CHAPTER I

Introduction

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1.1 BOUNDARY LAYER CONCEPT :

CHAPTER I

INTRODUCTION

In the eighteenth and nineteenth centur ies there developed two

schools of thought about fluid mechanics . One group, called the

hydraul icians looked at experimental da ta and a t tempted to generalise it in

to useful design equat ions. Their equat ions were generally empirical,

wi thout m u c h theoretical content . The other group, called the

hydrodynamicis ts started with Newton's equat ions of motion and tried to

deduce the necessary equat ions for fluid flow. It was quickly appai-ent to the

hydrodynamicists that , if they retained the viscous friction te rms or the

change of density te rms, then the result ing differential equat ions would be

so cumbersome tha t solutions would seldom, if ever, be possible. So, they

ignored the viscous friction and expansion te rms by hypothesizing a "perfect

fluid" with zero viscosity and cons tan t density. For th is perfect fluid they

were then able to calculate the complete behaviour of many kinds of flows.

For flows, which did not involve solid surfaces, such a s deep water waves or

tides, these mathemat ical solutions agreed very well with observed

behaviour. But hydraul icians found tha t the perfect fluid solutions did not

agree with observed behaviour in the problems which concerned them: flow

in channels , flow in pipes, forces on solid bodies caused by flow pas t them

etc. By 1900 the two schools had gone their separate ways, the

hydrodynamicists publishing learned mathemat ica l papers with little

bearing on engineering problems and hydraul ic ians solving engineering

problems by trial and error, intuition and experimental tes ts .

In 1904 Ludwig Prandtl suggested a way to bring the two schools together

by introducing a new concept, called the 'Boundary Layer'. If a fluid flows

past the leading edge of a flat surface, there will develop a velocity profile.

According to the lows of perfect fluid flow, the surface should not influence

the flow in any way; the velocity should be same everywhere in the

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flowing fluid. According to the ideas of viscous flow, there should exist a

velocity gradiant in the y- direction extending out to infinity. Prandtl's suggestion

to reconcile these views was that the flow be conceptually divided in to two

parts. In the region close to the solid surfaces the effects of viscosity are too

large so that its effect can not be ignored. However, this is a fairly small

region; outside it the effects of viscosity are small and can be neglected.

Thus, outside this region the laws of "perfect fluid" flow should be

satisfactory.

Prandtl called the region where the viscous forces can not be ignored the

boundary layer. He arbitrarily suggested that it be considered that region in

which the x-component of the velocity is less than 0.99 times the free-

stream velocity. Then, to obtain a complete solution to a flow problem in two

or three dimensions, one should use the viscous flow equations inside the

boundary layer and the equation of "perfect fluid" flow outside the boundary

layer. At the edge of the boundary layer the pressures and velocities of the

two solutions must be matched.

This is a very arbitrary division, which does not necessarily

correspond to any physically measurable boundary. The edge of the

boundary layer does not correspond to any sudden change in the flow but

rather corresponds to an arbitrary mathematical definition. Even with this

simplification the calculations are very difficult. The boundary layer has

become a standard idea in the minds of fluid mechanicians. Once it became

accepted in fluid mechanics, an analogous idea was tried in heat transfer

and in mass transfer, generally with useful results.

1.2 Development of boundary layer theory

The theoretical hydrodynamics developed from solutions of Euler's

equation of motion along with the equation of continuity for various flow

configurations of frictionless or non-viscous fluid (incompressible) flow past

obstacles like plates, cylinders, spheres and through pipes and channels

and against disks. However, the results of such studies did not agree with

the experimental results as regards to the pressure losses in tubes and

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channels, as well as that of the drag experienced b}' a body moved through a

fluid. The most glaring departure of the result of this subject from reality is

that leading to D' Alembert's Paradox, that is to the statement that a body

which moves uniformly through a fluid which extends to infinity experiences

no drag whereas a body experiences a drag in moving through any real fluid.

For this reason; engineers, developed the science of hydraulics. This relied

upon a large amount of experimental data and differed greatly from

theoretical hydrodynamics in both methods end objects.

The equation of motion of a real or viscous fluid was established in the

first half of the nineteenth century by Navier (1823), Poisson (1831), Saint-

Venant (1843) and Stokes (1845) and its components are known as Navier

Stokes equations. These equations are non-linear partial differential

equations, therefore there exist only a few exact solutions of these equations

for the cases where either the non-linear terms vanish automatically or

when the equations can be reduced to ordinary differential equations by

taking recourse to Laplace transform or some suitable similarity

transformations. Stokes in 1851 investigated the case of parallel flow past a

sphere for the limiting case when the viscous forces ai'e considerably greater

than the inertia forces and so the non-linear terms in the Navier stokes

equations are neglected. Oscen in 1910 gave an improvement on the Stokes

solution by taking partly into account the inertia terms in these equations.

But these types of solution are valid for small Reynolds number, which

corresponds to slow motion. These motions are called creeping motions and

do not occur often in practical applications. As a result, there was not much

progress, till the beginning of twentieth century in dealing with the flow

problems of real fluids by considering the full Navier-Stokes equations along

with the no slip condition at a solid wall.

An important characteristic of modern research into fluid mechanics

in general and more specifically into the branch of boundary layer theory is

the close connection between theory and experiment. The simplest example

of the application of the boundary layer equations is afforded by the flow

along a very thin semi-infinite flat plate. Historically this was the first

example illustrating the applications of Prandtl's boundary layer theory; it

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was discussed by Biasius (1908) and is often referred to a s Blasius problem.

Subsequently Bairstow (1925) and Goldstain (1930) solved the same

equation with the aid of a slightly modified procedure. Somewhat earlier,

Toepfer (1912) solved the Blasius equat ion numerically by Runge Kutta

method. The same was again solved by Howarth (1938) with increased

accuracy. A. Betz (1949) produced a review of the development of boundary-

layer theory, with part icular emphas i s on the m u t u a l fructification of theory

and experiment.

