Preface · Preface At the beginning of the 20th century Ludwig Prandtl introduced the concept of boundary-layer theory. He showed that the ﬂow past a body can be divided into two

Preface

At the beginning of the 20th century Ludwig Prandtl introduced the concept of boundary-

layer theory. He showed that the flow past a body can be divided into two regions: a

very thin layer close to the body where the viscosity is important, and the remaining

region outside this layer where the viscosity can be neglected. In fluid mechanics, a

boundary layer is that layer of fluid in the immediate vicinity of a bounding surface

where effects of viscosity of the fluid are considered in detail. The boundary layer effect

occurs at the field region in which all changes occur in the flow pattern. The boundary

layer distorts surrounding non-viscous flow. It is a phenomenon of viscous forces and its

effect is related to the Reynolds number. Initially boundary-layer theory was developed

mainly for the laminar flow of an incompressible fluid. The theory was extended to the

practically important turbulent incompressible boundary-layer flow. One of the most

important application of the boundary layer theory is calculation of the friction drag of

bodies in a flow. In the earth’s atmosphere, the planetary boundary-layer is the air layer

near the ground affected by diurnal heat, moisture or momentum transfer to or from the

surface. On an aircraft wing the boundary-layer is the part of the flow close to the wing.

In naval architecture, many of the principles that apply to aircraft also apply to ships and

submarines. Laminar boundary layers come in various forms and can be loosely classified

1

Preface 2

according to their structure and the circumstances under which they are created. The

thin shear layer which develops on an oscillating body is an example of a Stokes layer,

while the Blasius boundary layer refers to the well-known similarity solution for the steady

boundary layer attached to a flat plate held in an oncoming unidirectional flow. When

a fluid rotates, viscous forces may be balanced by coriolis effects, rather than convective

inertia, leading to the formation of an Ekman layer.

Boundary-layer behavior over a continuously moving solid surface has many important

applications. Some of these applications include aerodynamic extrusion of plastic sheets,

the boundary layer along material-handling conveyers, cooling of an infinite metallic plate

in a cooling bath, the boundary layer along a liquid film in condensation processes, and

heat-treated materials that travel between feed and wind-up rollers. It was Sakiadis [95]

who initiated the study of boundary-layer flow past a continuous solid surface moving

with constant speed. This boundary-layer flow is quite different from that in Blasius flow

past a semi infinite plate due to the entrainment of the ambient fluid. Tsou et al [104]

analyzed the effect of heat transfer in the boundary layer on a continuous moving surface

with a constant velocity and experimentally confirmed the numerical results of Sakiadis

[95]. In 1970, Crane [31] studied the steady two-dimensional boundary-layer flow caused

by a stretching sheet whose velocity is directly proportional to the distance from a fixed

point on the sheet. Grubka and Bobba [54] analyzed the heat transfer effect by considering

the power-law variation of surface temperature. Later on huge amount of literature has

been found on the this topic.

Transport processes through porous media play important role in many applications,

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such as geothermal operations, petroleum industries, thermal insulation, design of solid-

matrix heat exchangers, chemical catalytic reactors, and many others. Some studies were

based on Darcy law to incorporate the porous medium. But in many practical situations

the porous medium has huge flow rates and reveals irregular porosity distribution near

the wall region, cause inapplicability of Darcy’s law. Due to this reason it is necessary to

include the non-Darcian effects in the analysis of flow and heat transfer through a porous

medium. Representative studies dealing with the non-Darcy porous medium effects have

been reported by Hong et al [60], Kiwan and Ali [71],

The magnetic field play an important role in the process of purification of molten

metals from non-metallic inclusions. Many works have been reported on flow and heat

transfer of electrically conducting fluids over a stretched surface in the presence of a

magnetic field. In an ionized gas where the density is low/or the magnetic field is very

strong, the conductivity will be a tensor. The conductivity normal to the magnetic field

is reduced due to the free spiraling of electrons and ions about the magnetic lines of

force before suffering collisions and a current is induced in a direction normal to both

the electric and magnetic fields, and this phenomenon is called the Hall effect. When the

medium is rarefied or if the strong magnetic field is present, the conductivity of the fluid is

anisotropic and the effect of Hall current cannot be ignored. The study of MHD viscous

flows with Hall current has important applications in the problem of Hall accelerators

as well as flight magnetohydrodynamics. The current trend for the application of MHD

toward a strong magnetic field and towards a low density of the gas, and under this

condition Hall effect becomes important. With this understanding Sato [96], Yamanishi

[110] have studied the MHD flow of a viscous fluid through a channel by considering the

Preface 4

Hall effect. Then after, I Pop [89] studied the effects of Hall current on hydrodynamic

flow near a porous plate. Hall current and Ohmic heating effects on mixed convection

boundary layer flow of a micro polar fluid from a rotating cone with power-law variation

in surface temperature has investigated by Abd El Aziz et al [35]. Hydrodynamic free

convection and mass transfer of an electrically conducting viscous fluid past an infinite

vertical porous plate has been reported by Gorla et al [53]. Abd El Aziz [36] investigated

the effect of Hall currents on the flow and heat transfer of an electrically conducting fluid

over unsteady stretching sheet in the presence of strong magnetic field. Ali et al [10]

studied the influence of Hall current on MHD mixed convection boundary layer flow over

a stretched vertical flat plate.

Moreover, the magnetohydrodynamic rotating fluids are encountered in many impor-

tant problems in geophysics, astrophysics, and cosmical and geophysical fluid dynamics.

It can provide explanations for the observed maintenance and secular variations of the

geomagnetic field. It is also relevant in the solar physics involved in the sunspot devel-

opment, the solar cycle, and the structure of rotating magnetic stars. The effect of the

Coriolis force due to the earth’s rotation is found to be significant as compared to the iner-

tial and viscous forces in the equations of motion. The Coriolis and electromagnetic forces

are of comparable magnitude, the former having a strong effect on the hydromagnetic flow

in the earth’s liquid core, which plays an important role in the mean geomagnetic field.

Debnath [34] has studied the effect of Hall current on unsteady hydromagnetic flow past

a porous plate in a rotating fluid system. Prasada Rao and Krishna [90] have studied the

Hall effect on an unsteady hydromagnetic flow. Kanch and Jana [67] investigated Hall

effects on unsteady hydromagnetic flow past a rotating disk when the fluid at infinity

Preface 5

rotates about non-coincident axes. Effect of Hall current on unsteady MHD flow due to

a rotating disk with uniform suction or injection were investigated by Attia [13]. Several

investigations are carried out on the problem of hydrodynamic flow of a viscous incom-

pressible fluid in rotating medium considering various variations in the problem. Mention

may be made of the studies of Abelman et al [2], Hossain et al [61] and Hayat et al [56].

In recent years, a great deal of interest has been generated in the area of two di-

mensional boundary-layer flow over a stretching surface near a stagnation-point in view

of its numerous and wide range of applications in various technical and industrial fields

such as cooling of electronic devices by fans, cooling of nuclear reactors during emergency

shutdown, heat exchangers placed in a low-velocity environment, solar central receivers

exposed to wind currents, and many hydrodynamic processes. The two-dimensional flow

of a fluid near a stagnation point was first examined by Hiemenz [59]. Takhar et al [61]

studied unsteady mixed convection flow of a viscous incompressible, electrically conduct-

ing fluid in the vicinity of a stagnation-point adjacent to a heated vertical surface.

Flow through a cylinder is generally considered as two-dimensional as the radius of

the cylinder is large enough compared to the boundary layer thickness. The study of

viscous fluid flow and heat transfer outside a hollow stretching cylinder has importance

in extrusion processes. In view of this, Chen and Mucoglu [120] investigated the effects of

mixed convection over a vertical slender cylinder due to thermal diffusion with prescribed

wall temperature. Wang [73] obtained exact solution of the viscous flow and heat transfer

due to uniformly stretching cylinder and also he compared asymptotic solutions for large

Reynolds number to the numerical values. Chang [122] numerically investigated the flow

and heat transfer characteristics of natural convection in a micropolar fluid flowing along

Preface 6

a vertical hollow circular cylinder with conduction effects. The steady laminar flow caused

by a stretching cylinder immersed in an incompressible viscous fluid with prescribed sur-

face heat flux was investigated by Bachok and Ishak [16]. Mukhopadhyay [102] presented

a result on the distribution of a solute undergoing a first order chemical reaction in an

axisymmetric laminar boundary layer flow along a stretching cylinder. A boundary layer

analysis is presented for the warm, laminar nano fluid flow through a melting cylindrical

surface moving parallel to a uniform stream was done by Gorla et al [53].

The phenomena of heat transfer accompanied with melting or solidification effect has

recently attracted considerable attention due to its wide range of engineering applications

such as magma solidification, the melting of permafrost, silicon wafer process, energy stor-

age, frozen ground thawing. Roberts [93] was the first to describe the melting phenomena

of ice placed in a hot stream of air at a steady state. The steady laminar boundary layer

flow and heat transfer from a warm, laminar liquid flow to a melting surface moving paral-

lel to a constant free stream was studied by Ishak et al [63]. Chamkha et al [25] analyzed

hydromagnetic, forced convection, boundary-layer flow with heat and mass transfer of a

nanofluid over a horizontal stretching plate in the presence melting effect.

At high operating temperature, radiation effect is quite significant. Many processes

in engineering areas occur at high temperature and knowledge of radiation heat transfer

becomes very important for the design of the pertinent equipment. Nuclear power plants,

gas turbines and various propulsion devices for aircraft, missiles, satellites and space

vehicles are examples of such engineering areas. Hossain and Takhar [61] studied the effect

of thermal radiation using the Rosseland diffusion approximation on mixed convection

along a vertical plate with uniform free stream velocity and surface temperature. Cortell

Preface 7

[30] studied the effects of thermal radiation on the laminar boundary layer about a flat-

plate in a uniform stream of fluid, and about a moving plate in a quiescent ambient fluid

both under a convective surface boundary condition. The effects of thermal radiation

and magnetic field on flow and heat transfer over an unsteady stretching surface in a

micropolar fluid are studied by Aldawody and Elbashbeshy [8]. Bakier et al [18] simulated

hydromagnetic heat transfer by mixed convection along vertical plate saturated with

porous medium in the presence of melting and thermal radiation effects.

A fluid particle suspension is a mixture of fluid and fine dust particles. Its study is

important in areas like environmental pollution, smoke emission from vehicles, emission

of effluents from industries, cooling effects of air conditioners, flying ash produced from

thermal reactors and formation of raindrops, etc. Also it is useful in the study of lunar

ash flow which explains many features of lunar soil. In the recent years the attention

of researchers in fluid dynamics has been diverted towards the study of the influence of

dust particles on the motion of fluids. Other important applications of dust particles in

boundary layer, include soil erosion by natural winds and dust entrainment in a cloud

during nuclear explosion. Also such flows occur in a wide range of areas of technical

importance like fludization, flow in rocket tubes, combustion, paint spraying and more

recently blood flows in capillaries. The stability of the laminar flow of a dusty gas in

which the dust particles are uniformly distributed has been discussed by Saffman [94] and

formaulated the basic equations for the flow of dusty fluid.

Flow induced by the impulsive motion of an infinite flat plate in a dusty gas was consid-

ered by Liu [73]. Micheal and Miller [79] studied the flow of dusty gas past an impulsively

started infinite horizontal plate using the Laplace transform technique. Chakrabarti [22]

Preface 8

analyzed the boundary layer flow of a dusty gas. Datta and Mishra [33] have investigated

boundary layer flow of a dusty fluid over a semi-infinite flat plate. Datta [33] obtained

the solutions for pulsatile flow and heat transfer of a dusty fluid through an infinitely long

annular pipe. Evgeny and Sergei [41] have discussed the stability of laminar boundary

layer flow of a dusty gas over a flat plate. The laminar boundary layer flow of a second

order dusty fluid on an infinite plate were studied by Srivastava and Srivastava [101]. Here

the particles are assumed to be under the action of the Stokes drag only. Further, XIE

Ming-liang et al [108] have studied the hydrodynamic stability of a particle-laden flow in

growing flat plate boundary layer. Ezzat et al [42] studied an unsteady laminar free con-

vection flow of a dusty fluid through a porous medium. Unsteady flow and heat transfer of

dusty fluid between two parallel plates with variable viscosity and thermal conductivity

has been investigated by Makinde and Chinyoka [78]. Gireesha et al ([118], [48], [49],

[50], [51], [119], [88]) studied flow and heat transfer behavior of dusty fluid in different

regions like between two parallel plates, rectangular channel, in triangular cross section,

in cylinders, over a flat plate and over a linear/exponential/steady/unsteady stretching

surfaces with various aspects. They found important results, which are applicable in

several industrial and manufacturing processes. They also found that, presence of dust

particles enhances the heat transfer rate, which is better suited in cooling processes.

On the basis of these observations we have studied an unsteady boundary layer flow

and heat transfer of dusty fluid over a stretching sheet. This study has mainly divided

into following THREE chapters;

The FIRST chapter provides an overview that describes the basic definitions of fluid

Preface 9

mechanics and brief discussion about similarity transformations, dimensionless parame-

ters, boundary layer theory, Runge-Kutta Fehlberg fourth-fifth order method. etc., Fur-

ther it provides a basic definition of Laplace transform of a function and some important

properties.

The SECOND chapter concerned with the study of combined effects of Hall current

and rotation on MHD flow of an incompressible, viscous, conducting fluid with uniform

distribution of dust particles bounded by a semi-infinite plate. The fluid is acted upon

by a constant pressure gradient and an external uniform magnetic field which is applied

perpendicular to the plate. The governing nonlinear partial differential equations are

solved analytically by employing Laplace transform technique. This study is analyzed

for three cases namely impulsive motion, transition motion and motion for a finite times.

The effects of different flow parameters such as Ekman number, magnetic parameter, Hall

parameter and oscillation parameter, for both fluid and dust phase profiles are discussed

in detail through plotted graphs for all cases.

The THIRD chapter is focused on the study of two dimensional boundary layer flow

and heat transfer of a conducting dusty fluid due to a linearly stretching cylinder immersed

in porous media with the effect of radiation and heat source/sink. Governing partial differ-

ential equations are reduced into coupled non-linear ordinary differential equations using

suitable similarity transformations. Numerical solutions of these equations are obtained

with the help of efficient Runge-Kutta-Fehlberg-45 Method. Graphical display of the

obtained numerical solution is performed to illustrate the influence of various flow con-

trolling parameters like stretching parameter, magnetic parameter, Prandtl number, heat

Preface 10

source/sink parameter, fluid particle interaction parameter and Eckert number, on veloc-

ity and temperature distributions. The numerical results for the skin-friction coefficient

and Nusselt number are also obtained and discussed in detail.

