lungs, naturally occurring rocks, sand stones, lime stones etc. Examples of

unconsolidated media are beach sand, glass beds, catalyst pellets, soil gravel, etc. Flow

through porous media is prevalent in nature, and therefore the study of this type of

flow has attracted the attention of many researchers in scientific and engineering fields.

It is in particular, of great importance to petroleum engineers concerned with the

movement of gas, and to chemical engineers in connection with filtration processes etc.

For discussing the flow and heat transfer through porous media, as in hydrodynamics,

14

Introduction

the governing equations describing conservation of mass, momentum and energy are

written. Conservation of mass gives rise to the equation of continuity, conservation

of momentum gives rise to the equations of motion and conservation of energy gives

the energy equation. While writing equations describing conservation of momentum

for flow in a porous medium, different concepts can be made which results in different

forms of the momentum equations.

The flow through porous media is modeled mathematically, mainly in three models

(i) Darcy’s model (ii) Brinkman’s model (iii) Forchheimer’s model.

In fluid dynamics and hydrology, Darcy’s law is a phenomenologically derived

constitutive equation that describes the flow of a fluid through a porous medium and

it is empirical in nature. The law was formulated by Henry Darcy (1856) based on the

results of experiments on flow of water through beds of sand. The more details of Darcy

model, Forchhiemer model (Forchheimer, 1901) and Brinkman model (Brinkman, 1949)

and comprehensive literature on the subject is given in the books of Bejan (1984),

Pop and Ingham (2001), Vafai (2005), Vadasz (2008), Nield and Bejan (2013).

The problem of fluid flow and heat transfer in a channel filled with porous media

has been analyzed by several researchers (Beavers and Joseph, 1967; Irmay, 1958;

Neale and Nader, 1974; Poulikakos and Kazmierczak, 1987; Vafai and Kim, 1989).

Beji and Gobin (1992) numerically studied thermal dispersion effects on the Brinkman-

Forchheimer model. Mishra and Sarkar (1995) have provided comparative numerical

studies on a heated square cavity on Darcy, Brinkman and Brinkman-Forchheimer

models and confirmed that boundary friction and quadratic drag lead to a reduction

in heat transfer. Vafai and Kim (1995) have presented a numerical study based on the

Darcy-Brinkman-Forchheimer model for the forced convection in a composite system

containing fluid and porous regions. Knupp and Lage (1995) have studied generalized

Forchheimer – extended Darcy flow model to the permeability case via a variation

principle. Marpu (1995) have studied the Forchheimer and Brinkman extended Darcy

15

Introduction

flow model on natural convection in a vertical cylindrical porous medium.

Singh and Thorpe (1995) presented a comparative analysis of flow models on natural

convection in a fluid flow over a porous layer. Whitaker (1996) obtained a momentum

equation with a Forchheimer correction using the method of volume averaging. Nield

(1996) obtained a closed-form solution of the Brinkman-Forchheimer equation for

different values of Darcy number using the Brinkman-Forchheimer model and stress

jump boundary condition at the porous interface. Nakayama (1998) has presented

a unified treatment of Darcy-Forchheimer boundary layer flows. Paul et al. (1999)

studied this problem for natural convection flow in a vertical channel partially filled

with a porous medium using Brinkman–Forchheimer extended Darcy model.

The effect of viscous dissipation is significant on heat transfer inside channels.

There are three main models for accounting viscous dissipation in porous media:

“Darcy” model, “clear fluid compatible” model and “drag force” model. In “Darcy”

model the source of viscous dissipation is power needed to force the flow through

the porous medium while the “clear fluid compatible” model includes an additional

term which corresponds to the heat dissipation due to work done by viscous forces

(Al-Hadhrami et al., 2003). For “drag force” model, viscous dissipation arises from

the power of the drag force (Nield, 2000, 2002, 2007). The “drag force” model can

correctly predict viscous dissipation in the limit of small permeability, but when the

porosity gets values near unity then only the “clear fluid compatible” model is correct to

account for the viscous dissipation. Nield et al. (2004) analyzed that forced convection

is a parallel plate porous channel using a flow model with the above-mentioned three

different viscous dissipation models. Indeed, several momentum transfer models have

been proposed and used during the last few years, such as the Forchheimer-extended

Darcy model and the general extended Darcy model which is Brinkman–Forchheimer

extended Darcy model with an additional convectional inertial term. Umavathi et al.

(2005b) studied numerically mixed convection in a vertical channel filled with a porous

16

Introduction

medium including the effect of inertial forces by taking into account the effect of viscous

and Darcy dissipations. Dukhan and Chen (2007) investigated the forced convective

heat transfer in a parallel-plate channel filled with metallic foams both analytically

and experimentally. DeGroot et al. (2009) undertook a numerical study to explore

the details of forced convection in finned aluminum foam heat sinks. Chen and Tso

(2011) reported exact analytical solutions to a fully developed forced convection heat

transfer in a parallel-plate channel using a two-equation model in the presence of viscous

dissipation. Kumar et al. (2009) analyzed the fully developed combined free and forced

convective flow in a fluid saturated porous medium channel bounded by two vertical

parallel plates. Atul Kumar et al. (2011) studied the transient as well as non-Darcian

effects on laminar natural convection flow in a vertical channel partially filled with

porous medium. Recently, Umavathi and Veershetty (2012) and Umavathi et al. (2012b)

numerically studied the mixed convection in a vertical channel filled with a porous

medium, including the effects of inertia forces by taking the effects of viscous and Darcy

dissipations with and without heat source or sink. Very recently, Umavathi (2013)

investigated numerically steady natural convection flow through a fluid saturated porous

medium in a vertical rectangular duct.

1.2.4 Magnetohydrodynamics

In a pioneering work, Hartmann and Lazarus (1937) investigated steady laminar flow

between two parallel stationary and insulating plates under the influence of a transverse

magnetic force. They observed that the flow rate decreases with increase in the

magnetic field intensity. Since then, this work has received much attention and has

been extended in numerous ways; see Lock (1955). The monograph by Moreau (1990)

discusses some of these extensions and technological applications, and gives an ample

survey of the literature. Umavathi (1996) extended Barletta’s research with the effects

of viscous dissipation and magnetic field taken into consideration. Makinde and Alagoa

17

Introduction

(1999) studied the effect of magnetic field on steady flow through an indented channel.

Umavathi and Malashetty (2005) analyzed the combined free and forced convective

magnetohydrodynamic flow in a vertical channel by taking into account the effect of

viscous and ohmic dissipations. Xu et al. (2007) analyzed the MHD flow and heat

transfer in the boundary layer over an impulsively stretching plate. Barletta et al.

(2008a) considered the problem of buoyant magnetohydrodynamical flows with Joule

and viscous heating effects in a vertical parallel plate channel and the results have

important industrial applications in metallurgy and in growing of pure crystals, as

well as in energy engineering. Barletta and Celli (2008) studied the mixed convection

MHD flow in a vertical channel with the effects of Joule heating and viscous dissipation.

Saleh and Hashim (2010) analyzed the flow reversal phenomena of the fully-developed

laminar combined free and forced MHD convection in a vertical parallel-plate channel.