In the mid-fifties, mathemat ical methods into singular per turbat ion

theory were being systematically developed by S. Kaplun (1954); S. Kaplun,

P.A. Langerstrone (1957); M. Van Dyke (1964b); W. Schneider (1978). It

became clear tha t the Boundary-Layer Theory heuristically developed by

Prandtl was a classic example of the solution of a singular per turbat ion

problem. H. Schliching (1960), I. Tani (1977), A.D. Young (1989), K.Gersten

(1989) and A.Kluwick (1998) had made systematic s tudy on the development

of the boundary layer theory.

1.3 Role of suc t ion in boundary layer control

The problem of boundary layer control is very important in some

fields, in part icular in the field of aeronautical engineering. In actual

applications it is often necessar^^ to prevent sepai^ation in order to reduce

drag and to at tain high lift. Several methods have been developed for the

purpose of artificially controlling the behaviour of the boundary layer in

order to affect the whole flow in a desired direction. One important method

of controlling the boundary layer and hence of shifting the separat ion, which

reduces the drag is by applying suction at the solid boundary . The treatise

entitled "Boundary Layer and flow control", was presented by Lachman

(1961).

The effect of suction consis ts in removing the decelerated fluid

particles from the boundary layer which is again capable of overcoming a

certain adverse pressure gradient is alloxved to from in the region behind the

slit. With a suitable aiTangement of slits and unde r favourable condit ions

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separation can be prevented completely. The application of suction which

was first tried by Prandtl (1904) was later widely used in the design of

aircraft wings. By applying suction, considerably greater pressure increases

on the upper side of the aerofoil are obtained at large angle of incidence

and consequently, much large maximum lift values. Schrenk (1941)

investigated a large number of different arrangements of suction slits and

their effect on maximum lift.

Subsequently, suction was also applied to reduce drag. By the use of

suitable arrangements of suction slits it is possible to shift the point of

transition in the boundary layer in the down stream direction. This causes

the drag coefficient to decrease, because laminar drag is substantially

smaller than the turbulent drag.

A further way of preventing separation consists of supplying

additional energy to the particles in the fluid which are low in energy in the

boundary layer. This can be achieved by tangentially blowing higher velocity

fluid out from inside the body. The danger of separation is removed by the

supply of kinetic energy to the boundary layer. The separation of the

boundary layer can also be prevented by tangential suction. The low energy

fluid in the boundary layer is removed by suction before it can separate.

Behind the suction slit, a new boundary layer forms which can overcome a

certain pressure increase. If the slit is arranged suitably, in certain

circumstances the flow will not separate at all. A so-called boundary-layer

diverter is also based on the same principle. This is used in the entrance to

the engine on the fuselage of an airplane.

If a wall is permeable and can therefore let the fluid through, the

boundary layer can be controlled by continuous suction on blowing.

Separation can be prevented in the boundary layer is reinoved. In contrast,

the wall shear stress and therefore the friction drag can be reduced by

blowing.

1.4 Heat transfer in thermal boundary layer

The transfer of heat between a solid body and a liquid or gas flow is a

problem whose consideration involves the Science of fluid motion. On the

physical motion of the fluid there is superimposed a flow of heat and the two

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field interacts. In order to determine the temperature distribution and then

heat transfer co-efiicient, it is necessary to combine the equation of motion

with those of heat conduction. However a complete solution of the flow of a

viscous fluid about a body posses considerable mathematical difficulty for

all but the most simple flow geometries.

When observation are made of the behaviour of fluid convecting heat

to a body, it is noticed that the fluid immediately adjacent to the body

surface is brought to rest and attains the surface temperature, except in

those cases where the densities are exceeding low. Further the shearing

forces within the fluid retard additional layers of fluids above the surface. As

distance normal to the surface is increased, these forces eventually diminish

and the velocity parallel to surface approaches an asymptotic value. In the

limit, the shear force vanish and the fluid behaves as if it were inviscid.

Similarly, the temperature varies with distance from the surface and

approaches an asymptotic value. Thus the heat flux, normal to the surface

associated with the temperature gradiant in the direction, is relatively large

in the vicinity of the surface and diminishes to negligibly small value, away

from the surface, approaching the behaviour of a non conducting fluid. The

observation of this behaviour, early in the twenteeth century led prandtl

(1904) to introduce the boundary layer concept. Prandtl considered the flow

field about a body to be divided into two distinct regimes: the inner region

adjacent to the surface where shear and heat transmission are controlling

phenomena- the boundary layer, and the outer region where the gradients in

flow properties are so small as to render the effects of shear and heat

transfer negligible.

One predominant feature of the boundary layer is that the variation of

flow properties as effected by shear and heat conductions occurs; i.e. the

boundary layer is very thin compared with a characteristic dimension of the

body such as its length.

Since, the mass flux parallel to the surface is usually smaller in the

boundary layer than in the inviscid flow, the presence in the boundary layer

diverts or displaces the inviscid flow so that it behaves as if it were flowing

over a body usually slightly larger than the actual body. The outwards

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displacement of the effective body surface, defined as the "displacement

thickness" is negligibly small in most cases [kays 8& crowford(1980)]. The

inviscid region, therefore, generally can be evaluated independently of the

boundary layer growth and infact, is the first step in the evaluation of

convective heat transfer rates. An inter-dependence of these region does

occur, however, on slender bodies in high speed flight at low Reynolds

numbers and on the rear portion of bodies or near protruding air craft

control surfaces where rising surface pressures cause the boundary layer to

separate. Another equally important consequences of the "thinness" of the

boundary layer is the considerable mathematical simplification of the

general equations of the transport of mass momentum and energy within

this region [Goldstain (1965)]. This simplification made otherwise intractable

equations amenable to analysis and permitted the development of a

considerable amount of computed results based on boundary layer theory

[Schlichting (1968)] even prior to the advent of large electronic computers.

When there is a heat transfer or mass transfer between the fluid and

the surface, it is also found that in most practical applications the major

temperature and concentration change occur in a region very close to the

surface. This gives rise the concept of the thermal boundary layer and

concentration boundary layer. The influence of thermal conductivit}'^ and

mass diffusivity is confined within this regions. Outside the boundary layer

region, the flow is essentially non-conducting and non-diffusing. The

Thermal boundary layer may be larger than, smaller than or the saine size

as the velocity boundary layer.