Chapter 1

Preliminaries

1.1 Introduction to Fluid Dynamics

Fluid dynamics is a branch of mechanics, which deals with the study of fluid in motion

and the subsequent effect of the fluid on the boundaries. The essence of the subject of

fluid flow is that of judicious compromise between theory and experiment. Since fluid

dynamics is the branch of mechanics, its fundamental principles are based on Newton’s

laws of motion, the indestructibility of matter and conservation of energy. The earliest

significant contribution to the field were undoubtedly made by Archimedes, who lived in

Syracuse between the year 285-212 B.C., especially note worthy was his analysis of the

buoyancy of submerged bodies, which was applied successfully to the determination of

the gold content of the crown of king Hiero I.

The next significant advancement came at the end of the 19th century when the

science of fluid mechanics began to develop in two directions, which had practically no

point in common. On the one side there was the science of theoretical hydrodynamics,

which was evolved from Euler’s equation of motion for frictionless, non-viscous fluid and

which achieved a high degree of completeness. Since, however, the result of the so-called

11

Chapter-1: Preliminaries 12

classical science of hydrodynamics stood in glaring contradiction to experimental results-

in particular as regard the very important problem of pressure loss in pipes and channels,

as well as with regard to the drag of a body which moves through a mass of fluid- it had a

little practical importance. For this reason, practical engineers prompted by the need to

solve important problems arising from the rapid progress in technology, developed their

own highly empirical science of hydraulics.

1.2 Types of fluids

The fluid state is commonly divided into liquid, gaseous and plasma. The study of

former two states comes under fluid dynamics and the study of latter one comes under

plasma dynamics. Again, the two corresponding branches of fluid dynamics are called

hydrodynamics and aerodynamics, the former relating to water as well as other liquids

and the latter two air and other gases.

Another customary division of the subject depends on the practical importance of

fluid friction. The “prefect fluids” are treated as if all the tangential stresses caused by

friction were ignored. The “real fluids” refer to the cases in which friction is properly

taken into account.

1.2.1 Ideal fluid or Inviscid fluid

An Ideal fluid is one, which has no property other than density. No resistance is encoun-

tered when such a fluid flows or Ideal fluids or Inviscid fluids are those fluids in which two

contacting layers experience no tangential force (shearing stress) but act on each other


with normal force (pressure) when the fluids are in motion. This is equivalent to stating

that inviscid fluid offers no internal resistance to change in shape. The pressure at every

point of an ideal fluid is equal in all directions, whether the fluid is at rest or in motion.

Inviscid fluids are also known as effect fluids or frictionless fluids. In true sense, no such

fluid exists in nature. The assumption of ideal fluids helps in simplifying the mathematical

analysis. However fluids which have low viscosities such as water and air can be treated

as ideal fluids under certain conditions.

1.2.2 Viscous fluid or Real fluid

“Viscous fluid or real fluid are those, which have viscosity, surface tension and compress-

ibility in addition to the density” or viscous fluid or real fluid are those when they are in

motion the two contacting layers of those fluids experience tangential as well as normal

stresses. The property of exerting tangential of shearing stress and normal stress in a

real fluid when the fluid is in motion is known as viscosity of the fluid. In viscous fluid

internal friction plays an important role during the motion of the fluid. One of the im-

portant characteristics of viscous fluid is that it offers internal resistance to motion of the

fluid, viscosity, being the characteristic of the real fluids, exhibits a certain resistance to

alter the form also. Viscous or real fluids are classified into Newtonian fluids and non

Newtonian fluids.

1.2.3 Newtonian fluid

To understand the concept of Newtonian fluid, let us consider a thin layer of fluid between

two parallel plates at distance dy.


Figure 1.1 The flow diagram

Here one plate is fixed and a shearing force F is applied to the other. When conditions

are steady the force F is applied to the other and balanced by an internal force in the fluid

due to its viscosity. Newton, while discussing the properties of fluid, remarked that in a

simple rectilinear motion of a fluid two neighboring fluid layers, one moving over the other

with some relative velocity, will experience a tangential force proportional to the relative

velocity between the two layers and inversely proportional to the distance between the

layers, that is if the two neighboring fluid layers are moving with velocities u and u + δu

are at a distance δy, then the shearing stress, is given by

τ ∝ ∂u

∂yor τ = µ

∂u

∂y(1.2.1)

This is called Newtonian hypothesis and a fluid satisfying this hypothesis is called a

Newtonian fluid. It is clear from the Newton’s law that

• If τ = 0 then µ = 0, equation (1.2.1) will represent an ideal fluid.

• If ∂u∂y

= 0 then µ = ∞, equation (1.2.1) will represent an elastic bodies.


• A fluid for which the constant of proportionality µ does not change with rate of

deformation (shear strain ∂u∂y

) is said to be an Newtonian fluid and graph τ verses

∂u∂y

is a straight line.

Where µ is known as Newtonian viscosity. It will be seen that µ is the tangential force

per unit area exerted on layers of fluid a unit distance apart and having a unit velocity

difference between them.

The diagram relating shear stress and rate of shear for Newtonian fluids represents flow

curve of the type straight line. The example of Newtonian fluid are water, air, mercury,

benzene, ethyl alcohol, glycerin and oil etc.,

1.2.4 Non-Newtonian fluid

Non-Newtonian fluids are those fluids which do not obey Newtonian law. It can also be

stated as the non-Newtonian fluids are those for which flow curve is not linear, i.e., the

‘viscosity’ of a non-Newtonian fluid is not constant at a given temperature and pressure

but depends on other factors such as the rate of shear in the fluid. The typical examples

of these classes of fluids are paints, coal-tar, polymer solutions, condensed milk, paste,

lubricants, honey, plastics, molasses, molten rubber, printer ink, collides, macro/molecular

materials and so on.


1.2.5 Compressible and Incompressible fluid

Fluids undergo density changes when temperature and pressure variations occur in them

then they are consider compressible. For several flows situations, however density changes

are negligible and the fluid may be treated as incompressible. Therefore, flow of liquids

are treated as incompressible for small pressure and temperature variations.

1.2.6 Dusty fluid

The viscous fluid in which the dust particles are present and its aerodynamics resistance

is less than that of a clean gas is called dusty viscous fluid. Interest in the problems of

mechanics of fluids with more than one phase has developed very rapidly in recent years.

Situations, which occur frequently, are concerned with the motion of a liquid or gas which

contains a distribution of solid particles. e.g., the movement of dust laden air, fluidization

process, using dust in gas cooling system to enhance heat transfer process, formation

of rain drops by the coalescence of small droplets and the process of inhaling oxygen


in respiration. Scientists and technologists have taken keen interest in the study of the

problems of gas-solid particle flow being faced in many industries. Dusty fluid phenomena

are also important in sedimentation, pipe flows, gas purification and transport process.

The gas-particle flows are important in fallout of pollutants in air or water. It has

an important role in exhausting the gas through the nozzle of rockets with added metal

powders. In physiological science, motion of blood cells in the liquid plasma through

arteries can give vital information for cardiovascular problems. In micro ciliary transport

process of the lungs, the beating of cilia carries mucus up the bronchial tubes. The power

generation by MHD generator, as an alternative source of energy, also use the dusty fluid

phenomenon. The problem of two components fluids under the influence of temperature

difference is useful in soil science and geo-physics. The amount of solid particles present

in such systems is variable but definitely effective.

1.3 Some Important Types of Flow

1.3.1 Steady and Unsteady Flow

A flow in which the various parameters like velocity, pressure and density at any point do

not change with time is said to be a steady flow. For steady flow if u is the velocity at a

point then ∂u∂t

= 0.

A flow in which these parameter depend on time is called unsteady flow.


1.3.2 Laminar and Turbulent Flow

A flow in which each fluid particle traces out a definite curve and curves traced out by

any two different particle do not intersect, is said to be laminar. On the other hand, a

flow, in which each fluid particle does not trace out a definite curve and the curves traced

out by fluid particle intersect, is said to be turbulent flow. The most of the flows, which

occur in practical applications are turbulent, and this term denotes a motion in which an

irregular fluctuation (mixing, or eddying motion) is superimposed on the main stream.

1.3.3 Rotational and Irrotational Flow

The flow in which the fluid particles rotate about their own axis is called rotational and the

flow in which the fluid particle does not rotate about their own axis is called irrotational.

Mathematically,

If ∇× ~q = 0 ⇒ irrotational flow.

If ∇× ~q 6= 0 ⇒ rotational flow.

1.3.4 Uniform and Non-uniform Flow

A flow in which the velocities of fluid particles are equal at each section of the channel

is called uniform flow and a flow in which the velocities of fluid particles are different at

each section of the channel is called non-uniform flow.


1.3.5 Magnetohydrodynamics (MHD)

Magnetohydrodynamics is an important branch of fluid dynamics. It is concerned with the

interaction of electrically conducting fluids and electromagnetic fields. When a conductor

moves in magnetic field a current is induced in the conductor in a direction mutually at

right angles to both the field and the direction of motion. Conversely, when a conductor

currying an electric current moves in a magnetic field it experience a force tending to move

it at right angles to the electric field. These two statement first enunciated by Faraday.

MHD interactions occur both in nature and in manmade devices. MHD flow occurs in

the sun, earth interior, ionosphere, stars and their atmosphere, to mention a few. In the

laboratory many new devices have been made which utilize the MHD interaction directly,

such as propulsion units and power generators or which involve fluid-electromagnetic field

interactions, such as electron beam dynamics, traveling wave tubes, electrical discharges

and many others.

1.4 No-Slip condition of viscous fluid

When a viscous fluid flows over a solid surface, the fluid elements adjacent to the surface

attain the velocity of the surface; in other words, the relative velocity between the solid

surface and the adjacent fluid particles is zero. This phenomenon is known as the ‘no-slip’

condition.


1.5 Boundary-layer theory

At the beginning of the 20th century the Ludwig Prandtl introduced the concept of

boundary-layer theory. He showed that the flow past a body can be divided into two

regions: a very thin layer close to the body (boundary-layer) where the viscosity is im-

portant, and the remaining region outside this layer where the viscosity can be neglected.

In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate

vicinity of a bounding surface where effects of viscosity of the fluid are considered in de-

tail. The boundary layer effect occurs at the field region in which all changes occur in the

flow pattern.

Initially boundary-layer theory was developed mainly for the laminar flow of an incom-

pressible fluid. The theory was extended to the practically important turbulent incom-

pressible boundary-layer flow. One of the most important applications of the boundary-

layer theory is the calculation of the friction drag of bodies in a flow. In the Earth’s

atmosphere, the planetary boundary-layer is the air layer near the ground affected by

diurnal heat, moisture or momentum transfer to or from the surface. On an aircraft wing

the boundary-layer is the part of the flow close to the wing. In Naval architecture, many

of the principles that apply to aircraft also apply to ships and submarines.

1.5.1 Viscous boundary layer

The influence of boundary, due to no slip condition is confined to a very thin region in the

immediate neighborhood of the solid surface, known as viscous or momentum boundary

layer. In this thin layer there is rapid change in velocity of the fluid, from velocity of the

surface to its value that corresponds to external frictionless flow. This is the viscosity of


the fluid that gives rise to the boundary layer and for an inviscid fluid there exists no

boundary layer. For flow over a surface of finite or semi infinite length the thickness of

boundary layer increase in the down stream region. The thickness of the boundary layer

decreases with decrease in viscosity of the medium, but even for small viscosity, the shear

stress τw = µ∂u∂y

is important due to large velocity gradient. The injection or suction

across the porous surface also has a great effect on the size of the viscous boundary layer.

1.5.2 Thermal boundary layer

A thin region in which the temperature of the fluid particles changes from its free stream

value to body surface value is called “Thermal Boundary Layer”. The thermal boundary

layer strongly depends upon the thermal conductivity of the medium i.e., higher the

conductivity of the medium, thicker would be the thermal boundary layer. Like viscous

boundary layer, injection or suction across the porous surface also has a great effect on

the size of the thermal boundary layer.

1.6 Flow and Heat transfer

1.6.1 Temperature

The word temperature indicates a physical property on which depends the sense-impression

of hotness or coldness. Temperature has been defined as the “the state of a substance or

body with regard to sensible warmth referred to some standard of comparison”. Sense-

impressions can give only a crude estimate and temperature is usually measured by means

of a Thermometer.


1.6.2 Heat

The conception of heat which passes from the hotter to the colder body and is thought of as

bringing about the change of temperatures. According to Max Planck, the conception of

heat, like all other physical concepts originates in the sense-perception, but it acquires its

physical significance of the events which excite the sensation. So heat regarded physically,

has no more to do with the sense of hotness than color in the physical sense and has to

do with the perception of color.

The terms Heat and Temperature in older philosophy drew little or no distinction

between them and we still use words like blood-heat and summer heat, which introduce

the term heat in connection with the idea of temperature. Joseph Black was the first to

perceive clearly the necessity of removing this confusion and he pointed out that we must

distinguish between quantity and intensity of heat, quantity corresponding to the amount

of heat and intensity to temperature.

As we know that the knowledge of heat transfer is very important for construction

and designing of power plan, which will perform in the prescribed fashion, is the objective

of the engineer. This clearly requires detailed knowledge of the principles governing

heat transfer in the various components, which may be involved i.e., boilers, turbines,

condenser, pumps and compressors. Some of the other industrial fields of heat transfer

play an important role like heating and air conditioning, chemical reactions and process.

A detailed heat transfer analysis is essential, since the dimensions of boilers, heaters,

refrigerators and heat exchangers not only depend on the amount of heat to be transmitted

but also on the rate at which heat is to be transferred under given conditions.


1.6.3 Types of heat transfer

Heat transfer is a science that predicts the transfer of heat energy from one body to

another by virtue of temperature difference. Heat transfer occurs as a result of three

mechanisms.

• Conduction

• Convection

• Radiation

Conduction:

In conduction heat flows due to molecular interaction, molecules not being displaced

or due to the motion of free electrons.