Umavathi et al. (2010b) analyzed the Poiseuille-Couette flow of two immiscible fluids

between inclined parallel plates. One of the fluids is assumed to be electrically conduct-

ing while the other non-conducting fluid and channel walls were assumed to be electri-

cally insulating. Srinivas and Muthuraj (2010) analyzed the MHD mixed convection

flow through porous space in the presence of a temperature dependent heat source in

a vertical channel with radiation. Kumar et al. (2011a) analyzed the mixed convective

flow and heat transfer in a vertical channel with one region filled with conducting fluid

and another region with non-conducting fluid. Nikodijevic and Stamenkovic (2011)

investigated the MHD Couette flow of two immiscible fluids in a parallel plate channel

in the presence of an applied electric and inclined magnetic field. Pal et al. (2013)

analyzed the effects of thermal radiation and viscous dissipation on hydromagnetic

mixed convection flow over a non-linear stretching and shrinking sheets in nanofluids.

The use of electrically conducting fluids under the influence of magnetic field in

various industries has led to a renewed interest in investigating hydromagnetic flow

and heat transfer in different geometries. For example, Sparrow and Cess (1961)

18

Introduction

considered the effect of a magnetic field on the free convection heat transfer from

a surface. Raptis and Kafoussias (1982) analyzed flow and heat transfer through a

porous medium bounded by an infinite vertical plate under the action of a magnetic

field. Garandet et al. (1992) discussed buoyancy driven convection in a rectangular

enclosure with a transverse magnetic field. Chamkha (2002) studied the problem of

hydromagnetic fully developed laminar mixed convection flow in a vertical channel

with symmetric and asymmetric wall heating conditions in the presence or absence

of heat generation or absorption effects. Kumar et al. (2012) formulated the problem

of steady, laminar flow and heat transfer of an electrically conducting fluid through

vertical channel in the presence of uniform transverse magnetic field. Umavathi and Liu

(2013) analyzed the problem of steady, laminar mixed convective flow and heat transfer

of an electrically conducting fluid through a vertical channel with heat source or sink.

Kumar and Umavathi (2013) found the analytical solutions for the fully developed

MHD free convection flow in open-ended vertical wavy channel in the presence of

applied electric filed.

1.2.5 Nanofluid

The conventional heat transfer fluids, including oil, water and ethylene glycol mixture,

are poor heat transfer fluids, since the thermal conductivity of these fluids play an

important role on the heat transfer coefficient between the heat transfer medium

and the heat transfer surface. An innovative technique for improving heat transfer

by using ultra fine solid particles in the fluids has been used extensively during the

last years. Choi (1995) introduced the term nanofluid which refers to these kinds of

fluids by suspending nanoscale particles in the base fluid. Choi et al. (2001) showed

that the addition of a small amount (less than 1% by volume) of nanoparticles to

conventional heat transfer liquids increased the thermal conductivity of the fluid up to

approximately two times. Nanofluids have attracted enormous interest from researchers

19

Introduction

due to their potential for high rate of heat exchange incurring either little or no

penalty in the pressure drop. The convective heat transfer characteristics of nanofluids

depend on the thermophysical properties of the base fluid and the nanoparticles, the

flow pattern and flow structure, the volume fraction of the suspended particles, the

dimensions and the shape of these particles. In some works the Brownian motion and

thermophoresis effects are kept in the model of nanofluids. Especially Brownian motion

is suppressed when particle volume fraction is increased as reported by Kumar et al.

(2010b). Santra et al. (2009) numerically examined nanofluid laminar flow and heat

transfer in a two-dimensional (infinite depth) horizontal rectangular duct. They

observed that heat transfer in such a duct increases with increasing nanofluid volume

fraction. The comprehensive references on nanofluid can be found in the recent book

by Das et al. (2007) and in the review papers by Buongiorno (2006), Lee et al. (2010),

Eagen et al. (2010), Wong and Leon (2010), Fan and Wang (2011) and Mahian et al.

(2013). Xuan et al. (2005) have examined the transport properties of nanofluid and

have expressed that thermal dispersion, which takes place due to the random movement

of particles, takes a major role in increasing the heat transfer rate between the fluid and

the wall. Kuznetsov and Nield (2010) have examined the influence of nanoparticles on

natural convection boundary-layer flow past a vertical plate, using the model proposed

by Buongiorno (2006). The authors have assumed the simplest possible boundary

conditions, namely those in which both the temperature and the nanoparticle fraction

are constant along the wall. Memari et al. (2011) studied the problem of combined

forced and natural convection in a vertical porous channel for both regular fluids and

nanofluids by perturbation and numerical methods, taking into account the influences

of viscous heating and inertial force. Mixed convective heat transfer of nanofluids

in a vertical channel partially filled with highly porous medium was studied by

Hajipour and Dehkordi (2012). Cimpean and Pop (2012) presented the solutions of

the steady fully developed mixed convection flow of a nanofluid in a channel filled with

20

Introduction

a porous medium. Very recently, Matin and Pop (2013) studied the forced convection

heat and mass transfer flow of a nanofluid through a porous channel with a first

order chemical reaction on the wall. Hang et al. (2013) analyzed the effect of mixed

convection flow of a nanofluid in a vertical channel with the Buongiomo model.

1.2.6 Jeffery-Hamel flow

Jeffery (1915) and Hamel (1917) have worked on incompressible viscous fluid flow

through convergent–divergent channels, mathematically and so, it is known as Jeffery-

Hamel flow. They presented an exact similarity solution of the Navier–Stokes equations.

Discussions on Jeffery and Hamel flows may be found in Axford (1961), Rosenhead

(1940), Fraenkel (1962), Batchelor (1967), White (1991), Hamadiche et al. (1994),

McAlpine and Drazin (1998), Makinde and Mhone (2006, 2007), Esmaili et al. (2008)

and Ganji et al. (2009). Joneidi et al. (2010) proposed three analytical methods such

as Homotopy analysis method, Homotopy perturbation method and DTM to find

the solution for Jeffery-Hamel flow problem. Motsa et al. (2010) solved the MHD

Jeffery-Hamel problem by hybrid spectral-homotopy analysis technique and compared

their results with homotopy analysis method. Moghimi et al. (2011a) solved the Jeffery-

Hamel flow using the homotopy perturbation method. Moghimi et al. (2011b) also

investigated the MHD Jeffery-Hamel flows in non-parallel walls using homotopy analysis

method. Sheikholeslami et al. (2012) studied the effects of magnetic field and nano-

particle on the Jeffery-Hamel flow using Adomin decomposition method. More recently

Umavathi and Shekar (2013b) analyzed the problem of Jeffery-Hamel flow of nanofluid

with magnetic effect by DTM and compared the results with RKSM.

1.2.7 Micropolar fluid and couple stress fluid

The theory of non-Newtonian fluids has received great attention during the recent

years, because traditional viscous fluids cannot precisely describe the characteristics of

21

Introduction

many rheological fluids. The governing equations for such fluids are complicated and

more nonlinear than the Navier–Stokes equations and present interesting challenges to

physicists, computer scientists, mathematicians, and modelers.

Eringen (1966) proposed a theory of molecular fluids taking into account the inertial

characteristics of the substructure particles, which are allowed to undergo rotation.