1.5 MAGNETOHYDRODYNAMIC (MHD) BOUNDARY LAYER :

Magnetohydrodynamics (MHD) deals, as the name implies with the

dynamics of a fluid which interacts with a magnetic field. It has, infact been

used to describe a variety of configurations, which include incompressible or

compressible flow, liquid, or gaseous state, dynamic or static configurations,

particle atomic - physics or continuum fluid analysis. Several alternative

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wordings e.g. Magnetofluidynamics, hydromagnetics, magnetogasodynamics

are also widely used.

Only fluids having non-negligible electrical conductivity are capable of

magnetic interactions. As long as the research and engineering efforts in

fluidynamics concerned fluids of vanishingly small electrical conductivity,

such as air at ordinar>^ temperature, MHD received only limited attention.

However, considerably greater interest in the dynamics of conducting fluids

has arisen during the past decade in connection with attempts to harness

fusion energy and a consequence of problems in missile and spacecraft

dynamics, propulsion, and communication. This has also been applied to a

variety of astrophysics and geophysical phenomena.

The property of electrical conductivity implies that there are electric

charges in motion in the fluid. In turn, it is the magnetic force between

moving charged particles, the force exerted by a magnetic field on a charged

particle of fluid moving within the field that constitutes the magnetic part of

the MHD interaction. Such forces are governed by the basic laws of

electricity and magnetism [cowling (1957)]. It is the combined effect of these

forces and mechanical fluid forces that determine the ultimate dynamic

behaviour of the fluid and to which the term MHD implies. For order of

magnitude orientation purposes, we note that the magnetic pressure, given

B2 by , corresponds to a mechanical pressure P of one atom, when the

magnetic field B is 0.5 weber/(meter)2, (5000 gauss), a value which is

available with permanent magnets or electromagnets.

Conducting fluids with which one deals usually contain both neutral

particles and positive and negative charge. The latter tend to be well mixed

so that the fluid is neutral in the large; a gaseous fluid of this type is

referred to as a plasma. Thus the uniform motion of a plasma does not in

itself constitute an electric current.

Conducting fluids of common experience include mercury and

electrolytes such as sea water. Also, the core of the earth contains molten

metals, perhaps liquid iron, whose motion is connected with the existence of

the earth's magnetic field [Alfven H and Falthammar (1942)]. The more

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significant early experiments in MHD were, infact performed with mercury,

as the working fluid. An example is the work of Hartman and Lazarus (1937)

dealing with channel flow under the combined influence of mechanical

pressure head and magnetic field. Observed effects in this work and other

studies, such as the wave experiments of Lundquist (1947), show comforting

correspondence with the prediction of a theory based on the continuum

fluid - conservation relations combined with Maxwell's equations.

The description of an MHD interaction interms of conservation

equations for macroscopic physical variables, is justified only if these

variables represent good statistical averages over the component atomic

behaviour. This is possible, if the mean free path X is smaller than any

lengths characteristic of the structure of the flow. Such may often not be the

case when one deals with plasma, but the bulk fluid approach has been very

useful for a large class of MHD problems.

The governing equations of MHD are Navier - Stokes equations, the

equation of continuity and the Maxwell's equation of electrodynamics.

Analogous to the viscous boundary layer. Cowling (1957) and subsequent

workers assumed that the effect of magnetic field is confined to a thin layer

called MHD boundary layer.

1.7 Unsteady flow and hest transfer

The study of unsteady boundary layer flow has achieved importance

in view of its application in practical fields. The common examples of

unsteady boundary layers occur when the motion is started from rest or

when it is periodic. There are two main subdivisions of periodic flows to

consider, on the one hand periodic boundary layers in the absence of an

imposed mean flow, and on the other periodic boundaiy layers with an

imposed mean fl.ow.

In the case of a periodic boundary layer with an imposed mean flow,

the velocity at the edge of the layer fluctuates about a non-zero mean. For

small fluctuations, the mean flow in the boundary layer is unaffected by the

Reynolds stress of the oscillation and is given by the steady boundary layer

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equation. The oscillation flow is almost exactly like a periodic boundary layer

in the absence of an impressed mean flow, and may conveniently be referred

to as a secondary boundary layer. Periodic boundary layers occur when

either the body performs a periodic motion in a fluid at rest or when the

body is at rest and the fluid executes a periodic motion.

Very often, problems in non-steady boundary layers involve an

essentially steady flow on which there is superimposed a small non-steady

perturbation. If it is assumed that the perturbation is small compared with

the steady basic flow, it is possible to split the equations into a set of non

linear boundary layer equations for the perturbed quantities.

A well known example is that for which the extreme stream has the

from u(x,t) = u (x)+ eui(x,t), where e is a very small number. The most

important special case where the external perturbation is purely harmonic

was studied by M.J. Light Hill (1954). The same type of linearization can be

employed when the temperature at the wall is represented by the

expression.

Tw (x,t) = Tw(x)+ e Twi (x,t)

1.8 Porous Medium

Flows through a porous medium is a topic encountered in many

branches of engineering and science, e.g., ground water hydrology, reservoir

engineering, soil mechanics and chemical engineering. The aquifer and oil

reservoir are the porous medium domains treated by the ground water

hydrologist and by the reservoir engineer respectively. An aquifer (a ground

water basin) is a geologic formation, or a stratum, that (a) contains water,

and (b) permits significant amount of water to move through it under

ordinary field conditions. Ground water is a term used to denote all waters

found beneath the ground surface. However, the ground water hydrologist,

who is primarily concerned with the wafer contained in the zone of

saturation, used the term ground water to denote the water in this zone. In

drainage of agricultural lands, or agronomy, the term ground water is used

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also to denote the water in the partially saturated layers above the water

table. The portion of rock not occupied by the solid matter is the void space

or pore space. The space contains water or air or (sometimes) both water

and air. The oil or gas reservoir is a porous geological formation that

contains in its pore space, in addition to water at least one hydrocarbon (oil

or gas) in a liquid or gaseous phase. Examples of porous materials are

numerous, soil, porous or fissured rocks, filter paper, sand filters and a loaf

of bread ai'e just a few. We may also describe a porous medium as solid with

holes. We use Navier-Stokes equation for the flow of a viscous fluid to

determine the velocity distribution of the fluid in the void space, satisfying

specified boundary conditions. The first description of a porous body worth

mentioning are those of the ingenious Leonhard Euler (1762). In the

posthumously published Anleitung Zur Naturlehre, he attributed the

elasticity of a solid to a certain subtle matter in closed pores. His remark on

porous bodies are:

"All bodies in the world are composed of rough and subtile matter,

where of the first is called the characteristics matter whereas the other due

to its nearly infinitely small density contributes nothing to the increase of

their mass. Since the mixture of both matters extends to the smallest parts,

those parts of the space, in which no rough matter contained, are called the

pores of the bodies, and there are different kinds concerning the size,

because also the smallest parts are still filled up with pores. The most

distinct difference however, which much be considered for the pores of any

body, is that some from an open path to the others, whereas other ones are

surrounded by the rough matter in such a way that the subtle matter

therein contained cannot escape. In order to denote this difference they may

be called the first open pores and the last closed pores". In 1760 he gave an

example of this definition, the water saturated porous solid which is of

immediate relevance to the topic.

Reinhard Woltman was a harbor construction director (1757-1837)

from Humburg, he expanded his idea on soil mechanics and porous bodies

and introduced the volume fraction concept, an essential part of the theorj^

of porous media. Around the mid-19^' centur^^ fundamental effects

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concerning porous media were studied and described by Debsse, Fick and

Darcy, namely the equality of surface and volume fraction in porous media

with statistically distributed pores, the diffusion phenomenon, and the

interaction between the constituents. Darcy was the first scientist to study

the interaction between two constituents, i.e. between the skeleton (porous

soil body) and water.

The actual multiphase porous medium is replaced by a fictitious

continuum, a structureless substance. The variables and the parameters of

the various fictitious continua, averaged over a representative elementary

volume, enable us to describe flow and other phenomena within a porous

medium domain by means of partial differential equations. Such equations

describe what happens at every physical instant of time. The continuum

approach to the dynamics of fluids in porous media can be applied by

introducing macroscopic medium parameters or coefficients to accommodate

the observed phenomena which help to make the passage from the

microscopic to the macroscopic, continuum, leved. One such parameter is

porosity. Other parameters will be permeability, the dispersivity, etc. To a

certain extend, all such parameters may be called parameters of ignorance;

they are introduced because of our inability to solve the problem on the

microscopic level. In principle, it is possible to calculate these coefficients

from information supplied from the lower levels of treatment. In practice

however, this is not possible and they must be deduced from actual

experiments in which the various plenomena related to these parameters are

observed.

Darcy (1856) observed, in tests with natural sand, the proportionality

of the total volume of water running through the sand and the loss of

pressure. Although these investigations were of a purely experimental

nature, his results are essential for a continuous mechanical treatment of

the motion of a liquid in a porous solid. The porous medium is in fact a non

homogeneous medium. For sake of analysis it is possible to describe the flow

in terms of a homogeneous fluid with averaged dynamic properties having

some effects on the locally non-homogeneous continuum. Thus the flow

problems of non-homogeneous fluid under the action of the properly

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averaged external forces can be studied. On the basis of this hypothesis a

complicated problem of flow through a porous medium reduces to the flow

problem of a homogeneous fluid with some resistance. Today Darcy's law is

theoretically well founded by thermodynamics. For a homogeneous medium

the Darcy's law is expressed as

K q= • Vp (1.7.1)

where q is the volume low rate or filter velocity, K is the permeability

which is in general a 2"<i order tensor, i the coefficient of viscosity and p is

the pressure.

For the case of an isotropic medium the permeability is a scalars and

equation (1.7.1) simplifies to

Vp = - - q (1.7.2) K

Following Wooding (1957), many early authors on convection in porous media used an extension of equation (1.7.2) of the from

5 q + ( q. V ) q

.a - V p q (1.7.3)

K

This equation was obtained by analogy with the Navier-Stokes

equation.

An alternative to Darcy's equation is what is commonly known as

Brinkman's equation.

With inertial terms omitted this takes the from

Vp = q+[iV^ q (1.7.4) K

Where n is the coefficient of viscosity and JL' is an effective viscosity.

The first is the usual Darcy term and second is analogous to the Laplacian

term that appears in the Navier-Stokes equation. Brinkman's set \x and {[x)

equal to each other.

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Combining the equations (1.7.3) and (1.7.4) the Navier-Stokes

equation for flow of an incompressible viscous fluid through porous medium

takes the from

^ ^ 1 _ _ _ V p - q + ^ V 2 q + p F (1.7.5) — - + ( q. V ) q

.a J K

where (F) is the external force action on the fluid per unit mass.

1.9 Mass transfer in a porous medium

Like heat transfer, mass transfer considerations are also very

important in modern engineering design, particularly in Chemical

Engineering. The transport of a substance that is involved as a component

(constituent, species) in a fluid mixture is termed as "mass transfer". An

example is the transport of salt in saline water.

Let the subscript i refer to the ith component of fluid mixture of

volume v and mass m. The total mass is equal to the sum of the individual

masses mi, so

M = Z m i (1.9.1)

Hence if the concentration of component i is defined as

mi Ci= (1.9.2)

V then the aggregate density p of the mixture must be the sum of all the

"individual concentrations,

p = I C i (1.9.3)

When chemical reactions are of interest it is convenient to work in

terms of an alternative description, one involving the concept of mole. A

mole is the amount of substance that contains as many molecules as there

are in 12 grams of carbon 12. The molar mass of a substance is the mass of

one mole of that substance.