Heat conduction may be stated as the transfer of internal energy between the molecules.

Heat flows from a region of higher temperature to a region of lower temperature by kinetic

motion or direct impact of molecules whether the body is at rest or in motion.

Convection:

Heat transfer due to convection involves the energy exchange between a solid surface

and an adjacent fluid.

Convection is a mechanism in which heat flows or transferred between a fluid and a

solid surface as a consequence of motion of fluid particles relative to the solid surface

when there exists a temperature gradient. Convection heat transfer may be classified as

“Forced Convection” and “Free or Natural Convection”.


Forced convection:

If heat transfer between a fluid and a solid surface occurs by the fluid motion in-

duced by by external agencies or forces then the mode of heat transfer is termed as “

forced convection”. Heat transfer in all types of heat exchangers, nuclear reactor and air

conditioning apparatus is by forced convection.

Natural or Free convection:

If heat transfer between a fluid and a solid surface occurs by the fluid motion due to the

density differences caused by the temperature differences between the surface and the fluid,

then the mode of heat transfer is termed as “Free Convection or Natural Convection”.

Heat flows from a heated metal plate to the atmosphere, heat flows from hot water to

the container are certain examples of free convection.

Radiation:

The phenomenon or the mode of heat transfer in the form of electromagnetic waves

without the presence of any intervening medium is called Radiation. The transfer of heat

energy from the sun to the earth is an example of Radiation.

Heat Flux:

The heat transfer per unit area is called heat flux. If q is the amount of heat transfer

and A is area normal to the direction of the heat flow, then the heat flux is

Q = q/A.

Heat Dissipation:

The heat generated by internal friction within the volume element of the fluid per unit

time is called heat dissipation.


Thermal Conductivity:

The concept of thermal conductivity is that “The quantity of heat passing in unit time

through each unit of area when there is a difference of temperature of one degree between

the inside and outside face of a wall of unit thickness”.

To be more specific about discussion of thermal conductivity we consider two parallel

layers of a fluid, at distance d apart are kept at different temperatures T1 and T2 (One of

the layers may be a solid surface). Fourier noticed that a flow of heat is set up through

the layer such that the quantity of heat q transferred through unit area in unit time is

directly proportional to the difference of the temperature between the layers and inversely

proportional to the distance d. Thus he found q = k T1−T2

d, where k is the constant of

proportionality and is known as the coefficient of thermal conductivity.

If the distance d between the two layers of fluid is infinitesimal the above law can be

written in the differential form as q = −k dTdy

, where the negative sign indicated that the

heat flows in the direction of decreasing temperature.

The dimensions of the coefficient of thermal conductivity can be determined as follows

k = Heat fluxtemperature gradient

.

Thermal Diffusivity:

The effect of conductivity on the temperature field is determined by the ratio of k to

the product of density ρ and specific heat Cp rather than k alone. This ration is known

as the thermal conductivity and it is usually denoted by α = kρcP

.


1.7 Porous media

A porous medium is a material containing pores, like sponges, clothes wicks, paper, sand,

gravel, filters, concrete bricks, plaster walls, many naturally occurring rocks, packed beds

used for distillation, absorption etc. The skeletal portion of the material is often called

the matrix or frame. The skeletal material is usually a solid, but structures like foams

are often also usefully analyzed using concept of porous media. A porous medium is

most often characterised by its porosity. Other properties of the medium can sometimes

be derived from the respective properties of its constituents and the media porosity and

pores structure, but such a derivation is usually complex. Most of the studies of flow in

porous media assumes the Darcys law is valid. However this law is known to be valid

only for relatively slow flow through porous media. In general we must consider the

effect of fluid inertia as well as of viscous diffusion at boundaries which may become

significant for material with high porosities such as fibrous and foams. The concept of

porous media is of great interest in many areas of applied science and engineering due to

their important applications in the field of agricultural engineering to study the under-

ground water resource, seepage of water in river beds, in petroleum technology to study

the movement of natural gas, oil and water through oil reservoirs, in chemical engineering

for filtration and purification processes. The petroleum industry has been showing a

lot of interest in these problems in connection with the crude oil production from the

underground reservoirs. These problems are also of much interest in geophysics and

in the study of the interaction of the geomagnetic field with the fluid in the geothermal

region. The textile technologist is interested in fluid flow through fibers, whereas biologists

are interested in water movement through plant roots of the cells of living systems. On


the other hand, it has also encountered in the field of mechanics, engineering, geosciences,

biology and biophysics, material science, etc. Fluid flow through porous media is a subject

of most common interest and has emerged a separate field of study. The study of more

general behaviour of porous media involving deformation of the solid frame is called

poromechanics.

1.7.1 Darcy’s Law

Based on the experimental research of Darcy in flow through porous medium, Navier-

Stokes equation are replaced by linear partial differential equations. Suitable approxima-

tions are to be made to get the solution, as the governing equations of porous media are

partial differential equations. In 1856, Henri Darcy formulated the law which governs the

flow through a porous medium. Darcy’s law is given by,

q = constant(−∇p + ρg). (1.7.1)

where p is the pressure, ρ is the density and g is the acceleration due to gravity.

Equation (1.7.1) express that Darcy’s velocity q is proportional to the sum of pressure

gradient and the gravitational force. Moreover, q is inversely proportional to viscosity.

This Darcy’s law is macroscopic equation of motion for Newtonian fluid in porous media at

small Reynolds numbers. Many researchers verified this law experimentally. The constant

in the equation is replaced by the permeability k by Musket.

Now equation (1.7.1) becomes,

q =−k

µ∇p. (1.7.2)

This law is valid for the flow through isotropic porous media.


By using Darcy’s law various flows through porous media have been invistigated by

Musket, De Wiest, Bear and many other researchers. The most general form of Darcy’s

law is given by,

ρ

E=

dui

dt= −∂p

∂x+ ρxi −

µ

kui, (1.7.3)

where,

xi = The ith component of body force per unit mass,

ui = The ith component of velocity,

E = The Porosity,

ddt

= Substantial derivative.

The dimension of permeability is L2. The unit of permeability is Darcy which is used in

petroleum industry. The value of one darcy is 0.987×10−8cm2. The hydraulic conductivity

of the porous medium is measured in meinzers. If the porous medium has a permeability

of one Darcy, then it has the hydraulic conductivity 18.2 meinzers.

Darcy’s law is valid when the flow takes place at low speeds. But for high speed flows,

Darcy’s law is not valid. Also Darcy’s law fails to describe the flows with high speeds

or the flow near surfaces which are either permeable or rigid. In such cases, Brinkman

equation will be useful.

1.7.2 Brinkman Model

The following equation is proposed by Brinkman for the flow through porous media

∇P = ρ~g − µ

k~u + µ∇2~u, (1.7.4)

where ~u is the velocity vector.


This equation is valid when the permeability k is very high. In general, the particles of

the porous media are loosely packed so that k is small. Hence there exist a two boundary

layer very near to the surface.

In 1966, Tam supplemented a theoretical proof for this equation. Katto and Masuoka

experimentally found that Brinkman equation is valid up to the magnitude of kh2 of order

10 or so. If the porous medium is made up of spherical particles then kh2 corresponds to

considerably high values of dh

where d is the diameter of the fillings and h is the recital

thickness of the porous media. Yamamoto and Yoshida made improvements on Darcy’s

law by adding corrective terms. Saffman [94] gave the equations of motion for the flow

through porous medium by incorporating viscous stresses.

1.7.3 Non-Darcy Law

In many practical problems, the flow through porous media is curvilinear and the curva-

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

and the drag increase more rapidly. Lapwood was the first person who suggested the

inclusion of convective inertial term (q · ∇)q, in the momentum equation. Subsequently

many research articles have appeared on the non-Darcy model. Now the equation can be

written as,

1

δ2(q · ∇)q = −∇P + ρg − µf

kq + µe(∇2q). (1.7.5)

However, equation (1.7.5) does not take care of possible unsteady nature of velocity.

The flow pattern in a certain region may be unsteady and one has to consider the local


acceleration term 1δ2

∂q∂t

also. Adding this term equation (1.7.5) it becomes,

ρ

(1

δ2

∂q

∂t+

1

δ2(q · ∇)q

)= −∇ρ + ρg − µf

kq + µe(∇2q). (1.7.6)

This equation is known as Darcy-Lapwood-Brinkman equation. For an isotropic porous

medium equation (1.7.5) takes the form.

ρ

(1

δ2

∂q

∂t+

1

δ2(q · ∇)q

)= −∇ρ + ρg − µfQ + µe(∇2q). (1.7.7)

1.8 Dimensionless Parameters

Every physical problem involved some physical quantities, which can be measured in

different units. But the physical problem itself should not depend on the unit used for

measuring these quantities. In dimensional analysis of any problem we write down the

dimensions of each physical quantity in terms of fundamental units. Then by dividing

and rearranging the different units, we get some non-dimensional numbers.

Dimensional analysis of any problem provides information on qualitative behaviors of

the physical significance of a particular phenomenon associated with the problem. There

are usually two general methods for obtaining dimensionless parameters.

1. The inspection analysis

2. The dimensionless analysis

In this thesis the latter method has been used. The basic equations are made dimen-

sionless using certain dependent and independent characteristics values. In this process

certain dimensionless numbers appear as the co-efficient of various terms in these equa-

tions. The some of the dimensionless parameters used in this thesis are explained below.


1.8.1 Ekman Number

It is the ratio of viscous forces in a fluid to the fictitious forces arising from planetary

rotation.

1.8.2 Reynolds Number

The Reynolds number is the ratio of inertial forces to viscous forces and consequently it

quantifies the relative importance of these two types of forces for given flow conditions.

Thus, it is used to identify different flow regimes, such as laminar or turbulent flow.

It is one of the most important dimensionless numbers in fluid dynamics and is used,

usually along with other dimensionless numbers, to provide a criterion for determining

dynamic similitude. It is named after Osborne Reynolds (1842-1912), who first introduced

this number while discussing boundary layer theory in 1883. Typically it is given as

follows:

Re =ρU2/h

µU/h2=

ρUh

µ=

Uh

ν,

where

U - some characteristic velocity,

h - some characteristic length,

ν = µρ

- kinematic fluid viscosity,

ρ - fluid density.


1.8.3 Hartmann Number

Hartmann number is the ratio of electromagnetic force to the viscous force. It was first

introduced by Hartmann and is defined as:

Ha = BL√

σµ,

where

B - the magnetic field,

L - the characteristic length scale,

σ - the electrical conductivity,

µ - the viscosity.

1.8.4 Hall parameter

It is defined the product of cyclotron frequency of electrons and electron collision time

and is given by

m = ωeτe,

where ωe is cyclotron frequency of electrons and τe is electron collision time.

1.8.5 Prandtl Number

It is an important dimension parameter dealing with the properties of a fluid. It is defined

as the ratio of viscous force to thermal force of a fluid. Prandtl number physically means

or signifies the relative speed with which the momentum and heat energy are transmitted

through a fluid. It thus associates the velocity and temperature fields of a fluid. For gases

Prandtl number is of unit order and varies over a wide range in case of liquids.


Pr = viscous forceThermal force

= µCp

k,

where

µ - Coefficifent of viscosity,

Cp - Specific heat at constant pressure,

k - Coefficient of thermal conductivity.

1.8.6 Eckert Number

It is equal to the square of the fluid velocity far from the body divided by the product of the

specific heat of the fluid at constant temperature and the difference between temperatures

of the fluid and the body.

Ec =V 20

Cp(Tw−T∞),

where

Tw - Temperature near the plate,

T∞ - Temperature far away from the plate,

Cp - Specific heat at constant pressure,

V0 - Characteristic value of velocity.

1.8.7 Number Density

Number density is an intensive quantity used to describe the degree of concentration

of countable objects in the three-dimensional physical space, or Number density is the

number of specified objects per volume i.e., n = N/V.


1.8.8 Grashof Number

It plays a significant role in free convection heat transfer. The ratio of the product of the

inertial force and the buoyant force to the square of the viscous force in the convection

flow system is known as Grashof number. Grashof number in free convection is analogues

to Reynolds number in forced convection.

Gr = gβ(Tw − T∞)l3

ν2,

where

g - acceleration due to gravity,

β - volumetric coefficient of thermal expansion,

l - characteristic length,

Tw - temperature of the wall,

T∞ - constant temperature far away from the sheet.

1.8.9 Non-uniform heat source/sink parameter

It is defined as

q′′′ =

(kUw(x)

xν

)[A∗(Tw − T∞)f ′(η) + B∗(T − T∞)],

where A∗ and B∗ are the parameters of the space and temperature dependent internal

heat generation/absorption. It is to be noted that A∗ and B∗ are positive to internal heat

source and negative to internal heat sink, ν is the kinematic viscosity, Tw and T∞ denote

the temperature at the wall and at large distance from the wall respectively.


1.8.10 Radiation parameter

It is defined as

Nr =16σ∗T 3

∞3kk∗ ,

where

σ∗ - Stefan-Boltzman constant

k∗ - mean absorption co-efficient.

1.8.11 Melting parameter

M =Cp(T∞−Tm)

γ + Cs(Tm − T0),

where M is the dimensionless melting parameter, where Cp is the heat capacity of the

fluid at constant pressure. The melting parameter is a combination of the Stefan numbers

Cf (T∞−T0)

γand Cs(T∞−T0)

γfor the liquid and solid phases, respectively. We Take Tm is the

temperature of the melting surface, while the temperature in the free-stream condition is

T∞, where Tm > T∞.

1.8.12 Curvature parameter

γ =

√lν

ba2,

is the curvature parameter, γ = 0, corresponds to flat plate.

1.8.13 Shear stress/Skin Friction

Any real fluids moving along solid boundary will incur a shear stress on that boundary.

The no-slip condition dictates that the speed of the fluid at the boundary is zero, but at


some height from the boundary the flow speed must equal that of the fluid. The region

between these two points is aptly named the boundary layer. For all Newtonian fluids

in laminar flow the shear stress is proportional to the strain rate in the fluid, where the

viscosity is the constant of proportionality. However for Non-Newtonian fluids, this is

no longer the case as for these fluids the viscosity is not constant. The shear stress is

imparted onto the boundary as a result of this loss of velocity. The shear stress, for a

Newtonian fluid, at a surface element parallel to a flat plate, at the point y, is given by:

τ(y) = µ∂u

∂y

, where

µ is the dynamic viscosity of the fluid,

u is the velocity of the fluid along the boundary, and

y is the height of the boundary. Specifically, the wall shear stress is defined as:

τw ≡ τ(y = 0) = µ∂u

∂y

∣∣∣∣y=0

.