Physically, the micropolar fluid can consist of a suspension of small, rigid, cylindrical

elements such as large dumbbell shaped molecules. Micropolar fluids are those which

contain micro-constituents that can undergo rotation, the presence of which can affect

the hydrodynamics of the flow so that it can be distinctly non-Newtonian fluid. Eringen

(1972) developed the theory of thermomicropolar fluids by extending the theory of

micropolar fluids. The existing state of art can be seen in the excellent treatises of

Lyczkowski (1999) and Eringen (2001). Chamkha et al. (2002) analyzed numerical

and analytical solutions of the developing laminar free convection of a micropolar fluid

in vertical parallel plate channel with asymmetric heating. Bhargava et al. (2003)

obtained a numerical solution of a free convection MHD micropolar fluid flow between

two parallel porous vertical plates by means of the quasi-linearization method.

Bhargava et al. (2004) also made a sincere effort to study a fully developed mixed

convection flow of a micropolar fluid with heat sources in a vertical circular pipe

and discussed the effects of the sensitivity of various parameters like the micropolar

parameters, heat source parameter, surface parameter and dissipation parameter on

the velocity, microrotation velocity and temperature functions. Kamal et al. (2006)

studied the steady, laminar, incompressible and two-dimensional flow of a micropolar

fluid between two porous coaxial disks. The injective micropolar flow in a porous

channel was investigated by Joneidi et al. (2009b). Mohammad et al. (2009) analyzed

the two-layered blood flow through a flexible artery under stenotic conditions where

the flowing blood is represented by the two-fluid model, consisting of a core region of

suspension of all erythrocytes assumed to be Eringen’s micropolar fluid and a plasma

22

Introduction

layer free from cells of any kind as a Newtonian fluid. Zueco et al. (2009) studied the

unsteady free convection flow of an incompressible electrically conducting micropolar

fluid bounded by two parallel infinite porous vertical plates subject to an external

magnetic field and the thermal boundary condition of forced convection. Ashraf et al.

(2009b) presented a numerical study for the two-dimensional flow of a micropolar

fluid in a porous channel. Kumar et al. (2010a) analytically studied the problem of

fully-developed laminar free-convection flow in a vertical channel with one region filled

with micropolar fluid and the other region with a viscous fluid. Rashidi et al. (2011)

presented complete analytic solution to heat transfer of a micropolar fluid through a

porous medium with radiation by homotopy analytical method. Devakar and Iyengar

(2011) studied the run up flow of an incompressible micropolar fluid between two

horizontal infinitely long parallel plates. Recently, Umavathi and Sultana (2012)

investigated the problem of fully developed mixed convection for a laminar flow of a

micropolar fluid mixture in a vertical channel with a heat source/sink with boundary

conditions of third kind.

The couple stress fluid theory developed by Stokes (1966) represents the simplest

generalization of the classical viscous fluid theory that sustains couple stresses and

the body couples. The important feature of these fluids is that the stress tensor is

not symmetric and their accurate flow behavior cannot be predicted by the classical

Newtonian theory. The main effect of couple stresses will be to introduce a size

dependent effect that is not present in the classical viscous theories. The fluids consisting

of rigid, randomly oriented particles suspended in a viscous medium, such as blood,

lubricants containing small amount of polymer additive, electro-rheological fluids and

synthetic fluids are examples of these fluids. The constitutive equations in these fluids

can be very complex and involving a number of parameters, also the out-coming flow

equations lead to boundary value problems in which the order of differential equations

is higher than the Navier–Stokes equations. A review of couple stress (polar) fluid

23

Introduction

dynamics was reported by Cowin (1974) and Stokes (1984). Umavathi and Malashetty

(1999) investigated the flow and heat transfer characteristics of Oberbeck convection

of a couple stress fluid in a vertical porous stratum. Umavathi et al. (2005a) analyzed

analytically the problem of steady laminar fully developed flow and heat transfer in

a horizontal channel consisting of a couple-stress fluid sandwiched between two clear

viscous fluids. Patil and Kulkarni (2008) studied the combined effects of free convective

heat and mass transfer on the steady two-dimensional, laminar, polar fluid flow through

a porous medium in the presence of internal heat generation and chemical reaction of

the first order. Mekheimer (2008) have analyzed the MHD flow of a conducting couple

stress fluid in a slit channel with rhythmically contracting walls. An incompressible

laminar flow of a couple stress fluid in a porous channel with expanding or contracting

walls was considered by Srinivasacharya et al. (2009). Umavathi et al. (2009) presented

analytical solutions for fully developed laminar flow between vertical parallel plates

filled with 2 immiscible viscous and couple stress fluids in a composite porous medium.

Recently, Srinivasacharya and Kaladhar (2012a,b) discussed the convection flow of

couple stress fluid in vertical channel with Hall and Ion-slip effects.

Sarayanan and Premalatha (2012) studied the thermovibrational convection in a porous

layer permeated by a fluid exhibiting antisymmetric stress due to the presence of couple

stress fluid.

1.3 Basic Equations

The convective flow and heat transfer can be modeled mathematically by the set of

governing equations derived from the Conservation of Mass, Newton’s Second Law and

the First Law of Thermodynamics. They are widely used as mathematical models to

describe physical phenomena and valid only for continuous fluids. The resulting system

of partial differential equations referred to as the Navier-Stokes equations.

In this section we study the basic equations for the Newtonian fluid (viscous fluid,

24

Introduction

the fluid flow through a porous medium, magnetihydrodynamics or conducting fluid

and nanofluid), and non-Newtonian fluid (micropolar fluid and couple stress fluid).

1.3.1 Newtonian fluids

According to Newton’s law of viscosity, for laminar flows, the shear stress is directly

proportional to the strain rate or the velocity gradient,

τxy = µ∂u

∂y(1.1)

where µ is the constant of proportionality and is the dynamic viscosity of the fluid.

The shear stress is maximum at the surface, the fluid is in contact with, due to no

slip condition. The fluids obeying the Newton’s law of viscosity will be termed as

Newtonian fluids.

1.3.1.1 Equation of State

The equation of state for a viscous fluid can be derived by expanding the density ρ (T )

by Taylor’s series about T = T0 and doing so we get

ρ (T ) = ρ (T0) +

(∂ρ

∂T

)T=T0

(T − T0) +

(∂2ρ

∂T 2

)T=T0

(T − T0)2

2!+ . . . . . . (1.2)

Neglecting the second and higher order terms, we get

ρ (T ) = ρ (T0) +

(∂ρ

∂T

)T=T0

(T − T0) (1.3)

By the definition of thermal expansion coefficient we have

β = − 1

ρ0

(∂ρ

∂T

)T=T0

⇒(∂ρ

∂T

)T=T0

= −ρ0β (1.4)

25

Introduction

Therefore equation (1.3) becomes

ρ = ρ0 (1− β (T − T0)) (1.5)

where ρ is the density, β the thermal expansion coefficient, ρ0 the reference density

and T0 the reference temperature.

1.3.1.2 Continuity equation

The conservation of mass stated as ‘the amount of fluid flowing into a volume must

be equal to the amount of fluid flowing out that volume’ and it is mathematically

expressed by the continuity equation,

∂ρ

∂t+∇ · (ρq) = 0 (1.6)

where ρ is the density of the fluid, q = (u, v, w) velocity component and∇ =(

∂∂x, ∂∂y, ∂∂z

).