If there are n moles in a mixture of molar mass M and mass m, then

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m n = (1.9.4)

M

Similarly the number of moles ni of component i in a mixture is the

mass of that component divided by its molar mass Mi,

mi

ni = (1.9.5)

The mass fraction of component i is

mi (D= (1.9.6)

m

Then Z O, = 1

Similarly the mole fraction of component i is

ns Xi = (1.9.7)

n

To summarise, we have three alternative ways to deal with

composition - a dimensional concept (concentration) and two dimcnsionlcss

ratios (mass fraction and mole fraction). These quantities are related by

Mi Ci = p O i = p Xi (1.9.8)

M where the equivalent molar mass (M) of the mixture is given by

M = Im,Xi (1.9.9)

Using the notation pi instead of Ci for the concentration of component

i, the principal of mass conservation to each component in the mixture (in

the absence of component generation) takes the from

3 D — - - + V. (p i qi) = 0 (1.9.10) a

where qi is the intrinsic velocity of particles of component i. Summing

over i, we obtain

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dp + V . ( l p i q i ) = 0 (1.9.11)

a

This is the same a s

dp + V . ( p q ) - 0 (1.9.12)

a

Provided tha t we identify q with the m a s s averaged velocity,

1 q = I Pi qi (1.9.13)

P

If qi q is the diffusion velocity of component i, the diffusive flux of

component i is defined a s

Ji = Pi ( qi - q ) (1.9.14)

Equation (1.1.10) now gives

^ i _ _

+ V . ( p i q ) = - V . J (1.9.15) a

Reverting to the notation Ci for concentrat ion and assuming tha t the

mixture is incompressible, equation (1.9.15) becomes

DCi = - V . Ji (1.9.16)

Dt

D 5 _ where = + q. V

Dt a

For the case of a two component mixture. Pick's law of m a s s diffusion

i s

J i = - D , 2 VCz (1.9.17)

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where D12 is the mass diffusivity of component 1 into component 2 ,

and similarly for J2. In fact D12 = D2] = D. From equation (1.9.16) and

(1.9.17) we have

DCi V. (DVCi) (1.9.18)

Dt

If the migration of the first component is the only one of interest, then

the subscript can be dropped. For a homogenous solution (1.9.18) becomes

DC = D V2C (1.9.19)

Dt

Similarly the energy equation is

DT = amV2T (1.9.20)

Dt

where T is the temperature and am is the thermal diffusivity.

1.9.1 Mass Transfer :

In a system consisting of one or more components whose

concentrations vary from point to point, there is a natural tendency for the

transport of different species from the region of high to those of low

concentration. This process of transfer of mass as a result of the species

concentration difference in a system/mixture is called mass transfer. So

long as there is concentration difference mass transfer will occur.

Some examples of mass transfer are:

A. Examples of industrial importance

1. Refrigeration by the evaporation of liquid ammonia in the

atmosphere of H2 is electrolux refrigerator.

2. Humidification of air in cooling tower.

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B. Examples of day to day life

1. Dissolution of sugar added to a cup of coffee.

2. The separation of the components o a mixture by distillasion or

absorption.

1.9.2. MODES OF MASS TRANSFER

The mechanism of mass transfer depends greatly on the dynamics of

the system in which it occurs. Like those of heat transfer, there are different

modes of mass transfer, which are;

(i) Mass transfer by diffusion;

(ii) Mass transfer by convection;

(iii) Mass transfer by change of phase.

1. Mass transfer by diffusion (molecular or eddy diffusion)

The transport of water on a microscopic level as a result of diffusion

from a region of high concentration to a region of low concentration in a

system / mixture of liquids or gases is called molecular diffusion. It occurs

when a substance diffuses through a layer of stagnant fluid and may be due

to concentration, temperature or pressure gradients. In a gaseous mixture,

molecular diffusion occurs due to random motion of the molecules.

When one of the diffusing fluids is in turbulent motion, the eddy

diffusion tal<es place. Mass transfer is more rapid by eddy diffusion than by

molecular diffusion. An example of an eddying diffusion process is

dissipation of smoke from a smoke stack. Turbulence causes mixing and

transfer of smoke to the ambient air.

2. Mass transfer b> convection

Mass transfer by convection involves transfer between a moving fluid

nnd a surface, or between two relative^ immiscible moving fluids. The

convective mass transfer depends on the transport properties and in the

dynamic (laminar or turbulent) characteristics of the flowing fluid.

li:xample : The evaporation of ether.

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3. Mass transfer by change of phase

Mass transfer occurs whenever a change from one phase to another

takes place. The mass transfer in such a case occurs due to simultaneous

action of convection and diffusion. Some example are :

(i) Hot gases escaping from the chimney rise by convection and

then diffuse into the air above the chimney,

(ii) Mixing of water vapour with air during evaporation of water

from the lake surface (partly by convection and partly by

convection and partly by diffusion),

(iii) Boiling of water in open air - there is first transfer of mass from

liquid to vapour state and then vapour mass from the liquid

interface is transferred to the open air by convection as well as

by diffusion.

1.9.3 CONCENTRATIONS. VELOCITIES AND FLUXES

Concentration :

Mass concentration (or mass density) : The mass concentration or

mass density PA of species A in a multi-component mixture is defined at the

mass of A per unit volume of the mixture. It is expressed in kg/m^ units.

Molar concentration (or molar density). The molar concentration CA of

species A is defined as the number of moles of species A per unit volume of

the mixture. It is expressed in kg mole/m^ units.

1.9.4 CONVECTIVE MASS TRANSFER

When a medium deficient in a component flows over a medium having

an abundance of the component, then the component will diffuse into the

flowing medium. Diffusion in the opposite direction will occur if the mass

concentration levels of the component are interchanged.

In this case a boundary layer develops and at the interface mass

transfer occurs by molecular diffusion (in heat flow at the interface, heat

transfer is by conduction).

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Velocity boundaiy layer is used determine wall friction. Thermal

boundary layer is used to determine convectivc heat transfer. Similarly

concentration boundaiy layer is used to determine convective mass transfer.

1.9.5 SIMILARLY BETWEEN HEAT AND MASS TRANSFER

It is possible from similaiit3^ between the heat convection equation and

mass convection equation to obtain value of hm. (i.e. called as Lewis

number).

h = ^CpjLe^l^

hm a

Where Le =

D Many of the correlation in heat transfer can be applied to mass

transfer under similar condition, by replacing Nusselt number by Sherwood

number and Prandtl number by Schmidt number.