In case of wind, the shear stress at the boundary is called wind stress.

1.8.14 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 Fouriers 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 l is the length. Now Nusselt


number can be written as

Nu =Ch(Tw − T )

k(Tw − T )/l=

chl

k.

i.e., 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.

1.9 Laplace Transforms

Laplace Transform is an essential mathematical tool which can be used to solve several

problems in science and engineering. This technique becomes popular when Heaviside

function applied to the solution of an ODE representing a problem in electrical engineer-

ing. Transforms are used to accomplish the solution of certain problems with less effort

and in a simple routine way. The Laplace transform method reduces the solution of an

ODE to the solution of an algebraic equation. Also, when the Laplace transform technique

is applied to a PDE, it reduces the number of independent variables by one.

Definition 1.9.1. Let f(t) be a continuous and single-valued function of a real variable

t defined for all t, 0 < t < ∞, and is of exponential order. Then the Laplace transform of

f(t) is defined as a function F (s) denoted by the integral

L[f(t)] =

∞∫0

e−stf(t)dt. (1.9.1)

Definition 1.9.2. Error Function:

The error function is defined as,

erf(x) =2√π

x∫0

e−t2dt. (1.9.2)


and its compliment is

erfc(x) = 1 − erf(x) =2√π

∞∫x

e−t2dt. (1.9.3)

The Laplace transform of the error function is,

L[erf(x)] =1

ses2/4erfc(s/2).

Some of the inverse transforms are,

L−1

e(−k

√s+α)

s + a

=

e−at

2

[e−k

√α−aerfc(φ1) + e−k

√α−aerfc(φ2)

],

where φ1 =k

2√

t−√

(α − a)t, φ2 =k

2√

t+√

(α − a)t,

L−1

e(−k

√s+α)

(s + a)(s + b)

=

k

2√

π(b − a)

[e−atI(a, t) − e−btI(b, t)

],

where I(x, t) =

t∫0

τ−3/2exp

[−k2

4τ− (α − x)τ

]dτ

and x = a or b. On integration then the above inverse Laplace transform becomes,

L−1

e(−k

√s+α)

(s + a)(s + b)

=

1

b − a[T (a, t) − T (b, t)],

where T (x, t) =e−xt

2

[e(−k

√α−x)erfc

(k

2√

t−√

(α − x)t

)+ e(k

√α−x)erfc

(k

2√

t+√

(α − x)t

)],

L−1

e(−k

√s+α)

(s + a)(s + b)(s + c)

= −

∑ T (a, t)

(c − a)(a − b),

L−1

e(−k

√s+α)

(s + α)

= e−αterfc

(k

2√

t

),

L−1

e(−k

√s+α)

(s + α)(s + a)

=

T (a, t)

(α − a)− e−αt

(α − a)erfc

(k

2√

t

),

L−1

e(−k

√s+α)

(s + a)2

=

e−at

2

[(t − k

2√

α − a

)e−k

√α−aerfc(φ1)

+

(t +

k

2√

α − a

)ek

√α−aerfc(φ2)

].


1.9.1 Complex Inversion Formula/Mellin-Fourier integral

In solving partial differential equations using Laplace transform method, complex variable

theory may come in handy for finding inverse transform. Inverse Laplace transform can

be expressed as an integral which is known as inverse integral and this integral can be

evaluated by using contour integration methods.

The inverse Laplace Transforms of U , V are u, v respectively and are given by the

integrals

u =1

2iπ

r+i∞∫r−i∞

extUdt and v =1

2iπ

r+i∞∫r−i∞

extV dt. (1.9.4)

Which can be evaluated by means of contour integration. Since there is no branch point,

the contour chosen is the closed curve ABC formed by the line x = r and a semi circle C

with origin as center and radius R (See figure 1.1) so that

r+i∞∫r−i∞

extUdt = limR→∞

B∫A

extUdt

= limR→∞

∮ABC

extUdt −∫C

extUdt

.

Using Cauchy’s theorem of residues and Jordans lemma, we have

u =1

2iπ

r+i∞∫r−i∞

extUdt = sum of residues ofextU

at its poles.

Similarly,

v =1

2iπ

r+i∞∫r−i∞

extV dt = sum of residues ofextV

at its poles.

Table 1.1: Laplace transform of some important functions.


Sl.No. Function Laplace Transform

1 e−a2/4t√

πte−a

√s

√s

2 ae−a2/4t

2√

πt3e−a

√s

3 erfc(

a2√

t

)e−a

√s

s

4 2√

tπe−a2/4t − a erfc

(a

2√

t

)e−a

√s

s√

s

5 eateb2terfc(b√

t + a2√

t

)e−a

√s

√s√

s+b

6 eateb2terfc(b√

t + a2√

t

)+ erfc

(a

2√

t

)be−a

√s

√s√

s+b

1.10 Similarity Transformation

Birkhoff (1950) first recognized that Boltzmann’s method of solving the diffusion equation

with a concentration-dependant diffusion co-efficient is based on the algebraic symmetry

of the equation and special solutions of this equation can be obtained by solving a related

ordinary differential equation. Such solutions are called “similarity solutions” because

they are geometrically similar. He also suggested that the algebraic symmetry of the

partial differential equations can be used to find similarity solutions of other partial dif-

ferential equations by solving associated ordinary differential equations. Thus, the method

of similarity solutions has become a very successful dealing with the determination of a

group of transformation under which a given partial differential equation is invariant.

The simplifying feature of this method is that a similarity transformation of the form

u(x, t) = tp v(η), η = x t−q can be found which can, then, we used effectively to re-

duces the partial differential equations to an ordinary differential equations with η as the


independent variable. The resulting ordinary differential equations is relatively easy to

solve. In practice this method is simple and useful in finding solutions of both linear and

nonlinear partial differential equations.

1.11 Numerical Methods

Numerical methods are the way to do higher mathematics problems on a computer, a

technique widely used by scientists and engineers to solve their problems. A major ad-

vantage for numerical analysis is that a numerical answer can be obtained even when a

problem has no “analytical” solution. It is important to realize that a numerical anal-

ysis solution is always numerical. Analytical methods usually give a result in terms of

mathematical functions that can then be evaluated for specific instances. There is thus

advantage to the analytical results, in that the behavior and properties of the function are

often apparent. However, numerical results can be plotted to show some of the behavior

of the solution.

Another important distinction is that the result from numerical method is an approx-

imation, but results can be made as accurate as desired. To achieve high accuracy, many

separate operations must be carried out. Here are some of the operations that numerical

methods can do:

• Solve for the roots of a nonlinear equation.

• Solve large systems of linear equations.

• Get the solutions of a set of nonlinear equations.

• Interpolate to find intermediate values within a table of data.


• Solve ODE when given initial values for the variables.

• Solve boundary-value problems and determine eigenvalues and eigenvectors.

• Obtain numerical solutions to all types of partial differential equations and so on.

In connection with numerical analysis many symbolic algebraic programmes are avail-

able, namely Mathematica, DERIVE, Maple, MathCad, MATLAB, and MacSyma. In

this thesis the numerical solutions of the problem are solved by RKF -45 method with

the help of algebraic software MAPLE.

1.11.1 Runge Kutta Fehlberg Method

Runge-Kutta-Fehlberg is adaptive; that is, the method adapts the number and position

of the grid points during the course of the iteration in attempt to keep the local error

within some specified bound. Sketch of the ideas:

• Begin with two RK approximation algorithms, one with order p and with order

p + 1.

• Apply the algorithms to get two approximations at a given grid point tk.

• These approximations are used to approximate the local discretization error at the

grid point. This error approximations is then used to make several decisions.

• If the error approximation exceeds some prescribed maximum bound on accuracy,

then a smaller step size is assigned, a new grid point tk is assigned, and the preceding

steps are repeated.


• If the error approximation falls below some present minimum bound on accuracy,

then the step size is increased and the next step in the iteration is performed.

• If the error approximation falls in between some user-specified minimum and max-

imum values, then we may choose to leave the step size alone or we may compute

an optimal step size for the next step. The term optimal is used loosely because

there are some assumptions made and some approximations involved in getting this

value.

• Typically, the approximation given to the user is reported as the more accurate

p + 1st order approximation, even through, in the analysis, that approximation is

used to approximate the error in the pth order approximation.

The Runge-Kutta-Fehlberg method (denoted RKF45) is one way to resolve problem.

It has a procedure to determine, if the proper step size h is being used. At each step, two

different approximations for the solution are made and compared. If the two answers are

in close agreement, the approximation is accepted. If the two answers do not agree to a

specified accuracy, the step size is reduced. If the answers agree to more significant digits

than required, the step size is increased. Each step requires the use of the following six

values:

k1 = hf(tk, yk),

k2 = hf

(tk +

1

4h, yk +

1

4k1

),

k3 = hf

(tk +

3

8h, yk +

3

32k1 +

9

32k2

),


k4 = hf

(tk +

439

216h, yk +

1932

2197k1 −

7200

2197k2 +

7296

2197k3

),

k5 = hf

(tk + h, yk +

439

216k1 − 8k2 +

3680

513k3 −

845

4104k4

),

k6 = hf

(tk +

1

2h, yk −

8

27k1 + 2k2 −

3544

2565k3 +

1859

4104k4 −

11

40k5

).

Now the approximation solution to the given I.V.P. is made using a Runge-Kutta method

of order 4:

yk+1 = yk +25

216k1 +

1408

2565k3 +

2197

4101k4 −

1

4k5,

where the four function values f1, f3, f4 and f5 are used. Notice that f2 is not used in

the above formula. A better value for the solution is determined using a Runge-Kutta

method of order 5:

zk+1 = yk +16

135k1 +

6656

12825k3 +

28561

56430k4 −

9

50k5 +

2

55k6.

The optimal step size sh can be determined by multiplying the scalar s times the current

step size h. The scalar s is

s =

(tolh

2|zk+1 − yk+1|

) 14

≈ 0.84

(tolh

|zk+1 − yk+1|

) 14

.

Chapter 2

An analytical approach for thesolution of an unsteady MHD flow ofa rotating dusty fluid

2.1 Introduction

The flow of a binary mixture of fluid and solid particles bounded by a semi-infinite plate are

extremely useful in improving the design and operation of many industrial and engineering

devices. It has important applications in the fields of fluidization, combustion, use of

particles in gas cooling systems, centrifugal separation of matter from fluid, petroleum

industry, crude oil purification, electrostatic precipitation, polymer technology and fluid

droplet sprays. Saffman [94] carried out pioneering work on the stability of a laminar flow

of a dusty gas which describes the motion of a gas carrying small dust particles and he

derived the equations satisfied by small disturbances of a steady laminar flow.

Following the Saffman model, Liu [73] has studied the flow induced by an oscillating

infinite flat plate in a dusty gas. Michael and Miller [79] investigated the motion of

dusty gas with uniform distribution of the dust particles placed in the semi-infinite space

above a rigid plane boundary. Rao [?] obtained the analytical solutions for the dusty

45

Chapter-2: An analytical approach for the solution of an unsteady MHD flow · · · 46

fluid flow through a circular tube under the influence of constant pressure gradient, using

appropriate boundary conditions. Gupta and Gupta [55] studied the flow of a dusty gas

through a channel with arbitrary time varying pressure gradient. An unsteady flow of

a conducting dusty fluid through a rectangular channel with time dependent pressure

gradient was examined by Singh [98]. Recently, Gireesha et al [49] reported the exact

solutions for pulsatile flow of an unsteady dusty fluid through a rectangular channel.

In most of the cases, the effect of Hall current is ignored by applying Ohm’s law as

it has no marked effect for small magnetic fields. However, to study the effects of strong

magnetic fields on the electrically conducting fluid flow, one can see that, the influence

of the electromagnetic force is noticeable and causes anisotropic electrical conductivity

in the plasma. This anisotropy in the electrical conductivity of the plasma produces

a current known as the Hall current. The effects of Hall current on the fluid flow in

rotating frame of reference have many engineering applications in flows of laboratory

plasmas in MHD power generation, MHD accelerators, and in several astrophysical and

geophysical situations. In view of these applications, Seth and Ansari [97] considered

the magnetohydrodynamic convective flow in a rotating channel with Hall effect. Ghosh

and Pop [46] studied the effects Hall current on MHD plasma Couette flow in a rotating

frame of reference. Tiwari and Singh [103] studied the effect of Hall current on unsteady

hydromagnetic flow of an incompressible fluid with particle suspension bounded by a

semi infinite plate. An unsteady flow of a rotating dusty fluid past a porous plate in

the presence of Hall effect were studied by Debnath [34]. Recently, Gireesha et al [48]

presented a perturbation solution for viscoelastic fluid flow in non-uniform channel with

Hall current and chemical reaction.


Motivated by above researchers work, the current chapter is aimed at investigate the

influence of Hall current on an unsteady flow of fluid with uniform distribution of dust

particles embedded in the rotating system. The fluid and dust velocity fields are studied

subjected to three different boundary like impulsive motion, transition motion and motion

for finite times using an analytical framework. Further, effects of various parameters such

as Ekman number, magnetic parameter, time and Hall parameter etc., for both fluid and

dust velocity profiles are discussed in detail through graphs.

2.2 Mathematical Formulation

Consider an unsteady flow of electrically conducting incompressible viscous fluid with

uniform distribution of dust particles bounded by an infinite plate. A uniform magnetic

field B0 is applied to the flow and it is assumed that there is no applied or polarization

voltage exists.

Figure-2.1: A schematic representation of the flow diagram.


The fluid as well as the plate is in a state of solid body rotation with constant angular

velocity ~Ω about the z− axis and additionally non-torsional oscillation of frequency ω1 is

imposed on the plate in its own plane as shown in the figure 2.1.