For incompressible flow, the density of the fluid is assumed to be constant. It is

interesting to note that flow of a compressible fluid can be regarded as incompressible,

if the Mach number for the flow is less than 0.3. So, the continuity equation reduces

to

∇ · q = 0. (1.7)

1.3.1.3 Momentum equation

The Navier-Stokes equations, describe the momentum balance in the fluid flow.

Therefore, these equations are sometimes termed as momentum equation or simply the

equation of motion for the flow. The Navier-Stokes equations are differential equations

which, unlike algebraic equations, do not explicitly establish a relation among the

variables of interest (e.g. velocity and pressure), rather, they establish relation among

the rates of change of these quantities. A solution of the Navier-Stokes equations is

26

Introduction

called a velocity field or flow field, which is a description of the velocity of the fluid at

a given point in space and time. Once the velocity field is found, other quantities of

interest (such as flow rate, drag force, the path a particle of fluid etc.) may be found.

The momentum equation for the viscous incompressible fluid can be written in

terms of fluid velocity q = (u, v, w), body force g, pressure p, and dynamic viscosity µ

as follows

ρ

[∂q

∂t+ (q · ∇) q

]= −∇p+ ρg + µ∇2q (1.8)

1.3.1.4 Energy equation

The energy equation may be obtained by applying the first law of thermodynamics to

a controlled volume. The energy equation is written as

ρCp

[∂T

∂t+ (q · ∇)T

]= K∇2T + Φ (1.9)

where Φ is the viscous dissipation defined by

Φ = 2µ

((∂u∂x

)2+(

∂v∂y

)2+(∂w∂z

)2)+ 1

2

(∂u∂y

+ ∂v∂x

)2+ 1

2

(∂v∂z

+ ∂w∂y

)2+1

2

(∂w∂x

+ ∂u∂z

)2Due to the assumption of a homogeneous liquid, the temperature does not alter

the velocity profile and hence we do not see the effects of viscous dissipation, internal

heat generation/absorption on the velocity profiles unless buoyancy comes into play.

These effects are noticed only on the temperature profile if there is coupling between

temperature and velocity.

1.3.2 Different models of flows through porous media

While studying the problems involving the transport of momentum, a remarkable

degree of success have been achieved during the past century or so by neglecting the

27

Introduction

complexity of the internal geometry and adopting the concept of ‘equivalent continuum’

(see, Nield and Bejan, 2013; Rudraiah, 1986). In this thesis we adopt the well establi-

shed ‘macroscopic’ approach in studying different transport phenomena in viscous

fluid/porous media.

1.3.2.1 Darcy model (Darcy’s law)

Traditionally, the empirical Darcy’s (Darcy, 1856) law has been applied for flows

through porous media when the Reynolds number based on the pore size is very small.

Under this circumstance, the momentum equation for fluid flows passing through an

isotropic media is described by

−∇P − ρg =µ

κq (1.10)

where P is the pore pressure, ρ the density, g gravity, µ the fluid viscosity, κ the

intrinsic permeability, and q the Darcy velocity.

Values of κ for natural materials vary widely. Typical values for soils, in terms of

the unit m2, are: clean gravel 10−7−10−9, clean sand 10−9−10−12, peat 10−11−10−13,

stratified clay 10−13 − 10−16, and unweathered clay 10−16 − 10−20. Workers concerned

with geophysics often use as a unit of permeability, which equals 0.987× 10−12m2.

Darcy’s law has been verified by the results of many experiments. Theoretical

backing for it has been obtained in various ways, with the aid of either deterministic

or statistical models.

1.3.2.2 Darcy–Forchheimer model

More lately, engineering practices require the operation of flows in porous media at

high Reynolds number, such as those in packed-bed reactors. Experimental evidences

showed that equation (1.10) was unable to describe the flows at high Reynolds number.

By fitting to experimental data, a nonlinear term ρCp|q|q√κ

was added to equation (1.10)

28

Introduction

to correct for the advection inertia effect (Forchheimer, 1901). Thus, equation (1.10)

was modified empirically into

∇P − ρg = −µqκ

− ρCp |q| q√κ

(1.11)

where Cp is the drag coefficient and other quantities have the same meaning as defined

earlier. Equation (1.11) is known as Darcy–Forchheimer equation.

1.3.2.3 Darcy–Forchheimer–Brinkman model

There are many practical applications both experimental and theoretical, which suggest

that the Darcy model will sometime provide unsatisfactory explanation of

the hydrodynamic conditions. It is well known that Darcy’s law breaks down in

situations where in other effects like viscous shear and inertia force comes into play.

Beavers et al. (1970) and Rajashekar (1974) have experimentally demonstrated the

existence of shear within the porous medium, near the boundaries, thus forming a

region of shear influenced fluid flow. The Darcy model cannot predict the existence

of such a boundary region as no macroscopic shear term is included in the equation.

Brinkman (1947) and Tam (1969) have demonstrated that, to predict the boundary

effects, the Brinkman (1947) equation of the form

∇P − ρg = −µqκ

− ρCp |q| q√κ

+ µeff∇2q, (1.12)

is the most suitable governing equation for an incompressible creeping flow of a

Newtonian fluid within an isotropic and homogeneous porous medium. The equation

(1.12) is known as Darcy–Forchheimer–Brinkman equation (or model). The coefficient

µ is the fluid viscosity and µeff is the effective viscosity. Brinkman set µ and µeff

equal to each other, but in general they are only approximately equal.

29

Introduction

In many practical problems, the flow in porous media is curvilinear and the curvature

of the path yields the inertia effect, so that the streamlines become more distorted and

the drag increases more rapidly. Lapwood (1948) was the first person who suggested

for the inclusion of convective inertia term ρ (q.∇) q in the momentum equation.

Subsequently many research articles have been appeared on non-Darcy model (see

Al-Hadhrami et al., 2003; Nield and Bejan, 2013; Nield et al., 2004; Paul et al., 1999;

Umavathi and Veershetty, 2012; Umavathi et al., 2005b). Now equation (1.12) with

the usual inertia term ρ (q.∇) q can be written as

ρ (q.∇) q = −∇P + ρg − µq

κ− ρCp |q| q√

κ+ µeff∇2q (1.13)

This equation is known as Darcy–Lapwood–Forchheimer–Brinkman equation.

1.3.3 Basic equations of magnetohydrodynamics

The postulate of Faraday (1791-1867) states that solid or fluid material moving in

a magnetic field experiences an electromagnetic force. If the material is electrically

conducting and a current path is available, electric currents ensue. Alternatively,

currents may be induced by change of the magnetic field with time. There are two

consequences.

1. An induced magnetic field associated with these currents appears, perturbing the

original magnetic field.

2. An electromagnetic force due to the interaction of currents and field appears,

perturbing the original motion.

These are two basic effects of magneto-hydrodynamics (MHD), the science of the

motion of electrically conducting fluids under the action of magnetic fields. The

situation is essentially one of mutual interaction between the fluid velocity fields and

30

Introduction

the electromagnetic fields the motion affects the magnetic field and the magnetic field

affects the motion. The name MHD attempts to convey this relationship.

A charged particle such as an electron suffers forces of three kinds as follows.

1. It is repelled or attracted by other charged particles, the total force on the particle

per unit of its charge due to all the other charges present being the electrostatic

field Es (volt/m). From coulomb’s law it follows that Es is irrotational. i.e.,

∇×Es = 0. Es can also be represented as the negative gradient of an electrostatic

potential V (volts) where Es = −∇·V . It also follows that Es is solenoid in regions

devoid of charge while elsewhere.