1.10 Basic Equations ;

(a) Fundamental principles :

In fluid mechanics we consider three basic principles from which we

can derive the fundamental equations of motion of fluid. They are the

conservation of mass,, the conservation of momentura^ and the conservation

of energy; and they bring corresponding equations viz. the equation of

continuity, the equation of motion and the equation governing the

temperature distribution. In deriving the fundamental equations governing

the flow field and the temperature in MHD we modify the well know

conservation equations of classical fluid mechanics by incorporating into

them suitable momentum and energy terms obtained from Maxwell's

equation and Ohm's law.

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(b) Maxwell's Electromagnetic Equations:

The basic laws of electrodynamics are summarized by the Maxwell's

equations which define the properties of the electric and magnetic fields.

Under non-relativistic assumptions these equations are:

a B curl E = , (1.10.1)

at

dD curl H = J + , (1.10.2)

at

d i v B = 0, (1.10.3)

d i v b = / = ' (1.10.4)

B = JLle n, (1.10.5)

b = e E (1.10.6)

where E, B, H, J, ^, jUe , E and j° are respectively the electric

intensity, the magnetic flux density, the magnetic intensity, the current

density, the current displacement, the magnetic permeability, the dielectric

constant and electric charge density. In addition to these, we have current

conservation equation :

a/o div J + = 0 (1.10.7)

at

which follows directly from (1.10.2) and (1.10.4) .

In Magnetohydrodynamics low frequency electromagnetism, the

dD displacement current is neglected in the case when the fluid is in

at motion with a velocity veiy small compared to the velocity of light. Also for

fluids which are almost neutral, the charge density/^e is negligible this

djo A means that ( I must be omitted from (1.10.7). The Maxwell's

^ ^ ASSAM UfflVERSTFYiBRAfPy aLCHAR

Accession Nn

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equations under MHD approximation then take the forms :

d B curl E = , (1.10.8)

at

curl H - J (1.10.9)

div~B = 0, (1.10.10)

d i v J = 0 , (1.10.11)

B = //e H, (1.10.12)

b = e E (1.10.13)

(c) Ohm's Law :

In a conducting medium, the current density J is given by

J = a E (1.10.14)

where a is the electrical conductivity. For a moving medium, total

electric field intensity is the sum of the applied electric field intensity E and

the induced electric field intensity Ei.

If the conduced moves with velocity V in a magnetic field H, then the

induced electric field is given by

El = Vx B .

Hence the total electric field is

E ' = E+ El = E + Vx B

The Ohm's law then becomes

J = c r [ E + V X B ] (1.10.15)

(d) Magnetic Diffusion Equation :

Combining equations (1109) and (110.15), we obtain that

curl H = a [ E + V X B ] (1.10.16)

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Eliminating the electric field E by taking Curl on equation (1.10.16) and

using equation (1.10.8), we obtain for a fluid of uniform conductivity O and

constant magnetic permeability jUe .

dB Curl Curl B = -jUeG + / /e a Curl ( V X B ) (1.10.17)

at

Since div B = 0, the above equation (1.10.17) can be written as

a B = c u r l ( V x B ) + r| W B (1.10.18)

dt 1

which is the magnetic induction equation and T\ = is called the OjUe

magnetic diffusivity. In Magnetohydrodynamic problems, we often neglect

the induced magnetic field. This assumption is justified for flows where

magnetic Reynolds number is very small.

(e) Equation of Conservation of Mass :

If J° denotes the density of the fluid, the hydrodynamic equation of

conservation of mass is written as :

+ d i v ( / ^ v ) = 0 (1.10.19) a

which remain unchanged for conducting medium. For an incompressible

fluid the equation (1.10.19) reduces to :

divV = 0 (1.10.20)

(f) Equation of Conservation of Momentum :

The Navier-Stokes equation of motion for viscous fluid of density J°

and kinematic viscosity V can be written as

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5 V

8x + ( V. V ) V = - --3- V {3 + V V ( V. "V ) +V V^ V +

1

3 (1.10.21)

where {? is the fluid pressure, F is the local body force per unit volume and

V2 is the Laplacian derivative. If an electrically conducting fluid moves with

velocity V in presence of a magnetic field B, the body force F per unit

volume can be written as

F= /^e E + "Jx B (1.10.22)

The electric term J° . E can be omitted from the body force as it can be

shown that its ratio to the magnetic term is of the order of V^/C^ (c being

the velocity of light )which is very small. The momentum conservation

equation in MHD becomes :

8 V

a 1 1

+ ( V. V ) V = - --5- V t) + V V ( V. V ) +V V^ V + - - - ( J X B ) J° p

(1.10.23)

For an incompressible fluid the equation (1.23) can be written as :

a V _ 1 _ ^ ^ _ + ( V . V ) V = - - - 5 - V t ) + V V^ V + — ^ - ( J x B ) (1.10.24)

et

(g) Equation of Energy

J° /"

The equation of heat transfer arises from the principle of conseivation

of energy which states that the total time rate of change of kinetic and

internal energies is equal to the sum of the works done b}' the external

forces per unit time and the sum of the other energies supplied per unit

lime. P'rora this principle the equation of heat transfer can be written as (see

shercliff, 1965)

y°q. dT

at + V. VT k v^ T + y° V O (1.10.25)

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where O = 2

az J (1.10.26)

Where Cp, T, k being the specific heat at constant pressure, the temperature

and the thermal conductivity respectively. The components of the fluid

velocity vector V are u, V, w in the directions of x, y, z respectively.

Charges within a conductor move under the action of electromagnetic

forces, colliding and exchanging energy with the rest of the conductor,

means that electrical work can be done on or by the material. It has been

found the electromagnetic field puts energy into the material at a rate E. J

per unit volume and time (Shercliff, 1965). Now J can have three possible

forms conduction, convection and polarization. The contribution of

convection and polarization on the work done is negligible under MHD

approximation : only that of the conduction current plays a significant role.

Using equation (1.10.15) of Ohm's law, E.J can be written as

J 2 _ _ _ J . ( V x B ) E. J.

a J2 a

+ V. ( J X B )

The second term on the right hand side is the rate at which the

magnetic force J x B does work on the conductor as a whole.