The hydromagnetic dusty fluid flow in a rotating co-ordinate system for an unsteady

case is governed by the following equations [94]:

For fluid phase:

∂~u

∂t+ (~u · ∇)~u + 2~Ω × ~u = −1

ρ∇p +

1

ρ( ~J × ~B) + ν∇2~u +

KN

ρ(~v − ~u), (2.2.1)

∇ · ~u = 0. (2.2.2)

For dust phase:

m

[∂~v

∂t+ (~v · ∇)~v + 2~Ω × ~v

]= K(~u − ~v), (2.2.3)

∇ · ~v = 0. (2.2.4)

We have the following nomenclature:

~u = (u1, u2, u3) is velocity of fluid phase and ~v = (v1, v2, v3) is velocity of dust phase,

p is pressure field including the centrifugal term, ~J is electric current density, ~B is total

magnetic field, N is number density of dust particles, m is mass of dust particle, K = 6πaµ

is Stoke’s-co-efficient of resistance where a is radius of the dust particles, ρ is density and

ν is kinematic viscosity of the fluid.

Assuming magnetic Reynold s number to be small, we neglect induced magnetic field

in comparison with applied magnetic field. The generalized Ohm′ s law, in the absence

of the electric field is

~J +ωeτe

B0

( ~J × ~B) = σ

[~u × ~B +

1

ene

∇pe

], (2.2.5)


where ωe, τe, σ, e, pe and ne are respectively cyclotron frequency of electrons, electron

collision time, electrical conductivity, electron charge, electron pressure and number den-

sity of the electron. The ion-slip and thermoelecric effects are not included in equation

(2.2.5). Further, it is assumed that ωiτi 1, where ωi and τi are cyclotron frequency and

collision time for ions respectively.

We assume that the velocity field depends on z and t only, so that

~u(z, t) = [u1(z, t), u2(z, t), u3(z, t)], (2.2.6)

~v(z, t) = [v1(z, t), v2(z, t), v3(z, t)]. (2.2.7)

For the present problem

u3(z, t) = 0, v3(z, t) = 0 and N = N0 (constant). (2.2.8)

In the presence of constant pressure gradient, the equations of motion (2.2.1) and (2.2.3)

will takes the form;

∂u1

∂t− 2Ωu2 = −1

ρ

∂p1

∂z+ ν

∂2u1

∂z2+

σB20

ρ(1 + m2)(mu2 − u1) −

l

τ(u1 − v1), (2.2.9)

∂u2

∂t− 2Ωu1 = −1

ρ

∂p2

∂z+ ν

∂2u2

∂z2− σB2

0

ρ(1 + m2)(mu1 + u2) −

l

τ(u2 − v2), (2.2.10)

∂v1

∂t− 2Ωv2 =

1

τ(u1 − v1), (2.2.11)

∂v2

∂t− 2Ωv1 =

1

τ(u2 − v2), (2.2.12)

where l = mN0

ρis mass concentration and τ = m

Kis relaxation time.

Since we have assumed, a constant pressure gradient is impressed on the system for

t > 0, we can write;

C = −1

ρ

∂

∂z(p1 + p2). (2.2.13)


Introducing the notation p = u1 + iu2, q = v1 + iv2 and using the equation (2.2.13)

equations (2.2.9), (2.2.10), (2.2.11), (2.2.12) can be written as

∂p

∂t− 2iΩp = C + ν

∂2p

∂z2− σB2

0

ρ(1 + m2)(1 + im)p +

l

τ(q − p), (2.2.14)

∂q

∂t+ 2iΩq =

1

τ(p − q). (2.2.15)

In view of the imposed oscillation on the plate, equations (2.2.14) and (2.2.15) have to

be solved when subject to a no-slip boundary condition at the plate and no disturbance

at infinity for three different cases as follows;

2.3 Solution of the Problem

To make the above system dimensionless, introduce the following non-dimensional vari-

ables

z′ =zU∗

ν, t′ = Ωt, p′ =

p

U∗ , q′ =q

U∗ , C ′ = Cν

U∗3 Ω.

where U∗ is characteristic of velocity.

After non-dimensionalizing the equations (2.2.14) and (2.2.15) one can be written as

∂2p

∂z2− E

2

∂p

∂t−[iE +

M2

(1 − im)

]p + C − l

τ(p − q) = 0, (2.3.1)

E

2

∂q

∂t+ iEq − 1

τ(p − q) = 0, z > 0, (2.3.2)

where

E =2Ων

U∗2 , Ekman number,

m = ωeτe, Hall parameter,

M =

(σνB2

0

ρU∗2

) 12

, Magnetic parameter,


also ω = ω1/Ω is the non-dimensional frequency of oscillation and

τ ′ = τU∗2/ν = mU∗2/kν.

CASE-2.1 Impulsive Motion :

In the case of impulsive motion, the boundary conditions are considered as,

p = p0 + u0δ(t), (2.3.3)

q = q0 + u1δ(t) on z = 0, t > 0, (2.3.4)

p(z, t), q(z, t) → 0 as z → ∞, t > 0, (2.3.5)

where δ(t) is Dirac delta function and p0, q0, u0, u1 are complex constants so that p(z, t)

and q(z, t) become real on the plate.

The initial conditions of the problem are

p(z, t) = q(z, t) = 0 at t ≤ 0 for all z. (2.3.6)

After non-dimensionalizing the above initial and boundary condition one can get

p =p0

U∗ +U0

U∗ δ(t), (2.3.7)

q =q0

U∗ +U1

U∗ δ(t) on z = 0, Ωt ≥ 0, t > 0, (2.3.8)

p, q → 0 as z → ∞, t > 0, (2.3.9)

p, q = 0 at t ≤ 0 for all z > 0. (2.3.10)

To solve the initial and boundary value problem, we introduce the Laplace transforms

p(z, s) and q(z, s) of p(z, t) and q(z, t) respectively as

p(z, s) =

∞∫0

e−stp(z, t)dt and q(z, s) =

∞∫0

e−stq(z, t)dt. (2.3.11)


On applying the Laplace transform the equations (2.3.1) and (2.3.2) reduces to sec-

ond order ordinary differential equations. Then by solving these equations one can get

solutions for p(z, s), and q(z, s) under the transformed boundary conditions as follows

p(z, s) =

[P0

U∗1

s+

U0

U∗ − C

ks

]e−z

√k +

C

ks, (2.3.12)

q(z, s) =

[(q0

U∗ +U0

U1

)1

s

]e−z

√k −

[2

Eτs + 2iEτ + 2

(C

ks

)]e−z

√k (2.3.13)

+

(C

ks

)2

Eτs + 2iEτ + 2,

where

α =τ 2E2

4(1 − im),

λ = (1 − im)τE

2,

δ = τ(1 − im)(iEτ + 1),

a1 =

(2i +

2

Eτ

),

L =βδ − γλ

2γδ,

K =αs2 + βs + γ

λs + δ,

β = τ 2E(1 − im)

(iE +

1

2τ

)+

E

2τ(M2τ − l),

γ = −Eτ 2(1 − im)

(E − i

τ

)+ im2τ 2E + M2τ + ilEτ(1 − im).

a. Solutions for Small Times:

The nature of the flow fields p(z, t) and q(z, t) for small times can be determined by

the asymptotic behavior of their Laplace transforms p(z, s) and q(z, s) for the large value

of |s| which are given by

p(z, s) =

[P0

U∗1

s+

U0

U∗ − 2C

Es

]e−z

√E2

s +2C

Es, (2.3.14)


q(z, s) =

[(q0

U∗ +U0

U1

)1

s

]e−z

√E2

s −[

2

Eτs + 2iEτ + 2

(2C

Es

)]e−z

√E2

s,(2.3.15)

+

(2C

Es

)2

Eτs + 2iEτ + 2.

Taking inverse Laplace transform of equation (2.3.14) and (2.3.15) one can get

p(z, t) ∼[

p0

U∗ +U0

U∗

]erfc

z

2

√E

2t

+

(t +

z2E

4

)erfc

(z

2

√E

2t

)−

(z

√E

2

√t

π

)e

z2E8t +

2C

Et, (2.3.16)

q(z, t) ∼[

q0

U∗ +U1

U∗

]erfc

z

2

√E

2t

− 2Ce−a1t

E2τa21

e−z

√Ea1

2ierfc

(z

2

√E

2t− i

√a1t

)

+ e−z√

Ea12

ierfc

(z

2

√E

2t+ i

√a1t

)+

2C

E2a21τ

[−1 + a1t + e−a1t

]. (2.3.17)

From the above solution one can say that immediately after the impulsive motion is

imposed on the plate, an unsteady boundary layer flow builds up in the vicinity of the

plate. Further, the solution consists of Stoke’s layer of thickness of order√

νω

and the

Rayleigh layer of order√

νt. Also one can observe that the solution remains unaffected

by the dusty parameter as well as rotation and magnetic term. Similar discussion is true

for q(z, t).

b. Solutions for Large Times:

Solutions for large times can be determined for small values of |s|. Keeping |s| small

in equations (2.3.16) and (2.3.17) we have

p =

[1

s

(p0

u∗ +u0

u∗

)− δC

γLs(s + 1L)

]e−z

√γLδ

(s+ 1L

) +δC

γLs(s + 1L), (2.3.18)

q =

[1

s

(p0

u∗ +u0

u∗

)− 2

Eτs + 2iEτ + 2

δC

γLs(s + 1L)

]e−z

√γLδ

(s+ 1L

) +δC

γLs(s + 1L).(2.3.19)


where L = βδ−γλ2γδ

. Taking inverse Laplace transform of equations (2.3.18) and (2.3.19) we

get

p(z, t) ∼( p0

U∗ +u0

U∗

) 1

2

[ez√

γLδ erfc

z

2

√γ

δ

√L

t+

√t

L

+ e−z√

γLδ erfc

z

2

√γ

δ

√L

t−√

t

L

]

− Cδ

2γ

[e−z

√γδ erfc

(z

2

√γ

δ

√L

t−√

t

L

)

+ ez√

γδ erfc

(z

2

√γ

δ

√L

t+

√t

L

)]

+ e−tL Lerfc

(z

2

√γ

δ

√L

t

)+

Cδ

γ

[1 − e−

tL

], (2.3.20)

q(z, t) ∼( q0

U∗ +u1

U∗

) 1

2

[ez√

γLδ erfc

z

2

√γ

δ

√L

t+

√t

L

+ e−z√

γLδ erfc

z

2

√γ

δ

√L

t−√

t

L

]

− 2δC

γLEτ

[e−a1t

2a1(a1 − 1L)

×

e−z√

γLδ

( 1L−a1)erfc

z√

γLδ

2√

t−√

(1

L− a1)t

+

ez√

γLδ

( 1L−a1)erfc

z√

γLδ

2√

t+

√(1

L− a1)t

+2δC

γEτ

e−tL

(a1 − 1L)erfc

(z√

γδ

2√

t

)

− L

2a1

ez√

γδ erfc

z√

γLδ

2√

t+

√t

L

+ e−z

√γδ erfc

z√

γLδ

2√

t−√

t

L

+

2Cδ

γE2τ

[1

a1

+L

1 − La1

e−tL +

e−a1t

(La1 − 1)a1

]. (2.3.21)


CASE-2.2 Transition Motion :

Consider the case, transition Motion, in which the boundary conditions are

p = p0 + u0H(t)e−λt, (2.3.22)

q = q0 + u1H(t)e−λt on z = 0, t > 0, (2.3.23)

p(z, t), q(z, t) → 0 as z → ∞, t > 0, (2.3.24)

where H(t) is Heaviside’s unit step function and p0, q0, u0, u1 are complex constants so

that p(z, t) and q(z, t) become real on the plate.

The initial conditions of the present case are considered as


After non-dimensionalizing the above initial and boundary conditions one can obtained

as follows;

p =p0

U∗ +u0

U∗H(t)e−λt, (2.3.26)

q =q0

U∗ +u1

U∗H(t)e−λt on z = 0, t > 0, (2.3.27)

p, q → 0 as z → ∞, t > 0, (2.3.28)

p, q = 0 at t ≤ 0 forall z > 0, (2.3.29)

By applying the same procedure as in case-2.1, we obtain the expressions for p and q as

a. Solutions for Small Times

p(z, t) ∼ p0

U∗ erfc

z

2

√E

2t

+

U0

2U∗ e−λt

[e−z

√Eλ2

ierfc

z

2

√E

2t− i

√λt

+ ez√

Eλ2

i erfc

z

2

√E

2t+ i

√λt

]

+2C

E

(t +

z2E

4

)erfc

(z

2

√E

2t

)−

(z

√E

2

√t

π

)e

z2E8t +

2C

Et. (2.3.30)


q(z, t) ∼ q0

U∗ erfc

z

2

√E

2t

+

U0

2U∗ e−λt

[e−z

√Eλ2

ierfc

z

2

√E

2t− i

√λt

+ ez√

Eλ2

i erfc

z

2

√E

2t+ i

√λt

]

+2Ce−a1t

E2τa21

e−z√

Ea12

ierfc

z√

E2

2√

t− i

√a1t

+

(ez√

Ea12

i

)erfc

z√

E2

2√

t+ i√

(a1)t

+

4C

E2a21τ

[−1 + a1t + e−a1t

]. (2.3.31)

b. Solutions for Large Times

p(z, t) =( p0

U∗

) 1

2

[ez√

γδ erfc

z

2

√γ

δ

√L

t+

√t

L

+ e−z√

γδ erfc

z

2

√γ

δ

√L

t−√

t

L

]+

U0

2U∗ e−λt[e−z

√γLδ

( 1L−λ)

× erfc

z

2

√γ

δ

√L

t+

√(1

L− λ

)t

+ e−z√

γLδ

( 1L−λ) erfc

z

2

√γ

δ

√L

t+

√(1

L− λ

)t

+δC

2γ

[ez√

γδ erfc

(z

2

√γ

δ

√L

t+

√t

L

)

+ e−z√

γδ erfc

(z

2

√γ

δ

√L

t−√

t

L

)]

− δC

2γe−

tL erfc

(z

2

√γ

δ

√L

t

)+

cδ

γ(1 − e−

tL ), (2.3.32)


q(z, t) =( q0

U∗

) 1

2

[ez√

γδ erfc

z

2

√γ

δ

√L

t+

√t

L

+ e−z√

γδ erfc

z

2

√γ

δ

√L

t−√

t

L

]+

U1

2U∗ e−λt[ez√

γLδ

( 1L−λ)

× erfc

z

2

√γ

δ

√L

t+

√(1

L− λ

)t

+ e−z√

γLδ

( 1L−λ) erfc

z

2

√γ

δ

√L

t−

√(1

L− λ

)t

+4δC

γLE2τ

[e−a1t

2a1(a1 − 1L)

×

e−z

√γLδ

( 1L−a1)erfc

(z

2

√γ

δ

√L

t−

√(1

L− a1

)t

)

+

ez√

γLδ

( 1L−a1)erfc

(z

2

√γ

δ

√L

t+

√(1

L− a1

)t

)]

− 4Cδ

γLE2τ

Le−tL

(a1 − 1L)erfc

(z

2

√γ

δ

√L

t

)

+2Cδ

γE2τa1

[ez√

γδ erfc

(z

2

√γ

δ

√L

t+

√t

L

)

+ e−z√

γδ erfc

(z

2

√γ

δ

√L

t−√

t

L

)]

+4Cδ

γE2τ

[1

a1

+L

1 − La1

e−tL +

e−a1t

(La1 − 1)a1

]. (2.3.33)

CASE-2.3 Motion for Finite Time :

In view of the imposed oscillation on the plate equation (2.3.14) and (2.3.15) have to

be solved when subjected to the boundary condition at the plate and no disturbance at


infinity as

p = p0 + u0H[(t) − H(t − T )], (2.3.34)

q = q0 + u1H[(t) − H(t − T )] on z = 0, t > 0, (2.3.35)

p(z, t), q(z, t) → 0 as z → ∞, t > 0, (2.3.36)

where H(t) is Heaviside’s unit step function and p0, q0, u0, u1 are complex constants so

that p(z, t) and q(z, t) become real on the plate.