∇Es =q

ε0(1.14)

where q (coul./m3) is the net charge per unit volume and ε0 is a constant, which

equals 8.854× 10−12 (MKS units).

2. A charged particle moving with velocity q m/sec relative to a certain frame of

reference suffers a magnetic force q × B (Newton’s) per unit of its charge. The

force is perpendicular to q and B. The direction of B is that in which the particle

must travel to feel no magnetic force.

3. If the magnetic field B, so identified, is changing with time relative to a certain

frame of reference, then per unit of its charge a particle will suffer a further force

Ee the induced electric field, defined by (Faraday’s law)

∇× Ei = −∂B∂t

(1.15)

This implies

∂

∂t

(∇B

)= 0 (1.16)

31

Introduction

or ∇B = 0.

This shows that magnetic field lines can never end, though they do not in general

form closed loops.

The electric field E is defined equal to Es + Ei. It is a force per unit charge on a

particle due to its presence. Es and Ei are respectively the irrotational-divergent and

rotational-solenoidal parts of E.

The total force on a particle per unit of its charge is equal to (using equations (1.14)

and (1.15)) E + q × B called Lorentz force.

One more important law of electromagnetism remains to be specified is the Ohm’s

law. This is given by

J = σe

(E + q × B

)(1.17)

Ohm’s law includes the current induced by the motion of the conducting fluid

through the magnetic force lines.

The heat due to electrical dissipation in a conductor is given by Joule’s heating law

J2

σe= σe

(E + q × B

)(1.18)

Therefore the basic equations for electrically conducting fluids for motion and

energy are,

ρ

(∂q

∂t+ (∇ · q)q

)= −∇p− ρg + µ∇2q +

(J × B

), (1.19)

ρCp

(∂T

∂t+ (∇ · q)T

)= K∇2T ±Q+

J2

σe. (1.20)

32

Introduction

1.3.4 Nanofluid

1.3.4.1 Brinkman model

Brinkman (1952) proposed a model for viscosity in a concentrated suspension. The

viscosity µnf of the nanofluid is

µnf =µf

(1− ϕ)2.5(1.21)

The thermal expansion coefficient of the nanofluid (proposed by Xuan and Roetzel,

2000) can be determined by

(ρβ)nf = (1− ϕ) (ρβ)f + ϕ (ρβ)s , (1.22)

where subscript f indicates a base fluid, p a particle, ϕ is the solid volume fraction and

µf the viscosity of the base fluid.

1.3.4.2 Maxwell’s equation

Maxwell was the first person to investigate conduction analytically through a suspension

particle. Maxwell (1873) considered a very dilute suspension of spherical particles by

ignoring interactions among particles. The naofluid thermal conductivityKnf is defined

as

Knf = Kf + 3ϕ

(Ks −Kf

2Kf +Ks − ϕ (Ks −Kf )

)Kf . (1.23)

For low particle-volume concentrations,

Knf = Kf + 3ϕ

(Ks −Kf

2Kf +Ks

)Kf (1.24)

where Kf is the thermal conductivity of the base fluid and Ks the thermal conductivity

of the solid nano particles.

33

Introduction

1.3.5 Non-Newtonian fluids

1.3.5.1 Micropolar fluid

The law of conservation of mass as well as the Couchy’s law of conservation of linear

momentum as stated for general continuous media has the same form for both ordinary

and polar fluids.

The differences begin when one assumes different forms of laws of the conservation

of angular momentum and energy for ordinary and polar fluids. More general forms of

these laws for polar fluids come from phenomenological considerations of the (physical)

model when one has to take into account additional quantities such as body torques,

couple stresses and intrinsic angular momentum.

The laws of conservation of hydrodynamic, isotropic, polar fluids are given by

Dρ

Dt= −ρ (∇ · q) (1.25)

ρDq

Dt= ∇ · T + ρg (1.26)

ρIDw

Dt= ∇ · C + ρg + Tx (1.27)

ρDE

Dt= −∇ · q + T : (∇q) + C : (∇w)− Tx · w (1.28)

They are laws of conservation of mass, momentum, angular momentum, and energy

respectively.

We have to specify the stress tensor τ and the couple stress tensor C in the equations

of isotropic polar fluids, that is, to define the constitutive equations, as well as to specify

the equations of state.

We define a micropolar fluid as a polar, isotropic fluid with stress tensor τ and

34

Introduction

couple stress C as given by Eringen (1966).

τij = (−p+ λqkk)δij + µ(qij + qji) + µr(qji − qij)− 2µrεmijωm (1.29)

and

Cij = c0ωkkδij + cd(ωij + ωji) + ca(ωji − ωij) (1.30)

The symmetric part of the stress tensor τij in equation (1.29) is

τ(s)ij = (−p+ λqkk)δij + µ(qij + qji) (1.31)

which is just the stress tensor of classical hydrodynamics, where λ and µ are the usual

viscosity coefficients (second viscosity coefficient and dynamic Newtonian viscosity).

The positive constant µr in equation (1.29) represents the dynamic microrotation

viscosity and c0, ca, cd are constants called coefficients of angular viscosities.

Substituting the tensors τij and Cij from equations (1.30) and (1.31) into the system

of equations (1.25) to (1.28), we obtain the following system of field equations:

Dρ

Dt= −ρ (∇ · q) (1.32)

ρDq

Dt= −∇p+ (λ+ µ− µr)∇ (∇ · q) + (µ+ µr)∇q + 2µrrot w + ρg (1.33)

ρIDw

Dt= 2µr(rot q − 2w) + (c0 + cd − ca)∇(∇ · w) + (ca + cd)∇w + ρg (1.34)

ρDE

Dt= −p∇ · q + ρΦ−∇ · q (1.35)

where

ρΦ = λ(∇ · q)2 + 2µD : D + 4µr

(12rot q − w

)2+ c0(∇ · w)2

+(ca + cd)∇w : ∇w + (cd − ca)∇w : (∇w)T

is the dissipation function of mechanical energy per mass unit. In equation (1.35) D

35

Introduction

denotes the deformation tensor

D =1

2(qi,j + qj,i) (1.36)

Remark: If g, w, and the viscosity coefficients c0, ca, cd, µr are zero, the system

(1.32)-(1.35) reduces to the system of field equations of classical hydrodynamics:

Dρ

Dt= −ρ(∇ · q) (1.37)

ρDq

Dt= −∇p+ (λ+ µ)∇ (∇ · q) + µ∇q + ρg (1.38)

ρDE

Dt= −p∇ · q + ρΦ−∇ · q (1.39)

where ρΦ = λ(∇ · q)2 + 2µD : D.