J2 The first term is positive and is know as Joule heating term

c which is added in the energy equation (1.10.25) so that the

magnetohydrodynamic equation for energy conservation takes the from :

7°Cf dJ

a + V. V T - K V2 T +/^ V (j) +

J2

a (1.10.27)

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(h) BOUNDARY CONDITIONS OF MHD

The flow field and the electromagnetic field are to be determined by

solving the fundamental equations stated in previous section. Under

appropriate boundaiy conditions for the flow and the electromagnetic field.

The boundary conditions to be satisfied are the usu^U hydrodynamic

boundary conditions imposed on the velocity field (such as no slip

conditions at a solid surface for a viscous fluid), the continuity of

temperature field and the electromagnetic boundary conditions.

The electromagnetic properties change abruptly at a solid boundary.

Across such a surface of discontinuity, the electromagnetic variables satisfy

the following conditions.

(i) The normal component of magnetic field

B = jUe H is continuous, i.e.,

( B ? - B, ) . n = 0, (1.10.31)

Where n is the unit normal to the surface of discontinuity. Subscripts

'I'and '2'refer to the values on either side of the surface.

(iij The m.agnetic field H has T.OS satisfy the condition

n x ( H 2 - Hi) = Js (1.10.32)

where Js is the surface current density. When the electrical conductivity is

finite, a ^ oc we have Js = 0. But when a = oc, J s may be different from

zero.

(iii) The tangential component of electric field is continuous, i.e.

"nx ( E^ - El) = 0 (1.10. 33)

(iv) The behaviour of dielectric displacement D at the boundary is

n . ( D 2 - b , ) = y03 (1.10. 34)

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where/^s is the surface free chai'ge density. For most of our MHD problems

we may neglect the surface current density Js and the surface free charge

density y^s. Hence our boundary conditions becomes that both the

tangential components of H and E, and the normal components of B and

D are all continuous across a surface separating a bod}^ and a fluid on two

fluids.

(i) DIMENSIONLESS PARAMETERS IN MAGNETOHYDRODYNAMICS :

As in ordinary hydrodynamics by transforming the basic equations to

non-dimensional from it can be easily proved that the following parameters

govern the MHD flow :

(i) The magnetic Reynolds number Rm = CJ //e UL, which is a

measure of the ratio of the magnetic convection to magnetic diffusion.

(ii) The Hartmann number M = {G/ juY' Bo L which is a measure of

the ratio of magnetic force to the viscous force. It was introduced by

Hartmann to describe his experiments with viscous MHD channel flow.

Hence it is assumed that the Lorentz force »; 0 ( a Vo Bo^) which is true for

small conductivities. Thus the magnetude of the Hartmann number M

indicates the relative effects of magnetic and viscous drag for, say, the flow

of a conducting viscous fluid across magnetic lines of force (low Rm ).

(iii) The magnetic pressure number S = Me 1 o^/y^U ^, which is the

measure of the ratio of the magnetic pressure to the hydrodynamic pressure.

(iv) The magnetic prandil number Pm = cr /7<? V

which is the measure of the ratio of vorticity diffusion to magnetic diffusion.

] . l l Shooting Method :

The shooting method is a modern, advanced, sophisticated, computer

oriented method for solving boundary value problems [ Conte (1965),

Jacques, Ian and Judd, Colin (1987)).

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In all third order two-point boundary value problems two boundaiy

conditions are always prescribed at one end point and one boundary

condition at the other end point. Without loss of generality, let us assume

that two boundary conditions are prescribed at the initial point = xo and one

boundary condition at the final point x = X) (the problem prescribed with two

boundary conditions at the final point and one condition at the initial point

can be treated similarly by backward integration).

Let the general third order two-point boundary value problem be

y'" = f(xyy'y")

with boundary conditions

y(p)(xo) = yo'P). p is any two of 0, 1,2

and y(PHxi) = yi'p'. p is any two of 0, 1,2

where the upper index p denotes order of differentiation with respect to x

and the lower indices 0 and f are for initial point and final point respectively

i.e. we are considering the problem with two boundary conditions at the

initial point and one boundary condition at the final point.

We use here the initial value method for solving ordinary differential

equation. But in order to do so we must know all the initial conditions

needed. Since one of yo'P' (p = 0.1,2) is missing at x = xo let it be an unknown

parameter X (sa}' ) w^hich must be so determined that the resulting solutions

yield to prescribed final value y/P) (p = 0,1,2) at x = xi to some desired

accuracy.

Techniques based on this idea are known as shooting methods,

because the problem is analogous to that of the artilleryman to shoot a

cannonball at a fixed target (Jacques, Ian 8& Judd Colin 1987). Let the

cannon and the target be placed at the points (a, a) and (b, P) respectively.

The artilleryman has to choose the elevation of the cannon in such a way so

that the cannonball hits the target. Let us suppose that the trajectory of the

first shot be as shown in Fig 1.11.1(a). Clearly the cannonball has udershot

the target. So it becomes necessary to increase the elevation for the second

shot. In the second attempt, the hall passes over to the top of the target, as

is shown in Fig. 1.11.1(b). So it becomes clear that the elevation is

somewhere between the first two. The process is repeated until the target is hit.

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A

Fig. 1.11.1(a)

Fig. 1.11.1(b)

Let us come back to (1.11.1) and (1.11.2).

Let ho and X] be two guCvSsed values of the missing initial condition

y''' (xo) (p takes the value 0, 1 or 2 which is not in the given initial

conditions).

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Let y(p)(xf: Xo) and y(p)(xf: Xi) be the value of y(p)(xf) (p is given to be any

one of 0, 1, : for >, = Xo and X = Xi respectively) obtained on integrating the

initial value problem for (1.11.1) in which Xk, k = 0, 1 is taken as the missing

initial conditions. Then geometrically a better approximation A.2 of X come

obtained as follows.