The initial conditions of the problem are


By applying the same procedure as in case-2.1, we obtain the expressions for p and q as

a. Solutions for Small Times

p(z, t) ∼( p0

U∗ +u0

U∗

)erfc

z

2

√E

2t

+( u0

U∗

)erfc

z

2

√E√

2(t − T )

− 2C

E

(t +z2E

4

)erfc

z√

E2

2√

t

−

(z

√E

2

√t

π

)e

z2E8t + t

, (2.3.38)

q(z, t) ∼( q0

U∗ +u1

U∗

)erfc

z

2

√E

2t

+ (

u0

U∗ )erfc

z

2

√E√

2(t − T )

+2Ce−a1t

E2τa21

e−z√

Ea12

ierfc

z√

E2

2√

t− i

√a1t

+

[ez√

Ea12

i

]erfc

z√

E2

2√

t+ i

√a1t

+

4C

E2a21τ

[−1 + a1t + e−a1t

]. (2.3.39)


b. Solutions for Large Times

p(z, t) ∼( p0

U∗ +u0

U∗

) 1

2

[ez√

γδ erfc

z

2

√γ

δ

√L

t+

√t

L

+ e−z√

γδ erfc

z

2

√γ

δ

√L

t−√

t

L

]

+U0

2U∗

[ez√

γδ erfc

z

2

√γ

δ

√L

t − T+

√t − T

L

+ e−z√

γδ

erfc

(z

2

√γ

δ

√L

t − T−√

t − T

L

)]

− δC

2γ

[ez√

γδ erfc

(z

2

√γ

δ

√L

t+

√t

L

)

+ e−z√

γδ erfc

(z

2

√γ

δ

√L

t−√

t

L

)]

+δC

2γe−

tL erfc

(z

2

√γ

δ

√L

t

)+

cδ

γ(1 − e−

tL ), (2.3.40)

q(z, t) ∼( q0

U∗ +u1

U∗

) 1

2

[ez√

γδ erfc

z

2

√γ

δ

√L

t+

√t

L

+ e−z√

γδ erfc

z

2

√γ

δ

√L

t−√

t

L

]

+U1

2U∗

[ez√

γδ erfc

z

2

√γ

δ

√L

t − T+

√t − T

L

+ e−z√

γδ erfc

z

2

√γ

δ

√L

t − T−√

t − T

L

]

− 4δC

γLE2τ

[e−a1t

2a1(a1 − 1L)

×

e−z

√γLδ

( 1L−a1)erfc

(z

2

√γ

δ

√L

t−√

(1

L− a1)t

)

+

e−z

√γLδ

( 1L−a1)erfc

(z

2

√γ

δ

√L

t+

√(1

L− a1)t

)]

+4Cδ

γE2τ

e−tL

(a1 − 1L)erfc

(z

2

√γ

δ

√L

t

)


− 2Cδ

γE2τ(a1)

[ez√

γδ erfc

(z

2

√γ

δ

√L

t+

√t

L

)

+ e−z√

γδ erfc

(z

2

√γ

δ

√L

t−√

t

L

)]

+4Cδ

γE2τ

[1

a1

+L

1 − L(a1)e−

tL +

e−(a1)t

(L(a1) − 1)(a1)

]. (2.3.41)

2.4 Results and Discussion

An unsteady flow of an electrically conducting dusty fluid over a semi-infinite plate is

considered. The flow is due to the influence of uniform magnetic filed and Hall effect.

The governing non-linear partial equations of the flow problem are solved analytically by

employing Laplace transform technique. In order to get physical insight of the present

problem, numerical computations on the analytical solutions for third case are carried out

with the help of MATLAB program. Effect of various physical parameters on velocities

of both fluid and dust phases are studied with the help of plotted graphs.

Figures 2.2 and 2.3, depict the velocity profile for the case of large time for different

values of magnetic parameter M . We infer from these figures that the fluid velocity

decrease with increase in magnetic parameter. This is due to the fact that, the application

of transverse magnetic field plays an important role of a resistive type of force similar to

drag force, that acts in the opposite direction of the fluid motion, thereby fluid and dust

phase velocity reduces.

Figures 2.4 and 2.5, are plotted to elucidate the the effect of Hall parameter on both

fluid and dust particle velocity profiles in large time. It is observed that, velocity of

both fluid and particle velocity profile decreases for increasing values of Hall parameter.

Similar prediction for Hall current effect is also made by Debnath [34] and Tiwari and


Kamal Singh [103].

Figures 2.6-2.9 are plotted to depict the influence of Ekman number on velocity profiles

of both fluid and dust phase for both small time and large time cases respectively. It is

observed from these plots that, for increasing values of Ekman number (E) fluid and

particle phase velocity decreases. This shows that the effect of (E) also opposes the flow.

Figures 2.10-2.13, are respectively, plotted to depict the influence of time parameter

on velocity of both fluid and dust phase in both large time and small time cases. From

these figures, it reveals that the fluid and dust phase velocity decreases with increase in

the time. It is interesting to note that the thickness of boundary decreases with increase

in time for both the cases. The similar effects can be found in all the three cases.

2.5 Conclusion

A mathematical analysis is carried out on momentum characteristics in an incompressible

viscous unsteady hydromagnetic rotating dusty fluid flow in the presence of Hall current.

The governing equations are solved by applying the asymptotic behaviour of Laplace

transform treatment. The effect of various physical parameter like Ekman number (E),

magnetic parameter (M), and Hall current parameter (m), and time (t), are examined.

Some of the important findings of our analysis obtained by the graphical representation

are listed below:

• The solution remains unaffected by magnetic parameter and Hall current parameter

in small times where as these effect for large times.

• Effect of magnetic parameter (M) and Hall current parameter (m) decreases the


fluid and particle phase velocity.

• Effect of Ekman number (E) decreases the fluid and dust phase velocity.

• Effect of time is to decreases both fluid and dust phase velocity.

Figure - 2.2: Fluid velocity profile (large times) for the different values of M .

Figure - 2.3: Dust velocity profile (large times) for the different values of M .


Figure - 2.4: Fluid velocity profile (large times) for the different values of m.

Figure - 2.5: Dust velocity profile (large times) for the different values of m.


Figure - 2.6: Fluid velocity profile (small times) for the different values of E.

Figure - 2.7: Dust velocity profile (small times) for the different values of E.


Figure - 2.8: Fluid velocity profile (large times) for the different values of E.

Figure - 2.9: Dust velocity profile (large times) for the different values of E.


Figure - 2.10: Fluid velocity profile (large times) for the different values of t.

Figure -2.11: Dust velocity profile (large times) for the different values of t.


Figure - 2.12: Fluid velocity profile (small times) for the different values of t.

Figure - 2.13: Dust velocity profile (small times) for the different values of t.

Chapter 3

Hall effect on MHD boundary layertwo-phase flow and heat transfer industy viscous fluid

3.1 Introduction

Boundary layer flow and heat transfer of an electrically conducting viscous fluid induced

by a continuously moving or stretching surface is relevant to many manufacturing and

industrial processes such as polymers involving the cooling of continuous strips or fila-

ments by drawing them through a quiescent fluid. Further, glass blowing, continuous

casting of metals and spinning of fibers involve the flow due to a stretching surface in an

ambient fluid. The quality of the final sheeting material, as well as the cost of production

affected by the speed of collection and the heat transfer rate. Also, in several engineer-

ing processes, materials manufactured by extrusion processes and heat treated materials

traveling between a feed roll and a wind up roll on convey belts possess the characteristic

of a moving continuous surface. Due to these facts, Sakiadis [95] first who discussed the

boundary layer flow over a continuous solid surface moving with a constant speed. Later,

Crane [31] generalized the problem of [95] for a stretching sheet and obtained the closed

68

Chapter-3: Hall effect on MHD boundary layer two-phase flow · · · 69

form solution. Various direction of boundary layer flow over a stretching sheet problem

studied by several researchers [54]-[9].

Flow and transport of heat through a porous medium occurring in numerous practical

applications. Such investigations finds their applications over a broad spectrum of science

and engineering disciplines, especially in chemical catalytic reactors, grain storage, migra-

tion of moisture through the air contained fibrous insulations, heat exchange between soil

and atmosphere, salt leaching in soils, solar power collectors, electrochemical processes,

insulation of nuclear reactors, regenerative heat exchangers and geothermal systems and

many others. In fact, boundary layer flow and transport of heat through a porous medium

is abundant [?]. Representative studies dealing with boundary layer flow through porous

medium have been reported by many authors Cortell [29], Angel [12] and Alsaedi et al

[11].

On the other hand, in all above said studies the electrical conductivity of the fluid

was assumed to be uniform and low magnetic field intensity. Nevertheless, in an ionized

fluid where the density is low and thereby magnetic field intensity is very strong, the

conductivity normal to the magnetic field is reduced due to the spiraling of electrons and

ions about the magnetic lines of force before collisions take place and a current induced in

a direction normal to both the electric and magnetic fields, this phenomena is known as

Hall effect. The study of MHD flows with Hall current has important applications in the

problem of Hall accelerators as well as flight magnetohydrodynamics. The current trend

for the application of magnetohydrodynamics is towards a strong magnetic field and low

density of the gas. Under this condition Hall effect becomes significant. Watanabe and


Pop [106] investigated the MHD boundary-layer flow over a continuously moving semi-

infinite flat plate with Hall current. Hydrodynamic free convection and mass transfer

of an electrically conducting viscous fluid past an infinite vertical porous plate has been

reported by Singh and Gorla [99]. Later on, many authors like Fakhar [43], Aziz [35],

Ali et al [10] and Gireesha and Mahanthesh [50] shown their interest to investigate the

influence of Hall current on different fluids with different aspects.

A considerable interest has been shown in the study of thermal radiation on boundary

layer flow and heat transfer in fluids due to its significant effects in the surface heat

transfer. Further, thermal radiation effects on flow and heat transfer processes are of

major importance in the space technology and high temperature processes. Bataller [20]

studied the effects of thermal radiation on the Blasius flow. The heat and mass transfer in

stagnation point flow towards a stretching surface in the presence of thermal radiation have

been investigated by Pal [?]. Mukhopadhyay [81] reported the effects of thermal radiation

and variable fluid viscosity on stagnation point flow past a porous stretching sheet. Later,

Magyari and Pantokratoras [76] examined the effect of thermal radiation using in the

linearized Rosseland approximation on the heat transfer characteristics in boundary layer

flow. Also, the effect of heat source/sink play a significant role in controlling heat transfer

in the production of quality product as it depends on the heat controlling factor. Cortell

[29] investigated the effect of internal heat generation/absorption on flow and heat transfer

of a fluid through a porous medium over a stretching surface. Abo-Eldahab and Aziz

[6] analyzed the effect of internal heat source/sink on mixed convection boundary layer

flow and heat transfer over an inclined continuously stretching surface with transpiration

cooling. Then, a comprehensive survey of heat transport uniform or non-uniform heat


source/sink was conducted by [83], [4] and [1].

The presence of dust particles tend to thinning the thermal and momentum boundary

layer thickness. The study of boundary layer flow with fluid-particle suspension has

become of increasing importance in last few years. This is mainly due to their applications

in atmospheric fallout, powder technology, rain erosion, petroleum transport, nuclear

reactor cooling, dust collection, sedimentation, environmental pollution, guided missiles,

paint spraying, food technologies, fluidization transport of solid particles by a liquid and

liquid slurries in chemical and nuclear processing, soil pollution, control of the cooling

rate of sheets and etc. Saffman [94] first who studied the stability of laminar flow of dusty

gas in which dust particles are uniformly distributed. Datta and Mishra [33] discussed

boundary layer flow of an electrically conducting dusty viscous fluid over a semi-infinite

flat plate. Effect of uniform suction on hydrodynamic boundary layer flow of a dusty

fluid over a stretching sheet was numerically analyzed by Vajravelu and Nayfeh [105].

Srinivastava and Srinivastava [101] studied the stagnation point flow of a second order

dusty fluid near an oscillating plate. Very recently, Gireesha et al ([51], [119] and [88])

studied the boundary layer two-phase flow and heat transfer of dusty fluid over stretching

surface with different aspects.

The current study is a theoretical investigation of the fully developed two-phase bound-

ary layer flow and heat transfer of radiating dusty viscous fluid, using the Saffman model

[94], in the presence of non-uniform heat source/sink and Hall current. In this article,

we employ an extensively validated, highly efficient numerical method called, fourth-fifth

order Runge-Kutta-Fehlberg method to study this problem. The influence of various per-

tinent parameters on the different flow fields are presented and discussed though several


plots and tables.

3.2 Mathematical Formulation

Consider an unsteady two-dimensional laminar boundary layer flow and heat transfer of

an incompressible viscous dusty fluid (with electric conductivity σ0) over a stretching

sheet. The x-axis is taken along the stretching surface in the direction of the motion with

the slit as the origin, and the y-axis is perpendicular to the sheet in the outward direction

towards the fluid. The flow is assumed to be confined in a region y > 0. The flow is

considered to be generated by stretching of an elastic boundary sheet from a slit with the

application of two equal and opposite forces in such way that the velocity of boundary

sheet Uw(x, t) is linear function of the flow directional coordinate x and a function of

time. It is considered that the wall temperature Tw(x, t) of the sheet is suddenly raised

from T∞ to Tw(x, t > T∞) or there is suddenly imposed a heat flux qw(x, t) at the wall.