1.3.5.2 Couple stress fluid

The theory of couples stress fluid is given by Stokes (1966). This theory introduces a

second order gradient of velocity vector, instead of kinematically independent rotation

vector in the constitutive relationship between stress and strain rate. Stokes theory of

couple stress fluids is the simplest generalization of the classical theory of fluids which

allows for the polar effects such as the presence of a non-symmetric stress tensor, couple

stresses and body couples. The constitutive equations for couple stress fluids proposed

by Stokes (1966) are

T(ij) = (−P + λDkk) δij + 2µfDij, (1.40)

T[ij] = −2ηωij, kk −(ρ2

)εijsGs, (1.41)

Mij = 4ηωj,i + 4η′ωi,j, (1.42)

where Dij =12(qi,j + qj,i), Wij = −1

2(qi,j − qj,i), ωi =

12εijk qk,j, T(ij) is the symmetric

part and T[ij] is the anti-symmetric part of the stress tensor Tij, Mij the couple stress

36

Introduction

tensor, Dij the deformation rate tensor, Wij the vorticity tensor, qi the components

of velocity vector, ωi the vorticity vector, Gs the body couple, δij the Kronecker

delta, ρ the density, P the pressure, εijs the alternating unit tensor, λ and µf are

the material constants having the dimension of viscosity, η and η′ are the material

constants having the dimension of momentum. The ration of (η/µf ) has the dimension

of length squared and it characterizes the size of microstructure. In the flow of

fluid with microstructure, the departure from classical theory depends on the relative

size of substructure compared with the linear dimensions of the flow. In the case

of incompressible fluids, when the body forces and body moments are absent, the

momentum equation derived by Stokes (1966) in the vector notation become

ρ

[∂q

∂t+ (q · ∇) q

]= −∇p+ ρg + µf∇2q − η∇4q. (1.43)

1.4 Boundary Conditions

1.4.1 Boundary conditions on velocity

1.4.1.1 No-slip condition

When the fluid, in contact with the solid surface takes the velocity of the solid surface

then this condition at the solid surface is termed as no-slip condition. This is characteri-

stic of all viscous fluid flows.

1.4.2 Boundary conditions on temperature

1.4.2.1 Boundary condition of first kind

If the boundaries of the fluid layer have high heat conductivity and large heat capacity,

then their temperature would be specially uniform and unchanging in time. The

boundary temperature in this case would be unperturbed by any flow or temperature

37

Introduction

perturbations in the fluid, thus T = T1 or T2 at the boundaries. This is known as the

boundary condition of first kind (i.e. isothermal boundary condition).

1.4.2.2 Boundary conditions of third kind

The Robin boundary condition (or third kind boundary condition) is a type of boundary

condition, named after Victor Gustave Robin (1855-1897). When imposed on an

ordinary or a partial differential equation, it is a specification of a linear combination of

the values of a function and the values of its derivative on the boundary of the domain

for example Newton’s law of cooling, mathematically it can be expressed as follows

−K ∂T∂Y

∣∣Y=a

= h1 [T1 − T (X, a)]

−K ∂T∂Y

∣∣Y=b

= h2 [T (X, b)− T2](1.44)

where T1, T2 are the boundary temperatures at Y = a, b respectively and h1, h2 are

the external convection coefficient at Y = a, b respectively.

Robin boundary conditions are a weighed combination of Dirichlet boundary

conditions and Neumann boundary conditions. This contrasts to mixed boundary

conditions, which are boundary conditions of different types specified on different

subsets of the boundary. Robin boundary conditions are also called impedance

boundary conditions, from their application in electromagnetic problems.

Robin boundary conditions are commonly used in solving Sturm–Liouville problems

which appear in many contexts in science and engineering.

To obtain the required basic equation the following approximations have been made

in this thesis.

38

Introduction

1.5 Boussinesq Approximation

There are many situations of practical occurrence in which the basic equations can

be simplified considerably. These situations occur when the variability in the density

and in the various coefficients is due to variations in temperature amounts moderately.

The region of simplifications in these cases is due to the smallness of the coefficient of

volume of expansion: for gases and liquids we shall be most concerned with, thermal

expansion coefficient β is in the range of 10−3 to 10−4. For variations in temperature

not exceeding 100◦, the variations in density are at most 1 percent. The variations of

this small amount can, in general, be ignored. But there is one important exception;

this is because the acceleration resulting from ρg = β (∆T ) g can be large; larger than

the acceleration due to inertia term (q.∇) q in the equation of motion except the one

in the external force.

The Boussinesq approximation is assumed to be valid that is, density is constant in

all terms of the momentum equation except in the buoyancy force in which variations

in density brought about by temperature is valid only when the speed of the fluid is

much less than that of the sound (i.e. Mach number << 1) and accelerations are slow

compared with those associated with sound waves.

The other approximations used in the thesis are as follows

1. The fluid properties namely viscosity and the thermal conductivity are assumed

to be constant in some instant.

2. The gravity acts vertically downward in vertical channels.

3. At a given instant, the effective viscosity of the porous medium, is not same as

fluid viscosity.

4. The nanofluid is a two component mixture with the following assumptions:

(a) no-chemical reaction;

39

Introduction

(b) negligible radiative heat transfer; and

(c) nano-solid-particles and the base fluid which is chosen as water are in thermal

equilibrium and no slip occurs between them

1.6 Method of Solutions

1.6.1 Differential Transform method

Zhou (1986) employed the basic ideas of the DTM for ordinary differential equations.

The main advantage of DTM is that it can be applied directly to nonlinear differential

equations without requiring linearization, discretization, or perturbation. Another

important advantage is that this method is capable of greatly reducing the size of

computational work while still accurately providing the series solution with fast

convergence rate. The DTM is thus also free of discretization errors and yields closed

form solutions.

It is a semi-numerical-analytical method which does not need small parameters.

This method constructs an analytical solution in the form of a polynomial. It is different

from the traditional higher-order Taylor series method. The Taylor series method is

computationally expensive for large orders. The DTM is an alternative procedure for

obtaining an analytic Taylor series solution of differential equations.

DTM can easily be applied to linear and nonlinear problems and reduces the size of

computational work. With this method exact solutions may be obtained without any

need of cumbersome work and it is a useful tool for analytical and numerical solutions.

In this section we are presenting the basic idea of one-dimensional Differential

Transform Method.

40

Introduction

1.6.1.1 Basic idea of one-dimensional Differential Transform Method

An arbitrary function u (x) can be expanded in Taylor series about a point x = 0 as

u (x) =∞∑k=0

1

k!

dku (x)

dxk

∣∣∣∣x=0

xk (1.45)

The Differential Transform of u (x) is defined as

U (k) =1

k!

dku (x)

dxk

∣∣∣∣x=0

(1.46)

where u (x) and U (k) denote the original and transformed functions, orderly. The

inverse differential transform is

u (x) =∞∑k=0

U (k)xk (1.47)

In actual applications, the function u (x) is expressed by a finite series and equation

(1.47) can be rewritten as follows

u (x) =n∑

k=0

U (k) xk +Rn (x) (1.48)

which means that Rn (x) =∑∞

k=n+1 U (k) xk is small and negligible. The number of

fundamental operations of the one-dimensional DTM are given in Table 1.1.