X

/\

s

J \ /

cV

E

B

F

D

-^

Xo A,2 A, 1

Fig. 1.11.2

Let the points A and B in Fig. 1.11.2 represent the value of y'PHxf)

•when X - Xo and X = Xi respectively, when y(p'(xf) is plotted against X. Let SD

be the line y^P^x/) = y/'"''- Then a better approximation X2 of X can be obtained

by linear interpolation given by

y/<p) - y(P) (x/ : ?LO)

X - Xo + (Xi - Xo) y(p) (xj -.Xi)- y(P) (x/ : XG]

[from the similar triangles CAE and BAF ]

Now (1.11.1) can be integrated with the two initial conditions and X2

as the missing initial condition to obtain y(p)(x/ : ^ 2). Again using Unear

interpolation based on X2 and Xi, we can obtain the near approximation Xi.

The process is repeated until y(P)(x/ : A,k) = y/P* is satisfied to some desired

accuracy for some k.

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Convergence of the iteration process described above is not

guaranteed. But once the chosen value is nearer to the t rue value, the

convergence is very rapid.

In fact linear problems can be solved by shooting methods without

iteration whereas shooting methods for non-l inear problems are

characterized by interation. For non-l inear differential equat ions , shooting

me thods have certain advantages for the problem solver. Firstly, the

methods are quite general and are applicable to a wide variety of differential

equat ions. It is not necessary for the applicability of shooting me thods tha t

the equat ions be of special types such a s of even order, self-adjoint etc.

Secondly, shooting methods require a min imum of problem analysis and

preparat ion. It is relatively easy to implement shooting methods on digital

computers us ing s tandard subrout ines for the numerical integration of

ordinary differential equat ions , solut ions of linear algebraic equat ions etc.

With a properly written code, only one subrout ine needs to be altered from

problem to problem, the one in which the right h a n d side of the system of

differential equat ions written in a s tandard form is entered. All other pa r t s of

the code will handle automatically any problem from a broad class . For

some details on various shooting methods for the numerical solution of two-

point boundary value problems for linear and non-linear ordinary

differential equat ions, we may refer Bailey, Shampine and Wat tman (1968),

Goodman and Lance (1956), Robert and Shipman (1978).

1.12 Motivation of the thesis :

The thesis consists of six chapters. In the first chapter on introduction

we present a review of boundary layer concept, development of boundary

layer theory, role of suction in boundary layer control, heat transfer in

thermal boundary layer, magnetohydrodynamic boundary layer. We then

proceed to discuss unsteady flow and heat transfer, porous medium, mass

transfer in porous medium in the line of Euler (1762) and others.

Subsequently we present the basic equations used in our thesis and

describe the fundamental principal. Maxwell's Electromagnetic Equations,

Ohm's Law, Magnetic Diffusion equation. Equation of conservation of mass.

Equation of conservation of Momentum; Equation of energy together with

boundary conditions of MHD. The perturbation method is used in solving

problems where analytical solution is obtained. Whenever numerical

solution is sought shooting method is used. We present a brief description of

the shooting method.

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In the second chapter we discuss the Hydro-magnetic couette type flow with

small uniform suction at stationary plate. The governing equations are solved by

perturbation method. The longitudinal and transverse velocity profiles are drawn for

single value of the suction parameter (0.1) and Reynold number (100) but various

values of the Hartmann number. It is found that an adverse pressure gradient

develops which causes back flow.

The third chapter consists of two parts, in the first pari namely chapter III(A)

we discuss the flow non-coaxial rotation of porous disk and the fluid at infinity is

considered where the disk are rotating with the same angular velocity. The effect of

suction is studied. Near the plate the effect of suction is to increase the transverse

velocity while the radial velocity is decreased due to s^uction. It is observed that the

radial velocity where there is suction became greater than its value when there is no

suction. Then the corresponding value when there is no suction as to moves away

from the plate.

In the second part namely chapter III B, we consider two dimensional

unsteady convective MHD flow through a rotating porous medium with variable

suction. The basic equations of the problem with appropriate boundary condition

have been derived and an approximate solution is presented. The expression for the

skin friction at the wall is obtained. The amplitudes and phases of the fluctuating part

of the skin friction at the wall have been presented graphically for different values of

the parameters involved.

The fourth chapter is devoted to the idealized problem of the flow induced by

a uniformly heated vertical wall in fluid saturated porous medium with isotropic

properties is considered. Apart from the physical consideration of the problem, and

attempt has been made to investigate the solution numerically. The mathematical

problems have been reduced to a non-linear third order boundary value problem. The

governing non-linear differential equations are transformed in to a set of coupled

ordinary differential equations. Emphasis has been given to the numerical solution of

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the problem. The temperature and velocity function are computed by shooting

method.

In the fifth chapter we investigate the three dimensional free convective mass

transfer flow of all incompressible viscous fluid along a vertical porous plate with

transverse sinusoidal suction velocity distribution and uniform free stream velocity is

investigated. The governing equations are solved by regular perturbation technique.

The expressions for the velocity field, the wall shear stress in the direction of the

main flow and the rate of heat transfer and mass transfer from the plate to the fluid

are obtained and some of them are demonstrated in graphs for different values of the

parameters involved.

The six chapter is divided in two parts. The first chapter VI (A) is devoted to

the study of laminar boundaiy layers in oscillatory flow along a uniformly moving

infinite flat with variable suction. The problem of two dimensional convective flow

of a viscous incompressible fluid past a uniform moving infinite plate is discussed

when the suction velocity normal to the plate as well as the stream velocity varies

periodically with time. The expressions for the velocity and temperature field are

obtained in the non-dimensional form.

The amplitudes and the phases of the fluctuating parts of the skin friction and

the plate temperature are demonstrated graphically for different values of the

parameters involved.

In the second part namely chapter VI (B) the revolving flow of a viscous,

incompressible, electrically conducting fluid past over a fixed porous flat plate, is

studied in the presence of an external magnetic field. A series solution in descending

powers of a large suction parameter is developed. It is found that the effect of suction

is to induce an axial flow at infinity towards the plate, which increases as the suction

velocity increases. On the other hand the axial velocity along the direction apart from

the plate is seen to decrease as the suction parameter increases. The influence of the

magnetic field strength on the radial, tangential and axial velocity is analyzed.