Further, the flow field is exposed to the influence of an external transverse magnetic field

of strength B0 as shown in the figure 3.1. The dust particles are assumed to be spherical

in shape and are uniformly distributed throughout the flow.


Figure-3.1: A schematic representation of the physical model and coordinates system.

The flow is caused by stretching of the sheet which moves in its own plane with the

surface velocity Uw = bx, where b > 0 is initial stretching rate. It is considered that the

surface temperature Tw of the sheet is suddenly raised from T∞ to Tw (Tw > T∞) or there

is a suddenly imposed heat flux qw at the wall.

An external strong magnetic field is applied in the positive y-direction. In general, for

an electrically conducting fluid, Hall current affects the flow in the presence of a strong

magnetic field. The effect of Hall current gives rise to a force in z-direction, which induces

a cross flow in that direction and hence the flow becomes three-dimensional. To simplify

the problem, we assume that there is no variation of flow quantities in z-direction. This

assumption is considered to be valid if the surface be of infinite extent in z-direction. The

generalized Ohm’s law including Hall currents is given in the form [35];

~J +ωeτe

B0

× ( ~J × ~B) = σ

(~E + ~V × ~B +

1

ene

Pe

)(3.2.1)


where ~J = (Jx, Jy, Jz) is the current density vector, ~V = (u, v, w) is the velocity vector, ~E

is the intensity vector of the electric field, ~B = (0, B0, 0) is the magnetic induction vector,

σ, τe, e, ne and pe are respectively, electrical conductivity, electron collision time, charge

of electron, number density of electrons and electronic pressure. Since, no applied or

polarization voltage is imposed on the flow field, the electric field vector ~E = 0. For weakly

ionized gases, generalized Ohm’s law under above conditions gives Jy = 0 everywhere in

the flow. Hence, under these assumptions, equating the x and z components in (3.2.1)

and solving for the current density components Jx and Jz are readily read as,

Jx =σB0

1 + m2(mu − w), (3.2.2)

Jz =σB0

1 + m2(u + mw). (3.2.3)

Here, u, v and w are x, y and z components of the velocity vector ~V respectively and

m = ωeτe is Hall parameter.


Under these assumptions, two-phase boundary layer flow and generalized Ohm’s law

is governed by the following system of equations;

∂u

∂x+

∂v

∂y= 0, (3.2.4)

ρ

(u∂u

∂x+ v

∂u

∂y

)= µ

∂2u

∂y2+ KN(up − u)

− σB02

(1 + m2)(u + mw) − µ

k∗u, (3.2.5)

ρ

(u∂w

∂x+ v

∂w

∂y

)= µ

∂2w

∂y2+ KN(wp − w)

+σB0

2

(1 + m2)(mu − w) − µ

k∗w, (3.2.6)

∂up

∂x+

∂up

∂y= 0, (3.2.7)

ρp

(up

∂up

∂x+ vp

∂up

∂y

)= KN(u − up), (3.2.8)

ρp

(up

∂wp

∂x+ vp

∂wp

∂y

)= KN(w − wp), (3.2.9)

where, (u, v, w) and (up, vp, wp) are respectively, fluid and dust phase velocity components

along x, y and z-directions. ν = µρ

is kinematic viscosity of the fluid, µ is dynamic viscosity

of the fluid, ρ is density of the fluid, ρp = Nmp is density of dust particles, N is number

density of dust particles, mp is mass of the dust particles, K = 6πµr is the Stokes drag

constant, r is radius of dust particles, σ is electric conductivity and k∗ is permeability of

porous medium.

For the above bounadry layer equations, the relevant boundary conditions are;

u = Uw(x), v = Vw(x), v = 0, w = 0 at y = 0,

u → 0, up → 0, vp → v, w → 0, wp → 0 as y → ∞. (3.2.10)

where Uw is stretching sheet velocity, Vw = −√

bν(L − 1L), L is characteristic length. It

should be mention that L > 1 corresponds to suction (Vw < 0), L < 1 corresponds to


blowing (Vw < 0) and in case L = 1, the stretching surface is impermeable.

To solve the set of partial differential equations (3.2.5), (3.2.6), (3.2.8) and (3.2.9), we

adopt the similarity technique by introducing the following similarity variables;

u = bxf ′(η), v = −√

νbf(η), w = bxh(η), η =

√Uw

νxy,

up = bxF ′(η), vp = −√

bνF (η), wp = bxH(η).

(3.2.11)

In view of above transformations the equations, (3.2.4) and (3.2.7) are identically

satisfied and one can see that the equations (3.2.5), (3.2.6), (3.2.8) and (3.2.9) are reduced

to following set of non-linear ordinary differential equations,

f ′′′(η) + f ′′(η)f(η) − f ′(η)2+ lβv(F

′(η) − f ′(η)) − k0f′(η)

− M2

(1 + m2)(f ′(η) + mh(η)) = 0, (3.2.12)

h′′(η) + h′(η)f(η) − f ′(η)h(η) + lβv(H(η) − h(η)) − k0h(η)

+M2

(1 + m2)(mf ′(η) − h(η)) = 0, (3.2.13)

F ′′(η)F (η) − F ′(η)2+ βv (f ′(η) − F ′(η)) = 0, (3.2.14)

H ′(η)F (η) − H(η)F ′(η) + βv (h(η) − H(η)) = 0. (3.2.15)

The boundary conditions (3.2.10) will becomes,

f ′(η) = 1, f(η) = S, h(η) = 0 at η = 0,

f ′(η) → 0, F ′(η) → 0, F (η) = f(η), h(η) → 0, H(η) → 0 as η → ∞,

where prime denotes the differentiation with respect to η and l = mpN

ρis the mass con-

centration of the dust particles, τv = mp

Krelaxation time of dust particles, βv = 1

bτvis the

fluid-particle interaction parameter for velocity, M2 =σB2

0

ρbis the magnetic parameter,

k0 = νk∗b

is porous parameter and S = −vw(x)√bν

is suction/blowing parameter.


Physical quantities of interest in engineering point of view is the local skin friction

coefficient Cfx in x-direction and in Cfz in z-direction, are defined as,

Cfx =τwx

ρUw2 , and Cfz =

τwz

ρUw2 , (3.2.16)

where τwx and τwz are surface shear stress in x and z-directions respectively, which are

given by

τwx = µ

(∂u

∂y

)y=0

, and τwz = µ

(∂w

∂y

)y=0

. (3.2.17)

Using equations (3.2.11) and (3.2.17) in (3.2.16), we obtain

√RexCfx = f ′′(0),

√RexCfz = h′(0), (3.2.18)

where Rex = Uwxν

is the local Reynold’s number.

It is important to note that, tangential velocity equation (3.2.12) in the absence of

M2, k and l reduced to boundary-layer flow past a stretching sheet whose analytical

solution has been reported by Crane [31] as follows:

f(η) = 1 − e−η. (3.2.19)

In terms the equation (3.2.19), the skin friction co-efficient in x-direction is given by

f ′′(0) = −1. (3.2.20)


3.3 Heat Transfer Analysis

The fluid and dust particle phase boundary layer energy equations in the presence of

thermal radiation and non-uniform heat source sink are read as [94];

ρcp

[u∂T

∂x+ v

∂T

∂y

]= k

∂2T

∂y2+

ρpcp

τT

(Tp − T )

+ρp

τv

(up − u)2 − ∂qr

∂y+ q′′′, (3.3.1)

ρpcm

[up

∂Tp

∂x+ vp

∂Tp

∂y

]= −ρpcp

τT

(Tp − T ), (3.3.2)

where T and Tp are respectively the temperature of clean fluid and dust particles, cp

and cm respectively are the specific heat of clean fluid and dust particles. τT is thermal

equilibrium time i.e., the time required by the dust cloud to adjust its temperature to the

clean fluid. k is the thermal conductivity and q′′′ is the space and temperature dependent

heat generation/absorption, which can be expressed as [2];

q′′′ =kUw

xν[A∗(Tw − T∞)f ′(η) + B∗(T − T∞)] , (3.3.3)

where A∗ and B∗ are parameters of space and temperature dependent heat generation or

absorption. Here, if A∗ > 0 and B∗ > 0 correspond to internal heat generation, whereas

A∗ < 0 and B∗ < 0 correspond to internal heat absorption.

According to the Rosseland diffusion approximation, the radiative heat flux qr is of

the form,

qr = − 4σ∗

3k+

∂T 4

∂y, (3.3.4)

where σ∗ is the Stefan-Boltzmann constant and k+ is the mean absorption coefficient. It

is noted that the optically thick radiation limit is considered in this model. Assuming

that the temperature differences within the flow are sufficiently small such that T 4 may


be expressed as a linear function of temperature and then by neglecting the higher order

terms beyond the first degree in (T − T∞), one can get T 4 ∼= 4T∞3T − 3T∞

4. Using this

in (3.3.4), we obtain

∂qr

∂y= −16T∞

4σ∗

3k∗∂2T

∂y2. (3.3.5)

To solve the equations (3.3.1) and (3.3.2), the prescribed surface temperature and

prescribed surface heat flux boundary conditions are given by

T = Tw = T∞ + A(x/l∗)2 (PST case), at y = 0,

−k∂T/∂y = qw (PHF case) at y = 0,

T → T∞, Tp → T∞ as y → ∞,

where T = A( xl∗

)2θ(η) + T∞ (PST case), Tw − T∞ = Dk( x

l∗)2√

νb

(PHF case), Tw and T∞

and denote the temperature at the wall and at large distance from the wall respectively.

Now define the non-dimensional fluid phase temperature θ(η) and dust particle phase

temperature θp(η) as

θ(η) =T − T∞

Tw − T∞, θp(η) =

Tp − T∞

Tw − T∞. (3.3.6)

In view of equation (3.3.5) and using equation (3.3.6) into (3.3.1) and (3.3.2) and to

the boundary conditions (3.3.6), one can arrive at the following dimensionless system of

equations and boundary conditions:(1 +

4

3R

)θ′′(η) + Pr (θ′(η)f(η) − 2f ′(η)θ(η)) lP rβT (θp(η) − θ(η))

+EcPr (f ′′(η))2+ A∗f ′(η) + B∗θ(η) + lP rEcβv (F ′(η) − f ′(η))

2= 0, (3.3.7)

2F ′(η)θp(η) − F (η)θ′p(η) + γβT (θp(η) − θ(η)) = 0, (3.3.8)

θ(η) = 1 (PST ); θ′(η) = −1 (PHF ); at η = 0,

θ(η) → 0, θp(η) → 0 as η → ∞,

(3.3.9)


where Pr = µcp

kis the Prandtl number, βT = 1

bτTis the fluid-particle interaction parameter

for temperature, R = 4σ∗T∞3

k+kis the thermal radiation parameter, γ = cp

cmis the specific

heat ratio, Ec = l∗2bAcp

and Ec = l∗2b3/2kDcpν1/2 is the Eckert number for (PST) and (PHF)

respectively.

The important physical parameter for heat transfer co-efficient (Nusselt number) is

defined as

Nu =qw

k(Tw − T∞), (3.3.10)

where qw is the surface heat flux, which is given by

qw = −k

(∂T

∂y

)y=0

. (3.3.11)

In view of similarity variables and using equation (3.3.11) into (3.3.10), one can get

Re−1/2x Nu = −θ′(0). (3.3.12)

3.4 Numerical Method and Validation

The system of non-linear differential equations (3.2.12)-(3.2.15) and (3.3.7)-(3.3.8) un-

der the boundary conditions (3.2.16) and (3.3.9) have been solved numerically using

fourth-fifth order Runge-Kutta-Fehlberg method implemented on Maple. The algorithm

is proved to be precise and accurate in solving a wide range of mathematical and en-

gineering problems especially fluid flow and heat transfer cases. In this package, two

sub methods are available namely midpoint method and trapezoidal method. Midpoint

method is chosen as a sub method in our computation, since it is capable of handle harm-

less end point singularities. In accordance with the standard boundary layer analysis, the


asymptotic boundary conditions η → ∞ were replaced by those at η = 5, in which the

obtained solutions meet the far field condition asymptotically.

We have repeatedly confirmed in our previous publications to judge the accuracy

and robustness of the used numerical method. We compare our results of heat transfer

coefficient −θ′(0) with available results of Grubka and Bobba [54], Chen [120], Abel et al

[3], Ali [9] and El-Aziz [35] for different values of Prandtl number as a further check and

validation on the accuracy of our numerical computations with l = A∗ = B∗ = R = 0.

These comparisons are presented in table 3.1 and found to be excellent agreement.

3.5 Results and Discussion

Hydromagnetic two-phase boundary layer flow and heat transfer on a non-isothermal per-

meable stretching sheet in the presence of magnetic field, Hall current, suction/blowing,

thermal radiation and non-uniform heat source/sink effects have been investigated numer-

ically. Further, two different heating processes namely PST and PHF cases are considered

in heat transfer analysis. A parametric study on different flow fields is also made to an-

alyze the physical insight of the problem. It is worth mentioning that the present study

reduces to the classical viscous fluid flow problem if l = 0 i.e., in the absence dust particles

mass concentration.

Figures 3.2, 3.4, 3.6, 3.10, 3.12 and 3.3, 3.5, 3.7, 3.9, 3.11 are respectively present the

effects of m, M2, βv, l, S and k0 on axial and transverse velocity profiles of both fluid and

dust particle phases. It is apparent from figures 3.2 and 3.3 respectively that, the axial

and transverse velocity profiles for both fluid and dust phase are increases with increasing

values of Hall parameter m. Since as effective conductivity ρ(1+m2)

decreases with increase


in m and which reduces the magnetic damping force on axial and transverse velocity

profiles. Moreover, f ′(η) and F ′(η) profiles approach their classical hydrodynamic values

when the Hall parameter tends to infinity. Since the magnetic force terms approach zero

value for very large values of Hall parameter. It is observed from figure 3.4 that, for

increasing values of magnetic parameter results in flattening of f ′(η) and F ′(η) profiles.