1.6.2 Numerical method

1.6.2.1 Finite Difference method

The non-dimensional governing equations along with the boundary conditions are

descretized using finite difference technique. In numerical procedure computational

domain is divided into a uniform grid system. Both the second-derivative and the

squared first-derivative terms are discretized using the central difference of second-order

41

Introduction

Table 1.1: The operations for the one-dimensional Differential Transform Method.Original function Transformed functionsu (x) = g (x)± h (x) U (k) = G (k)±H (k)u (x) = αg (x) U (k) = αG (k)

u (x) = dg(x)dx

U (k) = (k + 1)G (k + 1)

u (x) = d2g(x)dx2 U (k) = (k + 1) (k + 2)G (k + 2)

u (x) = g (x)h (x) U (k) =∑k

r=0G (r)H (k − r)

f (u (x)) = un (x) F (U (k)) =∑k

rn−1=0 U (k − rn−1)∑rn−1

rn−2=0 U (rn−1 − rn−2)

. . .∑r2

r1=0 U (r2 − r1)U (r1)

u (x) = xm U (k) = δ (k −m) =

{1 if k = m0 if k = m

accuracy. The finite difference form of d2udx2 and du

dx, for example, were discretized as

d2udx2 = ui+1−2ui+ui−1

∆x2 + O (∆x2) and dudx

= ui+1−ui−1

2∆x+ O (∆x2), respectively, similarly

the corresponding boundary conditions are also discretized. Here i range from 1 to

Nx, where Nx denote the number of grids inside the computational domain in the

x directions. Until all the values of ui in the grid system are less than a prescribed

tolerance, the solutions are assumed to be sought.

1.6.2.2 Runge-Kutta shooting method

The shooting method is a boundary value problem (BVP) solver that works by conver-

ting the BVP into an initial value problem. The starting point is an assumed condition

for the unknown initial condition. The guess is improved through an iterative process

until a solution that satisfies all the given boundary conditions is achieved.

The boundary value problem is converted into an initial value problem,

by appropriately choosing the missed initial conditions, as follows

dU1

dy= U2,

dU2

dy= U3,

dU3

dy= U4,

dU4

dy= U5 (1.49)

with the initial conditions

U1 (0) = 0, U2 (1) = α, U3 (0) = 1, U4 (1) = β.

42

Introduction

Here, U1 (y) = u (y). The initial value problem (1.49) is integrated using the fourth

order Runge-Kutta method. Newton-Raphson method is implemented to correct the

guess values α and β. The initial values are differs for each set of parameter values.

1.6.3 Analytical method

1.6.3.1 Perturbation method

Perturbation methods are an alternative to numerical methods and are useful for finding

approximate analytic solutions of differential equations. Nonetheless, literature reveals

that the use of perturbation techniques in fluid dynamics has somewhat declined since

the advent of high-speed digital computers.

Perturbation techniques in general construct the solution for a problem involving a

small parameter ϵ, the perturbation parameter. The perturbation quantity may either

be part of the differential equation, the boundary conditions or both. In general, the

solution of the differential equation at ϵ = 0 should be known. The approximate

solutions are then generated using asymptotic expansions of suitable sequences of the

perturbation parameter. The accuracy of perturbation approximations does not depend

on the value of the independent variable but on the perturbation parameter. For smaller

values of ϵ, the accuracy of perturbation methods tends to improve.

The analytic solutions obtained through perturbation methods are often more

useful than numerical results as they provide a more qualitative and quantitative

representation of the solution compared to numerical solutions. They often provide

a clearer meaning of the physical parameters contained in the solutions. Choosing

suitable sequences of the perturbation parameter requires previous knowledge of the

general nature of the solution (Nayfeh, 1973).

43

Introduction

1.7 Non-dimensional Parameters

Dimensional analysis of any problem provides information on qualitative behavior of

the physical problem. The dimensionless parameters help us to understand the physical

significance of particular phenomenon associated with the problem. The following are

the dimensionless parameters appeared in this thesis. The general definitions of various

dimensionless numbers are given here.

1.7.1 Reynolds number

It is the ratio of the magnitude of the inertial force to viscous force in the flow.

Re =U b

v=

inertia force

visocus force

where U is some characteristic velocity, b is the characteristic length and v is the

kinematic viscosity.

This means, if Reynolds number (Re) is relatively small, the viscous force is high

compared to the inertial force and any disturbance that arise in the flow will be damped

by the action of the viscous force and the flow will tend to be laminar. On the other

hand if Re is high, the inertial force would be dominating the viscous force i.e., the

disturbances that arise in the flow will tend to grow and the turbulent flow will tend

to develop.

1.7.2 Grashof number

It may effectively be called as the ratio of the buoyancy force to the viscous force.

Gr =gβb3∆T

ν2=

buoyancy force

viscous forcex

inertia force

viscous force.

where β is the volumetric expansion coefficient.

44

Introduction

1.7.3 Brinkman number

Brinkman number (Br) is the ratio of the kinetic energy dissipated in the flow to

the thermal energy conducted into or away from the fluid.

Br =µU2

0

K∆T=

viscous force

energy conductedx

kinetic energy

thermal energy

When Brinkman number is very small (Br << 1) the energy dissipation can be

neglected relative to heat conduction in the fluid. For large Br the energy dissipated is

an important parameter in the heat transfer process and the kinetic energy can play a

significant role in determining the temperature distribution in the flow and the overall

heat transfer.

Brinkman number can also be defined as the product of Prandtl number and Eckert

number, that is Br = Pr Ec.

1.7.4 Darcy number

The Darcy number (Da) represents the relative effect of the permeability of the

medium versus its cross-sectional area. The number is named after Darcy (1856) and

is found from nondimensionalizing the differential form of Darcy’s Law. It is defined

as

Da =κ

d2

where κ is the permeability of the medium and d is the diameter of the particle.

The Darcy number is commonly used in flow and heat transfer through porous

media. The reciprocal of the Darcy number is the porous parameter.

1.7.5 Hartmann number

This is very important non-dimensional number in hydromagnetics and is the ratio

of magnetic force (Lorentz force) to viscous force. The hydromagnetic effects are

45

Introduction

important when the Hartmann number is significant.

Hartmann number is defined as

M = B0D

√σeµ

=pondenromotive force

viscous force

where ponderomotive force is the force on volume distributions of current in a magnetic

field.

1.7.6 Electric field load parameter

It is defined as

E =E0

UB0

=applield electric field

induced electric field

where U, E0 and B0 are the characteristic values of velocity, electric field and magnetic

field respectively. The sign of the parameter E is determined by the direction of the

applied field. It is negative when the applied field is in the opposite direction to the

induced field.

1.7.7 Prandtl number

It is defined as a measure of the ratio of the viscous diffusivity to the thermal

diffusivity

Pr =µCp

K=

viscous diffusivity

thermal diffusivity.

Prandtl number is a measure of the relative importance of viscosity and heat

conduction in a flow filed. The Prandtl number depends only on the physical properties

of the fluid but not on the flow conditions. Thus, in convection problems, a rich variety

of phenomena stems particularly from the Prandtl number dependence of the system.

46

Introduction

1.7.8 Biot number

The Biot number is a dimensionless group that compares the relative transport

resistances, external and internal. It arises when formulating and non-dimensionalizing

the boundary conditions for the typical conservation of species or energy equation for

heat/mass transfer problems.

The Biot number is defined as:

Bi =hD

K

where h is the heat transfer coefficient or convective heat transfer coefficient, D is the

characteristic length, which is commonly defined as the volume of the, body divided

by the surface area of the body, and K is thermal conductivity of the body.

The numerical value of Biot Number (Bi) is a criterion which gives a direct indication

of the relative importance of conduction and convection in determining the temperature

history of a body being heated or cooled by convection at its surface. Bi should always

be enumerated at the outset to identify transient conduction problems which may

be treated simply as lumped parameter problems, for which Bi < 0.1 and for which

it is seldom necessary to solve the conduction equation, i.e., convection is the rate

controlling process.

1.7.9 Eckert number

The Eckert number (Ec) is a dimensionless quantity useful in determining the

relative importance in a heat transfer situation of the kinetic energy of a flow. It is

the ratio of the kinetic energy to the enthalpy (or the dynamic temperature to the

temperature) driving force for heat transfer

Ec =U2

Cp ∆T

47

Introduction

where U is an appropriate fluid velocity (e.g., outside the boundary layer or along the

centerline of a duct) and other parameters have the same meaning as defined earlier. For

small Eckert number (Ec << 1) the terms in the energy equation describing the effects

of pressure changes, viscous dissipation, and body forces on the energy balance can be

neglected and the equation reduces to a balance between conduction and convection.

1.7.10 Vortex viscosity parameter

Vortex viscosity parameter (micropolar parameter) is defined as

R =µm

µ

where µm is the vortex viscosity and µ is the viscosity.

1.7.11 Couple stresses parameter

Couple stress parameter is defined as

a2 =ηcµ b2

where ηc is the material constant and has dimension of length squared and other

quantities have their usual meaning.

1.7.12 Skin friction

The dimensionless shear stress at the boundary of the channel is defined as the skin

friction, given by

Cf =2 τwρU2

48

Introduction

1.7.13 Nusselt number

The convective heat transfer from the surface will depend upon the magnitude

of Ch (Tw − T ), where, Ch is the heat transfer coefficient and Tw and T are the

temperatures of wall and fluid respectively. Also, if there was no flow, the heat transfer

was purely due to conduction. The Fourier’s law states that the quantity K (Tw − T ) /l

would be the measure of the heat transfer rate, where K is the thermal conductivity

and b the length. Now Nusselt number can be written as

Nu =Ch (Tw − T )

K (Tw − T ) /b=Ch b

K

That is, Nusselt Number is the measure of the ratio of magnitude of the convective

heat transfer rate to the magnitude of heat transfer rate that would exist when there

was pure conduction.

1.8 Nomenclature

A1 spin gradient viscosity parameter

a, b empirical constant

ac couple stress parameter

B microinertia density parameter

Bi1Bi2 Biot numbers

B0 uniform magnetic field

Br Brinkman number

b channel width

bv viscosity parameter

bk conductivity parameter

CF quadratic drag coefficient

49

Introduction

Cp specific heat at constant pressure

Cr conductivity ratio

D hydraulic diameter, 2b

DTM Differential Transform Method

Da Darcy number

Ec Eckert number

E0 uniform electric field

E electric field loading parameter

F velocity component

FDM Finite Difference method

Gr Grashof number

GR mixed convection parameter

g acceleration due to gravity

H non-dimensional microrotation velocity

h height of the channel

h1, h2 external heat transfer coefficients

I inertial parameter

j microinertia density

K thermal conductivity

Kt variable thermal conductivity

Knf effective thermal conductivity of nanofluid

K1 thermal conductivity in the micropolar fluid

L channel width

M Hartmann number

m wall temperature ratio

mr viscosity ratio

mw couple stress

50

Introduction

N buoyancy parameter

Nu1, Nu2 Nusselt numbers

P pressure

PM perturbation method

Pr Prandtl number

Q volumetric flow rate

Qh rate of internal heat generation/absorption

Qm mass flow rate

R micropolar parameter

Rt temperature difference ratio

Re Reynolds number

RKSM Runge-Kutta shooting method

T1, T2 wall temperatures

T temperature

T0 static temperature

U0 reference velocity

U,W velocity

U (k) transformed function

u non-dimensional velocity of the fluid along x direction

u1 average velocity

V0 injection/suction velocity

v3 microrotation velocity

(X, Y ) space co-ordinates

(r, θ0, z) polar co-ordinates

(x, y) non-dimensional space co-ordinates

51

Introduction

Greek symbols

α angle between the plates

β thermal expansion coefficient

βnf coefficient of thermal expansion of nanofluid

∆T reference temperature difference

γ spin-gradient viscosity

ϵ porosity

η similarity variable

ηc material constant

θ dimensionless temperature

Θ (k) transformed function

κ permeability of the porous media

µ viscosity

µm vortex viscosity

µnf effective viscosity of nanofluid

µt variable viscosity

µ1 viscosity in the micropolar fluid

ν kinematic viscosity

ρ density of fluid

ρ0 static density

ρnf effective density of nanofluid

σ porous parameter

σr conductivity of the fluid

ρnf density of nanofluid

τ shear stress

τ1, τ2 skin friction

ϕ solid volume fraction

52

Introduction

ϕh heat generation/absorption parameter

Φ contribution to viscous dissipation

Subscripts

nf nanofluid

f base fluid

s solid nanoparticles

i = 1, 2, 3 refers the region-I, region-II and region-III respectively.

1.9 Plan of Work

In order to achieve the objectives mentioned above, the plan of the thesis is as follows:

In Chapter 1, we presented the objectives and scope of the thesis. The literature

review relevant to the thesis is laid out. The relevant basic equations, boundary

conditions, approximations. We discuss the method of solution and the basic idea

of one-dimensional DTM and non-dimensional parameters are presented.

In Chapter 2 effect of mixed convection in a vertical channel filled with a porous

medium including the effect of inertial force in the presence of viscous and Darcy

dissipations is studied and the results obtained are discussed.

Chapter 3 presents the solutions of the coupled nonlinear equations governing the

flow for magnetoconvection in a vertical channel for open and short circuits and the

results obtained are reported.

An analysis is performed in Chapter 4 for fully developed free convection flow of a

viscous fluid in a vertical channel including the effect of viscous dissipation for the cases

of separated effect of variable viscosity, variable thermal conductivity and combined

effect of variable viscosity and variable thermal conductivity. The results are discussed

in detail.

53

Introduction

Chapter 5 presents the results of a numerical study on natural convection heat

transfer in a vertical channel filled with nanofluid and the results are discussed.

In Chapter 6, the steady fully developed non-Darcy mixed convection flow of

a nanofluid in a vertical channel filled with a porous medium with different viscous

dissipation models, is analyzed and the results are reported.

The problem of Jeffery-Hamel flow of nanofluid with magnetic effect is analyzed in

Chapter 7 and the results of nanofluid are discussed.

In Chapter 8, the flow and heat transfer in a porous medium saturated by a

micropolar fluid between two parallel permeable disks with uniform suction or injection

through the surface of the disks is studied and the results are reported.

An analysis of the viscous fluid sandwiched between micropolar fluids is presented

in Chapter 9 and the results are discussed.

Chapter 10 is focused on the study of mixed convective flow and heat transfer on

the steady, laminar, couple stress fluid flow in the presence of viscous dissipation with

Robin boundary conditions. The results obtained are reported.

The problem of fully developed laminar mixed convection flow of a couple stress

fluid in a vertical channel with the third kind boundary conditions in the presence or

absence of heat generation or absorption effects is studied in Chapter 11 and the

results are discussed.

The effect of free convection on fully developed electrically conducting micropolar

fluid flow between two vertical porous parallel plates is studied in Chapter 12 with the

presence of temperature dependent heat source and the applied electric and magnetic

field. The obtained results are reported.

Finally Chapter 13 concerns the general conclusions on the results obtained in

the thesis and the feature works of the thesis.

54