The transverse contraction of axial velocity boundary layer is due to the applied magnetic

field which results in the Lorentz force producing considerable opposition to the motion.

In the absence of the magnetic field (M2 = 0), there is no transverse velocity for both

phases (h(η) = 0, H(η) = 0)) and as magnetic field increases, a cross flow in the transverse

direction is greatly induced due to the Hall effect as depicted in the figure 3.3. In addition,

plot 3.5 elucidate that, close to the sheet surface an increase in the values of M2 leads to

an increase in lateral velocity profiles for both phases with shifting the maximum toward

the sheet. While for most part of the boundary layer at a fixed η position, the lateral

velocity profile along with boundary layer thickness decreases as M2 increases for both

phases.

Respectively, it is clear from figures 3.6 and 3.7 that, an increase in fluid-particle

interaction parameter leads to enhance the axial and transverse velocity of dust phase

whereas opposite phenomenon is observed for fluid phase. It is evident from plots 3.8

and 3.9 respectively that, an increase in the dust particles mass concentration parameter

used to decrease the axial and transverse velocity profiles for both the phases. Associate

with the axial and transverse momentum boundary layer thickness decreases. This is

because, presence of dust particles produces friction force in the fluid, which retards the

flow. Further, it is observed that the axial and transverse velocity profile is higher for


ordinary viscous fluid (l = 0) than that of dusty viscous fluid (l 6= 0). Figures 3.10 and 3.11

elucidate that, as expected the opposite results are found for suction and blowing. Blowing

causes an increase in axial and transverse velocity profile for both phases, although by

increasing suction parameter the axial and transverse velocity profiles notably decreases

near the boundary for both phases. This is due to the fact that, while stronger blowing

is provided, the heated fluid is pushed farther away from the sheet, where due to less

effect of viscosity, the flow is accelerated. This effect acts to increase maximum axial

and transverse velocity within the boundary layer. The same principle is acted but in

reverse direction in the case of suction. It is noted from figures 3.12 and 3.13 respectively

that, both axial and transverse velocity profiles are strictly decreases for increasing the

porous parameter. This is due to fact that, the presence of porous medium is to increase

the resistance to the flow, which causes the fluid velocity to decrease, associated with a

decrease in momentum boundary layer thickness.

Figures 3.14-3.23 are respectively plotted to show the influence of A∗, B∗, m, l, Ec, Pr,R,

S, M2 and k0 on both fluid and dust particles temperature distribution in PST and PHF

cases. It is important to note that, A∗ > 0, B∗ > 0 corresponds to internal heat generation

and A∗ < 0, B∗ < 0 corresponds to internal heat absorption. It is the cumulative effect

of the space-dependent and temperature-dependent heat source/sink parameter that de-

termines the extent to which the temperature falls or rises in the boundary layer region.

From the plots 3.14 and 3.15, it is observed that, the energy is released for increasing val-

ues of A∗ > 0 & B∗ > 0 and this causes the magnitude of temperature to increase both in

PST and PHF cases, where as energy is absorbed for increasing values of A∗ < 0 & B∗ < 0

resulting in temperature dropping significantly near the boundary layer. Further, It is


possible to see that thermal boundary layer thickness for heat source is thicker than heat

sink in both PST and PHF cases. The fluid and dust particles temperature profile notably

decreases for increasing the hall parameter in both PST and PHF cases and it is depicted

in the figure 3.16.

Figure 3.17 represents that increase in the dust particles mass concentration parameter

decreases the heat transfer in thermal boundary layer of both fluid and dust phase in PST

and PHF cases. The central reason for this effect that, the presence of dust particles in

clean viscous fluid tend to absorb the heat, when they come into contact, which has the

tendency to diminish the temperature profile. It is also note that, temperature of clean

fluid i.e., l = 0 is higher than that of dusty fluid i.e., l 6= 0, corresponding heat transfer

rate is higher in dusty fluid than clean viscous fluid. Thus, dusty fluid is more preferable

in the context of cooling processes. Figure 3.18 elucidates that increase in the Eckert

number used to strictly increase the fluid and particle temperature profiles. This trend

is quiet opposite for the effect of increasing values of Prandtl number on temperature

distributions of both phases in PST and PHF cases, which is illustrated in the figure 3.19.

The reason is a higher Prandtl number fluid has relatively low thermal conductivity, which

reduce the conduction and thermal boundary layer thickness and as a result temperature

profile trim downs.

Figure 3.20 displays that increasing the radiation parameter increase the both fluid

and dust particles temperature profiles in PST and PHF cases, consequently brings about

decrease in rate of heat transfer. This is due to fact that, as radiation parameter increases,

mean Rosseland absorption co-efficient k+ decreases, this cause increase in temperature

profiles. Thus radiation should be at its minimum in order to facilitate the cooling process.


Figure 3.21 shows that, both fluid and dust particles temperature profile retards for the

influence blowing (S < 0) whereas enhances for influence the suction (S > 0). As expected

that, the temperature profile of both fluid and dust particles increases for increasing the

magnetic parameter, which is illustrated in plot 3.22. Figure 3.23 elucidates that increase

in porous parameter enhances the temperature profile of both phases, and corresponding

thermal boundary layer thickness is decreases. Physically speaking, the higher value of

porous parameter produces higher restriction on the fluid motion, which is responsible

for increasing the temperature profile. Most interestingly, it is observed from the all plots

that, velocity and temperature profile of dust particles are parallel to that of dusty fluid

and fluid phase velocity and temperature profiles are higher than that of dust phase.

Moreover, It is noted from the graphs 3.14-3.23 that, the temperature profile is higher in

PHF heating process as compared with PST, thus PHF thermal heating process is better

suited for cooling processes.

We now move over to a discussion on the skin friction at the stretching a sheet along

x-direction and z-direction. Figures 3.24 and 3.25 are plotted to depict the effect of dust

particles mass concentration parameter l and Hall parameter m on f ′′(0) and h′(0) profiles

versus suction/blowing parameter S respectively. It is shown from plot 3.24 that, f ′′(0)

and h′(0) profiles decreases as increase in l. Further, f ′′(0) and h′(0) profiles are higher

for clean viscous fluid (l = 0) than that of dusty viscous fluid (l 6= 0). On the other hand,

as m increases f ′′(0) and h′(0) profiles also increases, which is presented in plot 3.25. It

is also observed that, in the absence of Hall parameter, f ′′(0) profile is smaller and h′(0)

profile vanishes. Furthermore, we can see from both plot 3.24 and 3.25 that, higher values

of suction/blowing parameter retards the f ′′(0) and h′(0) profiles. Influence of l and m


on Nusselt number profile has been illustrated in figure 3.26. It is shown that, the rate of

heat transfer is high for higher values of Hall parameter and rate of heat transfer is lower

in ordinary viscous fluid (l = 0) than that of dusty viscous fluid (l 6= 0). The rate of heat

transfer −θ′(0) is decreases for increasing values of M2 and R is as shown in figure 3.27.

Table 3.2 presents the numerical values of Nusselt number −θ′(0) and wall temperature

θ(0) for various values of M2, R, A∗ and B∗ for clean fluid (l = 0) and dusty fluid (l 6= 0).

It reveals that, the rate of transfer is decreases for increasing of M2, R, A∗ and B∗. This

means that, in order to facilitate the cooling process M2, R, A∗ and B∗ are should be at

its minimum. Further it is noticed that rate of heat transfer is higher for dusty fluid than

that of clean fluid. Finally, table 3.2 presents the numerical values of f ′′(0) and h′(0)

profiles for various values of k0, M2, l and S for with Hall current (m 6= 0) and without

Hall current (m = 0). The f ′′(0) profile higher in the presence of Hall current and h′(0)

profile vanishes in the absence of Hall current.

3.6 Conclusion

LARGE

Two-phase boundary layer flow of radiating dusty fluid past a non-isothermal stretch-

ing sheet in the presence of non-uniform internal heat source/sink Hall current has been

investigated. The resulting mathematical problems have been solved numerically and ob-

tained results are compared with existing one, an excellent agreement is noticed. There is

decrease in velocity and boundary layer thickness with an increase in dust particles mass

concentration (l). However, the temperature of dusty fluid and dust particles are found

to be decrease with an increase in l. It is observed that, the rate of heat transfer is higher


for dusty viscous fluid than that of clean fluid, thus effect of suspended dust particles

play an very important role. Dusty fluid velocity suppresses for increase in fluid-particle

interaction parameter but dust particles velocity accelerates. Moreover, both axial and

transverse velocity increases and momentum boundary layer thickens when the strength of

Hall effect where as opposite trend has been observed for increase in M2, l, S and k0. The

temperature profile of both phase and its corresponding thermal boundary layer thickness

are increasing function of M2, k0, A∗, B∗, RandEc and decreasing function of l,mandPr.

The rate of transfer is decreases for increasing of M2, R, A∗ and B∗, thus in order to

facilitate the cooling process M2, R, A∗ and B∗ are should be at its minimum. In the

absence of Hall current h′(0) profile get vanishes. Finally conclude that, momentum and

thermal boundary layer are thinner due to the influence of suspended dust particles and

the prescribed heat flux boundary condition is better suited for effective cooling of the

stretching sheet. If l = 0, M2 = 0, R, A∗, B∗ and m = 0, then our results coincides with

the results of Abel et al [3] and Grubka and Bobba [54] for different values of Prandtl

number.

Table-3.1: Comparison results for surface temperature gradient −θ′(0) for ordinary vis-

cous fluid with A∗ = B∗ = R = 0.

Pr Grubka Chen Abel Ali et al El-Aziz et al Present

et al [54] [120] et al [3] [9] [35] Study

0.72 1.0885 1.0885 1.0885 - 1.0885 1.0886

1.0 1.3333 1.3333 1.3333 1.3333 1.3333 1.3333

3.0 2.5097 2.5097 - 2.5097 2.5097 2.5097

10.0 4.7969 4.7968 4.7968 4.7969 4.7968 4.7969


Table-3.2: Numerical values of −θ′(0) and θ(0) for different values of M2, R, A∗ and B∗

for ordinary fluid and dusty fluid.

l = 0 l = 1

M2 R A∗ B∗ −θ′(0) θ(0) −θ′(0) θ(0)

0.0 1.0 0.1 0.1 0.50046 1.94489 0.56042 1.72543

1.0 0.47805 2.03446 0.54154 1.78402

2.0 0.45154 2.15205 0.51795 1.86307

0.5 0.0 0.1 0.1 0.84916 1.16651 0.95397 1.04403

0.5 0.60605 1.61262 0.68466 1.42362

2.0 0.37856 2.56677 0.42095 2.28572

0.5 1.0 -0.5 0.1 0.65494 1.66641 0.70029 1.50252

0.0 0.51777 1.93132 0.57668 1.70978

0.5 0.38061 2.19622 0.45306 1.91705

0.5 1.0 0.1 -0.5 0.75525 1.31447 0.79518 1.24539

0.0 0.54821 1.78632 0.60325 1.61365

0.5 0.02220 17.16481 0.20106 4.13557

Table-3.3: Numerical values of f ′′(0) and h′(0)for different values of k0, M2, l and S for

with and without Hall current.

m = 0 m = 1

k0 M2 l S f ′′(0) h(0) f ′′(0) h(0)

0.0 0.5 0.5 -0.1 -1.23908 0.0 -1.14558 0.12397

1.0 -1.58075 0.0 -1.50500 0.08799

2.0 -1.86242 0.0 -1.79747 0.07193

0.5 0.0 0.5 -0.1 -1.23908 0.0 -1.23908 0.00000

0.5 -1.41985 0.0 -1.33638 0.10149

1.5 -1.72716 0.0 -1.52501 0.25983

0.5 0.5 0.0 -0.1 -1.36510 0.0 -1.27838 0.10537

0.7 -1.44112 0.0 -1.35886 0.10005

1.4 -1.51300 0.0 -1.43460 0.09547

0.5 0.5 0.5 -0.1 -1.41985 0.0 -1.33638 0.10149

0.0 -1.47196 0.0 -1.38840 0.10108

+0.1 -1.52542 0.0 -1.44188 0.10052


Figure - 3.2: Variation of axial velocity profile for different values of m.

Figure - 3.3: Variation of transverse velocity profile for different values of m.


Figure - 3.4: Variation of axial velocity profile for different values of M2.

Figure - 3.5: Variation of transverse velocity profile for different values of M2.


Figure - 3.6: Variation of axial velocity profile for different values of βv.

Figure - 3.7: Variation of transverse velocity profile for different values of βv.


Figure - 3.8: Variation of axial velocity profile for different values of l.

Figure - 3.9: Variation of transverse velocity profile for different values of l .


Figure - 3.10: Variation of axial velocity profile for different values of S.

Figure - 3.11: Variation of transverse velocity profile for different values of S.


Figure - 3.12: Variation of axial velocity profile for different values of k0.

Figure - 3.13: Variation of transverse velocity profile for different values of k0.


Figure - 3.14: Variation of temperature profile for different values of A∗ for both PST

and PHF cases.

Figure - 3.15:Variation of temperature profile for different values of B∗ for both PST

and PHF cases.


Figure - 3.16: Variation of temperature profile for different values of m for both PST

and PHF cases.

Figure - 3.17: Variation of temperature profile for different values of l for both PST and

PHF cases.


Figure - 3.18: Variation of temperature profile for different values of Ec for both PST

and PHF cases.

Figure - 3.19: Variation of temperature profile for different values of Pr for both PST

and PHF cases.


Figure - 3.20: Variation of temperature profile for different values of R for both PST

and PHF cases.

Figure - 3.21: Variation of temperature profile for different values of S for both PST and

PHF cases.


Figure - 3.22: Variation of temperature profile for different values of M2 for both PST

and PHF cases.

Figure - 3.23: Variation of temperature profile for different values of k0 for both PST

and PHF cases.


Figure - 3.24 Variation of skin friction co-efficient versus S for different values of l

respectively.

Figure - 3.25: Variation of skin friction co-efficient versus S for different values of m

respectively.


Figure - 3.26: Variation of Nusselt number versus S for different values of m&l

respectively.

Figure - 3.27: Variation of Nusselt number versus S for different values of M2&R

respectively.

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Documents

Preface · Preface At the beginning of the 20th century Ludwig Prandtl introduced the concept of boundary-layer theory. He showed that the ﬂow past a body can be divided into two